Keldysh Field Theory for Driven Open Quantum Systems

L. M. Sieberer,1 M. Buchhold,2 and S. Diehl2, 3, 4 1Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001, Israel 2Institute of Theoretical Physics, TU Dresden, D-01062 Dresden, Germany 3Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA 4Institut f¨urTheoretische Physik, Universit¨atzu K¨oln,D-50937 Cologne, Germany Recent experimental developments in diverse areas - ranging from cold atomic gases over light-driven semi- conductors to microcavity arrays - move systems into the focus, which are located on the interface of quantum optics, many-body physics and statistical mechanics. They share in common that coherent and driven-dissipative quantum dynamics occur on an equal footing, creating genuine non-equilibrium scenarios without immediate counterpart in condensed matter. This concerns both their non-thermal flux equilibrium states, as well as their many-body time evolution. It is a challenge to theory to identify novel instances of universal emergent macro- scopic phenomena, which are tied unambiguously and in an observable way to the microscopic drive conditions. In this review, we discuss some recent results in this direction. Moreover, we provide a systematic introduction to the open system Keldysh functional integral approach, which is the proper technical tool to accomplish a merger of quantum optics and many-body physics, and leverages the power of modern quantum field theory to driven open quantum systems.

CONTENTS II Applications 32

I. Introduction2 III. Non-equilibrium stationary states: spin models 32 A. New phenomena2 A. Ising spins in a single-mode cavity 32 B. Theoretical concepts and techniques3 1. Large-spin Holstein-Primakoff representation 33 C. Experimental platforms5 2. Effective Ising spin action for the single-mode cavity 35 1. Cold atoms in an optical cavity, and microcavity arrays: driven-dissipative spin-boson models5 B. Ising spins in a random multi-mode cavity 36 1. Keldysh action and saddle point equations 37 2. Exciton-polariton systems: driven open interacting bosons7 2. Non-equilibrium glass transition 38 3. Thermalization of photons and atoms in the glass 3. Cold atoms in optical lattices: heating phase 39 dynamics9 4. Quenched and Markovian bath coupling in the D. Outline and scope of this review 10 Keldysh formalism 40

IV. Non-equilibrium stationary states: bosonic models 40 I Theoretical background 11 A. Density-phase representation of the Keldysh action 42 II. Keldysh functional integral for driven open systems 11 B. Critical dynamics in 3D 44 A. From the quantum master equation to the Keldysh 1. Effective action for driven-dissipative functional integral 11 condensation 45 B. Examples 16 2. Non-equilibrium FRG flow equations 46 1. Single-mode cavity, and some exact properties 16 3. Scaling solutions and critical behavior 48 2. Driven-dissipative condensate 20 C. Absence of algebraic order in 2D 49 C. Semiclassical limit of the Keldysh action 21 D. KPZ scaling in 1D 51 D. Symmetries of the Keldysh action 23 1. Thermodynamic equilibrium as a symmetry of V. Universal heating dynamics in 1D 52 the Keldysh action 24 A. Heating an interacting Luttinger liquid 53 2. Semiclassical limit of the thermal symmetry 26 B. Kinetic equation 54 3. The Noether theorem in the Keldysh formalism 27 C. Self-consistent Born approximation 56 4. Closed systems: energy, momentum and particle 1. Kinetics for small momenta 58 number conservation 28 2. Scaling of the self-energy 58 5. Extended continuity equation in open systems 29 D. Heating and universality 58 6. Spontaneous symmetry breaking and the 1. Phonon densities 58 Goldstone theorem 30 2. Non-equilibrium scaling 59 E. Open system functional renormalization group 31 3. Observability 59 2

VI. Conclusions and outlook 60 techniques to be used, and calls for the development of new theo- retical tools. The physical framework sparks broader theoretical VII. Acknowledgements 60 questions on the existence of new phases of bosonic [18, 19] and fermionic [20–22] matter, the nature of phase transitions in such A. Functional differentiation 61 driven systems [10, 23–26], and the observable consequences of quantum mechanics at the largest scales [27]. Beyond stationary B. Gaussian functional integration 61 states, a fundamental challenge is set by the time evolution of interacting quantum systems, which is currently explored theo- References 62 retically [28–33] and experimentally in cold atomic [34–38] and photonic systems [39]. A key goal is to identify universal dy- namical regimes that hold beyond specific realizations or precise I. INTRODUCTION initial conditions. Combining idealized closed system evolution with the intrinsic open system character of any real world ex- periment takes this setting to the next stage, and exhibits emer- Understanding the quantum many-particle problem is one of gent dynamics markedly different from closed systems both for the grand challenges of modern physics. In thermodynamic short [40, 41] and long evolution times [42–47]. equilibrium, the combined effort of experimental and theoretical The interplay of coherent and driven-dissipative dynamics can research has made tremendous progress over the last decades, be a natural consequence of the driving necessary to maintain revealing the key concepts of emergent phenomena and univer- a certain many-body state. Going one step further, it is possi- sality. This refers to the observation that the relevant degrees ble to exploit and further develop the toolbox of quantum optics of freedom governing the macrophysics may be vastly differ- for the driven-dissipative manipulation of many-body systems. ent from those of the microscopic physics, but on the other Recently, it has been recognized that the concept of dissipative hand are constrained by basic symmetries on the short distance state preparation in quantum optics [48, 49] can be developed scale, restoring predictive power. Regarding the role and power into a many-body context, both theoretically [50–53] and exper- of these concepts in out-of-equilibrium situations, there is a imentally [54–56]. Suitably tailored dissipation then does not large body of work in the context of classical near equilibrium necessarily act as an adversary to subtle quantum mechanical and non-equilibrium many-body physics and statistical mechan- correlations such as phase coherence or entanglement [57–62]. ics [1–5]. However, analogous scenarios and theoretical tools In contrast, it can even create these correlations, and dissipa- for non-equilibrium quantum systems are much less developed. tion then represents the dominant resource of many-body dy- This review addresses recent theoretical progress in an impor- namics. In particular, even topologically ordered states in spin tant and uprising class of dynamical non-equilibrium phases of systems [63] or of fermionic matter [64, 65] can be induced dis- quantum matter, which emerge in driven open quantum systems sipatively ([52] and [66–68], respectively). These developments where a Hamiltonian is not the only resource of dynamics. This open up a new arena for many-body physics, where the quan- concerns both non-equilibrium stationary states, but also the dy- tum mechanical microscopic origin is of key importance despite namics of such ensembles. Strong motivation for the exploration a dominantly dissipative dynamics. of non-thermal stationary states comes from a recent surge of Summarizing, this is a fledging topical area, where first results experiments in diverse areas: in cold atomic gases [6–8], hy- underpin the promise of these systems to exhibit genuinely new brid light-matter systems of Bose-Einstein condensates placed physics. In the remainder of the introduction, we will discuss in optical cavities are created [9, 10], or driven Rydberg ensem- in some more detail the three major challenges which emerge in bles are prepared [11]; light-driven semiconductor heterostruc- these systems. The first one concerns the identification of novel tures realize Bose condensation of exciton-polaritons [12, 13]; macroscopic many-body phenomena, which witness the micro- coupled microcavity arrays are pushed to a regime demonstrat- scopic driven open nature of such quantum systems. Second, we ing both strong coupling of light and matter [14] and scalabil- anticipate the theoretical machinery which allows us to perform ity [15]; large ensembles of trapped ions implement varieties of the transition from micro- to macrophysics in a non-equilibrium driven spin models [16, 17]. All those systems are genuinely context in practice. Third, we describe some representative ex- made of quantum ingredients, and share in common that coher- perimental platforms, which motivate the theoretical efforts, and ent and driven-dissipative dynamics occur on equal footing. This in which the predictions can be further explored. creates close ties to typical setups in quantum optics. But on the other hand, they exhibit a continuum of spatial degrees of free- dom, characteristic for many-body physics. Systems located at this new interface are not guaranteed to thermalize, due to the A. New phenomena absence of energy conservation and the resulting breaking of detailed balance by the external drive at the microscale. They As pointed out above, one of the key goals of the research re- rather converge to flux-equilibrium states of matter, creating viewed here is the identification of new macroscopic many-body scenarios without counterpart in condensed, equilibrium matter. phenomena, which can be uniquely traced back to the micro- This rules out conventional theoretical equilibrium concepts and scopic driven open nature of such systems, and do not have an 3 immediate counterpart in equilibrium systems. cal behavior is characterized by a fine structure in a new and in- The driven nature common to the systems considered here can dependent critical exponent, which measures decoherence, and always be associated to the fact that the underlying microscopic whose value distinguishes equilibrium and non-equilibrium dy- Hamiltonian is time dependent, with a time dependence relating namics, see Sec.IVB. to external driving fields such as lasers. When such an ensemble Instead of emergent thermal behavior indicating the fadeout of (i) in addition has a natural partition into a “system” and a “bath” non-equilibrium conditions upon coarse graining, also the oppo- — a continuum of modes well approximated by harmonic oscil- site behavior is possible. For example, low dimensional (d ≤ 2) lators with short memory —, (ii) the system-bath coupling is bosonic systems at low noise level, such as exciton-polaritons weak compared to a typical energy scale of the “system” Hamil- well above threshold, are not attracted to the equilibrium fixed tonian and (iii) linear in the bath operators, then an effective point as their three dimensional counterparts, but rather flow to (still microscopic) description in terms of combined Hamilto- the non-equilibrium fixed point of the Kardar-Parisi-Zhang [69] nian and driven-dissipative Markovian quantum dynamics of the universality class, see Sec.IVC. This can be interpreted as a “system” ensues. The “system” dynamics obtains by tracing out universal and indefinite increase of the non-equilibrium strength, the bath variables. which is triggered even if the violation of detailed balance at the The resulting effective microdynamics is “non-equilibrium”, microscopic level is very small. in a sense sharpened Sec. II D 1. This not only concerns the Universal non-equilibrium phenomena can also occur in the time evolution, but also holds for the stationary flux equilibrium time evolution of driven open systems. For example, intriguing states. More precisely, the above situation implies an explicit scaling laws describing algebraic decoherence [43], anomalous breaking of detailed balance, since the “system” energy is not diffusion [70], or glass-like behavior [42, 71–73] have been iden- conserved. In an operational sense, this principle states that there tified in the long time asymptotics of driven spin systems close is a partition invariance for the temperature (or, more generally, to the stationary states. Conversely, the short time behavior of the noise level) present in the system: any arbitrary bipartition driven open lattice bosons shows universal scaling laws directly of the system can be chosen, one part can be traced out, and the witnessing the non-equilibrium drive, see Sec.V. This scaling resulting subsystem will be at the same temperature (noise level) can be related to a strongly pronounced non-equilibrium shape as the total system. This situation is the condition for a globally of the time-dependent distribution function in the early stages of well-defined temperature characteristic for systems in thermal evolution, and be traced back to conservation laws of the driven- equilibrium. More formally, thermal equilibrium can be detected dissipative generator of dynamics. by means of so-called fluctuation-dissipation relations (FDRs). The above discussion mainly focuses on the difference be- These connect the two fundamental observables in physical sys- tween equilibrium and non-equilibrium systems on the macro- tems — correlation and response functions (see Sec.IIB). In scopic level of observation. Another direction, still much less the case of thermal equilibrium, the connection is dictated by developed, concerns the distinction between classical and quan- the particle statistics alone. It is then given by the Bose- and tum effects. Again, although the quantum mechanical descrip- Fermi-distributions, respectively. Deviations from this univer- tion is necessary at a microscopic level, the persistence of quan- sal form, which has only two free parameters (temperature and tum effects at the macroscale is not guaranteed. This is mainly chemical potential, relating to the typical conserved quantities due to the Markovian noise level inherent to such quantum sys- energy and particle number), provide a necessary requirement tems. Nevertheless, systems with suitably engineered driven- for non-equilibrium conditions. dissipative dynamics show typical quantum mechanical phe- In non-equilibrium stationary states, no such general form ex- nomena such as phase coherence [50, 51, 57, 59, 60, 74], en- ists. We will encounter a concrete and simple example in the tanglement [54–56, 58], or topological order [52, 62, 66–68]. context of the driven Dicke model (a cavity photon coupled to Especially fermionic systems, which do not possess a classical a collective spin) in Sec. III A, where the form of the FDR de- limit, are promising in this direction. pends on the observable we are choosing (e.g., the position or momentum correlations and responses). On the other hand, thermal FDRs can emerge at long wave- B. Theoretical concepts and techniques length, even though the microscopic dynamics manifestly breaks detailed balance [26]. In particular, in three dimensions and The development of theoretical tools needed to perform the close to a critical point, a degeneracy of critical exponents in- transition from micro- to macro-physics in driven open quantum dicates a universal asymptotic thermalization, in the sense of an systems is still in progress, as a topic of current research. A rea- emergent thermal FDR. A similar phenomenology is observed son for the preliminary status of the theory lies in the fact that in a disordered multimode extension of the Dicke model, see two previously rather independent disciplines — quantum op- Sec. III B. This underlines the strongly attractive nature of the tics and many-body physics — need to be unified on a technical thermal equilibrium fixed point at low frequencies. Still in these level. systems, non-equilibrium conditions leave their traces in the dy- Quantum optical systems are well described microscopically namical response of the system, in terms of information that in terms of Markovian quantum master equations, which treat does not enter the FDR at leading order. For example, the criti- coherent Hamiltonian and driven-dissipative dynamics on equal 4 footing. To solve such equations both for their dynamics dation in terms of the renormalization group already mentioned and their stationary states, powerful techniques have been de- above, yet another tool developed and most clearly formulated vised. This comprises efficient exact numerical techniques for in a functional language. Finally, the functional integral based small enough systems, such as the quantum trajectories ap- on a single scalar quantity — the system’s action which encodes proach [45, 75, 76]. But it also includes analytical approaches all the dynamics on the microscopic scale — is a convenient such as perturbation theory for quantum master equations, e.g., framework when it comes to the classification of symmetries and in the frame of the Nakajima-Zwanzig projection operator tech- associated conservation laws, and their use in devising approxi- nique [77, 78], or mappings to P, W, or Q representations [79], mation schemes respecting them. casting the problem from a second quantized formulation into To put it short, while the driven open many-body systems partial differential equations. are well described by microscopic master equations, the tradi- A characteristic feature of traditional quantum optical sys- tional techniques of quantum optics cannot be used efficiently tems is the finite spacing of the few energy levels which play – at least not in the case where the generic complications of a role. When considering systems with a spatial continuum of many-body systems start to play a role. Conversely, their driven degrees of freedom instead, the energy levels become contin- open character makes it impossible to approach these problems uous. This does not mean that the microscopic modelling in in the framework of equilibrium many-body physics. This sit- terms of a quantum master equation is inappropriate: for the uation calls for a merger of the disciplines of quantum optics driven-dissipative terms in such an equation, the assumption of and many-body physics on a technical level. On the numerical spatially independent dissipative processes (such as atomic loss side, progress has been made in one spatial dimension recently or spontaneous emission) is still valid as long as the emitted by combining the method of quantum trajectories with power- wavelength of radiation is well below the spatial resolution at the ful density matrix renormalization group algorithms [83–86], scale where the microscopic model is defined. Indeed, in this sit- see [45] for an excellent review on the topic. For more analyti- uation, destructive interference of radiation justifies the descrip- cal approaches, the Keldysh functional or path integral [87–92] tion of driven dissipation in terms of incoherent processes. How- is ideally suited (but see [93] for a recent systematic perturbative ever, under these circumstances the smallness of a microscopic approach to lattice Lindblad equations with extensions to sophis- expansion parameter no longer guarantees the smallness of the ticated resummation schemes [94], and [95–97] for advanced associated perturbative correction. Here the reason is that in per- variational techniques). Conceptually, the latter captures the turbation theory, one is summing over intermediate states with most general situation in many-body physics — the dynamics propagation amplitudes down to the longest wavelengths. This of a density matrix under an arbitrary temporally local generator can lead to infrared divergences in naive perturbation theory. of dynamics. We refer to [98] for an introduction to the Keldysh This circumstance found its physical interpretation and techni- functional integral. In our context, it can be derived by a di- cal remedy in equilibrium in terms of the renormalization group rect functional integral quantization of the Markovian quantum [80, 81]. We emphasize that it is precisely this situation of long master equation. This procedure results in a simple translation wavelength dominance which underlies much of the universality, table from the master equation to the key player in the associated i.e., insensitivity to microscopic details, which is encountered Keldysh functional integral, the Markovian action. At first sight, when moving from the microscale to macroscopic observables the complexity of the non-equilibrium Keldysh functional inte- in many-particle systems. gral is increased by the characteristic “doubling” of degrees of The modern framework to understand many-particle prob- freedom compared to thermal equilibrium. However, it should lems in thermodynamic equilibrium is in terms of the functional be noted that it is precisely this feature which relates the Keldysh integral formulation of quantum field theory. The spectrum of functional integral much closer to real time (Heisenberg-von its application ranges over a remarkable range of energy scales, Neumann) evolution familiar from quantum mechanics. We will from ultracold atomic gases over condensed matter systems with demonstrate this in Sec.IIA, making it a rather intuitive object strong correlations to quantum chromodynamics and the quan- to work with. Furthermore, when properly harnessing symme- tum theory of gravity. It provides us with a well-developed tool- tries, the complexity of calculations can often be made compara- box of techniques, such as diagrammatic perturbation theory ble to thermodynamic equilibrium. Most importantly however, including sophisticated resummation schemes. But it also en- the powerful toolbox of quantum field theory is opened in this compasses non-perturbative approaches, which often capitalize way. It is thus possible to leverage the full power of sophisti- on the flexibility of the functional integral when it comes to pick- cated techniques from equilibrium field theory to driven open ing the relevant degrees of freedom for a given problem. This is many-body systems. the challenge of emergent phenomena, whose solution typically Relation to classical dynamical field theories — The quan- is strongly scale dependent [82]. Familiar examples include tum mechanical Keldysh formulation reduces to the so-called an efficient description of emergent Cooper pair or molecular Martin-Siggia-Rose (MSR) functional integral in the (semi-) degrees of freedom in interacting fermion systems, or vortices classical limit, in turn equivalent to a stochastic Langevin equa- which conveniently parameterize the long-wavelength physics tion formulation [98]. This statement will be made precise and of interacting bosons in two dimensions. The description of the discussed in Sec.IIC. A large amount of work has been dedi- change of physics with scale was given its mathematical foun- cated to this limit in the past. On the one hand, this concerns 5 dynamical aspects of equilibrium statistical mechanics, and we bosonic degrees of freedom. However, it is also possible to ad- refer to the classic work by Hohenberg and Halperin [1] for an dress spin systems in terms of functional integrals [106], and overview. This work shows that, while the static universal crit- simple models systems have been analyzed in this way, see [107] ical behavior is determined by the symmetries and the dimen- and Sec. III A. More sophisticated approaches to spin systems sionality of the problem, the dynamical critical behavior is sen- were elaborated in the context of multimode optical cavities sitive to additional dynamical conservation laws. This leads to in [108], and systematically for various symmetries for lattice a fine structure, defining dynamical universality classes spanned systems in [109]. Fermi statistics is also conveniently imple- by models A–J. These models also provide a convenient frame- mented in the functional integral formulation. This is relevant, work to describe the statistics of work, summarized in Jarzyn- e.g., for driven open Fermi gases in optical cavities [21, 22, 110] ski’s work theorem [99, 100] and Crooks’ relation [101]. On or lattices [111], or dissipatively stabilized topological fermion the other hand, non-equilibrium situations are captured as well. matter [66–68]. Here, we highlight in particular genuine non-equilibrium univer- sality classes, which are not smoothly connected to the equilib- rium models. Among them is the problem of reaction-diffusion C. Experimental platforms models [102] including directed percolation [4], which is rele- vant to certain chemical processes (for an implementation of this The progress in controlling, manipulating, detecting and scal- universality class with driven Rydberg gases, see [47]). Another ing up driven open quantum systems to many-body scenarios has key example is surface growth, described by the Kardar-Parisi- been impressive over the last decade. Here we sketch the basic Zhang equation [69], giving rise to a non-equilibrium universal- physics of three representative platforms, and indicate the rele- ity class which is at the heart of driven phenomena such as the vant microscopic theoretical models in the frame of the Marko- growth of bacterial colonies or the spreading of fire. vian quantum master equation. In later sections, we will trans- In the same class of approaches ranges the so-called Doi-Peliti late this physics into the language of the Keldysh functional inte- functional integral [103, 104], which is a functional representa- gral. For each of these platforms, excellent reviews exist, which tion of classical master equations, and may be viewed as an MSR we refer to at the end of Sec.ID together with further litera- theory with a specific, highly non-linear appearance of the field ture on open systems. The purpose of this section is to give variables. It reduces to the conventional MSR form in a leading an overview only, and to put the respective platforms into their order Taylor expansion of the field non-linearities. A compre- overarching context as driven open quantum systems with many hensive overview of models, methods, and physical phenomena degrees of freedom. in the (semi-)classical limit is provided in [5]. We also note that the usual mean field theory, where corre- lation functions are factorized into products of field amplitudes 1. Cold atoms in an optical cavity, and microcavity arrays: and which is often used as an approximation to the quantum driven-dissipative spin-boson models master equation in the literature, corresponds to a further for- mal simplification of the semi-classical limit. Here the effects of Cavity quantum electrodynamics (cavity QED), with its fo- noise are neglected completely. Conversely, the semi-classical cus on strong light-matter interactions, is a growing field of re- limit represents a systematic extension of mean field theory, search, which has experienced several groundbreaking advances which includes the Markovian noise fluctuations. This level of in the past few years. Historically, these systems were devel- approximation is referred to as optimal path approximation in oped as few or single atom experiments, detecting the radiation the literature on MSR functional integrals [5, 98]. properties of atoms, which are strongly coupled to a quantized In many cases, even though the microscopic description is light field. The focus has recently been shifted towards loading in terms of a quantum master equation, at long distances the more and more atoms inside a cavity. Thereby, not only single Keldysh field theory reduces to a semi-classical MSR field the- particle dynamics in strong radiation fields can be probed, but ory. The reason is the finite Markovian noise level that such also collective, macroscopic phenomena, which are driven by systems exhibit generically, as explained in Sec.IIC. The prefix light-matter interactions. For an excellent review on this topic, “semi” refers to the fact that phase coherence may still persist covering important experimental and theoretical developments, in such circumstances — the situation is comparable to a Bose- see [10]. In the experiments, cold atoms are loaded inside an Einstein condensate at finite temperature. Recently however, sit- optical or microwave cavity, for which the coherent interaction uations have been identified where the drastic simplifications of between the atomic internal states and a single cavity mode dom- the semi-classical limit do not apply. In particular, this occurs inates over dissipative processes [112]. The atoms absorb and in systems with dark states — pure quantum states which are emit cavity photons, thereby changing their internal states. Due dynamical fixed points of driven-dissipative evolution [50, 105]. to this process, the spatial modulation of the intra-cavity light In these cases, classical dynamical field theories are inappropri- field induces a coupling of the cavity photons to the atomic in- ate, which calls for the development of quantum dynamical field ternal state as well as their motional degree of freedom [113]. In theories [27]. These developments are just in their beginnings. this way, cavity photons mediate an effective atom-atom interac- In this review, we concentrate on systems composed of tion, which leads to a back-action of a single atom on the motion 6 of other atoms inside the cavity. This cavity mediated interaction of the pump laser was chosen such that the atoms effectively re- is long-ranged in space and represents the source of collective ef- main in the internal ground state, but acquire a characteristic re- fects in cold atomic clouds within cavity QED experiments. One coil momentum when scattering with a cavity photon. This scat- hallmark of collective dynamics in cavity QED has been the ob- tering creates a collective, motionally excited state, which re- servation of self-organization of a Bose-Einstein condensate in places the role of an individual, internally excited atom. The ex- an optical cavity. This is accompanied by a a Dicke phase tran- perimental realization of a superradiance transition in the Dicke sition via breaking of a discrete Z2-symmetry of the underlying model is usually inhibited, since the required coupling strength model [9, 114–116]. by far exceeds the available value of the atomic dipole coupling. Although the dominant dynamics in these systems is coherent, However, for the BEC in the cavity, the energy scales of the ex- there are dissipative effects which cannot be discarded. These cited modes are much lower than the optical scale of the atomic are the loss of cavity photons due to imperfections in the cav- modes. In this way, the superradiance transition indeed became ity mirrors, or spontaneous emission processes of atoms, which experimentally accessible. This was inspired by a theoretical emit photons into transverse modes. These effects modify the proposal using two balanced Raman channels between different dynamics of the system on the longest time scales. Therefore, internal atomic states inside an optical cavity, which reduced the they become relevant for macroscopic phenomena, such as phase effective level splitting of the internal states to much lower en- transitions and collective dynamics. For instance, dissipative ef- ergy scales [124]. fects have been shown theoretically [117, 118] and experimen- In addition to the unitary dynamics represented by the Dicke tally [119] to modify the critical exponent of the Dicke transi- model, the cavity is subject to permanent photon loss due to im- tion compared to its zero temperature value. This illustrates that perfections in the cavity mirrors. For high finesse cavities, the for the analysis of collective phenomena in cavity QED experi- coupling of the intra cavity photons to the surrounding vacuum ments, the dissipative nature of the system has to be taken into radiation field is very weak, and the latter can be eliminated in a account properly [107, 120]. Born-Markov approximation [77]. This results in a Markovian Typically, for a cavity field which has a very narrow spectrum, quantum master equation for the system’s density matrix the atomic internal degrees of freedom can be reduced to two − L internal states, whose transitions are near resonant to the pho- ∂tρ = i[HD, ρ] + ρ, (2) ton frequency. The operators acting on these two internal states where ρ is the density matrix for the intra cavity system and L can be represented by Pauli matrices, making them equivalent adds dissipative dynamics to the coherent evolution of the Dicke to a spin-1/2 degree of freedom. A very important model in the model. For a vacuum radiation field, it is given by framework of cavity QED is the Dicke model [121–123], which  † 1 †  describes N atoms (i.e., two-level systems) coupled to a single Lρ = κ LρL − 2 {L L, ρ} . (3) quantized photon mode. This is expressed by the Hamiltonian (here and in the following we set ~ = 1) The Lindblad operator L acting on the density matrix describes pure photon loss (L = a) with an effective loss rate κ; the lat- N N † ωz X z g X x  †  ter depends on system specific parameters, but is typically the HD = ω0a a + σ + √ σi a + a . (1) 2 i N smallest scale in the master equation (2)[10]. i=1 i=1 In generic cavity experiments, there are also atomic sponta- z Here, ω0 is the photon frequency, σ = |1ih1| − |0ih0| repre- neous emission processes. The atoms scatter a laser or cavity sents the splitting of the two atomic levels with energy differ- photon out of the cavity, and this represents a source of deco- x ence ωz. σ = |0ih1| + |1ih0| describes the coherent excitation herence. This process has been considered in Ref. [107], and and de-excitation of the atomic state proportional to the atom- leads to an effective decay rate of the atomic excited state and photon coupling strength g. The Dicke model features a dis- therefore to a dephasing of the atoms, described by additional crete Z Ising symmetry: it is invariant under the transforma- z 2 Lindblad operators Li = σi [125]. However, these losses are † x † x tion (a , a, σi ) 7→ (−a , −a, −σi ). In the thermodynamic limit, at least three orders of magnitude smaller than the cavity de- for N → ∞, it features a phase transition, which spontaneously cay rate and typically not considered [9, 120]. The basic model breaks the Ising symmetry. Crossing the transition, the system for cavity QED with cold atoms is therefore represented by the enters a superradiant phase, characterized by finite expectation Dicke model with dissipation, formulated in terms of the master x values hai , 0, hσi i , 0. This describes condensation of the equation (2). Its dynamics is discussed in Sec. III, including the cavity photons, i.e., the formation of a macroscopically occu- Dicke superradiance transition. pied, coherent intra-cavity field, and a “ferromagnetic” ordering The Dicke model Hamiltonian takes the form HD = Hc + Hs + of the atoms in the x-direction. Hcs, where Hc,s,cs represent the bosonic cavity, spin, and spin- Although the Dicke model is a standard model√ for cavity QED boson sectors, respectively. There are many directions to go be- experiments in the ultra-strong coupling limit Ng > ωzω0, it yond the Dicke model with single collective spin, still keeping has been realized only very recently in cold atom experiments, the basic feature of coupling spin to boson (cavity photon) de- where an entire BEC was placed inside an optical cavity [9]. It grees of freedom. One direction – relevant to future cold atom has been shown that this setup maps to a Dicke model, with a experiments – is to consider multimode cavities instead of a sin- “collective” spin degree of freedom [115]. Here, the detuning gle one. In particular, in conjunction with quenched disorder, 7 intriguing analogies to the physics of quantum glasses can be the excited spin states, the latter degrees of freedom can be established in this way [108, 126]. Here, the global coupling of integrated out. In this limit, their physical effect is to gen- all spins to a single mode gi ≡ g is replaced by random couplings erate Kerr-type bosonic non-linearities, giving rise to driven gi,`, where the index ` now refers to a collection of cavity modes. open variants of the celebrated Bose-Hubbard model. These The many-body physics of such an open system is discussed in models can even be brought into the correlation dominated Sec. III. regime [24, 127, 128, 133]. Oftentimes, these approximations on The basic building block of the Dicke model is the spin-boson the spin sector actually apply, and it is both useful and interesting † x term of the form Hsc ∼ (a + a )σ , i.e. a Rabi type non- to study these effective low frequency bosonic theories instead linearity that preserves the Z2 symmetry of Hc,s. In the con- of the full many-body spin-boson problems, see Refs. [134–138] text of circuit quantum electrodynamics, a natural many-body for recent work in this direction. generalization of Hamiltonians with a spin-boson interaction is to consider entire arrays of microcavities (instead of consider- ing many modes within a single cavity). These cavities can be 2. Exciton-polariton systems: driven open interacting bosons coupled to each other by single photon tunnelling processes be- tween adjacent cavities, giving rise to Hubbard-type hopping † Exciton-polaritons are an extremely versatile experimental terms ∼ Jai a j + h. c., where i, j now label the spatial index platform, which is documented by the richness of physical phe- of the cavities. In cirquit QED, strong non-linearities can be nomena that have been studied in these systems both in the- generated, e.g., by coupling to adjacent qubits made of Cooper ory and experiment. For a comprehensive account of the sub- pair boxes [127, 128]. This gives rise many-body variants of ject, we refer to a number of excellent review articles [13, 139, the Rabi model [129], whose phase diagrams have been studied 140]. A Keldysh functional integral approach is discussed in recently [130]. Furthermore, for the implementation of lattice Refs. [141, 142], which provides both a microscopic deriva- Dicke models with large collective spins, the use of hybrid quan- tion of an exciton-polariton model and a mean field analysis tum systems consisting of superconducting cavity arrays cou- including Gaussian fluctuations. At this point, we content our- pled to solid-state spin ensembles have been proposed [131]. selves with a short introduction, with the aim of showing that In many physical situations (away from the ultra-strong cou- in a suitable parameter regime, exciton-polaritons very natu- pling limit), a form of the spin-cavity interaction alternative rally provide a test-bed to study Bose condensation phenom- to the Rabi term is more appropriate, with non-linear building ena out of thermal equilibrium. Similar physics can also arise + † − block Hcs ∼ aσ + a σ . In fact, this form results naturally in a variety of other systems, including condensates of pho- from the weak coupling rotating wave approximation of a driven tons [143], magnons [144], and potentially excitons [145]. Re- spin-cavity problem. For a single cavity mode and spin, the re- markably, even cold atoms could be brought to condense in a sulting model is the Jaynes-Cummings model, which in contrast non-equilibrium regime, where continuous loading of atoms bal- to the Dicke model possesses a continuous U(1) phase rotation ances three-body losses [146], or in atom laser setups [147–149]. − − symmetry under a → eiθa, σ → eiθσ . A basic experimental setup for exciton-polaritons consists of Clearly, when such systems are driven coherently via a Hamil- a planar semiconductor microcavity embedding a quantum well † tonian Hd = Ω(a + a ) or suitable multimode generalizations (see Fig.1 (a)). This setting allows for a strong coupling of cav- thereof, both the U(1) and even the Z2 symmetries of the above ity light and matter in the quantum well, as originally proposed models are broken explicitly. Coherent drive is usually the sim- in [150]. The free dynamics of the elementary excitations of this plest way to compensate for unavoidable losses due to cavity system — i.e., of cavity photons and Wannier-Mott excitons — leakage, although incoherent pumping schemes are conceivable is described by the quadratic Hamiltonian [13] using multiple qubits [132]. An advantage of such schemes is that the symmetries of the underlying dynamics are preserved H0 = HC + HX + HC−X, (4) or less severely corrupted in this way (cf. the discussion in Sec.IID). In other platforms, such as exciton-polariton systems where the parts of the Hamiltonian involving only photons and (cf. the subsequent section), incoherent pumping is more natural excitons, respectively, take the same form, which is given by from the outset. (here the index α labels cavity photons, α = C, and excitons, All the systems discussed here represent genuine instances of α = X, respectively)1 driven open many-body systems. Besides the coherent drive, Z dq X they undergo dissipative processes, which have to be taken into H ω q a† q a q . α = 2 α( ) α,σ( ) α,σ( ) (5) account for a proper understanding of their time evolution and (2π) σ stationary states. A generic feature of these processes is their locality. For the effective spin degrees of freedom, the typical − processes are qubit decay (local Lindblad operators Li = σi ) and z dephasing (Li = σi ). For the bosonic component, local single- 1 In Ref. [141, 142], a different model for excitons is used: they are assumed photon loss is dominant, Li = ai. to be localized by disorder, and interactions are included by imposing a hard- Under various circumstances, such as a low population of core constraint. 8

† Field operators aα,σ(q) and aα,σ(q) create and destroy, respec- (a) tively, a photon or exciton (note that both are bosonic excita- photon tions) with in-plane momentum q and polarization σ (there are exciton two polarization states of the exciton which are coupled to the e cavity mode [13]). For simplicity, we neglect polarization ef- fects leading to an effective spin-orbit coupling [13]. Due to the h confinement in the transverse (z) direction, i.e., along the cav- ity axis, the motion of photons in this direction is quantized as q = πn/l , where n is a positive integer, and l is the length z,n z z Braggmirror QW Braggmirror of the cavity. In writing the Hamiltonian (5), we are assuming that only the lowest transverse mode is populated, which leads to (b) upper ! a quadratic dispersion as a function of the in-plane momentum polariton q q = |q| = q2 + q2: x y photon q q2 excitation ω (q) = c q2 + q2 = ω0 + + O(q4). (6) relaxation C z,1 C 2m C exciton 0 Here, c is the speed of sound, ω = cqz, , and the effective mass C 1 lower of the photon is given by mC = qz,1/c. Typically, the value of the photon mass is orders of magnitude smaller than the mass of the polariton exciton, so that the dispersion of the latter appears to be flat on emission the scale of Fig.1 (b). q Upon absorption of a photon by the semiconductor, an exciton is generated. This process (and the reverse process of the emis- Figure 1. (a) Schematic of two Bragg mirrors forming a microcavity, sion of a photon upon radiative decay of an exciton) is described in which a quantum well (QW) is embedded. In the regime of strong by light-matter interaction, the cavity photon and the exciton hybridize and form new eigenmodes, which are called exciton-polaritons. (b) Energy Z dq X   H a† q a q . . , dispersion of the upper and lower polariton branches as a function of in- C−X = ΩR 2 X,σ( ) C,σ( ) + H c (7) (2π) σ plane momentum q. In the experimental scheme illustrated in this figure (cf. Ref. [12]), the incident laser is tuned to highly excited states of the where ΩR is the rate of the coherent interconversion of photons quantum well. These undergo relaxation via emission of phonons and into excitons and vice versa. The quadratic Hamiltonian (4) can scattering from polaritons, and accumulate at the bottom of the lower be diagonalized by introducing new modes — the lower and polariton branch. In the course of the relaxation process, coherence is upper exciton-polaritons, ψLP,σ(q) and ψUP,σ(q) respectively, quickly lost, and the effective pumping of lower polaritons is incoher- which are linear combinations of photon and exciton modes. The ent. dispersion of lower and upper polaritons is depicted in Fig.1 (b). In the regime of strong light-matter coupling, which is reached when ΩR is larger than both the rate at which pho- peculiarity of exciton-polaritons is that they are excitations with tons are lost from the cavity due to mirror imperfections and the relatively short lifetime. In turn, this necessitates continuous re- non-radiative decay rate of excitons, it is appropriate to think of plenishment of energy in the form of laser driving in order to exciton-polaritons as the elementary excitations of the system. maintain a steady population. In Fig.1 (b), we consider the In experiments, it is often sufficient to consider only lower case in which the excitation laser is tuned to energies well above polaritons in a specific spin state, and to approximate the disper- the lower polariton band. The thus created high-energy excita- sion as parabolic [13]. Interactions between exciton-polaritons tions are deprived of their excess energy via phonon-polariton originate from various physical mechanisms, with a dominant and stimulated polariton-polariton scattering. Eventually, they contribution stemming from the screened Coulomb interactions accumulate at the bottom of the lower polariton band. As a con- between electrons and holes forming the excitons. Again, in the sequence of multiple scattering processes, the coherence of the low-energy scattering regime, this leads to an effective contact incident laser field is quickly lost, and the effective pumping of interaction between lower polaritons. As a result, the low-energy lower polaritons is incoherent. description of lower polaritons takes the form (in the following A phenomenological model for the dynamics of the lower po- we drop the subscript indices in ψLP,σ)[13] lariton field, which accounts for both the coherent dynamics gen- Z " 2 ! # erated by the Hamiltonian (8) and the driven-dissipative one de- † 0 ∇ † 2 2 HLP = dx ψ (x) ω − ψ(x) + ucψ (x) ψ(x) . (8) scribed above, was introduced in Ref. [151]. It involves a dissi- LP 2m LP pative Gross-Pitaevskii equation for the lower polariton field that While this Hamiltonian is quite generic and arises also, e.g., in is coupled to a rate equation for the reservoir of high-energy ex- cold bosonic atoms in the absence of an external potential, the citations. However, for the study of universal long-wavelength 9 behavior in Sec.IV, this degree of microscopic modeling is ac- reached, for which the dissipative coupling to the environment tually not required: indeed, any (possibly simpler) model that becomes visible in experimental observables. Such dissipation possesses the relevant symmetries (see the discussion at the be- may even hinder the system from relaxation towards a well de- ginning of Sec.IV) will yield the same universal physics. Such fined steady state. a model can be obtained by describing incoherent pumping and A relevant example of a dissipative coupling is decoherence losses of lower polaritons by means of a Markovian master equa- of an atomic cloud induced by spontaneous emission of atoms tion:2 in the lattice [158, 159]. In this way, the many-body system is heated up, and therefore driven away from the low entropy ∂tρ = −i[HLP, ρ] + Lρ, (9) state in which it was prepared initially. A detailed discussion of the microscopic physics and its long time dynamics can be where Lρ encodes incoherent single-particle pumping and found in [42–44], see also the review article [45]. For bosonic losses, as well as two-body losses: atoms in optical lattices, the coherent dynamics is described by Z the Bose-Hubbard Hamiltonian [160, 161]  † 2  Lρ = dx γpD[ψ(x) ]ρ + γlD[ψ(x)]ρ + 2udD[ψ(x) ]ρ , X U X H = −J b†b + n (n − 1), (12) (10) BH l m 2 l l hl,mi l where which models bosonic atoms in terms of the creation and annihi- † † 1 † D[L]ρ = LρL − {L L, ρ} (11) lation operators [bl , bm] = δlm in the lowest band of a lattice with 2 site indices l, m. The atoms hop between neighboring lattice sites with an amplitude J, and experience an on-site repulsion U. The reflects the Lindblad form, and γp, γl, 2ud are the rates of single- particle pumping, single-particle loss, and two-body loss, re- lattice potential V(x), which leads to the second quantized form spectively. The inclusion of the non-linear loss term ensures of the Bose-Hubbard model, is created by the superposition of saturation of the pumping. An analogous mechanism is im- counter-propagating laser beams in each spatial dimension. The plemented in the above-mentioned Gross-Pitaevskii description. coherent laser field couples two internal atomic states via stimu- More precisely, in the spirit of universality, the above quantum lated absorption and emission, which leads to the single particle master equation (9) and the above mentioned phenomenologi- Hamiltonian pˆ 2 ∆ Ω(xˆ) cal model reduce to precisely the same low frequency model for H = − σz − σx , (13) bosonic degrees of freedom upon taking the semiclassical limit atom 2m 2 2 in the Keldysh path integral associated to Eq. (9) (see Sec.IIC where pˆ is the atomic momentum operator, ∆ is the detuning of for its implementation), and integrating out the upper polariton the laser from the atomic transition frequency, and Ω(xˆ) is the reservoir in the phenomenological model. laser field at the atomic position. For large detuning, the excited state of the atom can be traced out, which leads to the lattice Hamiltonian 3. Cold atoms in optical lattices: heating dynamics pˆ 2 |Ω(xˆ)|2 Heff = + . (14) atom 2m 4∆ In recent years, experiments with cold atoms in optical lat- tices have shown remarkable progress in the simulation of many- This describes a lattice potential for the ground state atoms body model systems both in and out of equilibrium. A par- generated by the spatially modulated AC Stark shift. Adding ticular strength of cold atom experiments is the unprecedented an atomic interaction potential and expanding both the single tuneability of model parameters, such as the local interaction particle Hamiltonian (14) and the interaction in terms of Wan- strength and the lattice hopping amplitude. This becomes possi- nier states, the leading order Hamiltonian is the Bose-Hubbard ble by, e.g., manipulation of the lattice laser and external mag- model (12)[161]. netic fields. It comes along with a very weak coupling of the In the semi-classical treatment of the atom-laser interaction, system to the environment, such that the dynamics can often be spontaneous emission events are neglected, as their probabil- seen as isolated on relevant time scales for typical measurements ity is typically very small (see below for a more precise state- of static, equilibrium correlations. However, more and more ex- ment). They can be taken into account on the basis of optical periments start to investigate the realm of non-equilibrium phe- Bloch equations [158], which leads, after elimination of the ex- nomena with cold atoms, e.g., by letting systems prepared in a cited atomic state, to an additional, driven-dissipative term in non-equilibrium initial condition relax in time towards a steady the atomic dynamics. It describes the decoherence of the atomic state [34, 36, 155–157]. With these experiments, time scales are state due to spontaneous emission, i.e., position dependent ran- dom light scattering. The leading order contribution to the dy- namics in the basis of lowest band Wannier states is captured by the master equation 2 X While this approach captures the universal behavior, we note that non- ∂ ρ = −i[H , ρ] − γ [n , [n , ρ]], (15) Markovian effects can be of key importance for other properties [152–154]. t BH l l l 10

Sec. II B 2: there we discuss the mean field theory of conden- sation in a bosonic many-body system with particle losses and pumping. The semi-classical limit of this model and its valid- ity are the content of Sec.IIC. This is followed by a discussion of symmetries and conservation laws in the Keldysh formalism in Sec.IID. In particular, we point out a symmetry that allows one to distinguish equilibrium from non-equilibrium conditions. Finally, an advanced field theoretical tool — the open system Figure 2. Illustration of the coherent (a) and the incoherent (b) con- functional renormalization group — is introduced in Sec.IIE. tribution to the dynamics stemming from the coupling of the atoms to a laser with amplitude |Ω| and large detuning ∆. (a) Via the AC Stark PartII harnesses this formalism to generate an understanding effect, stimulated absorption and emission lead to a coherent periodic of the physics in different experimental platforms. We begin in |Ω|2 Sec. III with simple but paradigmatic spin models with discrete potential with amplitude 4∆ . (b) Stimulated absorption and subsequent spontaneous emission lead to effective decoherence of the atomic state Ising Z2 symmetry in driven flux equilibrium states. In partic- 2 with rate Γ |Ω| , where Γ is the microscopic spontaneous emission rate. ular, we discuss the physics of the driven open Dicke model in 4∆ Sec. III A. This is followed by an extended variant of the lat- ter in the presence of disorder and a multimode cavity, which hosts an interesting spin and photon glass phase in Sec. III B. where n = b†b is the local atomic density, and ρ is the many- l l l Sec.IV is devoted to the non-equilibrium stationary states of |Ω|2 body density matrix. For a red detuned laser the rate γ = Γ 4∆2 is bosons with a characteristic U(1) phase rotation symmetry: the proportional to the microscopic spontaneous emission rate Γ and driven-dissipative condensates introduced in Sec. II B 2. After the laser amplitude |Ω|. Note the suppression of the scale γ by some additional technical developments relating to U(1) sym- a factor Γ/∆  1 for large detuning, compared to the strength metry in Sec.IVA, we discuss critical behavior at the Bose eff of Hatom. The coherent and incoherent contribution of the atom- condensation transition in three dimensions. In particular, we laser coupling to the dynamics is illustrated in Fig.2. show the decrease of a parameter quantifying non-equilibrium The dissipative term in the master equation (15) leads to an strength in this case (Sec.IVB). The opposite behavior is ob- energy increase hHBHi(t) ∼ t linear in time, and thus to heat- served in two (Sec.IVC) and one (Sec.IVD) dimensions. Fi- ing. Furthermore, it introduces decoherence in the number state nally, leaving the realm of stationary states, in Sec.V we dis- basis, i.e., it projects the local density matrix on its diagonal in cuss an application of the Keldysh formalism to the time evolu- Fock space and leads to a decrease of the coherences in time [42– tion of open bosonic systems in one dimension, which undergo 45]. Starting from a low entropy state at t = 0, the heating leads number conserving heating processes. We set up the model in to a crossover from coherence dominated dynamics at short and Sec.VA, derive the kinetic equation for the distribution func- intermediate times [40] to a decoherence dominated dynamics at tion in Sec.VB, and discuss the relevant approximations and long times [42–44]. In one dimension, both regimes have been physical results in Secs.VC andVD, respectively. Conclusions analyzed extensively both numerically (with a focus on deco- are drawn in Sec.VI. Finally, brief introductions to functional herence dominated dynamics) as well as analytically, and dis- differentiation and Gaussian functional integration are given in play several aspects of non-equilibrium universality, see Sec.V. AppendicesA andB. Therefore, heating in interacting lattice systems represents a cru- Reflecting the bipartition of this review, the scope of it is cial example for universality in out-of-equilibrium dynamics, twofold. On the one hand, it develops the Keldysh functional in- which can be probed by cold atom experiments. tegral approach to driven open quantum systems “from scratch”, in a systematic and coherent way. It starts from the Marko- vian quantum master equation representation of driven dissipa- D. Outline and scope of this review tive quantum dynamics [77, 78, 162], and introduces an equiv- alent Keldysh functional integral representation. Direct contact The remainder of this review is split into two parts. is made to the language and typical observables of quantum op- PartI develops the theoretical framework for the e fficient de- tics. It does not require prior knowledge of quantum field the- scription of driven open many-body quantum systems. In Sec.II, ory, and we hope that it will find the interest of — and be use- we begin with a direct derivation of the open system Keldysh ful for — researchers working on quantum optical systems with functional integral from the many-body quantum master equa- many degrees of freedom. On the other hand, this review doc- tion (Sec.IIA). We then discuss in Sec. II B 1 in detail a simple uments some recent theoretical progresses made in this concep- example: the damped and driven optical cavity. This allows the tual framework in a more pedagogical way than the original lit- reader to familiarize with the functional formalism. In particu- erature. We believe that this not only exposes some interesting lar, the key players in terms of observables — correlation and re- physics, but also demonstrates the power and flexibility of the sponse functions — are described. We also point out a number of Keldysh approach to open quantum systems. exact structural properties of Keldysh field theories, which hold This work is complemetary to excellent reviews putting more beyond the specific example. Another example is introduced in emphasis on the specific experimental platforms partially men- 11 tioned above: The physics of driven Bose-Einstein condensates the norm tr(ρ) of the system density matrix, this term must be ac- in optical cavities is reviewed in [10]. A general overview of companied by fluctuations. The corresponding term, where the driven ultracold atomic systems, specifically in optical lattices, Lindblad operators act from both sides onto the density matrix, is provided in [45], and systems with engineered dissipation are is referred to as recycling or quantum jump term. Dissipation described in [53]. Detailed accounts for exciton-polariton sys- occurs at rates γα which are non-negative, so that the density tems are given in [13, 139, 140]; specifically, we refer to the matrix evolution is completely positive, i.e., the eigenvalues of review [141] working in the Keldysh formalism. The physics of ρ remain positive under the combined dynamics generated by H microcavity arrays is discussed in [14, 24, 129, 133], and trapped and D [168]. If the index α is the site index in an optical lattice ions are treated in [17, 163]. We also refer to recent reviews or in a microcavity array, or even a continuous position label (in on additional upcoming platforms of driven open quantum sys- which case the sum is replaced by an integral), in a translation tems, such as Rydberg atoms [164] and opto-nanomechanical invariant situation there is just a single scale γα = γ for all α settings [165]. For a recent exposition of the physics of quan- associated to the dissipator. tum master equations and to efficient numerical techniques for The quantum master equation (16) provides an accurate de- their solution, see [45]. scription of a system-bath setting with a strong separation of scales. This is generically the case in quantum optical systems, which are strongly driven by external classical fields. More pre- Part I cisely, there must be a large energy scale in the bath (as com- pared to the system-bath coupling), which justifies to integrate Theoretical background out the bath in second-order time-dependent perturbation the- ory. If in addition the bath has a broad bandwidth, the combined II. KELDYSH FUNCTIONAL INTEGRAL FOR DRIVEN Born-Markov and rotating-wave approximations are appropri- OPEN SYSTEMS ate, resulting in Eq. (16). A concrete example for such a set- ting is provided by a laser-driven atom undergoing spontaneous emission. Generic condensed matter systems do not display such In this part, we will be mainly concerned with a Keldysh field a scale separation, and a description in terms of a master equa- theoretical reformulation of the stationary state of Markovian tion is not justified. However, the systems discussed in the intro- many-body quantum master equations. As we also demonstrate, duction, belong to the class of systems which permit a descrip- this opens up the powerful toolbox of modern quantum field the- tion by Eq. (16). We refer to them as driven open many-body ory for the understanding of such systems. quantum systems. The quantum master equation, examples of which we have Due to the external drive these systems are out of thermody- already encountered in Sec.IC, describes the time evolution of namic equilibrium. This statement will be made more precise a reduced system density matrix ρ and reads [77, 78] in Sec. II D 1 in terms of the absence of a dynamical symme- ! try which characterizes any system evolving in thermodynamic X † 1 † ∂tρ = Lρ = −i[H, ρ] + γα LαρL − {L Lα, ρ} , (16) equilibrium, and which is manifestly violated in dynamics de- α 2 α α scribed by Eq. (16). Its absence reflects the lack of energy con- servation and the epxlicit breaking of detailed balance. As stated where the operator L acts on the density matrix ρ “from both in the introduction, the main goal of this review is point out the sides” and is often referred to as Liouville superoperator or Li- macroscopic, observable consequences of this microscopic vio- ouvillian (sometimes this term is reserved for the second con- lation of equilibrium conditions. tribution on the RHS of Eq. (16) alone). There are two contri- butions to the Liouvillian: first, the commutator term, which is familiar from the von Neumann equation, describes the coher- ent dynamics generated by a system Hamiltonian H; the second A. From the quantum master equation to the Keldysh functional part, which we will refer to as the dissipator D 3, describes the integral dissipative dynamics resulting from the interaction of the sys- tem with an environment, or “bath.” It is defined in terms of a In this section, starting from a many-body quantum master set of so-called Lindblad operators (or quantum jump operators) equation Eq. (16) in the operator language of second quantiza- Lα, which model the coupling to that bath. The dissipator has a tion, we derive an equivalent Keldysh functional integral. We characteristic Lindblad form [166, 167]: it contains an anticom- focus on stationary states, and we discuss how to extract dynam- mutator term which describes dissipation; in order to conserve ics from this framework in Sec.V. Our derivation of the Keldysh functional integral applies to a theory of bosonic degrees of free- dom. If spin systems are to be considered, it is useful to first per- form the typical approximations mapping them to bosonic fields, 3 We use the term “dissipation” here for all kinds of environmental influences and then proceed along the construction below (see Sec. III and on the system which can be captured in Lindblad form, including effects of Refs. [107, 109]). Clearly, this amounts to an approximate treat- decay and of dephasing/decoherence. ment of the spin degrees of freedom; for an exact (equilibrium) 12 functional integral representation for spin dynamics, taking into a) account the full non-linear structure of their commutation rela- (t0) tions, we refer to Ref. [169]. For fermionic problems, the con- t t t0 | i struction is analogous to the bosonic case presented here. How- U ever, a few signs have to be adjusted to account for the fermionic b) anticommutation relations [98, 169]. The basic idea of the Keldysh functional integral can be de- ⇢(t0) t t veloped in simple terms by considering the Schrodinger¨ vs. the U U † t von Neumann equation, c) + contour

i∂t|ψ(t)i = H|ψ(t)i ⇒ |ψ(t)i = U(t, t0)|ψ(t0)i, ⇢(tf ) ⇢(t0)

† tf =+ - contour t = ∂tρ(t) = −i[H, ρ(t)] ⇒ ρ(t) = U(t, t0)ρ(t0)U (t, t0), 1 0 1 (17) Figure 3. Idea of the Keldysh functional integral. a) According to the −iH(t−t0) where U(t, t0) = e is the unitary time evolution opera- Schrodinger¨ equation, the time evolution of a pure state vector is de- −iH(t−t0) tor. In the first case, a real time path integral can be constructed scribed the unitary operator U(t, t0) = e . In the Feynman func- along the lines of Feynman’s original path integral formulation tional integral construction, the time evolution is chopped into infinites- of quantum mechanics [170]. To this end, a Trotter decomposi- imal steps of length δt, and completeness relations in terms of coherent tion of the evolution operator states are inserted in between consecutive time steps. This insertion is signalled by the red arrows. b) In contrast, if the state is mixed, a den- −iH(t−t0) N e = lim (1 − iδtH) , (18) sity matrix must be evolved, and thus two time branches are needed. As N→∞ explained in the text, the dynamics need not necessarily be restricted to

t−t0 unitary evolution. The most general time-local (Markovian) dynamics with δt = N , is performed. Subsequently, in between the is generated by a Liouville operator in Lindblad form. c) For the anal- factors of the Trotter decomposition, completeness relations in ysis of the stationary state, we are interested in the real time analog of terms of coherent states are inserted in order to make the (nor- a partition function Z = tr ρ(t), starting from t0 = −∞ and running until mal ordered) Hamilton operator a functional of classical field t f = +∞. The trace operation connects the two time branches, giving variables. This is illustrated in Fig.3 a), and we will perform rise to the closed Keldysh contour. these steps explicitly and in more detail below in the context of open many-body systems. Crucially, we only need one set of field variables representing coherent Hamiltonian dynamics, So far, we have concentrated on closed systems which evolve which corresponds to the forward evolution of the Schrodinger¨ according to purely Hamiltonian dynamics. However, we can state vector. It is also clear that — noting the formal analogy of allow for a more general generator of dynamics and still proceed the operators e−iH(t−t0) and e−βH — this construction can be lever- along the two-branch strategy. The most general (time local) aged over to the case of thermal equilibrium, where the “Trotter- evolution of a density matrix is given by the quantum master ization” is done in imaginary instead of real time. equation (16). Its formal solution reads In contrast to these special cases, the von Neumann equation (t−t0)L N ρ(t) = e ρ(t0) ≡ lim (1 + δtL) ρ(t0). (19) for general mixed state density matrices cannot be rewritten in N→∞ terms of a state vector evolution, even in the case of purely co- herent Hamiltonian dynamics.4 Instead, it is necessary to study The last equality gives a meaning to the formal solution in terms the evolution of a state matrix, which transforms according to of the Trotter decomposition: at each infinitesimal time step, the the integral form of the von Neumann equation in the second exponential can be expanded to first order, such that the action line of Eq. (17). Therefore, we have to apply the Trotter formula of the Liouvillian superoperator is just given by the RHS of the and coherent state insertions on both sides of the density matrix. quantum master equation (16); At finite times, the evolved state This leads to the doubling of degrees of freedom, characteristic is given by the concatenation of the infinitesimal Trotter steps. of the Keldysh functional integral. Moreover, time evolution can If we restrict ourselves to the stationary state of the system 5 now be interpreted as occurring along the closed Keldysh con- , but want to evaluate correlations at arbitrary time differences, tour, consisting of a forward and a backward branch (cf. Fig.3 we should extend the time branch from an initial time in the dis- b)). Indeed, this is an intuitive and natural feature of evolving tant past t0 → −∞ to the distant future, t f → +∞. In analogy to matrices instead of vectors. thermodynamics, we are then interested in the so-called Keldysh partition function Z = tr ρ(t) = 1. The trace operation contracts the indices of the time evolution operator as depicted in Fig.3

4 Of course, the evolution of an M × M matrix, where M is the dimension of the Hilbert space, can be formally recast into the evolution of a vector of length M × M. While such a strategy is often pursued in numerical approaches [45], 5 We assume that it exists. We thus exclude scenarios with dynamical limit it does in general not allow for a physical interpretation. cycles, for simplicity. 13 c), giving rise to the closed time path or Keldysh contour. Con- first on a single time step, as in the usual derivation of the co- servation of probability in the quantum mechanical system is herent state functional integral [171]. That is, we decompose the reflected in the time independent normalization of the partition time evolution from t0 to t f into a sequence of small steps of du- function. In order to extract physical information, again in anal- ration δt = (t f − t0)/N, and denote the density matrix after the ogy to statistical mechanics, below we introduce sources in the n-th step, i.e., at the time tn = t0 + δtn, by ρn = ρ(tn). We then partition function. This allows us to compute the correlation and have response functions of the system by taking suitable variational derivatives with respect to the sources. δtL 1 2 ρn+1 = e ρn = ( + δtL) ρn + O(δt ). (20) After this qualitative discussion, let us now proceed with the explicit construction of the Keldysh functional integral for open As anticipated above, we proceed to represent the density matrix systems, starting from the master equation (16). As mentioned in the basis of coherent states. For instance, ρn at the time tn can above, the Keldysh functional integral is an unraveling of Li- be written as ouvillian dynamics in the basis of coherent states, and we first collect a few important properties of those. Coherent states are Z ∗ ∗ dψ+,ndψ dψ−,ndψ− ∗ ∗ +,n ,n −ψ ψ+,n−ψ− ψ−,n defined as (for simplicity, in the present discussion we restrict ρn = e +,n ,n ourselves to a single bosonic mode b) |ψi = exp(ψb†) |Ωi, where π π |Ωi represents the vacuum in Fock space. A key property of co- × hψ+,n|ρn|ψ−,ni |ψ+,ni hψ−,n| . (21) herent states is that they are eigenstates of the annihilation oper- ator, i.e., b |ψi = ψ |ψi, with the complex eigenvalue ψ. Clearly, As a next step, we would like to express the matrix element † ∗ 6 this implies the conjugate relation hψ| b = hψ| ψ . The overlap hψ+,n+1|ρn+1|ψ−,n+1i, which appears in the coherent state repre- ψ∗φ of two (non-normalized) coherent states is given by hψ|φi = e , sentation of ρn+1, in terms of the corresponding matrix element 1 R dψdψ∗ −ψ∗ψ and the completeness relation reads = π e |ψi hψ|. at the previous time step tn. Inserting Eq. (21) in Eq. (20), we The starting point of the derivation is Eq. (19), and we focus find that this requires us to evaluate the “supermatrixelement”

hψ+,n+1|L(|ψ+,ni hψ−,n|)|ψ−,n+1i = −i hψ+,n+1|H|ψ+,ni hψ−,n|ψ−,n+1i − hψ+,n+1|ψ+,ni hψ−,n|H|ψ−,n+1i " # X 1   + γ hψ |L |ψ i hψ |L† |ψ i − hψ |L† L |ψ i hψ |ψ i + hψ |ψ i hψ |L† L |ψ i . (22) α +,n+1 α +,n −,n α −,n+1 2 +,n+1 α α +,n −,n −,n+1 +,n+1 +,n −,n α α −,n+1 α Without loss of generality, we assume that the Hamiltonian is normal ordered. Then, a matrix element hψ|H|φi of the Hamiltonian between coherent states can be obtained simply by replacing the creation operators by ψ∗ and the annihilation operators by φ. The same is true for matrix elements of L, L†, and L†L, after performing the commutations which are necessary to bring these operators to the form of sums of normal ordered expressions (see Ref. [172] for a detailed discussion of subtleties related to normal ordering). Then we obtain by re-exponentiation

Z ∗ ∗   dψ+,ndψ+,n dψ−,ndψ−,n iδ −ψ i∂ ψ∗ −ψ∗ i∂ ψ −iL(ψ∗ ,ψ ,ψ∗ ,ψ ) 2 hψ |ρ |ψ i = e t +,n t +,n −,n t −,n +,n+1 +,n −,n+1 −,n hψ |ρ |ψ i + O(δ ), (23) +,n+1 n+1 −,n+1 π π +,n n −,n t

where we are using the shorthand suggestive notation ∂tψ±,n = laps: (ψ±,n+1 − ψ±,n)/δt. The time derivative terms emerge from the overlap of neighboring coherent states at time steps n and n + 1, ∗ ∗ hψ+,n+1|L(|ψ+,ni hψ−,n|)|ψ−,n+1i combined with the weight factor in the completeness relation for L(ψ+,n+1, ψ+,n, ψ−,n+1, ψ−,n) = . step n; the quantity L(ψ∗ , ψ , ψ∗ , ψ ) is the superma- hψ+,n+1|ψ+,ni hψ−,n|ψ−,n+1i +,n+1 +,n −,n+1 −,n (24) trixelement in Eq. (22), divided by the above-mentioned over- By iteration of Eq. (23), the density matrix can be evolved from ρ(t0) at t0 to ρ(t f ) at t f = tN . This leads in the limit N → ∞ (and hence δt → 0) to

6 Note that the creation operator cannot have eigenstates due to the fact that Z ρ t e(t f −t0)Lρ t there is a minimal occupation number of a bosonic state. In particular, the t f ,t0 = tr ( f ) = tr ( 0) coherent states are not eigenstates. We rather have the relations b† |ψi = Z (25) ∗ ∗ ∗ iS (∂/∂ψ) |ψi , hψ| b = (∂/∂ψ ) hψ|. = D[ψ+, ψ+, ψ−, ψ−] e hψ+(t0)|ρ(t0)|ψ−(t0)i , 14 where the integration measure is given by functional integral is used to derive equations of motion for a given set of correlation functions. The initial values of the cor- N ∗ ∗ ∗ ∗ Y dψ+,ndψ+,n dψ−,ndψ−,n relation functions have to be taken from the physical situation D[ψ+, ψ+, ψ−, ψ−] = lim , (26) N→∞ π π under consideration. For interacting theories, the set of corre- n=0 lations functions typically corresponds to an infinite hierarchy. and the Keldysh action reads The possibility of truncating this hierarchy to a closed subset usually involves approximations, which have to be justified from Z t f ∗ ∗ ∗ ∗
case to case. S = dt ψ+i∂tψ+ − ψ−i∂tψ− − iL(ψ+, ψ+, ψ−, ψ−) . (27) The Keldysh partition function is normalized to 1 by construc- t 0 tion. As anticipated above, correlation functions can be obtained ∗
The coherent state representation of L in the exponent in by introducing source terms J± = j±, j± that couple to the ∗
Eq. (23) comes with a prefactor δt, so that to leading order for fields Ψ± = ψ±, ψ± ,

δt → 0 it is consistent to ignore the difference stemming from Z R  † †  iS +i J+Ψ+−J−Ψ− the bra vector at n + 1 and the ket vector at n in Eq. (24). As- Z[J+, J−] = D[Ψ+, Ψ−] e t,x suming all operators are normally ordered in the sense discussed  R   (31) i J† Ψ −J† Ψ above, we obtain = e t,x + + − − ,

∗ ∗ L(ψ , ψ+, ψ−, ψ−) = −i (H+ − H−) where we abbreviate (switching now to a spatial continuum of + R R R d " # fields) t,x = dt d x, the average is taken with respect to the X ∗ 1  ∗ ∗  + γ L L − L L + L L − , (28) action S , and we have the normalization Z[J = 0, J = 0] = 1 α α,+ α,− 2 α,+ α,+ α,− α, + − α in the absence of sources. Physically, sources can be realized, e.g., by coherent external fields such as lasers; this will be made where H = H(ψ∗ , ψ ) contains fields on the ± contour only, ± ± ± more concrete in the following section. The source terms can be and the same is true for L . We clearly recognize the Lindblad α,± thought of as shifts of the original Hamiltonian operator, justify- superoperator structure of Eq. (16): operators acting on the den- ing the von Neumann structure indicated above. sity matrix from the left (right) reside on the forward, + (back- Keldysh rotation — With these preparations, arbitrary correla- ward, -) contour. This gives a simple and direct translation ta- tion functions can be computed by taking variational derivatives ble from the bosonic quantum master equation to the Markovian with respect to the sources. However, while the representation in Keldysh action (28), with the crucial caveat of normal ordering terms of fields residing on the forward and backward branches to be taken into account before performing the translation. allows for a direct contact to the second quantized operator for- Keldysh partition function for stationary states — When we malism, it is not ideally suited for practical calculations. In fact, are interested in a stationary state, but would like to obtain infor- the above description contains a redundancy which is related to mation on temporal correlation functions at arbitrarily long time the conservation of probability (this statement is detailed below). differences, it is useful to perform the limit t → −∞, t → +∞ 0 f This can be avoided by performing the so-called Keldysh rota- in Eq. (25). In an open system coupled to several external baths, tion, a unitary transformation in the contour index or Keldysh it is typically a useful assumption that the initial state in the in- space according to finite past does not affect the stationary state — in other words, there is a complete loss of memory of the initial state. Under this 1 1 φc = √ (ψ+ + ψ−) , φq = √ (ψ+ − ψ−) , (32) physical assumption, we can ignore the boundary term, i.e., the 2 2 matrix element hψ (t )|ρ(t )|ψ (t )i of the initial density matrix + 0 0 − 0 c q in Eq. (25), and obtain for the final expression of the Keldysh and analogously for the source terms. The index ( ) stands for partition function “classical” (“quantum”) fields, respectively. This terminology signals that the symmetric combination of fields can acquire a Z (classical) field expectation value, while the antisymmetric one ∗ ∗ iS Z = D[ψ+, ψ+, ψ−, ψ−] e = 1, (29) cannot. In terms of classical and quantum fields, the Keldysh Z ∞ partition function takes the form ∗ − ∗ − L ∗ ∗
S = dt ψ+i∂tψ+ ψ−i∂tψ− i (ψ+, ψ+, ψ−, ψ−) . (30) Z R  † †  −∞ iS +i Jq Φc+Jc Φq Z[Jc, Jq] = D[Φc, Φq]e t,x This setup allows us to study stationary states far away from  R   (33) i J†Φ +J†Φ thermodynamic equilibrium as realized in the systems intro- = e t,x q c c q ; duced in Sec.I, using the advanced toolbox of quantum field ∗
theory. For the discussion of the time evolution of the system’s note in particular the coupling of the classical field Φc = φc, φc  ∗ initial state, the typical strategy in practice is not to start directly to the quantum source Jq = jq, jq , and vice versa. Apart from from Eq. (25) — strictly speaking, this would necessitate knowl- removing the redundancy mentioned above, a further key ad- edge of the entire density matrix of the system, which in a gen- vantage of this choice of basis is that taking variational deriva- uine many-body context is not available. Rather, the Keldysh tives with respect to the sources produces the two basic types 15 of observables in many-body systems: correlation and response where ν0 = q for ν = c and vice versa. Switching to these new functions. Of particular importance is the single particle Green’s variables is accomplished by means of a Legendre transform fa- function, which has the following matrix structure in Keldysh miliar from classical mechanics, space (for a brief introduction to functional differentiation see Z   AppendixA), ¯ ¯ † ¯ † ¯ Γ[Φc, Φq] = W[Jc, Jq] − Jc Φq + Jq Φc . (38) t,x ∗ 0 0 ∗ 0 0 ! hφc(t, x)φc(t , x )id hφc(t, x)φq(t , x )id We now proceed to show that the difference between the action ∗ 0 0 ∗ 0 0 hφq(t, x)φc(t , x )id hφq(t, x)φq(t , x )id S and the effective action Γ lies in the inclusion of both statis-  δ2Z δ2Z  tical and quantum fluctuations in the latter. To this end, instead  δ j∗ t,x δ j t0,x0 δ j∗ t,x δ j t0,x0  −  q( ) q( ) q( ) c( )  of working with Eq. (38), we represent Γ as a functional inte- =  δ2Z δ2Z   ∗ 0 0 ∗ 0 0  δ jc(t,x)δ jq(t ,x ) δ jc(t,x)δ jc(t ,x ) gral [173], with the result Jc=Jq=0 (34) K 0 0 R 0 0 ! G (t − t , x − x ) G (t − t , x − x ) Z ¯ ¯ δΓ δΓ d d ¯ ¯ iS [Φc+δΦc,Φq+δΦq]−i T δΦq−i T δΦc = i A 0 0 . iΓ[Φc,Φq] δΦ¯ c δΦ¯ q Gd (t − t , x − x ) 0 e = D[δΦc, δΦq] e . (39) In the last equality, in addition to stationarity (time translation in- R/A/K On the right hand side, we have used the property of the Legen- variance) we have assumed spatial translation invariance. G dre transformation are called retarded, advanced, and Keldysh Green’s function, and — in the terminology of statistical mechanics — the index ¯ ∗ ¯ ∗ δΓ/δΦc = −Jq, δΓ/δΦq = −Jc . (40) d stands for disconnected averages obtained from differentiating the partition function Z; the zero component is an exact prop- Equation (39) is obtained by exponentiating (i times) Eq. (38), erty and reflects the elimination of redundant information. We and using the explicit functional integral representation of the anticipate that the retarded and advanced components describe Keldysh partition function in W = −i ln Z, Eq. (33). We have in- ¯ responses, and the Keldysh component the correlations. The troduced the notation Φν = Φν+δΦν. This implies hδΦνi|Jc,Jq = 0 physical meaning of the Green’s function is discussed in the sub- by construction, and moreover allows us to use D[Φc, Φq] = sequent subsection by means of a concrete example. D[δΦc, δΦq] in the functional measure. Note the appearance of Keldysh effective action — At this point we need one last the field fluctuation δΦν in the last term in the functional inte- technical ingredient: the effective action, which is an alternative gral (39). This is due to the explicit subtraction of the source way of encoding the correlation and response information of a term in Eq. (38). non-equilibrium field theory [173]. We first introduce the new Equation (39) simplifies when we consider vanishing exter- generating functional nal sources Jc = Jq = 0. This case is particularly intuitive, as it shows that the effective action Γ[Φ¯ c, Φ¯ q] corresponds to sup- W[Jc, Jq] = −i ln Z[Jc, Jq]. (35) plementing the action S [Φ¯ c, Φ¯ q] by all possible fluctuation con- tributions, expressed by the functional integration over δΦ¯ c and Differentiation of the functional W generates the hierarchy δΦ¯ q. of so-called connected field averages, in which expectation The variational condition on Γ, Eq. (40) (technically reflect- values of lower order averages are subtracted; for example, ing the change of active variables via Eq. (38)), precisely takes ∗ 0 0 ∗ 0 0 ∗ 0 0 hφc(t, x)φc(t , x )i = hφc(t, x)φc(t , x )id − hφc(t, x)idhφc(t , x )id the form of the equation of motion derived from the principle of describes the density of particles which are not condensed. In a least action (in the presence of an external source or force term) compact notation, where W(2) denotes the second variation as in familiar form classical mechanics. Here, however, it governs the Eq. (34), dynamics of the full effective action. In light of the above in- terpretation of the effective action, Eq. (40) thus promotes the 0 0 W(2)(t − t , x − x ) conventional classical action principle to full quantum and sta- Jc=Jq=0 tistical status. 0 0 0 0 ! GK(t − t , x − x ) GR(t − t , x − x ) Conversely, neglecting the quantum and statistical fluctua- = − A 0 0 . (36) G (t − t , x − x ) 0 tions in Eq. (39), we directly arrive at Γ[Φ¯ c, Φ¯ q] = S [Φ¯ c, Φ¯ q]. This approximation is appropriate at intermediate distances7 in The effective action Γ is obtained from the generating functional the case where there is a macroscopically√ occupied conden- W by a change of active variables. More precisely, the active sate: we may then count Φ¯ c = O( N), with N the exten- variables of the functional W, which are the external sources sive number of particles in the condensate, while fluctuations Jc,q, i.e., W = W[Jc, Jq], are replaced by the field expectation values Φ¯ c,q, i.e., Γ = Γ[Φ¯ c, Φ¯ q]. The field expectation values are defined as (the notation indicates that these expectation values are taken in the presence of the sources) 7 At very long wavelengths, gapless fluctuations of the Goldstone mode lead to infrared divergences in perturbation theory and invalidate this power counting ¯ ∗ argument [174, 175]. Φν = hΦνi|Jc,Jq = δW/δJν0 , (37) 16

δΦc, δΦq = O(1), as well as Φ¯ q = O(1) for the noise field, which also in the presence of interactions, and are thus exact properties cannot acquire a non-vanishing expectation value. This crude of non-equilibrium field theories. We point this out alongside approximation of the full effective action reproduces the stan- the examples. dard Gross-Pitaevski mean-field theory, if we consider a generic In Sec. II B 2, we consider an example for an interacting Hamiltonian with kinetic energy and local two-body collisions, bosonic many-body system, which contains non-linearities not and drop the dissipative contributions to the action. only due to particle-particle interactions, but also in the dissi- The functions generated by the effective action functional (via pative contribution to the dynamics. These features are realized taking variational derivatives with respect to its variables Φ¯ c, Φ¯ q) experimentally in exciton-polariton systems. In the present sec- are called the one-particle irreducible (1PI) or amputated vertex tion, we restrict ourselves to a mean-field analysis of this system functions [173]. A crucial and useful relation between the 1PI and defer a discussion of the role of fluctuations to Sec.IV. vertex functions and connected correlation functions (generated by W) is 1. Single-mode cavity, and some exact properties Z  δ2Γ   0 δφ¯∗ t,x δφ¯ t00,x00  −  c( ) q( )  †  δ2Γ δ2Γ  The master equation describing the decay of photons a, a in a t00,x00  ¯∗ ¯ 00 00 ¯∗ ¯ 00 00  δφq(t,x)δφc(t ,x ) δφq(t,x)δφq(t ,x ) single-mode cavity takes the general form of Eq. (16), with H =  δ2W δ2W  † ω0a a, where ω0 is the frequency of the cavity mode. Assuming  δ j∗ t00,x00 δ j t0,x0 δ j∗ t00,x00 δ j t0,x0  ×  q( ) q( ) q( ) c( )  − 0 − 0 1  δ2W  = δ(t t )δ(x x ) . (41) the external electromagnetic field to be in the vacuum state, there  ∗ 00 00 0 0 0  δ jc(t ,x )δ jq(t ,x ) is only a single term in the sum over α with L = a, describing the decay of the cavity field at a rate 2κ (the factor of 2 is chosen This relation follows directly from Eqs. (37) and (40), using the for convenience). The corresponding Keldysh action is given chain rule. Combined with Eq. (36), it states that the second vari- by [107] ation of the effective action is precisely the full inverse Green’s function. Z  ∗ ∗ Typically, an exact evaluation of the effective action is not S = a+ (i∂t − ω0) a+ − a− (i∂t − ω0) a− possible — it would constitute the full solution of the interacting t  ∗ ∗ ∗
 non-equilibrium many-body problem. However, powerful ana- −iκ 2a+a− − a+a+ + a−a− , (42) lytical tools have been developed for the analysis of such prob- where a , a∗ represent the complex photon field. Performing lems, ranging from systematic diagrammatic perturbation the- ± ± the basis rotation to classical and quantum fields as in Eq. (32) ory over the efficient introduction of emergent degrees of free- in Sec.IIA, and going to Fourier space, the action becomes dom to genuine non-perturbative approaches such as the func- Z A ! ! tional renormalization group. The latter technique is discussed  ∗ ∗  0 P (ω) ac(ω) in Sec.IIE. Examples in which such strategies were put into S = ac(ω), aq(ω) R K , (43) ω P (ω) P aq(ω) practice are discussed in PartII of this review. R R ∞ ≡ dω where we used the shorthand ω −∞ 2π . Furthermore, we have B. Examples R A ∗ K P (ω) = P (ω) = ω − ω0 + iκ, P = 2iκ. (44) In this section, we bring the rather formal considerations of PR/A are the inverse retarded and advanced Green’s functions, the previous one to life by considering explicit examples for and PK is the Keldysh component of the inverse Green’s func- driven-dissipative systems. To be specific, in Sec. II B 1 we con- tion. To see this, we evaluate the generating functional (33) sider the decay of a single-mode cavity. This is arguably one by Gaussian integration (cf. AppendixB). Then, the generating of the simplest examples that combines coherent and dissipative functional (35) for connected correlation functions is given by dynamics: the system itself consists of a single bosonic mode Z K R ! ! (the cavity photon), has a quadratic Hamiltonian and linear Lind-  ∗ ∗  G (ω) G (ω) jq(ω) W[Jc, Jq] = − jq(ω), jc(ω) A . (45) blad operators. Hence, the Keldysh action is quadratic and the ω G (ω) 0 jc(ω) functional integral can be solved exactly. Hence, we are able ob- tain an explicit expression for the generating functional defined According to Eq. (36), the second variation (i.e., the matrix in in Eq. (33), which allows us to conveniently study several prop- the above equation, see AppendixA) represents the Green’s erties of the Keldysh formalism: what is known as the causality function. It is obtained by inversion of the matrix in the action structure, the analyticity properties of the Green’s functions, and in Eq. (43), the intuitive and transparent way in which spectral and statistical ∗ 1 1 properties are encoded in the formalism. Moreover, we compare GR(ω) = GA(ω) = = , (46) PR(ω) ω − ω + iκ these findings with the case of a bosonic mode in equilibrium. 0 i2κ The presence of the properties discussed in this section is not GK(ω) = −GR(ω)PKGA(ω) = − , (47) 2 2 restricted to the non-interacting case. Much rather, they prevail (ω − ω0) + κ 17

† i.e., the matrix in the action indeed represents the inverse Green’s under the trace an element of Lρn transforms as, e.g., L+ρL− 7→ † function. We now summarize a few key structural properties that L−ρL+, where we introduced a fictitious index ± to track the be- can be gleaned from this explicit discussion. Indeed, as we argue havior of operators under Hermitian conjugation. This gives a below, these properties are valid in general. prescription of how to perform the complex conjugation of the Conservation of probability — There is a zero matrix entry partition function in the functional integral representation: in Eq. (43), or, equivalently, in Eq. (45). Technically, as antic- Z !∗ ipated above, this property reflects a redundancy in the ± basis ∗ ∗ ∗ ∗ ∗ iS [a+,a+,a−,a−] and eliminates it. This simplifies practical calculations in the Z = D[a+, a+, a−, a−]e Keldysh basis. Physically, this property ensures the normaliza- Z (49) ∗ ∗ ∗ ∗ −iS [a−,a−,a+,a+] tion of the partition function (Z = tr ρ(t) = 1), and can thus be = D[a+, a+, a−, a−]e . interpreted as manifestation of the conservation of probability (∂t tr ρ(t) = 0), which is an exact property of physical problems. This can be seen as follows: consider the more general property, Hence, the operation to be carried out is complex conjugation which implies the vanishing matrix element in the quadratic sec- of the fields, and contour exchange. The functional measure ∗ ∗ is invariant, and in the action we indicated the transformation tor, S [ac, ac, aq = 0, aq = 0] = 0. Any Keldysh action associated to the Liouville operator Eq. (16), has this property, as can be explicitly. In the basis of classical and quantum fields, this is implemented by complex conjugation and ac 7→ ac, aq 7→ −aq. seen from setting a+ = a− (i.e., aq = 0) in Eq. (28). Indeed, this operation on the Keldysh action may be interpreted as tak- To keep the exponent iS invariant under this transformation, the ing the trace in the operator based master equation Eq. (16): in Hermitian (real) part of the action must be odd in the quantum this way, using the cyclic property of the trace allows us to shift field, the anti-Hermitian (imaginary) part must be even. Note all operators to one side of the density matrix, leading to the can- that this argument also holds in the presence of sources, pro- cellation of terms such that ∂ tr ρ(t) = 0. We still need to argue vided that these are transformed as the fields. t R that the above property of the classical action also holds for the Analytic structure — The poles of G (ω) are located at ω = full theory, i.e., the effective action, ω0 − iκ, i.e., in the lower half of the complex plane (accordingly, the poles of GA(ω) are in the upper half). In the real time domain, ∗ ∗ GR retarded Γ[¯ac, a¯c, a¯q = 0, a¯q = 0] = 0. (48) this implies that describes the response (and ac- cordingly, GA the advanced): indeed, taking the inverse Fourier This can be done using an RG continuity argument. The func- transform, we obtain tional renormalization group equation (cf. Eq. (129)) smoothly connects the classical action S to the full effective action Γ. At GR(t) = −iθ(t)e−(iω0+κ)t, (50) the level of S , the property holds. Performing the first RG step, there is no way of generating a nonvanishing matrix element, where θ(t) is the Heaviside step function. Hence, the response of due to the zero in the Green’s function associated to the clas- the system, if it is perturbed at t = 0, is retarded (non-zero only sical action. Iterating this argument down to the full effective for t > 0) and decays. action establishes the property also for the latter object. Some- The analytic structure of retarded and advanced Green’s func- times, the property of conservation of probability is referred to tions is a general property of Keldysh actions, too; one may then as “causality structure” in the literature. think of these Green’s functions as renormalized, full single par- (Anti-)Hermiticity and even/odd sectors of the action — As ticle Green’s functions connected to the bare, microscopic ones R A can be read off from Eqs. (46) and (47), G (ω) and G (ω) are via renormalization, and thus preserving the analyticity proper- K Hermitian conjugates, and G (ω) is anti-Hermitian. Equiva- ties. We note, however, that the pole of the retarded bosonic lently, these properties hold for the corresponding components Green’s function can approach the real axis from below via tun- of the inverse Green’s function. Moreover, the R/A contributions ing of microscopic parameters. The touching point typically sig- are odd in the number of quantum fields, while the K contribu- nals a physical instability: beyond that point, the description of tions are even. the system must be modified qualitatively. Such a scenario is Also these properties hold more generally. In particular, the discussed in the next subsection. sum of the Hermitian conjugates provides a Hermitian contri- Connection to the operator formalism — It is sometimes use- bution to the (effective) action, while the remaining part is anti- ful to restore the precise relation between the operator formalism Hermitian. Clearly, any scalar, as the effective action, can be and the functional integral description at the level of the Green’s written as a sum of a Hermitian (real) and an anti-Hermitian functions. At the single particle level, they read (imaginary) component, Γ = Γh + Γa. We now argue that the Hermitian component is always odd in the number of quan- GR(t, t0) = −iθ(t − t0)h[a(t), a†(t0)]i, (51) tum fields, while the anti-Hermitian is even. To see this, we K 0 † 0 note that Z = Z∗. In the operator formalism, Z = tr ρ(t) = G (t, t ) = −ih{a(t), a (t )}i, (52) N limN→∞ (1 + δtL) ρ(t0) (cf. Eq. (19)). Complex conjugation is realized under the trace as Hermitian conjugation of the oper- and we note again that in stationary state GR/A/K(t, t0) = ators, and complex conjugation of all numbers. For example, GR/A/K(t − t0). These relations are exact and can be obtained 18 from going back to the ± basis: the retarded Green’s function is the field hc can, in principle, be different from zero and we will see in the following what the physical meaning of this classical R 0 ∗ 0 G (t, t ) = −ihac(t)aq(t )i source field is, and in which respect it generates the response i function. = − h(a (t) + a (t)) (a∗ (t0) − a∗ (t0))i 2 + − + − Responses: The retarded Green’s function, often called syn- i   onymously response function, describes the linear response of a = − hTa(t)a†(t0)i + h[a(t), a†(t0)]i − hTa˜ (t)a†(t0)i 2 system which is perturbed by a weak external source field. As an illustrative example, consider a driven cavity system consisting = −iθ(t − t0)h[a(t), a†(t0)]i, of atoms and photons, which is considered to be in a stationary (53) state. Due to imperfections in the cavity mirrors, there is a finite rate with which a photon is escaping the cavity, or an external where T, T˜ are the time-ordering, anti-time-ordering operators, photon is entering the cavity. This process is expressed by the which lead to a cancellation of the commutator for t0 > t (see Hamiltonian Ref. [98] for a more detailed discussion of time ordering on the √  † †  Keldysh contour). The Keldysh Green’s function in the operator Hp = 2κ b a + a b , (58) formalism is obtained the same way, where the operators b, b† represent photons outside the cavity. K 0 ∗ 0 † G (t, t ) = −ihac(t)ac(t )i Shining a laser through the cavity mirror, the operators b, b can ∗ i ∗ 0 ∗ 0 be replaced by the coherent laser field h(t), h (t), which oscillates = − h(a+(t) + a−(t)) (a+(t ) + a−(t ))i 2 with the laser frequency ωh. The corresponding Hamiltonian, i   describing the complete system is (for the present purposes we = − hTa(t)a†(t0)i + h{a(t), a†(t0)}i + hTa˜ (t)a†(t0)i 2 can leave the Hamiltonian H of photons and atoms in the cavity = −ih{a(t), a†(t0)}i. unspecified) √ (54)  ∗ † Hh = H + 2κ h (t)a + h(t)a . (59) Response vs. correlation functions — In order to generate re- Since the fields h, h∗ are classical external fields, they are equal sponse and correlation functions directly from the partition func- on the plus and minus contour and the Keldysh action in the tion, it is convenient to introduce source fields h+, h− and express presence of the laser is the partition function as the average (cf. Eq. (31)) Z Z  ∗ ∗  S h = S − h (t)aq(t) + h(t)aq(t) . (60) −iS h ∗ ∗ iS −iS h Z[h+, h−] = he i = D[a+, a−, a+, a−]e , (55) t This action is very similar to the action in Eq. (56), with a lin- where the source action is defined as ear source term, which is however h+ = h− = h. In the present Z Z example, the meaning of the source term is physically very trans- ∗ ∗
 ∗ ∗  S h = h+a+ − h−a− + c.c. = hcaq + hqac + c.c. . (56) parent: it is nothing but the coherent laser field that is coupled to t t the cavity photons. For a weak source field, one is interested in In the second step, we have performed a Keldysh rotation. Due the first order correction of observables induced by the coupling to the normalization of the Keldysh path integral Z[hq = 0, hc = to the source. In the present case, this is the coherent light field 0] = 1, and as a consequence, expectation values of n-point func- inside the cavity hac(t)i. Up to first order in h(t) it is given by tions of the fields a∗, a can be expressed via n-th order functional Z √ ∗ 0 0 derivatives of the partition function with respect to the source hac(t)ih = hac(t)ih=0 − i κ hac(t)aq(t )ih=0h(t ) fields (see AppendixA). For example t0 √ Z R 0 0 = hac(t)ih=0 + κ G (t − t )h(t ). δZ(h , h ) 0 h i c q t ac(t) = i ∗ , δhq(t) hc=hq=0 The retarded Green’s function is therefore a measure of the sys-

δ2Z(h , h ) tem’s response to an external perturbation. An experimental K 0 − h ∗ 0 i c q G (t, t ) = i ac(t)ac(t ) = i ∗ 0 , (57) setup, with which one can measure the coherent light field and δhq(t)δhq(t ) hc=hq=0 therefore the response function of the cavity via so-called homo-

δ2Z(h , h ) dyne detection, is illustrated in Fig.4. A closely related function R 0 − h ∗ 0 i c q G (t, t ) = i ac(t)aq(t ) = i ∗ 0 . of interest is the spectral function δh (t)δhc(t ) q hc=hq=0 A(ω) = −2 Im GR(ω). (61) Due to causality, the field hq has to be zero in any physical setup and the introduction of this field only serves as a technical tool to It is the distribution of excitation levels of the system, i.e., when compute expectation values via derivatives. On the other hand, adding a single photon with frequency ω to the system, A(ω) is 19

In the absence of interactions (as in our example), it is straight- forward to show that

|GK(t, τ + t) + GR(t, τ + t) − GA(t, τ + t)|2 g(2)(t, τ) = 1 + . (65) |iGK(t, t) − 1|2

For equal times, the retarded and advanced Green’s functions satisfy GR(t, t) − GA(t, t) = −i, indicating the expected photon- bunching at τ → 0. On the other hand, in the presence of in- teractions, Eq. (64) is modified and contains additional higher Figure 4. Illustration of a homodyne detection measurement which order terms, as well as off-diagonal contributions. However, it R 0 determines the response function G (t, t ) of the cavity photons, see remains a measure of the photonic statistics in the cavity due to Ref. [108]. The system of atoms and photons inside the cavity is per- the physical relation to the intensity fluctuations. turbed by the laser field η(t), entering the cavity through the left mirror. A particularly relevant and instructive limit of the two-time The response of the system is encoded in the light field which is leaking 0 out of the cavity on the right mirror with a rate κ. It can be measured Keldysh Green’s function Eq. (52) concerns equal times t = t , by a standard homodyne detection measurement in which the reference which in general describes static correlation functions, or covari- laser β(t) and a beam splitter are used in order to obtain information on ances in the quantum optics language. In the cavity example, it the system’s coherence hac(t)i. Figure copied from Ref. [108]. (Copy- yields the mode occupation number, right (2013) by The American Physical Society.) iGK(t, t) = 2ha†ai + 1. (66) the probability to hit the system at resonance. Indeed, it can be The appearance of the combination 2ha†ai + 1 is rather intuitive shown [169] that the spectral function is positive, A(ω) > 0, and when taking into account the relation to the operatorial formal- fulfills the sum rule ism. Indeed, in operator language, 2a†a + 1 = {a, a†} = {a†, a} is † Z invariant under permutation of the operators a, a , and therefore A(ω) = h[a, a†]i = 1. (62) lends itself to a direct functional integral representation, which 8 ω in fact carries no information on operator ordering. At the same time, the anticommutator carries all the physical information on For our example of a single-mode cavity, the spectral function is state occupation, while the commutator carries none, [a, a†] = 1 given by † 1 † † (note a a = 2 ({a, a } − [a, a ])). Then, the explicit form of the Keldysh Green’s function stated 2κ A(ω) = , (63) in Eq. (47) leads to the result 2 2 (ω − ω0) + κ ! 1 Z i.e., it is a Lorentzian, which is centered at the cavity frequency ha†ai = i GK(ω) − 1 = 0, (67) 2 ω ω0 and has a half-width at half-maximum given by κ. Note that for κ → 0, the photon number states become exact eigenstates showing that the cavity is empty in the steady state—which is and the spectral density reduces to a δ-function peaked at ω , 0 not surprising in the absence of external pumping. A(ω) = 2πδ(ω − ω ). As these considerations illustrate, the re- 0 We note that in the master equation formalism, information on tarded Green’s function contains essential information on the the static or spatial correlations is most easily accessible — only system’s response towards an external perturbation, and on the the state (density matrix) has to be known, but not the dynamics spectral properties. acting on it. Temporal correlation functions can be extracted Correlations: The Keldysh Green’s function contains elemen- using the quantum regression theorem [176]. In the Keldysh tary information on the system’s correlations and the occupation formalism, spatial and temporal correlation functions are treated of the individual quantum mechanical modes. A prominent ex- on an equal footing. ample of a correlation function in quantum optics is the photonic We can concisely summarize the above discussion of the re- g(2) correlation function. It is defined as the four-point correlator sponse and Keldysh Green’s functions of the system, at the risk of oversimplifying: ha†(t)a†(t + τ)a(t + τ)a(t)i g(2)(t, τ) = (64) |ha†(t)a(t)i|2 and it is proportional to the intensity fluctuations of the intra- cavity radiation field g(2)(t, τ) ∝ hI(t)I(t + τ)i. In the limit of (2) τ → 0, the g correlation function reveals the statistics of the 8 The same structural argument based on operator ordering insensitivity of the cavity photons, i.e., it demonstrates bunching (g(2) > 1) or anti- functional representation demonstrates why an ordinary Euclidean functional bunching (g(2) < 1) as an effect of the light-matter interactions. integral yields the time ordered Green’s functions, cf., e.g., Ref. [171]. 20

be applied in the interacting case. Then, the construction ensures Responses GR/A ↔ spectral information on excitations: that the free Green’s functions (i.e., the ones obtained by ignor- which excitations are there? ing the interactions) obey an FDR. If the non-linear terms in the Correlations GK ↔ statistical information on excitations: action obey the equilibrium symmetry discussed in Sec. II D 1 how are the excitations occupied? (which is the case for generic interaction terms), this property is shared by the full Green’s functions of the non-linear system. General Fluctuation-Dissipation Relations — While Eq. (71) Relation to thermodynamic equilibrium — Before we move is valid only in thermodynamic equilibrium, it is always possible on, let us briefly discuss how we have to modify the formal- to parameterize the (anti-Hermitian) Keldysh Green’s function ism in order to describe a cavity in thermodynamic equilibrium. in terms of the retarded and advanced Green’s functions (which, In general, a system is in thermodynamic equilibrium at a tem- −βH as discussed above, are Hermitian conjugates of each other) and perature T = 1/β, if it is in a Gibbs state ρ = e /Z where a Hermitian matrix F = F† in the form [98, 169] Z = tr e−βH, and its dynamics is coherent and generated by the Hamiltonian H. (In particular, dissipative dynamics described GK = GR ◦ F − F ◦ GA, (72) by a term in the Lindblad form in Eq. (16) is not compatible with equilibrium conditions: see Refs. [177–179], and the dis- where ◦ denotes convolution. In this parametrization, F is the cussion in Sec. II D 1.) Specifying the thermal density matrix distribution function, which describes the distribution of (quasi- at t0 in the Keldysh functional integral in Eq. (25) explicitly in ) particles over the modes of the system. For a non-equilibrium terms of its matrix elements is rather inconvenient, especially if steady state, F is time-translational invariant and its Fourier one is interested in steady state properties and wants to take the transform in frequency space F(ω) represents the energy re- limit t0 → −∞. An alternative is suggested by the observation solved occupation of (quasi-) particle modes. On the other hand, that thermodynamic equilibrium can be established in the system as we discuss in detail in Sec.VB, for the case of a time- if it is weakly coupled to a thermal bath. Then, the finite decay evolving system, for which time translation invariance is absent, rate κ in the retarded Green’s function in Eq. (46) should be re- the Wigner transform of F corresponds to the instantaneous local placed by an infinitesimally weak system-bath coupling δ → 0. distribution function. For the important case of thermodynamic Additionally, the Keldysh Green’s function has to be modified in equilibrium (of a bosonic system), it is F(ω) = coth(βω/2) = such a way that the integral in Eq. (67) yields the Bose distribu- 2n(ω) + 1, and Eq. (72) reduces to (71). For the case where the tion function n(ω0) implying thermal occupations of the cavity bosonic Green’s function has a matrix structure in Nambu space, mode, a subtlety concering the preservation of the symplectic structure of that space arises, cf. Sec.VB. There, also a time-dependent † 1 variant of the non-equilibrium fluctuation-dissipation relations ha ai = n(ω0) = . (68) eβω0 − 1 is discussed. This can be achieved by replacing the Keldysh component of the inverse Green’s function in Eq. (43) by PK(ω) = i2δ (2n(ω) + 1). 2. Driven-dissipative condensate We emphasize the key structural difference of the thermal Keldysh component, which is strongly frequency dependent, to the Markovian case discussed previously, where this entry is fre- In the previous section, we discussed the simple case of quency independent — this gives a strong hint that Markovian a quadratic Keldysh action, which allowed us to perform the systems can behave quite differently from systems in thermal Keldysh functional integral explicitly. An additional simplifi- R A ∗ cation resulted from the fact that we were considering only a equilibrium. With P (ω) = P (ω) = ω − ω0 − iδ we obtain the equilibrium Green’s functions single bosonic mode. Let us now consider a genuine many-body problem, which is non-linear and in which the system consists 1 of a continuum of modes. To be specific, in this section we dis- GR(ω) = GA(ω)∗ = , (69) ω − ω0 + iδ cuss the model introduced in Sec.IC2 for a bosonic many-body GK(ω) = −i2πδ(ω − ω ) (2n(ω) + 1) . (70) system with interactions and non-linear loss processes (i.e., with 0 a loss rate that is proportional to the density) in addition to the Note that the Green’s functions obey a thermal fluctuation- linear dissipative terms which were already present in the exam- dissipation relation (FDR), which for the present example of a ple of the single-mode cavity. Then, for a specific value of the single bosonic mode reads mean-field density ρ0, non-linear loss and linear pump exactly balance each other and the system reaches a stationary state. If K  R A  G (ω) = (2n(ω) + 1) G (ω) − G (ω) . (71) ρ0 is different from zero, this signals the presence of a conden- sate, which is accompanied by the breaking of a specific phase This is discussed in more detail in Sec. II D 1. For the time being, rotation symmetry as will be discussed in detail in Sec.IID, and let us mention that this construction to describe thermodynamic the establishment of long-range order. In Sec.IV, we give a equilibrium by adding infinitesimal dissipative terms does not detailed account of the influence of fluctuations on this driven- only work in the present case of a quadratic action but can also dissipative condensation transition. Here, we content ourselves 21 with a mean-field analysis, which serves to illustrate some of the In particular, we have in the frequency and momentum domain field theoretical concepts introduced in Sec.IIA — in particular, 2 ! the effective action and field equations — in a simple setting. R ω − Kcq − (uc − iud) ρ0 − (uc − iud) ρ0 P (ω, q) = 2 , In the basis of classical and quantum fields, the Keldysh action − (uc + iud) ρ0 −ω − Kcq − (uc + iud) associated with the quantum master equation (9) reads PA(ω, q)) = PR(ω, q)†, K 1 Z P = iγ . n ∗  2  (76) S = φq i∂t + Kc∇ − rc + ird φc + c.c. t,x h   i The excitation spectrum is obtained from the condition − − ∗ ∗ 2 ∗ ∗ 2 (uc iud) φqφcφc + φqφcφq + c.c. det PR(ω, q) = 0. Indeed, this is the condition for the field equa- ∗
∗ o tion of the fluctuations PR(ω, q)δΦ (ω, q) = 0 to have nontrivial +i2 γ + 2udφcφc φqφq , (73) c solutions. This yields [151] where K = 1/(2m ) and r = ω0 ; as additional parameters, we q c C c LP R 2 2
2 ω = −iudρ0 ± Kcq Kcq + 2ucρ0 − (udρ0) . (77) introduced the noise level γ = (γl + γp)/2 and the spectral mass 1,2 or gap r = (γ − γ )/2. Hence, the rates of losses and pump- d l p We note that due to the tree-level shifts ∝ ρ the above described ing add up to the total noise level; in contrast, the difference of 0 instability for r < 0 is lifted: both poles are consistently lo- these rates enters in the spectral gap r , which becomes negative d d cated in the lower complex half-plane, indicating a physically when the rate of incoherent pumping exceeds the single-particle stable situation with decaying single-particle excitations. For loss rate, signaling the physical instability against condensation. u = 0, Eq. (77) reduces to the standard Bogoliubov result [180], In a mean-field analysis of the condensation transition, we per- d where for q → 0 the dispersion is phononic, ωR = ±cq form a saddle-point approximation of the functional integral in p 1,2 Eq. (19). To leading order, fluctuations around the field expecta- (c = 2Kcucρ0 the speed of sound), whereas particle-like be- R 2 tion values are completely neglected. The expectation values are havior ω1,2 ∼ Kcq is obtained at high momenta. Here, due to then obtained as spatially homogenous and stationary solutions the presence of two-body loss ud , 0, the dispersion is qualita- to the classical field equations (here, “classical” refers to the fact tively modified: while at high momenta the dominant behav- R 2 that these field equations are derived from the classical (or: bare, ior is still ω1,2 ∼ Kcq , at low momenta we find purely in- microscopic) action S discarding fluctuations, in contrast to the coherent diffusive, non-propagating modes ωR ∼ −i Kcuc q2 and 1 ud field equations in Eq. (40), which involve the effective action) R R ω2 ∼ −i2udρ0. In particular, for q = 0 we have ω1 = 0: this is a dissipative Goldstone mode [151, 181, 182], associated with δS δS the spontaneous breaking of the global U(1) symmetry in the or- ∗ = 0, ∗ = 0. (74) δφc δφq dered phase. The existence of such a mode is not bound to the mean-field approximation, but rather is an exact property guar- As already mentioned above in Sec. II B 1, there are no terms anteed by the U(1) invariance of the effective action, even in the ∗ present case of a driven-dissipative condensate. We will come in the action Eq. (38) with zero power of both φq and φq, and ∗ back to this point in Sec. II D 6. the same is clearly true for δS/δφc. Therefore, the first equation in (74) is solved by φq = 0. Inserting this condition into the second equation, we have C. Semiclassical limit of the Keldysh action h i −r + ir − (u − iu ) |φ |2 φ = 0. (75) c d c d 0 0 The theoretical description of a many-body system depends crucially on the scale (this could be a length, time, or energy The solution φc = φ0 is determined by the imaginary part of scale — in practice these scales can be expressed in terms of Eq. (75): for rd ≥ 0, in the so-called symmetric phase, the clas- each other) on which it is observed and, in particular, on the re- 2 sical field expectation value is zero, ρ0 = |φ0| = 0, whereas for lation between this observation scale and the intrinsic scales of rd < 0 we have a finite condensate density ρ0 = −rd/ud. The the system. For example, at finite temperature T above a quan- parameter rc is then determined by the real part of Eq. (75) as tum critical point, non-trivial quantum critical behavior can be rc = ucrd/ud. This corresponds to a choice of a rotating frame. observed at moderate energy scales which are larger than T but In a first step beyond mean field theory, quadratic fluctu- smaller than the Ginzburg scale where fluctuations start to dom- ations around the mean-field order parameter can be investi- inate over the mean field effects [106]. On the other hand, clas- gated within a Bogoliubov or tree-level expansion: we set φc = sical thermal critical behavior, which is perfectly described by φ0 +δφc, φq = δφq in the action Eq. (73) and expand the resulting taking the semiclassical limit of the underlying quantum theory, expression to second order in the fluctuations δφc,q. The inverse sets in at energy scales below T. Out of thermodynamic equilib- retarded, advanced and Keldysh Green’s functions now become rium, the analogue of a finite temperature is Markovian noise. ∗
2 × 2 matrices in the space of Nambu spinors δΦν = δφν, δφν . In the Keldysh action, this corresponds to a constant term in 22 the Keldysh sector of the inverse Green’s function, i.e., a con- other words, the Keldysh component of the inverse Green’s func- stant noise vertex. Such a term is present in the action given in tion has vanishing canonical scaling dimension and classifies as Eq. (73) and, therefore, we expect that in the long-wavelength marginal, PK = i2γ ∼ q0. We furthermore use the natural scal- limit this action can actually be simplified by taking the semi- ing ddx ∼ q−d and dt ∼ 1/ω ∼ q−2, and the condition of scale classical limit [98, 169]. We refer to it as the semiclassical limit, invariance of the action, S ∼ q0 — this is a requirement stem- as, e.g., effects of quantum mechanical phase coherence are not ming from the fact that the action appears in the exponent of the necessarily suppressed in this limit, as we will see. A useful functional integral (25), and thus must be dimensionless. We equilibrium analogy is the physics of Bose-Einstein condensates then find the scaling dimensions of the fields from the Gaussian, at finite but low temperatures, where phase coherence still per- quadratic part of the action to be sists. φ ∼ q(d−2)/2, φ ∼ q(d+2)/2. (78) Formally, the suitability of the semiclassical limit can be un- c q derstood in terms of ideas that lie at the basis of the renormal- This in turn allows us to derive the scaling dimensions of the ization group (RG). The latter provides a recipe for finding an quartic terms, and to classify their degree of relevance as pointed effective description that is valid on large length scales, starting out above. In particular, we find that in three spatial dimensions, from a microscopic theory. Various RG schemes exist, which al- any quartic term that includes more than a single quantum field low to systematically eliminate fluctuations on short scales and is irrelevant; the only non-irrelevant term of higher order in the infer their influence on the effective description of the system on noise field is the quadratic noise vertex discussed above. There- large scales; cf. Sec.IIE. The most basic but still informative fore, omitting irrelevant terms amounts to keeping only the clas- RG approach however consists in simply ignoring the effect of sical vertex in the action Eq. (73), i.e., to taking the semiclassical short-scale fluctuations: one subdivides momentum space into a limit [98, 169]. Then, the Keldysh action takes the form slow and a fast region, with qs ∈ [0, Λ/b] and q f ∈ [Λ/b, Λ] where Λ is the UV cutoff (given by the inverse of some mi- Z n ∗ h 2 i croscopic length scale below which the theoretical description S = φq i∂t + (Kc − iKd) ∇ − rc + ird φc + c.c. t,x is not valid any more, e.g., a lattice spacing) and b > 1, and h ∗ ∗ 2 i ∗ o omits all contributions to the action which involve fluctuations − (uc − iud) φqφcφc + c.c. + i2γφqφq , (79) with fast momenta. In a second step, one rescales all momenta ∈ where in addition to omitting irrelevant contributions we have with b to restore the original range of momenta q [0, Λ], and 2 the thus obtained effective long-wavelength description can be added an effective diffusion term ∝ iKd∇ . Such a term can compared to the original one. As a result, due to this simple RG and will be generated upon integrating out short-scale fluctua- transformation couplings in the action are rescaled as g 7→ gbdg , tions [26, 172]. Moreover, a complex prefactor of the term in- volving the derivative with respect to time, which also emerges where dg is known as the canonical scaling dimension of g. Ev- upon renormalization, can be absorbed into a redefinition of the idently, for dg > 0, g grows under renormalization and hence g is a relevant coupling, while it shrinks under renormalization fields. Strictly speaking, the above analysis is valid in the vicinity of and, hence, is irrelevant at long wavelength for dg < 0. For the the critical point only, where the inverse retarded and advanced case of a marginal coupling with dg = 0, this level of approx- imation is not conclusive as to whether the coupling becomes Green’s function show scaling. However, as long as one is close larger or smaller under renormalization. Classifying the cou- enough to threshold |γl − γp|/(γl + γp)  1, the canonical power pling parameters of an action according to this scheme is known counting is expected to give a useful orientation in the problem. as canonical scaling analysis, or power counting. A useful ap- Apart from providing a significant simplification, taking the proximation consists in neglecting all irrelevant couplings. On semiclassical limit also allows us to establish the connec- the other hand, relevant couplings which are compatible with the tion [183, 184] between the Keldysh functional integral for- symmetries of the model – even those, which are not present in malism and the more traditional formulation of dynamics close the microscopic description — should be included in the long- to a continuous phase transition in terms of Langevin equa- wavelength effective action. tions [1,5]. In fact, the action in Eq. (79) is fully equivalent to the following Langevin equation: Let us perform such an analysis for the Keldysh action in h 2 2i Eq. (73). We can anticipate the canonical scaling dimensions i∂tφc = − (Kc − iKd) ∇ + rc − ird + (uc − iud) |φc| φc + ξ, by noting that they are just the physical dimensions, measured (80) in powers of the momentum. We first focus on the vicinity of where ξ is a Markovian Gaussian noise source with zero mean, the threshold for condensation, i.e., when rd ∼ γl − γp → 0. hξ(X)i = 0, and second moment hξ(X)ξ(X0)i = 2γδ(X − X0). Then the retarded and advanced inverse Green’s functions scale This equivalence can be established by means of a Hubbard- R/A 2 2 as P ∼ q (note that ω ∼ q for low-momentum excitations). Stratonovich transformation of the noise vertex [98, 169], i.e., The canonical dimension is thus positive, d R/A = 2. On the ∗ P the term i2γφqφq. To wit, in the Keldysh functional integral other hand, as anticipated above, Markovian noise analogous to a finite temperature corresponds to a momentum-independent Z iS [Φc,Φq] ∗ Z = D[Φc, Φq] e , (81) noise vertex, i.e., the term ∝ φqφq in the Keldysh action. In 23

quantum quantum Keldysh functional microscopic Langevin equation master equation integral

semiclassical limit

Langevin MSR functional quantum mesoscopic Fokker-Planck equation equation integral problem

renormalization group

long-wavelength macroscopic effective action scale

Figure 5. Equivalent descriptions on varying length scales. The quantum and classical Langevin equations are stochastic equations of motions for the field operators, or for classical field variables. In contrast, the descriptions in the middle column are deterministic equations of motion, where either a density operator or a probability distribution (diagonals of a density matrix) are evolved. In the functional integral formulation, the basic object in both quantum and classical cases is an action, which is averaged over all possible realizations of field configurations. The semiclassical limit is valid at mesoscopic scales as dicsussed in Sec.IIC. An e ffective description at macroscopic scales can be obtained by means of renormalization group methods (see Sec.IIE). Generically in Markovian systems, in order to reduce the complexity of the problem it is useful to first perform the semiclassical limit before doing a renormalization group computation. However, in driven open quantum systems there are also circumstances where this is inappropriate, and the full quantum problem has to be analyzed, cf. [27]. where S is the semiclassical action in Eq. (79), we write the dynamics of the order parameter, and only later a functional in- noise vertex as a Gaussian integral over an auxiliary variable ξ tegral approach — known as the MSR approach — has been (cf. AppendixB), developed [185–188]. The action derived in these references is Z formally equivalent to the one in Eq. (79), with φc taking the role R ∗ 1 R ∗ R ∗ ∗ −2γ φqφq ∗ − ξ ξ−i (φqξ−ξ φq) e t,x = D[ξ, ξ ] e 2γ t,x t,x . (82) of the order parameter field, while the field that corresponds to φq is known as the response field. As a result, the exponent in the functional integral (81) becomes In addition to the descriptions of semiclassical dynamical linear in the quantum field, and the corresponding integration models in terms of a semiclassical Keldysh (or MSR) func- can be performed, resulting in a δ-functional, tional integral or a Langevin equation, there exists yet another equivalent approach, in which the stochastic Langevin equation Z R for the classical field or order parameter field is replaced by − 1 ∗ ∗ ∗ 2γ t,x ξ ξ Z = D[φc, φc, ξ, ξ ] e a deterministic evolution equation for the probability distribu- tion of the latter, known as the Fokker-Planck equation. The h 2 2i  × δ i∂t + (Kc − iKd) ∇ − rc + ird − (uc − iud) |φc| φc − ξ derivation of the Fokker-Planck equation can be found, e.g., in h i  ×δ −i∂ + (K + iK ) ∇2 − r − ir − (u + iu ) |φ |2 φ∗ − ξ∗ . Refs. [5, 98, 169, 189]. Figure5 illustrates the equivalence of t c d c d c d c c these approaches, which are valid on a (coarse-grained) meso- (83) scopic scale, and the relations to microscopic and macroscopic This expression can be interpreted as follows: for a given descriptions. realization of the noise field ξ, the δ-functional restricts the functional integral over φc to the manifold of solutions to the Langevin equation (80). The statistics of the noise field is de- D. Symmetries of the Keldysh action termined by the Gaussian weight factor in Eq. (83). Correlation functions of classical fields can then be calculated by picking a Symmetries take center stage in field theories. Translating the random realization of ξ and calculating the corresponding φc that physics of the quantum master equation to the Keldysh func- solves the Langevin equation; evaluating the correlation function tional integral allows us to leverage the power of symmetry con- for this solution and finally averaging the result according to the siderations over to the context of open systems. In this section, Gaussian distribution of the noise field. Equation (80) is, there- we discuss three different aspects of symmetries. fore, equivalent to the functional integral (83) in that it allows The presence of a first, discrete symmetry, considered in for the evaluation of arbitrary correlation functions. Sec. II D 1, allows us to conclude whether or not a quantum or Originally, Langevin equations like Eq. (80) have been intro- classical system is situated in the realm of thermodynamic equi- duced as a phenomenological description of the coarse-grained librium. This symmetry is equivalent to the validity of thermal 24

0 fluctuation-dissipation relations (cf. Sec. II B 1) for correlation matrix, ρ = e−βH /Z0, with a Hermitian operator H0. On the functions of any order and can be connected to energy conser- other hand, static (i.e., purely momentum- or space-dependent) vation. Thermal equilibrium can thus be diagnosed by means a properties do not allow to discriminate whether the generator simple symmetry test on the Keldysh action. In particular, we of dynamics coincides with H0. In sharp contrast, any dynami- argue that Markovian quantum master equations explicitly vi- cal (i.e., frequency- of time-dependent) observable is manifestly olate this symmetry, indicating non-equilibrium conditions. In governed by this generator, as is easily seen in the Heisenberg the semiclassical limit, we obtain a simple and intuitive geomet- picture (or a suitable generalization to open systems). Response ric interpretation of the symmetry and its absence under non- functions at finite frequency are such dynamical observables, equilibrium drive in terms of the location of the coupling con- and the equilibrium conditions formulated above are reflected in stants of the effective action in the complex plane. the fact that in thermodynamic equilibrium the response of the A second fundamental consequence of the presence of sym- system to an external perturbation at a frequency ω is related to metries — now, continous global symmetries — is the Noether thermal fluctuations within the system at the same frequency by theorem, stating that such symmetries imply conserved charges. what is known as a fluctuation-dissipation relation [190] (FDR; In Sec. II D 3, we discuss the Noether theorem in the context for a discussion in the context of non-equilibrium Bose-Eintsein of the Keldysh formalism. Working on the Keldysh contour, condensation see Ref. [191]). Such a relation, which is equiv- we have to distinguish between two symmetry transformations, alent to the combination of the Kubo-Martin-Schwinger condi- for each the forward and backward branches of the closed time tion [192, 193] with time reversal [194], is valid for any pair of path. In the basis of classical and quantum fields, we can iden- operators. In particular, for the case that these are the basic field tify “classical” symmetry transformations, which act in the same operators of a single-component Bose system at the temperature way on fields on the forward and backward branches, and “quan- T = 1/β, the FDR reads (note that this is a generalization of tum” transformations, for which the transformation of the fields Eq. (71) to the case of a spatial continuum of degrees of free- on the backward branch is the inverse of the transformation on dom) the forward branch. Non-trivial conservation laws follow only  ω  from the symmetry of the Keldysh action under quantum trans- GK(ω, q) = coth (GR(ω, q) − GA(ω, q)) (84) formations. We illustrate this point with the example of the sym- 2T metry of the action of a closed system with respect to space-time (in the presence of a chemical potential µ, in the argument of the translations and phase rotations in Sec. II D 4, which entails con- trigonometric function we have to shift ω → ω − µ). In quan- servation of energy and momentum and the number of particles, tum field theory, relations among Green’s functions often follow respectively. On the other hand, in open systems only classical as consequences of a symmetry of the action. Then, they are phase rotations are a symmetry of the action, and the continuity known as Ward-Takahashi identities associated with the symme- equation that is implied by particle number conservation has to try [82, 195].9 This raises the question, whether FDRs and hence be extended as detailed in Sec. II D 5. the presence of thermodynamic equilibrium conditions are also Finally, another consequence of a global continuous symme- connected to a symmetry of the Keldysh action. try is the existence of gapless modes in case that this symmetry The existence of such a symmetry has first been realized in the is broken spontaneously, as in a Bose condensation transition. context of classical stochastic models [187, 188, 196–198], and In a driven-dissipative condensate in which the number of parti- recently these considerations have been extended to the realm cles is not conserved (cf. Sec. II B 2), we show that the symme- of quantum systems [179, 199]. It turns out, that the FDR (84) try which is broken spontaneously at the condensation transition for single-particle Green’s functions and corresponding relations is the classical phase rotation symmetry, and we work out the for higher order correlation functions can be derived from the Goldstone theorem, which guarantees the existence of a mass- condition that the Keldysh action is symmetric with respect to less mode, for this case in Sec. II D 6. the following transformation of the fields (σ = +(−) for fields on the forward (backward) branch):

∗ 1. Thermodynamic equilibrium as a symmetry of the Keldysh action Tβψσ(t, x) = ψσ(−t + iσβ/2, x), ∗ (85) Tβψσ(t, x) = ψσ(−t + iσβ/2, x) A system is in thermodynamic equilibrium at a temperature T = 1/β, if (i) the density matrix is given by ρ = e−βH/Z where (in the presence of a chemical potential µ, we have to multiply Z = tr e−βH, and (ii) the very same Hamiltonian operator H ap- the RHS of the first (second) line by eσβµ/2(e−σβµ/2)). Thus, the pearing in ρ generates the unitary time evolution of the system, symmetry of the Keldysh action under this transformation is a U = e−iHt. Condition (ii) implies that static correlations are in general not sufficient to prove that a system is in thermal equi- librium; however, dynamical correlations are: this can be in- ferred from the fact that the static correlations of a physical sys- 9 Usually, the term “Ward-Takahashi identity” is reserved for relations that fol- tem can always be encoded in a density matrix ρ, and the latter low from a continuous symmetry. Here, we use it also in the context of dis- can always be parameterized formally as an equilibrium densiy crete symmetry transformations. 25 direct proof of the presence of thermodynamic equilibrium con- and ditions, and therefore provides us with a convenient means to X Z   iωlt ∗ −iωlt find out whether a given Keldysh action corresponds to a system S l = Ω σ dt aσ(t)e + aσ(t)e in thermodynamic equilibrium. σ (87) X ∗
The transformation Tβ specified in Eq. (85) is a composition = Ω σ aσ(ωl) + aσ(ωl) . of complex conjugation of the fields ψσ, inversion of the sign σ of the time t, and translation of the value of t by an amount Ω is the amplitude of the driving field, and due to the harmonic iσβ/2. In particular, the shift of t takes opposite signs depend- time dependence of the drive it affects only the component a(ω ) ing on whether a field on the forward or backward branch is l in frequency space. The action for the bath is expressed most transformed. After analytic continuation of the time shifts to conveniently in the basis of classical and quantum fields (cf. real values, iσβ/2 → σβ/2, the symmetry of an action under Eq. (32)). In addition to the coherent part stemming from the such time shifts implies the conservation of energy as detailed in oscillator frequencies, it involves infinitesimal dissipative regu- Sec. II D 4. Indeed, any action associated to a time-independent larization terms specifying the thermal equilibrium state of the (time-translation invariant) Hamiltonian is symmetric with re- bath at inverse temperature β = 1/T as discussed in Sec. II B 1, spect to Tβ, thus expectedly allowing for equilibrium condi- tions [179]. In contrast, it is straightforward to check that the dis- Z X  ∗ ∗  sipative contribution to the Keldysh action Eq. (30) associated to S b = bµ,c(ω), bµ,q(ω) the quantum master equation (16) does not possess this symme- µ ω ! ! try. The violation of the symmetry directly indicates that quan- 0 ω − ω − iδ b (ω) × µ µ,c . (88) tum master equations describe genuine non-equilibrium condi- ω − ωµ + iδ i2δ coth(βω/2) bµ,q(ω) tions [177–179]. Physically, this is due to the fact that a sys- tem for which such a description is appropriate is necessarily Finally, the system-bath interaction with coupling strength λ cor- driven — there is a microscopic scale, at which the time trans- responds to the following contribution to the action: lation invariance of the Hamiltonian is broken explicitly by a X X Z time-dependent classical field, such as a laser.  ∗ ∗  S sb = λ σ aσ(ω)bµ,σ(ω) + aσ(ω)bµ,σ(ω) . (89) In some cases, the Markov and rotating-wave approximations σ µ ω leading to a description in terms of a quantum master equation or To check whether the transformation (85) is a symmetry of the equivalent Keldysh action are applied in the absence of external action even in the presence of the driving term in Eq. (87), it driving fields. Then, these approximations might still be justified is most convenient to rewrite the transformation in frequency to study the behavior of specific observables, even though they space. Moreover, for future reference, we give the form of the explicitly break the symmetry [179]. To give an example, if one transformation including a chemical potential: attempts to study thermalization of a system due to the coupling with a heat bath, and one integrates out the bath using the above- −σβ(ω−µ)/2 ∗ Tβ,µaσ(ω) = e a (ω), σ (90) mentioned approximations, the resulting dynamics of the system ∗ σβ(ω−µ)/2 will still lead to a thermal stationary state. Correspondingly, all Tβ,µaσ(ω) = e aσ(ω). static properties of the system will appear thermal. However, as It is straightforward to check that both S b and S sb are invariant discussed at the beginning of the present section, to unambigu- under this transformation with µ = 0 [179]; the same holds true ously prove the presence of thermal equilibrium conditions, one for the contribution S 0 to the action of the system. However, the has to consider dynamical signatures such as FDRs. Then, as a driving part (87) becomes after the transformation consequence of the explicit violation of the thermal symmetry by X   the Markov and rotating-wave approximations, the description σβωl/2 ∗ −σβωl/2 S l = Ω σ aσ(ωl)e + aσ(ωl)e , (91) of the system dynamics in terms of a quantum master equation σ will lead to the wrong prediction that fluctuations in the system do not obey an FDR. where the appearance of the exponentials shows that the sym- metry is violated. One might wonder whether it is possible to A simple example that illustrates the breaking of the thermal restore the symmetry in a rotating frame in which the explicit symmetry by external driving fields is given by a single bosonic time dependence of the action is eliminated, i.e., by introduc- mode a with energy ω , driven coherently at a frequency ω , 0 l ing new variablesa ˜ (t) and b˜ (t) rotating at the frequency of the and coupled to a bath of harmonic oscillators b in thermal equi- σ σ µ driving field,a ˜ (t) = a (t)eiωlt and analogously for b˜ (t). In librium. The associated Keldysh action can be decomposed as σ σ σ terms of these variables, the driving part of the action becomes S = S s + S sb + S b, where the action for the system consists of Z two parts, S = S 0 + S l, which read X ∗
S l = Ω σ dt a˜σ(t) + a˜σ(t) Z σ (92) X ∗ X ∗
S 0 = σ aσ(ω) (ω − ω0) aσ(ω), (86) = Ω σ a˜σ(0) + a˜σ(0) , σ ω σ 26 i.e., the drive couples to the zero-frequency component ofa ˜σ(ω). where σx is the usual Pauli matrix. “High temperatures” thus Clearly, applying again the same transformation Eq. (90) with means that the typical frequency scale of the field is much µ = 0 to the new variables, the driving term, as well as S 0 in smaller than temperature; indeed this recovers the intuition of Eq. (86) and the system-bath coupling S sb in Eq. (89) are in- a semiclassical limit. If we replace the quantum field Φq by variant. However, in the action for the bath (88), as a conse- the response field Φ˜ = −iσzΦq, Eq. (94) takes the form of the quence of the transformation to the rotating frame the distribu- classical symmetry introduced in Ref. [197]. The FDR in the tion function acquires an effective chemical potential and be- semiclassical limit, which may be derived as a Ward-Takahashi comes coth(β(ω − ωl)/2). Hence, to leave this part of the ac- of the symmetry Eq. (94), simplifies to the Raleigh-Jeans form tion invariant, the transformation Eq. (90) has to be applied with 2T   µ = ωl, and again the full action is not invariant under the equi- GK(ω, q) = GR(ω, q) − GA(ω, q) . (95) librium transformation. ω While this simple example demonstrates explicitly that gener- The thermal FDR is just one consequence of the presence of ically external driving takes a system out of thermal equilib- the symmetry Eq. (94). A second one concerns the possible val- rium, and how this is manifest in the symmetry properties of ues of the coupling constants defining the action in the semiclas- the Keldysh action, it is interesting to note that there are sur- sical limit. This allows us to state precisely in which sense the prising exceptions to this rule. One of them has been identified driven-dissipative systems represent a genuine non-equilibrium in Ref. [200]. Along the lines of this reference, we discard the situation. To this end, it is most convenient to discuss the equiv- driving term S l and instead consider a parametric system-bath alent Langevin equation Eq. (80). We rewrite it by splitting the 10 coupling of the form deterministic parts on the RHS into reversible (coherent) and ir- X X Z   reversible (dissipative) contributions according to (here we re- ∗ −iωpt ∗ iωpt S sb = λ σ dt a (t)bµ,σ(t)e + aσ(t)b (t)e σ µ,σ place φc = ψ) σ µ X X Z δHc δHd  ∗ ∗  i∂ ψ = − i + ξ (96) = λ σ aσ(ω + ωp)bµ,σ(ω) + aσ(ω + ωp)bµ,σ(ω) . t δψ∗ δψ∗ µ σ ω (93) with effective coherent and dissipative Hamiltonians (α = c, d)

Then, the full action S = S 0 + S sb + S b, where S 0 and S b are Z   2 2 uα 4 as above (Eqs. (86) and (88)), is invariant if the system fields are Hα = Kα |∇ψ| + rα |ψ| + |ψ| . (97) t,x 2 transformed with Tβ,µ=ωp and the bath oscillators with Tβ,µ=0. In other words, the parametric coupling acts to thermalize the sys- It can be shown [172, 201] that the presence of the symme- tem at the temperature of the bath while at the same time shifting try (94), or, in physical terms, relaxation of a system to ther- the chemical potential by ωp with respect to the bath chemical modynamic equilibrium (a state with global detailed balance, potential. Concomitantly, the FDR for the system variables takes where arbitrary subparts are in equilibrium with each other) re- the form of Eq. (84) with ω → ω − ωp, while for the bath it is quires the condition Eq. (84) without modification. Kc uc Hc = rHd ⇔ r = = . (98) Kd ud 2. Semiclassical limit of the thermal symmetry (Note that there is no condition on the ratio rc/rd since the ef- fective chemical potential r can always be adjusted by a gauge For many applications, as discussed in Sec.IIC, the Keldysh c transformation ψ 7→ ψe−iωt such that r 7→ r − ω without chang- action in the semiclassical limit (79) is appropriate. Correspond- c c ing the physics.) In equilibrium dynamics, the ratio of real (re- ingly we should consider the semiclassical limit of the transfor- versible) and imaginary (dissipative) parts is thus locked to one mation (85). In the limit of high temperatures β = 1/T → 0, we common value for all couplings. This is illustrated in Fig.6 (a). can perform an expansion of the transformed fields, with their In the complex plane spanned by real and imaginary parts of the arguments shifted by iσβ/2, in terms of derivatives. To leading couplings K = K + iK and u = u + iu , they lie on one single order,11 this yields c d c d ray. The intuition behind this seeming fine-tuning is the follow- TβΦc(t, x) = σxΦc(−t, x), ing: a microscopically reversible dynamics starts from a Hamil-  i  (94) tonian functional Hc alone, i.e., all couplings are located on the TβΦq(t, x) = σx Φq(−t, x) + ∂tΦc(−t, x) , real axis. Coarse graining the system from the microscopic to 2T the macroscopic scales introduces irreversible dynamics in the form of finite imaginary parts, however preserving their loca- 10 In realistic systems the system-bath coupling usually also contains terms of the tion on a single ray: the ray just rotates under coarse graining, form aσ(t)bσ(t) that are neglected in the rotating-wave approximation [200]. but does not spread out. The geometric constraint is thus not 11 Note that a contribution ∼ ∂tΦq in the first line is suppressed according to due to fine tuning, but results from the microscopic “initial con- canonical power counting, cf. Sec.IIC, Eq. (78). dition” for the RG flow, in combination with the presence of a 27 symmetry. In stark contrast, in a driven non-equilibrium system, any such global continuous symmetry entails the existence of a the microscopic origins of reversible and irreversible dynamics current j with components jµ, which obeys a continuity equation µ are independent, as illustrated in Fig.6 (b). For example, in the on average, i.e., h∂µ j i = 0 (here and in the following summation microscopic description of Eq. (73) the rates can be tuned fully over repeated indicies is implied), where ∂0 = ∂t, and ∂1,2,...d are independently from the Hamiltonian parameters — they have derivatives with respect to spatial coordinates. Then, the integral R 0 completely different physical origins. over space Q = x j — the Noether charge — is an integral of motion, and we have hdQ/dti = 0. (a) (b) (c) In order to prove the Noether theorem, it is sufficient to con- sider infinitesimal transformations. Then, for α  1 we expand Im Im Im u u the transformation as 2 u TαΨ = Ψ + αGΨ + O(α ), (100) K ⇤ K K where G is called the generator of the transformation. In general, ⇤ Re Re Re G is a 4 × 4 matrix with derivative operators as entries. We equilibrium non-equilibrium fixed point consider specific examples below in Sec. II D 4. In the following we assume that as in Eq. (30) the action SR can be written in Figure 6. Location of the couplings in the Langevin equation (96) in the terms of a Lagrangian density L as S = t,x L , and that the complex plane. Real (imaginary) parts describe reversible (irreversible) Lagrangian density is a local function of the fields and their first contributions to the dynamics. (a) An equilibrium system is charac- derivatives with respect to time and space. This assumption is terized by the location of couplings on a single ray, reflecting detailed appropriate for most practical purposes. balance. The irreversible dynamics is not independent of the underlying In an expansion of the LHS of Eq. (99) in powers of α, each reversible Hamiltonian dynamics, but rather generated by it. (b) In con- coefficient has to vanish individually. The first-order contribu- trast, in a driven non-equilibrium system, generically there is a spread in the location of couplings, because the reversible and dissipative dy- tion yields the relation Z ! namics have different physical origins. (c) Near a critical point in three ∂L ∂L G G dimensions, the couplings flow strongly with scale and approach the T Ψ + T ∂µ Ψ = 0, (101) imaginary axis (decoherence). The RG fixed point is purely dissipative. t,x ∂Ψ ∂∂µΨ (Figure adapted from [26].) ∗ ∗
T where ∂L /∂Ψ ≡ ∂L /∂ψ+, ∂L /∂ψ+, ∂L /∂ψ−, ∂L /∂ψ− . In the cases we consider, Eq. (101) holds true because the in- Finally, we note that, while an explicit violation of the sym- tegrand can be written as the divergence of a vector field f µ, i.e., metry is present on the microscopic scales on which the quantum we have master equation (16) or the corresponding Keldysh action (27) ∂L ∂L provide a suitable description of the system, in several cases it µ G G ∂µ f = T Ψ + T ∂µ Ψ. (102) has been found that driven-dissipative systems appear as approx- ∂Ψ ∂∂µΨ imately thermal at low frequencies [25, 30, 50, 105, 107, 117, To proceed we consider local transformations, i.e., we con- 181, 202, 203]. In driven-dissipative condensates in three spa- sider Tα with α = α(t, x). We perform a change of integration tial dimensions, this thermalization at low frequencies is partic- variables Ψ → TαΨ in the partition function. Then, assuming ularly sharply reflected via the emergence of the thermal sym- that the functional measure is invariant with respect to the local metry in the RG flow in this regime [26, 172], cf. Fig.6 (c); this transformation, we have is discussed in more detail in Sec.IV. On the other hand, there Z Z are also cases where the opposite behavior occurs, and the non- Z = D[Ψ]eiS [Ψ] = D[Ψ]eiS [TαΨ]. (103) equilibrium character becomes more pronounced as one coarse grains to the macroscale. This occurs in driven two dimensional As before, we expand the RHS of this equality in a power se- systems, as explained in Sec.IV. ries in α. Since by assumption the latter is a function of (t, x), Eq. (99) does not hold true any more. Instead, we find Z ! ∂L 3. The Noether theorem in the Keldysh formalism µ − G 2 S [TαΨ] = S [Ψ] + α∂µ f T Ψ + O(α ), (104) t,x ∂∂µΨ A global continuous symmetry of the Keldysh action is a ∗ ∗
T where we used Eq. (102) and integration by parts to write the transformation Tα of the fields Ψ = ψ+, ψ+, ψ−, ψ− that leaves RHS in a form that does not contain derivatives of α explicitly. the value of the action invariant, i.e., Inserting Eq. (104) in the exponential on the RHS of Eq. (103), and expanding the latter to first order in α, we obtain the condi- S [TαΨ] = S [Ψ]. (99) tion Z * + α is a real parameter and we assume that for α = 0 the transfor- ∂L α∂ f µ − G . 1 µ T Ψ = 0 (105) mation is the identity, T0 = . The Noether theorem states that t,x ∂∂µΨ 28

Since there are no restrictions on the choice of α, this equation where implies that the divergence of the expectation value vanishes. ∂Lσ ∂Lσ The latter is just the quantum Noether current, µ g g ∂µ fσ = T Ψσ + T ∂µ Ψσ. (111) ∂Ψσ ∂∂µΨσ ∂L µ G − µ j = T Ψ f . (106) ∂∂µΨ With this definition, j− can be obtained from j+ simply by re- placing all instances of Ψ appearing in j by Ψ . Then, the µ µ + + − Note that j is not unique: for constant a, b , the combination symmetry of the action under a quantum transformation yields µ µ a j + b is also a conserved current. The associated Noether a classical Noether current and vice versa. These currents are charge Q corresponds to the integral over space of the zeroth given by (as noted above we are free to choose a convenient component of the Noether current, multiplicative normalization of the currents) Z Z ! ∂L Q = j0 = GΨ − f 0 . (107) 1 ∂∂ ΨT jc = ( j+ + j−) , jq = j+ − j−. (112) x x t 2 While this derivation is quite general, a more concrete expres- As pointed out in Sec. II B 2, due to causality only the clas- sion for the Noether current can be obtained by restricting the sical component of the field can acquire a finite expectation form of the generator G in Eq. (100). In particular, in the next value. The same is true for the currents defined in Eq. (112): section we consider the specific cases of classical and quantum in presence of an external field that induces a current, we have transformations. h j+i = h j−i = h jci , 0, whereas h jqi = 0. Hence, as anticipated µ above, the continuity equation h∂µ jqi = 0, which follows from 4. Closed systems: energy, momentum and particle number the symmetry under a classical transformation, does not entail conservation a non-trivial Noether charge. In the next section, we show that such a symmetry nevertheless does have physical consequences, as it can be broken spontaneously in a condensation transition. Energy and momentum conservation – This property is ex- Before, let us derive the Noether charge associated with pected for system whose classical action is determined by a the symmetry of a Hamiltonian Keldysh action under contour- translationally invariant Hamiltonian alone. Indeed, the con- dependent — i.e., quantum – translations in space-time. To struction of the Keldysh action for a Markovian master equation be specific, we consider the transformation Ψ (X) 7→ Ψ (X + in Sec.IIA shows that terms which couple the forward and back- σ σ σαe ), where X = (t, x), and e is the basis vector in direc- ward branches are only contained in the dissipative parts of the ν ν tion of the νth coordinate of d + 1-dimensional space-time. This action (30). In other words, they arise upon coupling the system is a symmetry of the action (108): the shift of the coordinates to a bath and integrating out the latter. On the other hand, the can be absorbed by performing a change of integration variables Keldysh action for a closed system with unitary dynamics does X → X − σαe in the integral over the Lagrangian density . not contain such terms and can be written as ν Lσ Clearly, the symmetry is violated in the presence of dissipative Z X terms that coupled the forward and backward branches. S = σ Lσ, (108) For space-time translations, the generators g± in Eq. (109) ac- σ t,x quire an additional index ν and are given by g±,ν = ±∂ν. It µ where L+ and L− are the Lagrangian densities evaluated with follows that the vector field fσ,ν in Eq. (111) takes the form µ µ fields on the forward and backward branches, respectively. fσ,ν = δν Lσ, and the classical Noether current, which we de- Given this structure of the action, let us consider a transforma- note as Tc,ν, is given by tion Tα which does not mix fields on the forward and backward branches. Then, the generator G has the block-diagonal struc- µ 1 X µ µ ∂Lσ µ T = T , T = ∂νΨσ − δ Lσ, (113) ture c,ν 2 σ,ν σ,ν ∂∂ ΨT ν σ µ σ ! g 0 G = + . (109) µ 0 g− where T±,ν are the components of the energy-momentum ten- sor [82, 195] on the forward and backward branches. In particu- 0 0 The cases of physical interest are the classical and quantum lar, the component T±,0 is the energy density, whereas T±,i is the transformations mentioned above, corresponding to the choices density of momentum in the spatial direction i = 1, 2,..., d. The g+ = g− and g+ = −g−, respectively. In both cases, the Noether spatial integrals over these densities yield the associated Noether current can be represented as superposition of currents on the charges, e.g., if we express the Lagrangian density Lσ in terms forward and backward branches, which we define in terms of of a Hamiltonian density (as in Eq. (30), where L is given by g ≡ g+ as Eq. (28) with γα = 0) as

µ ∂Lσ µ 1 jσ = gΨσ − fσ , (110) † T Lσ = Ψσiσz∂tΨσ − Hσ, (114) ∂∂µΨσ 2 29 we obtain the conserved energy density E , which is as expected rotations. In an open system, this symmetry is absent, and the given by the sum of the Hamiltonian densities on the forward continuity equation has to be extended to account for the ex- and backward branches, change of particles with the environment, as we discuss in the Z Z following. 1 X 0 1 E = T = (H + H−) . (115) 2 σ,0 2 + To be specific, we consider the Keldysh action (73) for a sys- σ x x tem with single-particle pump as well as single-particle and two- Along the same lines, conservation of angular momentum fol- body loss. In order to derive the extended continuity equation, lows from the quantum symmetry of the Keldysh action with as in Eq. (103) we perform a change of integration variables respect to rotations of the spatial coordinates. Ψ → UqΨ, where Uq is a local quantum phase rotation, in the Number conservation – As another example, we consider Keldysh partition function. Since the partition function is invari- phase rotations and their relation to particle number conser- ant under this transformation, the coefficients in an expansion of vation. In this case the classical transformation reads ψ± 7→ the partition function in powers of the phase shift must vanish. iα Ucψ± = e ψ± and the quantum transformation is ψ± 7→ Uqψ± = In linear order, we find the condition ±iα e ψ±. Classical phase rotations are a symmetry of the Keldysh ∗ µ ∗ ∗ ∗
2 action, if the fields appear only in the combinations ψσψσ0 . The h∂µ jc i − γphψ+ψ−i + γlhψ−ψ+i + 4udh ψ−ψ+ i = 0. (117) quantum transformation is more restrictive: in this case only products with σ = σ0 are allowed. Hence, the symmetry with This should be compared to Eq. (105): if Uq were a symme- regard to quantum phase rotations implies the classical symme- try of the action, we would have found an ordinary continuity try. The generators of Uq are given by g± = ±iσz. Then, with equation. This would have been the case in the absence of the Eq. (10) and inserting the representation of the Lagrangian den- pumping and loss terms proportional to γp, γl, and ud. The inter- sity in Eq. (114) in the expression for the Noether current (110), pretation of these terms is as follows: in a system that is perfectly T we find that the latter can be written as jσ = (ρσ, jσ) , where isolated from its environment, the temporal change of the den- 2 ρσ = |ψσ| is the density on the contour with index σ and the sity at a given point is due to the motion of particles towards to spatial components of the current are given by or away from this point, i.e., the rate of change of density is a source or sink for the mass flow as measured by the divergence 1 ∗ ∗
jσ = ψσ∇ψσ − ψσ∇ψσ , (116) of the current h∇ · jci. In an open system, on the other hand, i2m there is in addition a dissipative current [204, 205] due to the which is just the ordinary quantum mechanical current for par- exchange of particles between the system and its environment. ticles with quadratic dispersion relation, evaluated on the closed In particular, in Eq. (117) this dissipative current has contribu- time path. What is the physical meaning of the classical and tions from the pumping and loss terms. quantum components, which are defined in Eq. (112), of the cur- It is interesting to note that Eq. (117) can also be ob- rent given in Eq. (116)? If we introduce in the Hamiltonian an tained from the equation of motion of the local density n(x) = externally imposed gauge field, in the Keldysh action it would ψ†(x)ψ(x) in the operator formalism. From the master equation quantum couple to the current, in analogy to the source terms ∂tρ = Lρ with Liouvillian L in Lindblad form given in Eq. (9), in the generating functional Eq. (33). However, the observable it follows that the time evolution of n(t, x) in the Heisenberg rep- classical ∗ effect of such a gauge field is that can induce a current resentation is given by ∂tn(t, x) = L n(t, x) with the adjoint Li- h jci , 0. ouvillian defined by tr(ALB) = tr(BL∗A). We find We close this section with a comment on the status of the Uq(1) symmetry. It will be present on the microscopic level for ∂ n(t, x) = −∇ · j(t, x) + γ ψ(t, x)ψ†(t, x) an action corresponding to a Hamiltonian which commutes with t p † 2 2 the number operator. Since the symmetry considerations are − γln(t, x) − 4udψ (t, x) ψ(t, x) , (118) properly applied to the microscopic action (and the functional measure), the conservation law ensues. However, this does not where first term encodes coherent dynamics and corresponds to the Heisenberg commutator i[H , n(t, x)], whereas the remain- imply that the effective action must manifestly preserve a Uq(1) LP symmetry. In fact, classical models of number conserving dy- ing contributions incorporate the dissipative parts. Taking the namics [1] do not exhibit this symmetry. This leads to the pic- average of this relation and expressing the expectation values as ture that this symmetry is broken spontaneously under RG. The Keldysh functional integrals yields Eq. (117). precise workings of such a mechanism are, to the best of our Finally, we can obtain some intuition on the dissipative cor- knowledge, not settled so far. rection to the current expectation value in mean field theory, where the correlators in Eq. (117) factorize into products of sin- gle field expectation values hψ+i = hψ−i = ψ0. We then recog- nize in the non-derivative terms (ψ∗ times) the LHS of Eq. (75) 5. Extended continuity equation in open systems √ 0 (note that due to the factor of 1/ 2 in the Keldysh√ rotation (32) The conservation of particle number in a closed system fol- the expectation values are related as hφci = φ0 = 2 hψ±i, and lows from the continuity equation for the classical Noether cur- that rd = (γl − γp)/2), which equals zero in a homogeneous sit- rent that is associated with the symmetry under quantum phase uation within mean field theory. In this way, we see that there is 30 no particle number current on average in a homogeneous driven- tions of fields which are invariant under classical or quantum dissipative system in stationary state, as expected. phase rotations. In the basis of classical and quantum fields, ∗ these Uc-symmetric combinations are ρνν0 = φνφν0 , where the indices ν and ν0 can take the values c and q. In the following, we 6. Spontaneous symmetry breaking and the Goldstone theorem show that Uc invariance is sufficient to guarantee the existence of a massless mode. For convenience we switch to a basis of real fields χ = Even though the pumping and loss terms in the Keldysh ac- (χ , χ , χ , χ ) which correspond to the real and imagi- tion in Eq. (73) break the symmetry with respect to quantum c,1 c,2 q,1 q,2 nary parts of the complex classical and quantum fields, i.e., phase rotations Uq, the action is still symmetric under Uc trans- 1
φν = √ χν, + iχν, for ν = c, q. In this basis the mass ma- formations. As a result, even in the absence of particle num- 2 1 2 ber conservation there is a possibility of spontaneous symmetry trix reads breaking. In particular, a finite average value hφci , 0 breaks the 2 " 2 2 # ∂ U ∂ ρa ∂U ∂ρa ∂ρb ∂ U classical phase rotation symmetry in a non-equilibrium conden- Mi j = = + , (120) ∂χ ∂χ ∂χ ∂χ ∂ρ ∂χ ∂χ ∂ρ ∂ρ sation transition. Here we show that this spontaneous symmetry i j ss i j a i i a b ss breaking is accompanied by the appearance of a massless mode, where the subscript ss indicates that the fields should be set to i.e., a mode with vanishing frequency and decay rate for zero their average values in the stationary state. a and b are double in- momentum, which is known as the Goldstone boson [173]. dices, taking the values cc, cq, qc, qq, and summation over these The single-particle excitation spectrum is encoded in the poles indices is implied. Let us consider the first term on the RHS of of retarded and advanced Green’s functions, and such a pole at Eq. (120): in the ordered phase, the classical field has a finite ex- ω = q = 0 is dubbed a massless mode. Poles of the Green’s pectation value in the stationary state. Without loss of generality functions correspond to roots of the determinant of inverse prop- we assume that this value is real. Then, the field equations agator (see Sec.IIA). Hence, the presence of a massless mode can be detected by checking whether the determinant of the mass ∂U ∂ρa ∂U matrix M vanishes. The latter is determined by the inverse prop- = = 0, (121) ∂χi ss ∂χi ∂ρa ss agator which actually determine the average value of the fields χi in A ! p 0 P (ω, q) −1 stationary state, have the solution χi|ss = 2ρ0δi,1. Performing P(ω, q) = R K = G (ω, q), (119) P (ω, q) P (ω, q) the derivatives ∂ρa/∂χi in Eq. (121) explicitly and inserting χi|ss, we obtain the following conditions: − given exemplarily in Eq. (76), at ω = q = 0, i.e., M = P(0, 0). Therefore, it is sufficient to consider frequency- and momentum- ∂U ∂U ∂U = = = 0. (122) independent field configurations or, in other words, homoge- ∂ρcc ss ∂ρcq ss ∂ρqc ss neous field configurations, so that instead of the full effective action Eq. (39) we only have to deal with the effective potential Therefore, only the derivative uqq = [∂U/∂ρqq]ss contributes U = − Γ|hom. /Ω, where Ω is the quantization volume in space- to the first term on the RHS of Eq. (120). Denoting mixed time. Since the effective potential U inherits the symmetries of second derivatives of the effective potential as uab = uba = 2 the effective action, it is a function of precisely these combina- [∂ U/∂ρa∂ρb]ss, we find that the mass matrix can be written as

    0 0 0 0   0 0 ucc,cq + ucc,qc i ucc,cq − ucc,qc         0 0 0 0  0 0 0 0        M =   + ρ0  1 i  . (123) 0 0 uqq 0   ucc,cq + ucc,qc 0 ucq,cq + 2ucq,qc + uqc,qc ucq,cq − uqc,qc       2    2  0 0 0 uqq  i 1  i ucc,cq − ucc,qc 0 2 ucq,cq − uqc,qc − 2 ucq,cq − 2ucq,qc + uqc,qc

An entry ucc,cc is not ruled out by symmetry, however, it must (i.e., the complex dispersion relation has the property ω(q = vanish due to conservation of probability, cf. Sec. II B 1. Cru- 0) = 0), it does not provide a statement on the functional de- cially, the retarded and advanced sectors feature one zero eigen- pendence ω(q). The latter could be inferred along the lines of value. This proves the existence of a massless mode as a conse- Refs. [206–210]. As we found in Sec. II B 2 in Bogoliubov ap- quence of spontaneous symmetry breaking for a theory with Uc proximation, in an open system without (i.e., broken) Uq sym- invariance. metry and spontaneously broken Uc symmetry, the leading be- havior at low momenta is diffusive, ω(q) ∼ −iDq2, with a real While this analysis guarantees the existence of a zero mode 31 diffusion coefficient D. This has to be contrasted to closed sys- work. tems, in which microscopically both Uq and Uc symmetry are The transition from the action S to the effective action Γ con- present. There, the leading behavior in a phase with sponta- sists in the inclusion of both statistical and quantum fluctuations neously broken Uc symmetry is that of coherent sound waves, into the latter (cf. Eq. (39)). In the functional renormalization ω(q) ∼ cq, with real speed of sound c. A phenomenological group approach based on the Wetterich equation [214], the func- justification of this behavior is given in Ref. [1], where this can tional integral over fluctuations is carried out stepwise. To this be understood as a consequence of a coupling to additional slow end, an infrared regulator is introduced, which suppresses the modes relating to particle number conservation. In these phe- fluctuations with momenta less than an infrared cutoff scale Λ. nomenological models, Uq symmetry is absent. This suggests a This is achieved by adding to the action in (33) a term scenario of an additional spontaneous breakdown of U symme- Z ! ! q   0 R φ try upon coarse graining, but this issue seems not to be settled to ∆S = φ∗, φ∗ Λ c , (124) Λ c q R∗ 0 φ date. t,x Λ q with a cutoff or regulator function RΛ. Some key structural prop- erties are indicated below, but apart from these properties the E. Open system functional renormalization group choice of the cutoff is flexible and problem-specific. The re- sulting cutoff-dependent generating functional and its logarithm (cf. Eq. (35)) are denoted by, respectively, Z and W . Then, the In Sec.IVB, we investigate dynamical criticality of the Λ Λ scaled-dependent effective action Γ is defined by modifying the Bose condensation transition in driven open systems, moti- Λ Legendre transform in Eq. (38) according to vated by many-body ensembles such as exciton-polariton con- densates [12, 13, 151, 182, 211]. There again, coherent dy- ΓΛ[Φ¯ c, Φ¯ q] = WΛ[Jc, Jq] namics naturally competes with dissipation in the form of in- Z  † ¯ † ¯  ¯ ¯ coherent particle losses and pumping. The situation parallels a − Jc Φq + Jq Φc − ∆S Λ[Φc, Φq]. (125) laser threshold [212, 213], however with a continuum of spa- t,x tial degrees of freedom. This ingredient, however, causes the The subtraction of the ∆S Λ on the RHS guarantees that the only characteristic long-wavelength divergences of many-body prob- difference between the functional integral representations for Γ lems in their symmetry broken phase, or at a critical point. It and ΓΛ is the inclusion of the cutoff term in the latter, implies that perturbation theory necessarily breaks down, even Z ¯ ¯ ¯ ¯ when the interaction constants are small, due to a continuum of iΓΛ[Φc,Φq] iS [Φc+δΦc,Φq+δΦq] e = D[δΦc, δΦq] e modes without a gap, which form the intermediate states and δΓ δΓ are summed over in many-body perturbation theory. This calls −i δΦ −i δΦ +i∆S [δΦ ,δΦ ] δΦ¯ T q δΦT c Λ c q for the development of efficient functional integral techniques × e c q . (126) able to cope with these problems. Our method of choice is the Physically, ΓΛ can thus be viewed as the effective action for av- functional renormalization group based on the Wetterich equa- erages of fields over a coarse-graining volume with a size ∼ Λ−d, tion [214], which we briefly introduce here in its Keldysh for- where d is the spatial dimension. mulation. This approach offers the particularly attractive feature Note that we chose the form of the cutoff action ∆S Λ such of not being restricted to the critical point. that it modifies the inverse retarded and advanced propagators The functional renormalization group equation (for reviews in Eq. (124) only. This is sufficient to regularize possible in- see [215–219]) constitutes an exact reformulation of the func- frared divergences, which result from poles of the retarded and tional integral representation of a quantum many-body prob- advanced propagators being located at the origin of the complex lem in terms of a functional differential equation. In this, it is frequency plane. A typical choice in practical calculations is strongly distinct from, e.g., perturbative field theoretical renor- 2 2 malization group approaches, which concentrate exclusively on RΛ(q ) ∼ Λ , q/Λ → 0, (127) the critical surface of a given problem. Instead, it may be viewed giving the inverse propagators a mass ∼ Λ2. In this way, fluctu- as an alternative and potentially more tractable tool for the anal- ations with wavelength & Λ−1 are effectively cut off. Therefore, ysis of the complete many-body problem, also on length scales for any finite Λ, the technical problem of infrared divergences is well below the correlation length near criticality. Indeed, it has under control. proven a very versatile tool in many different physical context, The main usefulness of the so-modified effective action, how- ranging from quantum dots [220–222], ultracold atoms [219], ever, lies in the fact that it smoothly interpolates between the strongly correlated electrons [223], classical stochastic mod- action S for Λ → Λ0, where Λ0 is the ultraviolet cutoff of the els [224, 225], quantum chromodynamics [218, 226], to quan- problem (e.g., the inverse lattice spacing), and the full effective tum gravity [227]. Here we give a brief overview of the gen- action Γ for Λ → 0. This is ensured by the following require- eral concept adapted to non-equilibrium systems [26, 172, 220– ments on the cutoff [230]: 222, 228–240]. It is used for the discussion of critical behavior 2 ∼ 2 → in driven open quantum systems in Sec.IV, with applications RΛ(q ) Λ0, Λ Λ0, 2 (128) to a broader non-equilibrium many-body context left for future RΛ(q ) → 0, Λ → 0. 32

Under the condition that Λ0 exceeds all energy scales in the ac- A. Ising spins in a single-mode cavity tion, for Λ → Λ0 we may evaluate the functional integral (126) in the stationary phase approximation. Then, we find to lead- As for the equilibrium path integral, the Keldysh field theory ing order Γ ∼ S — in the absence of fluctuations (suppressed Λ0 for spin models which obey the standard Ising Z2 symmetry, is 2 by the cutoff mass gap ∼ Λ0), the effective action approaches formulated in terms of real fields, fluctuating in time and space. the classical, or microscopic one. The evolution of ΓΛ from this These models have become rather important in the field of quan- starting point in the ultraviolet to the full effective action in the tum optics, where the typical situation consists of a set of atoms, infrared for Λ → 0 is described by an exact flow equation — the modeled as two-level systems, coupled to the radiation field of Wetterich equation [214] — which was adapted to the Keldysh a high finesse cavity. One important model in this context is the framework in [220, 221, 230, 232]. It reads Dicke model, which has been introduced in Eq. (1) as the generic model for cavity QED experiments with strong light-matter cou- i  −1  ∂ Γ = Tr Γ(2) + R ∂ R , (129) pling. Its Hamiltonian is Λ Λ 2 Λ Λ Λ Λ ω XN g XN   (2) † z z x † where Γ and R denote, respectively the second variations of H = ωpa a + σ + √ σi a + a . (130) Λ Λ 2 i i=1 N i=1 the effective action and the cutoff action ∆S Λ. As anticipated above, the flow equation provides us with an alternative but fully The Dicke Hamiltonian represents an effective model which de- equivalent formulation of the functional integral (126) as a func- scribes the dynamics of a set of two-level atoms, labeled by the tional differential equation. Like the functional integral, the flow index i, inside a cavity, which is pumped by a transverse laser. equation can not be solved exactly for most interesting prob- In this scenario, the coupling constant g describes the scattering lems. It is, however, amenable to numerous systematic approxi- of laser photons into the cavity, as well as the reverse process, mation strategies. For example, in the vicinity of a critical point and therefore rotates in time with the laser frequency ωl. In a ro- it is possible to perform an expansion of the effective action ΓΛ tating frame, this time dependence is gauged away in the Hamil- in terms canonical scaling dimensions, keeping only those cou- tonian, resulting in an effective shift ωp = ωc − ωl of the bare plings which are — in the sense of the renormalization group — cavity frequency ωc. In addition to this external drive, the cavity relevant or marginal at the phase transition, cf. the discussion in is subject to permanent photon loss, due to imperfections in the Sec.IIC, and see Sec.IVB for applications. cavity mirrors. The photon loss is described by a weak coupling of the intra-cavity photons to the environment, i.e., the vacuum radiation field outside the cavity. For typical experimental pa- Part II rameters, this coupling, which represents the rate with which the intra-cavity field and the environment exchange photons, is Applications much smaller than the typical relaxation time of the environment and the latter can be traced out under the Born and Markov ap- III. NON-EQUILIBRIUM STATIONARY STATES: SPIN proximation. This results in a Markovian master equation for the MODELS system’s density matrix, which reads (cf. Eqs. (2,16) repeated for convenience)

In this section, we discuss the steady state properties of many- ∂tρ = −i[H, ρ] + Lρ. (131) body systems consisting of atoms, which are coupled to the radi- ation field of a cavity, in turn subject to dissipation in the form of In this equation, ρ is the density matrix, H is the Dicke Hamilto- permanent photon loss. The corresponding low frequency field nian (130) and L is the dissipative Liouvillian, acting as on the theory, a 0 + 1-dimensional path integral for real valued, Ising density matrix as type fields corresponds to the simplest, non-trivial field theoretic ! † 1 † models and is therefore particular useful to get used to applica- Lρ = κ aρa − {a a, ρ} (132) 2 tions of the Keldysh formalism for relevant physical setups. The basic model describing the dynamics of atoms in a cavity is the with loss rate κ for the cavity photons. Dicke model (1) with dissipation, as described in Sec.IC1, as Given the Markovian master equation (131), there are two well as its extension to multiple cavity photon modes. Despite common ways to derive a corresponding path integral descrip- the simplicity of the underlying field theory, it is a non-trivial tion for the dissipative Dicke model, which are based on different task to solve it for its rich many-body dynamics. This includes representations of the atomic degree of freedom in terms of field the critical behavior of a single mode cavity at the superradi- variables: the representation of the atomic degrees of freedom ance transition as well as universal dynamics in the formation of as a collective spin and subsequent bosonization in terms of a spin glasses in multi-mode cavities. We discuss these and fur- Holstein-Primakoff transformation [117, 118, 123], and the rep- ther features of the Dicke model here by putting a focus on the resentation of the atomic variables in terms of individual Ising theoretical framework of solving the corresponding Ising model fields. We discuss both approaches in the following, putting a on the Keldysh contour. focus on the more general Ising representation, which can also 33 be applied in the case of multiple cavity modes and individual nian (133), these spin operators can be transformed into bosonic atomic loss processes, where a representation of the atoms in operators via the common Schwinger-boson or Holstein- terms of a single, collective spin is no longer possible. In the Primakoff transformations. For weak coupling g, the system re- regimes for which both approaches can be applied, they are com- mains strongly polarized |hS zi|  1 and the Holstein-Primakoff pletely equivalent, as we demonstrate. transformation around the non-interacting ground state, which has the eigenvalue S z = −N/2, is the most natural choice. It reads 1. Large-spin Holstein-Primakoff representation N  p   p  S z = b†b − , S x = N − b†b b + b† N − b†b (134), 2 The single-mode Dicke Hamiltonian (130) has the property that all atomic variables couple to the cavity photon mode via where b, b† are bosonic operators. In the thermodynamic limit the same coupling constant g. Consequently, the coupling of N → ∞, the square root in the S x operator is expanded in pow- the photons to the sum of the individual Pauli matrices can be ers of 1/N, and the Hamiltonian is subsequently formulated on replaced by the coupling of the cavity photon mode to a large the Keldysh contour. To zeroth order in 1/N, the corresponding spin, Keldysh action is g     −1  † z 2 x † Z  A  H = ωpa a + ωzS + √ S a + a , (133) †  0 G4×4  N S = Φ (ω)   −1  Φ(ω), (135) ω  R K  G4×4 Σ4×4 where S z, S x are spin operators in a spin-N/2 Hilbert space. In order to find a path integral representation for the Hamilto- with the combined Nambu-Keldysh spinor

 ∗ ∗ ∗ ∗ T Φ(ω) = ac(ω), ac(−ω), bc(ω), bc(−ω), aq(ω), aq(−ω), bq(ω), bq(−ω) , (136) the inverse retarded Green’s function in Nambu space

 ω − ω + iκ 0 −g −g   p  −1    R   0 −ω − ωp − iκ −g −g  G4×4 =   (137)  −g −g ω − ωz 0    −g −g 0 −ω − ωz

and the Keldysh self-energy modified compared to their non-interacting values, see Fig.7. For small coupling strength, the elementary excitations can still K Σ = 2iκ diag(1, 1, 0, 0). (138) be seen as weakly dressed atoms and photons, with excitation energies close to the non-interacting values, while they strongly Due to the photon decay terms ∼ κ and the atom-photon cou- hybridized, inseparable degrees of freedom for strong coupling pling g, the above theory is regularized even without infinites- strengths. imal imaginary contributions in the atomic sector. When inte- grating out the photons, the latter infinitesimal contributions are overwritten in any case by the finite imaginary part of the photon Green’s function, and it is therefore reasonable to leave them out from the start. Above a critical coupling strength gc, the ground state of the The excitation spectrum of the atom-photon system is en- system breaks the Z2 symmetry and the atoms form a macro- coded in the retarded Green’s function, and the poles, marking scopic spin aligned in the x-direction, hS xi = O(N), which is ac- the excitation energies, fulfill the requirement companied with a coherent macroscopic population of the intra- cavity mode hai , 0. This symmetry broken phase is called the  −1 ! R 2 2 2 2 2 superradiant phase of the cavity system. In the case of a macro- 0 = det G4×4 = (ω − ωz )[(ω + iκ) − ωp] − 4g ωpω(139)z. scopic expectation of S x, the orthogonal S z component can no In the absence of atom-photon coupling, i.e., for g = 0, these longer be macroscopically large, which renders any expansion are the non-interacting poles ω1,2 = ±ωz, corresponding to the of the Holstein-Primakoff operators (134) in 1/N invalid. This atomic transition frequencies and ω3,4 = ±ωp+iκ, corresponding is expressed by an instability of the quadratic theory at the su- to the energy of a photon ωp and its decay rate κ. For g > 0, the perradiance transition, revealed by the presence of a zero energy modes begin to hybridize and the excitation energies are slightly mode, i.e., a pole at ω = 0 in the excitation spectrum. According 34 to Eq. (139), this happens at s 2 2 (κ + ωp)ωz gc = . (140) 4ωp

The mode structure in the vicinity of the transition has been ana- lyzed in Ref. [107], where it has been found that in the presence of photon decay, the critical mode becomes purely imaginary before the transition happens, and the corresponding critical dy- Figure 7. Illustration of the pole structure of the system’s eigenmodes. namics corresponds to a classical finite temperature transition, The poles represent the location of the mode frequencies ω in the com- see also Fig.7. This is mirrored by the fact that the photons plex plane, the real part Re(ω) represents the energy of the mode, while effectively thermalize in the low energy regime and their corre- the imaginary part Im(ω) is the mode’s decay rate. Without interactions, g = 0, the system is described by the bare atomic, ω = ±ω , and pho- lations can be described by a low energy effective temperature z tonic modes ω = ±ωp−iκ. For finite interactions, the atoms and photons Teff. The effective temperature can be obtained via a fluctua- hybridize and form polaritonic modes, two of which move closer to the tion dissipation relation, as discussed in Eq. (72) but promoted imaginary axis. At the critical point g = gc, one single mode becomes to Nambu space (see Sec.VB for a discussion of the FDR in critical at ω = 0 and the system undergoes a phase transition from a Nambu space). Solving this equation yields the photon distribu- disordered phase to the Z2 symmetry breaking superradiant phase. The tion function in Nambu representation classical nature of the transition in the presence of dissipation is ex- pressed by the fact that the two zero energy modes Re(ω) = 0 become 2g2 ω2 purely dissipative for couplings g < gc already (indicated with the red F σz σx z . ph = + 2 2 (141) arrows). ωzω ω − ωz

2 For large frequencies ω  g , this corresponds to the zero by frequency integration over the photonic correlation function ωz temperature distribution of the non-interacting system F = σz, Z ! K −1 which is diagonal in the photon modes. The approach to this 2n = iGph (ω) − 1 ∼ |g − gc| . (142) limit is, however, algebraic ∼ ω−3, in contrast to an exponen- ω tial approach according to large frequency behavior of the Bose- The exponent νx = 1 found for the present non-equilibrium distribution function in equilibrium. On the other hand, for small transition coincides with the classical, finite temperature expo- 2 frequencies ω  g , the occupation becomes essentially ther- nent for the equilibrium Dicke model in line with the discussion ωz above. On the other hand, at zero temperature, the exponent mal and purely off-diagonal F = 2Teff σx, with low energy effec- ω of the corresponding quantum phase transition is found to be g2 tive temperature Teff = , cf. Eq. (84). In this low frequency ωz νx = 1/2. limit, the elementary excitations are strongly hybridized polari- The scaling behavior of the critical mode at the transition is ton modes, which are diagonal in the photon quadratures x and expressed in terms of the dynamical critical exponent νt. In the p. The absence of a global temperature scale in the present presence of dissipation, the excitations in the system decay ex- system reflects the fact that due to the Markovian dissipation, ponentially in time, which results in an asymptotic exponential this interacting system with a discrete set of degrees of freedom decay of real-time correlation functions is not able to achieve detailed balance between the particular h{a(t), a†(0)}i = GK(t) ∼ e−t/ξt . (143) modes, i.e. the x and p quadratures, for all energy scales. ph

One should note that Teff is not proportional to the decay rate The characteristic decay time ξt is determined by the slowest κ and consequently the zero temperature equilibrium limit is not decaying mode in the system, i.e., by the mode with the smallest simply obtained by taking the limit κ → 0. The reason is that in imaginary part in the frequency spectrum. In the vicinity of the the present setting, the atom-photon coupling g leads to a com- phase transition, this is the critical mode , which shows scaling petition of unitary and dissipative dynamics, which is reflected 1 in the fact that for g = 0, the eigenstate of the Hamiltonian is a ξt ∼ , (144) steady state of the dynamics while this is no longer the case for |g − gc| finite g. As a consequence, the low energy effective temperature i.e., a dynamical critical exponent νt = 1. This behavior is in depends on g and vanishes in the limit g → 0. stark contrast to a zero temperature quantum phase transition, A further consequence of the dissipative nature of the tran- where correlation functions do not decay over time but oscillate sition is that the critical exponents correspond to the classical with characteristic frequencies ωc, the smallest of which indi- finite temperature equilibrium transition. One such critical ex- cates the distance to the phase transition and encodes the critical ponent of the superradiant transition is the so-called photon flux scaling behavior. The fact that in the present setting the critical exponent νx, which describes the divergence of the cavity pho- mode becomes purely imaginary is a generic feature of dissipa- −ν ton number at the transition, i.e., n ∼ |g − gc| x . It is obtained tive phase transitions, which is discussed further in Sec.IV. 35

All of the above discussed results are encoded in the quadratic which introduces dynamical Lagrange parameters λi(t) for each Keldysh action, described by Eqs. (135-138). The resulting crit- spin variable. In this framework, the purely atomic part of the ical behavior is then that of a non-interacting system, and is action on the (±)-contour is associated accordingly to a Gaussian fixed point of the renor- Z 1 X h  2  2i malization group flow. This is in contrast to critical behavior S at = α λαi φ − 1 − (∂tφαi) . (148) ωz αi in interacting systems, which are described by an interacting ω α=± i=1,..,N Wilson-Fisher fixed point, not smoothly connected to the pre- vious one (cf. the discussion in Sec.IVB). This action describes The action for the atom-photon coupling and the bare photonic the problem up to 1/N corrections resulting from the expansion part is straightforwardly derived according to the previous sec- of the Holstein-Primakoff bosons in 1/N. It contains correctly tions, and the corresponding Keldysh action is the physics in the thermodynamic limit N → ∞ and captures Z X 2! ! the essential dynamics of the superradiance transition. For finite 1 λqi(t) λci(t) + ∂t φci S = (φci, φqi) 2 systems, the higher order terms in 1/N can not be neglected and ω λci(t) + ∂ λqi(t) φqi z t i t have to be taken into account properly. Analytical approaches, Z g X  ∗ ∗  relying on an expansion of the Holstein-Primakoff operators up + √ φci(aq + aq) + φqi(ac + ac) to first order in the ratio 1/N, have shown that the first order cor- N t i Z ! ! rection term leads to a quartic contribution in the action, whose ∗ ∗ 0 i∂t − ωp − iκ ac self-consistently determined one-loop corrections are already in + (ac, aq) . (149) t i∂t − ωp + iκ 2iκ aq very good agreement with numerical results [107]. The latter have been obtained from a Monte-Carlo wave function (MCWF) It is quadratic in the fields φ, a, a∗ as it was the case for the simulation of the master equation and agreed with the analytical Holstein-Primakoff action in the large-N limit. However, the results even on the level of N = 10 atoms. non-linear constraint, i.e., the fact that λq,c is a dynamic vari- able, introduces higher order terms in the Ising fields. The ap- proach corresponding to the large-N expansion of the Holstein- 2. Effective Ising spin action for the single-mode cavity Primakoff fields in the present formalism (in the sense that the action becomes quadratic) is to treat the Lagrange parameters as The Dicke model defined by the master equation (131) obeys static mean field variables, i.e., λci(t) → λc and λqi(t) → λq. Due the common Ising Z2 symmetry, which is spontaneously bro- to causality, a static mean quantum field must be zero, λq = 0. ken by the ground state of the superradiant phase. Another ap- On the other hand, the value of λc has to be chosen such that the proach to express the Dicke model in a Keldysh path integral non-linear constraint is preserved on average approach is therefore to represent the spin operators in terms Z x 2 2 K ! of real Ising field variables, σx(t) → φ(t). In contrast to the h(σi (t)) i = hφc(t)i = iGat (ω) = 1. (150) Holstein-Primakoff transformation applied in the previous sec- ω tion, the Ising representation does not require a single, large spin This is equivalent to a vanishing variation of the action with re- to have a well defined field theory representation, and is there- spect to the Lagrange parameters λc,q and is known as the “soft- fore also applicable in situations in which the atom-photon cou- spin” approach to spin models (i.e. the constraint is treated on pling is spatially dependent g → g(x) and a large spin represen- the mean field level). It becomes exact in the thermodynamic tation becomes impossible. Furthermore, we use this approach limit N → ∞ [242, 243]. to describe the symmetry broken phase of the problem. Integrating out the N individual atoms leads to the effective The atomic sector of the Hamiltonian (130) describes Ising photon action spins in a transverse field and the corresponding universal low   −1  energy description for frequencies below the level spacing ω is 1 Z  0 GA  z †  2×2  obtained by transforming the spin operators to real fields accord- S = A (ω)   −1  A(ω), (151) 2 ω  GR 2iκ diag(1, 1)  ing to 2×2

x with the Nambu spinor σi (t) → φi(t). (145)  a (ω)   c  From the standard quantum to classical mapping [241], one ob-  a∗(−ω)  tains for the transfer matrix A(ω) =  c  (152)  aq(ω)   ∗  z 2 2 aq(−ω) σ (t) → 1 − 2 (∂tφi(t)) . (146) i ωz and the inverse retarded Green’s function x 2 The dynamic constraint (σi (t)) = 1 is implemented via the non-  2g2ω 2g2ω  linear constraint  −1 − − z − z R  ω ωp + iκ ω2−λ ω2−λ  G =  2 2  . Z 2×2  2g ωz 2g ωz  2 iλ (t)(φ2(t)−1) − 2− ω − ωp − iκ − 2− − D l i ω λ ω λ δ(φi (t) 1) = λl(t) e , (147) (153) 36

The poles of the Green’s function are again determined by the of the transition, as well as the mode structure and the critical zeros of the determinant of the inverse Green’s function, i.e., by scaling behavior have been directly derived from the quadratic the roots of the equation Keldysh action, which illustrates the strength of the present field theory approach. In the following section, we discuss the exten- ! 2  2 2  2 0 = (λc − ω ) (ω + iκ) − ωp + 4ωpωzg . (154) sion of the Dicke model to multiple cavity modes, which con- tains the possibility of frustrated atom-atom interactions and the For the non-interacting theory, i.e., for g =√ 0, this determines formation of a spin glass in the cavity system. the poles of the atomic sector to be ω = ± λc and in turn fixes 2 λc = ωz to be consistent with the microscopic theory. As long as the system is not superradiant, the atom-photon interaction does not modify the integral (150), and for the entire param- eter range of the normal phase, we find λ = ω2 as for the c z B. Ising spins in a random multi-mode cavity non-interacting case. This again yields the critical value of the coupling strength (140) we obtained above from the Holstein- Primakoff approach. Consequently, the results from the previous In the previous section, we saw that dissipation renders the su- section are recovered in the Ising representation of the Keldysh perradiant transition essentially a classical phase transition with path integral. critical exponent corresponding to a Z2 symmetry breaking, fi- We now turn to the description of the ordered phase. Di- nite temperature transition. However, no signature of the non- rectly at the superradiant transition, the system becomes unsta- equilibrium nature of the steady state could be found in the long- ble, which is indicated by a critical mode with frequency ω = 0. wavelength dynamics governing the transition since the phonon In order to stabilize the system, for coupling strengths g > gc distribution function shows a classical Rayleigh-Jeans diver- the steady state breaks the Z2 symmetry of the action, which is gence F ∼ 1/ω at small frequencies. This changes drastically expressed in terms of a finite, symmetry breaking order parame- when the setup allows for multiple dissipative cavity modes, ter φc(ω) → ψδ(ω) + φc(ω), and equivalently in the photon basis which couple atoms and photons via a random interaction po- ac(ω) → aδ(ω) + ac(ω). These order parameters correspond to tential. The random interaction potential is realized by freezing x finite expectation values hσi (t)i = ψ and ha(t)i = a in the sys- the atoms on random positions inside the cavity and by exclud- tem’s ground state. They modify the soft-spin condition (150) ing atomic self-organization with a large set of modes. according to The effective, cavity mediated interactions in the case of mul- Z x 2 2 K ! tiple cavity modes are able to induce frustration in the effective h(σi (t)) i = ψ + iGat(ω) = 1, (155) ω atom-atom interactions. At a critical frustration level, these will K drive the system into an Ising spin glass phase. Due to the pres- where Gat is the Keldysh Green’s function of the fluctuating vari- ence of drive an dissipation in the quantum optical realization of able φ. As a consequence, in order to fulfill Eq. (155), the La- this spin glass phase, it features no analogue in condensed matter grange multiplier λc becomes a continuous function of the order physics and the corresponding spin glass transition does not cor- parameter, and therefore an implicit function of the coupling g in respond to the classical finite temperature transition [108]. The the superradiant phase. The dependence of λc can be determined corresponding model is motivated by recent works on ultracold by from Eq. (154). At the superradiant transition at g = gc, this atoms in cavities [108, 126, 244–246] and is described by the equation has one critical solution, i.e., the propagator has a pole master equation (131), where now the Hamiltonian at ω = 0, see Fig.7. For larger coupling strengths, this pole crosses the origin and obtains a positive imaginary part, thereby rendering the system unstable. In order to compensate for this N M ωz X z X † X † x mechanism, λc is modified such that the pole does not cross the H = σ + ω a a + g (a + a )σ (156) 2 s l l l sl l l s real axis, i.e., remains at its value ω = 0 in the superradiant s=1 l=1 s,l 2 λ 4g ωpωz g > g phase. Consequently c = κ2+ω2 for c. Via Eq. (155), this p p reveals the scaling of ψ ∼ |g − gc| when the transition is ap- describes the energy of M photon modes with frequency ωl and proached from inside the superradiant phase, which is the same N Ising spins with level spacing ωz, interacting via coupling con- critical behavior of the order parameter as for the equilibrium stants gsl = g0 cos(kl xs), which depend on the atomic position xs transition. and the photon mode function kl, such that −g0 ≤ gsl ≤ g0. The The results on the critical properties of the Ising field theory at dissipator describes the decay of each of the M photon modes the superradiance transition conclude the discussion of the single with a uniform decay rate κ, mode Dicke model in the framework of the Keldysh path inte- gral. We have shown that both the Holstein-Primakoff approach in terms of complex bosonic variables as well as the Ising ap- M ! X † 1 † proach in terms of real Ising variables are equivalent on the level Lρ = κ alρa − {a al, ρ} . (157) l 2 l of the quadratic, large-N limit of the theory. The critical point l=1 37

1. Keldysh action and saddle point equations where 2 N X Jsi The corresponding Keldysh action is similar to Eq. (149) and iS P = − (163) 2 K reads s,i Z X 2! ! 1 0 λc + ∂t φcs is the action of the random couplings Jsi, with correlations S= (φcs, φqs) 2 ω λc + ∂ 0 φqs z t s t K Z hJsi Jlmi = (δs,lδi,m + δs,mδi,l) . (164) X  ∗ ∗  N + gsl φcs(aql + aql) + φqs(acl + acl) (158) t sl In this sense, the coupling of the atomic degrees of freedom to Z ! ! the random variables J has the same structure as the coupling of X ∗ ∗ 0 i∂t − ωl − iκ acl si + (acl, aql) . the atoms to an external bath. However, the significant difference i∂t − ωl + iκ 2iκ aql t l to a Markovian bath, which would be δ-correlated in space and Here, the soft spin approximation has been performed and the time is that the quenched disorder, while correlated locally in scaling of the coupling constants gsl in the thermodynamic limit space, has infinite correlation time. The latter is expressed by − 1 the fact that the variables J are time-independent. is implicit, i.e. gsl ∼ N 2 . The thermodynamic limit is reached si by taking an extensive number of atoms N → ∞ but leaving the Averaging over all realizations of the couplings is done by a number of photon modes M < ∞ finite since the consideration of Gaussian integration (cf. App.B), and transforms the disorder an extensive number of photon modes is physically unrealistic. part, i.e., the second line of Eq. (159), to a time non-local quartic In order to obtain a large-N effective action, the photon degrees contribution of freedom are integrated out, leading to the atomic action in iK Z X S = ΦΛ (ω)ΦΛ (ν), (165) frequency space at-at N l,m m,l ω,ν l,m Z 2! ! 1 X 0 λc − ω φcs S= (φcs, φqs) 2 with ωz ω λc − ω 0 φqs s A ! ! Z A ! ! Λ 0 Λ (ω) φci X 0 Λ (ω) φci Φl,m(ω) = (φcs, φqs) R K . (166) + Jsi (φcs, φqs) R K . (159) Λ (ω) Λ (ω) φqi Λ (ω) Λ (ω) φqi si ω The double sum in (165) is decoupled via a Hubbard- The frequency dependent terms are the symmetrized photon Stratonovich transformation, which introduces the macroscopic Green’s functions 0 fields Qαα0 , with α, α = c, q being Keldysh indices [108, 126]. −ωp This transformation is in general not unique but can be made ΛR(ω) = , 2 2 unique by requiring the equivalence (ω + iκ) − ωp 2 2 2 (160) 2iκ(ω + κ + ω ) δ 1 X K p ⇔ 0 h 0 i ω Z = 0 Qαα (ω, ν) = N φαl(ω)φα l(ν) . (167) Λ ( ) = 2 δQ 0 2 2 αα l (ω + iκ) − ωp This identifies the Hubbard-Stratonovich field with the average with an average photon frequency of the cavity ω . The effective p atomic correlation function. Since one is interested in the sta- coupling tionary, time translational invariant state, M X gslgli J = (161) Qαα0 (ω, ν) = 2πδ(ω + ν)Qαα0 (ω). (168) si 4 l=1 After the Hubbard-Stratonovich decoupling, the resulting action Mg2 Mg0 0 is quadratic in the atomic fields and they are integrated out, lead- fluctuates between the values − ≤ Jsi ≤ and can be ei- 4 4 ing to the macroscopic action ther ferromagnetic Jsi < 0 or antiferromagnetic Jsi > 0 depend- " # ing on the s, i. Assuming gsl to be independently and equally 1 distributed for each atom, the couplings J are for sufficiently S = iNTr KΛQΛQ − ln G˜ (ω). (169) si 2 large M distributed according to a Gaussian distribution func- tion. We set their average value J¯ ≡ hJsii = 0 as it lifts the frus- Here Λ and Q are matrices in Keldysh space and G˜ is defined as tration in the system caused by fluctuating couplings but does not  −1 modify the universal behavior of the system at the glass transi- ˜ −1 G(ω) = G0 (ω) − 2KΛ(ω)Q(ω)Λ(ω) . (170) tion [108, 126]. The averaged partition function of the system, including the In the thermodynamic limit, the partition function is determined probability distribution for the couplings Jsi is by the saddle point value of the action, i.e. by the condition Z δS iS +iS P ! Z = D[φs]D[Jsi]e , (162) = 0 (171) δQαα0 38

0 for all α, α . This yields the values of the fields Qαα0 (ω) at the saddle-point, which can be identified with the atomic response and correlation functions. The saddle-point equations for the atomic response and correlation functions are !−1 2(λ − ω2)  2 QR(ω) = − 4K ΛR(ω) QR(ω) (172) ωz and

K 4K|QR|2ΛK (QAΛA+QRΛR) Q (ω) = 1−4K|QRΛR|2 . (173) Additionally, due to Eq. (167), the soft-spin constraint maps to the Q-fields according to Z K 2 X i Q (ω) = hφcl(−ω)φ (ω)i = 2. (174)  R A  N cl Figure 8. Spectral density A(ω) = i Q (ω) − Q (ω) of the atoms in ω l the glass phase for parameters K = 0.01, ωz = 2 and varying photon√ pa- In the glass phase, the spins attain temporally frozen configu- rameters ωp, κ. For frequencies below the crossover ω < ωc, A ∼ ω rations, expressed by an infinite correlation time of the atomic is overdamped, corresponding to Eq. (179), while it recovers the typi- cal thermal glass behavior A ∼ ω at intermediate frequencies ω > ωc. correlator, which is expressed by a non-zero Edwards-Anderson Figure copied from Ref. [108]. (Copyright (2013) by The American parameter Physical Society) 1 X x x qEA ≡ lim hσ (t)σ (0)i. (175) t→∞ N l l critical modes are described by poles in the complex plane which Consequently, the correlation function QK(ω) consists of a regu- are neither purely real as in the zero temperature (or quantum) lar part, describing the short time dynamics and a non-vanishing case nor purely imaginary as for the finite temperature transition. contribution at ω = 0. It can be expressed via the modified fluc- In the presence of dissipation, a photon that is emitted from tuation dissipation relation [247] an atom can either be absorbed by another atom, leading to an K K effective atom-atom interaction, which is infinite range in space, Q (ω) = 4iπqEAδ(ω) + Qr (ω) (176) or decay from the cavity. The latter happens on a characteristic in terms of the order parameter and a regular contribution time scale τp = 1/κ, which is as well the time scale for which K atoms can interact with each other by exchanging a photon be- Qr (ω). The soft-spin condition (174) together with Eq. (173) fixes the value of the Lagrange parameter λ and therefore fully fore the photon decays. As a consequence, the photon decay determines the spectrum of the system. Similar to the superradi- reduces the effective range of the atom-atom interaction in time ant transition, the Edwards-Anderson parameter becomes non- and reduces the strength of frustration in the system. Close to zero when the poles of the system become critical (approach the the glass transition and in the glass phase, this leads to the emer- −1 real axis) and its emergence is a mechanism to stabilize the crit- gence of a crossover scale ωc = τc , above which the spectral ical modes in the ordered phase. properties of the system correspond exactly to a zero tempera- ture spin glass. Below this frequency, the spectrum reveals the breaking of time reversal symmetry by the dissipation and for- 2. Non-equilibrium glass transition mally corresponds to the dynamical universality class of dissipa- tive quantum glasses, other examples of which are spin glasses coupled to an ohmic bath [247–249] or a bath of metallic elec- The variance of the effective, long-range atom-atom interac- trons [243, 245]. The crossover frequency is tion K is a measure of the frustration in the system and for suf- ficiently strong frustration, the system enters a glass phase, de-   2 −1  2 1 2 ω2 + κ2  scribed by an infinite autocorrelation time of the spins and the ωp + 2 κ p  ω = κ  + √  . (177) emergence of a non-zero Edwards-Anderson parameter qEA > 0. c  2 2   ωp + κ 2 Kω2  This goes hand in hand with a critical continuum of modes z reaching zero, which is the characteristic feature of a critical phase of matter. The phase diagram for the fully coherent model and the modifications of the spectrum below this scale are due to has been analyzed in [126] and in [108] it was shown that the the damping introduced by the Markovian bath. On the normal glass phase persists in the presence of dissipation. However, or disordered side of the phase diagram, the corresponding low the dissipative model shows new universal features of the glass frequency propagator is phase, which correspond neither to a zero nor to a finite temper- R  2 2−1 ature equilibrium transition. This is attributed to the fact that the Qn (ω) = Z (ω + iγ) − α , (178) 39 describing the Ising spins as damped harmonic oscillators with model in the normal as well as in the superradiant phase, the decay rate γ > 0 and frequency α, with the physical meaning LET for the photonic and the atomic subsystem did not coincide, of an inverse lifetime and energy gap of the atomic excitations. i.e., the subsystems have not thermalized but are held at differ- When the glass transition is approached, the gap, the inverse ent temperatures corresponding to their individual coupling to a lifetime and the residue Z scale to zero simultaneously and the bath [107]. This dramatically changes in the present system with atomic response in the glass phase multiple photon modes. As frustration is increased by driving the variance K towards the critical value Kc, atoms and photons  − 1 R ¯ 2 2 begin to thermalize towards the same, shared LET. The distri- Qg = Z ω + γ¯|ω| (179) bution function F of photonic or atomic modes is obtained by is non-analytic and can no longer be interpreted as a harmonic solving the fluctuation dissipation relation oscillator. The broken time reversal invariance manifests itself K R − A in both parameters γ, γ¯ , 0 being non-zero, which modifies the Q (ω) = Q (ω)Fat(ω) Fat(ω)Q (ω) (181) low frequency dynamics in the entire glass phase towards a dis- for the atomic Green’s functions. It leads to an atomic LET sipative quantum glass. This is for instance expressed by a spec- 2 2 tral density, which features√ an anomalous square root behavior ωp + κ A − R ∼ | | Teff = (182) (ω) = 2Im(Q (ω)) ω for small frequencies and there- 4ω fore has a non-analytic response at zero frequency, see Fig.8. p The latter leads to a characteristic algebraic decay of the photon in the glass phase, which coincides with the photonic LET [108]. correlation function, as discussed below. The universality class The thermalization of the two subsystems is a consequence of of the present dissipative quantum glass transition is determined the disorder induced effective long range interactions, which re- by the critical exponents at the transition, which describe the distribute energy between the different modes and enable equili- scaling of the parameters α, γ, Z as a function of δ = |K − Kc|. It bration even in the presence of the Markovian dissipation in the is summarized in the equations photon sector. In the paramagnetic phase, the distribution func- √ tions of atoms and photons are identical for frequencies ω > α 2 2 3 2(ωp+κ ) δ 2 αδ = √ larger than the gap, but deviate from each other for lower fre- 8 K3κ log(δ) 2 2 2 quencies. The elementary excitations above this frequency are ωp+κ δ , (180) γδ = 16K2κ log(δ) strongly correlated and can not be seen as weakly dressed pho- 2 2 3 ωp(ωp+κ ) δ tons or atoms. As a consequence, the observables in the critical Zδ = √ 8 K5κ2 log(δ) glass phase are dominated by the universal low energy behavior which show the typical logarithmic correction to scaling at a of the glass propagator (179) and thermal statistics of the excita- quantum glass transition and identify the critical exponents. In- tions. dicated by the scaling behavior, dissipative and coherent dynam- The strong light-matter interactions not only lead to thermal- ics rival each other when approaching the transition, separating ization of the atomic and photonic subsystems, they also fea- this glass transition from equilibrium transitions and the previ- ture a complete imprint of the glass dynamics of the atoms onto ously discussed Dicke transition, which are either fully coherent the cavity photons. This results in universal spectral proper- (quantum) with γ = 0 or fully dissipative (classical) with α = 0 ties of the photonic response, as has been shown in Ref. [108], sufficiently close to the transition. The present glass transition and concerns a complete locking of both subsystem’s low en- has therefore no counterpart in static equilibrium physics and is ergy response properties. The photons form a photon glass state, termed dissipative quantum glass transition. Similar behavior is which features a large, incoherent population of photon modes, present in system bath settings in equilibrium, where the bath signaled by a finite Edwards-Anderson parameter in the photon however not only imprints a finite temperature to the system but correlation function as well modifies its spectral properties [245, 247, 248]. What GK(ω) = 4πδ(ω)˜q + GK (ω), (183) these situations share in common is that the effective theory, after ph EA reg elimination of the bath variables, obeys no time reversal symme- which shows the same scaling behavior as the atomic Edwards- try, which ensures the same asymptotic universal behavior as in Anderson parameter close to the glass transitionq ˜EA ∼ qEA.A the case of the Markovian photon loss in the present system. finiteq ˜EA implies long temporal memory in photon autocorrela- tion functions and an extensive number of photons permanently occupying a continuum of high energy modes. The latter is re- 3. Thermalization of photons and atoms in the glass phase vealed by an algebraic decay of the system’s correlation func- tions, as for instance for the four point (or g(2)) correlation func- As typical for many open quantum systems (for exceptions, tion, which is shown in Fig.9. see the discussion in Sec.IV), the statistical properties of the The photon response and correlation functions are easily ac- excitations are described by a low energy effective temperature cessible via cavity photon output measurements, and therefore (LET) and a corresponding thermal distribution of the excita- represent the natural observables for the detection of the glass tions. However, as has been found for the single mode Dicke dynamics in cavity QED experiments. In fact, in Ref. [108], it 1 1

S(ν) Sinc(ν) Sc(ν) ν

S(ν) Sinc(ν) Sc(ν) ν

2κ 1 (ω) ω ωξδ ω ξδ A ω0√2Kδκ ∝ − 1 (ω) ω ωξδ ω ξδ A ω0√Kδ ∝ − S(ν) Sinc(ν) Sc(ν) ν 1 S(ν) Sinc(ν) Sc(ν) ν 3 κ =0.03,ω0 =0.9 κ =0.1,ω0 =1 κ = 10− ,ω0 =0.8 3 κ =0.03,ω0 =0.9 κ =0.1,ω0 =1 κ = 10− ,ω0 =0.8

2κ S(ν) Sinc2κ(ν) Sc(ν) ν (ω) ω (ω)ωξω δ ω ωξξδ δ ω ξδ A ω0√AKδ ω∝0√Kδ−ω ∝ − ∝ ω ∝

3 κ =0.03,ω0 =0.9 2κκ=0.1,ω0 =1 κ = 10− ,ω0 =0.8 3 (ω) ωκ =0.03,ω0ωξ=0.9 κ =0.1ω,ω0 =1ξ κ = 10− ,ω0 =0.8 J √Kκδ =0.2 κ =0δ.05 κ =0 A ω0√Kδ ∝ − J √Kκ=0 .2 κ =0.05 κ =0 ω ∝ ω ∝ 3 3 c κ =0.03,ωκ0=0=0.1.,ω9 0 κ=0=0.7 .1κ,ω=00 .=103,ω0 κ=0=.9 10−κ =,ω 100 −=0,ω.08=0.6 √ωωQG 3 ∝ c κ =0.1,ω =0.7 κ =0.03,ω =0.9 κ = 10− ,ω =0.6 √ωω 0 0 0 ∝ QG J √Kκ=0.2 κ =0.05 κ =0 ω2 + ω2 + κ2 ωF(ω) 0 J =0.1 J =0.11 J =0.14 J =0.17 ω2 +2ωωJ2 +√κKκ2 =0.2 κ =0.05 κ =0 ωF(ω) 00 J =0.1 J =0.11 J =0.14 J =0.17 2ωω0 ∝ 3 c κ =0.1,ω =0.7 κ =0.03,ω =0.9 κ = 10− ,ω =0τ .6 √ωω 0 0 1+0 0 ∝ QG ττ0 1+  3 c κ2=0.21,ω02=0.7 κ =0.03,ω0 =0.9τκ = 10− ,ω0 =0.6 √ωωQG ω + ω0 + κ ∝ ωF(ω) J =0.1 J =0.11 J τω=0.14 J =0.17 2√ω0 0 J Kκ=0.2 κ =0.05 κ =0τω 40 ω2 + ω2 + κ2 2π 0 =1, =-6, =0.4, J=0.4,0 Wurzel(K)=0.4 ωF(ω0 ) τ0 J =0.21π J =0.11 J =0.14 J =0.17 1.2 1+2ω0 tem correlation time. As a consequence, at each single system- τ (2)   g (τ) bath interaction event, the Markovian bath immediately equi- g(2)(τ) librates and looses all its memory on the interaction. On the τω0 3 τ0 c κ =0.1,ω =0.7 κ =0.03,ω =0.9 κ = 10−1+,ω =0.6 √otherωω hand, the quenched bath equilibrates infinitely slowly and 0 0 2π 0 2κτ QG 1+2e− τ ∝ keeps memory on each single system-bath interaction. As a con-  2κτ 1+2e− sequence, it introduces effective temporally long range interac- (2) tions between the system variables. These tend to slow down the 2 2 2 g (τ) τω0 ω + ω0 + κ 2π system, pronouncing its ω = 0 correlations as expressed by the ωF(ω) 1.1 J =0.1 J =0.11τc =J2=0π .14 J =0.17 2ω0 ω2cπ Edwards-Anderson parameter. 2κτ τ = 1+2e− c ωc In an equilibrium formalism, disordered systems have to be g(2)(τ) dealt with a computationally demanding replica method [242, τ20π 243], which is furthermore physically far less transparent than 1+τc = the Keldysh formalism, which thus represents a convenient ap- τωc 2κτ proach to disordered systems. In open quantum systems, where   1+2e− 1 disorder, i.e., the coupling to a quenched bath, comes together 0 10 τω0 20 30 with the dissipation introduced by a Markovian bath, both types 2π 2π of bath couplings compete with each other, leading in the present τc = glassy system to strongly modified spectral properties, which ωc Figure 9. Algebraic decay of the photon four point correlation function mirror the effect of the Markovian bath by a crossover from non- (2) † † 2 equilibrium to equilibrium scaling behavior. g (τ) = ha (0)a (τ)a(τ)a(0)i/ng(2)at( longτ) times, for parameters ωp = 1, κ = 0.4, ωz = 6, K = 0.16, see also Ref. [108]. The algebraic decay sets in at the inverse crossover frequency ωc, given by Eq. (177). The corresponding exponential decay of g(2) in the normal phase is plotted 2κτ IV. NON-EQUILIBRIUM STATIONARY STATES: BOSONIC for comparison. Figure copied1+2 frome Ref.− [108]. (Copyright (2013) by MODELS The American Physical Society) Recent years have seen tremendous progress in experiments 2π on exciton-polaritons in semiconductor microcavities (for re- has been demonstrated thatτ ac complete= characterization of the glass phase can be performed by diωffcerent but standard photon views see, e.g., Refs. [13, 140]), making these systems the output measurements, such as homodyne detection and inten- prime candidates for studying condensation phenomena under sity correlation measurements. Formally, the photon Green’s non-equilibrium conditions. As we have already mentioned in functions can be obtained from the Keldysh action formalism by Sec.IC2, the fundamental di fference from conventional con- densates is due to the finite life-time of exciton-polaritons, which introducing source fields µαα0 in the microscopic action, which couple to the photon variables. After integrating out the pho- makes continuous driving necessary to maintain a steady popu- ton variables in the action, the photon Green’s functions are then lation. obtained via functional derivatives with respect to these source What makes these systems genuinely driven-dissipative, is fields. This standard field theory technique relates the photon that they are coupled to several baths, or to time-dependent driv- Green’s functions to the solution for the atomic response and ing fields, with which they exchange particles and energy. In correlation functions [108]. the case of exciton-polaritons, which are hybrid quasiparticles composed of a Wannier-Mott exciton and a photon, the photonic component can leak out through the mirrors forming the cav- 4. Quenched and Markovian bath coupling in the Keldysh formalism ity. Thus, the electromagnetic vacuum outside the cavity serves as a reservoir into which particles are lost. These losses have to be compensated by laser driving. In many experiments, the The Keldysh path integral formalism represents a theoretical driving laser is far blue-detuned from the bottom of the lower approach which incorporates in a very straightforward way the polariton band, and thus coherently creates high-energy excita- coupling of the system to a non-thermal bath. Here we com- tions. During the relaxation of the latter, which is caused by pare quenched disorder and Markovian baths in more detail. As phonon-polariton and stimulated polariton-polariton scattering, shown above in the present section, the coupling of the system coherence is quickly lost. In other words, lower polaritons can variables to quenched disorder, realized by the variables Jlm, is be regarded as being pumped not directly by the laser but rather equivalent to the coupling to a bath with infinite correlation time, by a reservoir of high energy excitons. Several approaches have been used to model this effectively incoherent pumping mecha- 0 K hJlm(t)Jl0m0 (t )i = (δll0 δmm0 + δlm0 δml0 ). (184) nism [151, 182, 250]. In this context, the description of a driven- N dissipative condensate introduced in Sec. II B 2 might be con- This represents the opposite limit of a Markovian bath, which sidered as a toy model, which hides all the microscopic details has a typical correlation time that is much shorter than the sys- associated with coherent excitation and subsequent relaxation of 41 high energy excitons in the coupling of the system to several different from its equilibrium counterpart. Which of these sce- independent baths. narios is realized depends crucially on the spatial dimensionality In the following, we review recent investigations of the uni- of the system: in 3D the long-wavelength regime is effectively versal long-wavelength scaling properties of driven-dissipative thermal, whereas in one- and two-dimensional systems the devi- condensates [19, 26, 172, 251–253]. Then, the precise details ation from equilibrium is relevant in the RG sense, and these sys- of the chosen model cease to matter: according to the power- tems are governed by strongly non-equilibrium RG fixed points. counting arguments given in Sec.IIC, for any choice of a mi- To see how this comes about, we require a quantitative mea- croscopic model the effective long-wavelength description is sure for the deviation from equilibrium conditions in a driven- given by the semiclassical Langevin equation for the conden- dissipative condensate. In Sec. II D 1 we discussed that thermal sate field (80). In fact, the latter can be derived in the spirit of equilibrium requires the ratios of coherent to corresponding dis- Ginzburg-Landau theory for continuous phase transitions [180] sipative couplings to take a common value. This is illustrated by writing down the most general equation that is compatible in Fig.6 and formalized in Eq. (98). In turn, it implies that any with the symmetries of the problem. The reasoning behind such deviation Kc/Kd , uc/ud in the values of these ratios directly an approach is that the universal properties are fully determined indicates a deviation from equilibrium conditions. Hence, the by the spatial dimensionality and symmetries of a physical sys- quantity λ defined by [19] tem [80, 82, 173]. ! Kduc A comparison in terms of symmetries of systems exhibiting λ = −2Kc 1 − (185) driven-dissipative Bose-Einstein condensation with systems in Kcud thermal equilibrium can be given most conveniently on the basis can serve as a quantitative measure for the departure from ther- of the semiclassical limit of Sec.IIC: indeed, the equation of mal equilibrium. Note that λ = 0 in equilibrium, and that λ is motion (80) for the classical field takes the form of the Langevin well-defined also for K = 0, which is the microscopic value equations that are used to model universal dynamics in ther- d of the diffusion constant K in the model for driven-dissipative mal equilibrium phenomenologically [1]. Structurally, equa- d condensates introduced in Sec. II B 2. It turns out to be most tion (80) is similar to model A for a non-conserved order pa- convenient to combine λ with the quantities [19] rameter, with the additional inclusion of reversible mode cou- plings [254]. However, the equilibrium symmetry discussed in !  2  Kd uc γ  uc  Sec. II D 1, which is present in all models of Ref. [1], is violated D = Kc + , ∆ = 1 +  (186) K u ρ  2  in driven-dissipative systems. This violation is due to pumping c d 2 0 ud and loss terms, which moreover lead to the absence of parti- cle number conservation. The latter typically is associated with (see Sec. II B 2 for the definition of the noise strength γ and the the symmetry under quantum phase rotations (see Sec. II D 4). mean-field condensate density ρ0) to define the dimensionless This is another crucial difference to Bose-Einstein condensation non-equilibrium strength as [5]: in equilibrium: there, particle number conservation entails the λ2∆ existence of a slow dynamical mode that modifies the universal g = Λd−2 , (187) properties, and is taken into account in model F of Ref. [1]. To 0 D3 summarize, driven-dissipative condensation differs from Bose- where Λ is the UV momentum cutoff. Thus, the answer to Einstein condensation in equilibrium by the absence of the equi- 0 the question of whether equilibrium vs. non-equilibrium uni- librium symmetry and the symmetry under quantum phase ro- versal behavior is realized in driven-dissipative condensates, is tations. This difference lies at the heart of the novel universal encoded in the RG flow of g. behavior out of equilibrium discussed in the following. In the condensed phase, the RG flow of g is driven domi- In Sec. II B 2, we analyzed driven-dissipative condensates nantly by fluctuations of the gapless Goldstone mode discussed within mean-field theory, disregarding fluctuations around the in Sec. II D 6, i.e., by fluctuations of the phase of the conden- homogeneous condensate mode. However, in order to access sate field. The latter were shown [255–258] to be governed by universal aspects such as the behavior of correlations of the con- the Kardar-Parisi-Zhang (KPZ) equation [69], in which λ de- densate field at large distances or dynamical critical phenomena fined in Eq. (185) appears as the coefficient of the characteristic at the condensation transition, one has to carefully include the non-linear term, see Eq. (196). Below in Sec.IVA, we present effect of fluctuations. This can be done gradually — first inte- an alternative mapping of the long-wavelength condensate dy- grating out short-scale fluctuations and moving on to account for namics to the KPZ equation, starting from the Keldysh action in fluctuations on larger and larger scales — by means of RG meth- Eq. (73) and integrating out the gapped density mode within the ods, such as the FRG discussed in Sec.IIE. Then, an intrigu- Keldysh functional integral. As a consequence of the mapping ing question is, whether the non-equilibrium nature of driven- to the KPZ equation, the RG flow of g in d spatial dimensions is dissipative condensates at the microscopic scale becomes more at the one-loop level given by [5] or less pronounced under renormalization. In the former case, effective equilibrium is established at large scales, while in the (2d − 3) C ∂ g = − (d − 2) g + d g2, (188) latter case, the universal physics is expected to be profoundly ` 2d 42

g where ` = ln(Λ/Λ0), Λ is the running momentum cutoff, and 1−d −d/2 Cd = 2 π Γ(2 − d/2) is a geometric factor. The key role that is played by spatial dimensionality becomes manifest in the canonical scaling of g, which is encoded in the first term on the RHS of the flow equation: to wit, g is relevant in 1D where d−2 < 0, marginal in 2D, and irrelevant in 3D since then d−2 > 0. In 2D, the loop correction — the second term on the RHS of Eq. (188) — is positive, making g marginally-relevant. This has far-reaching consequences for a driven-dissipative condensate in which the microscopic value of g is small, i.e., which is close to 1 equilibrium: upon increasing the scale at which the system is ob- served, the non-equilibrium nature is more pronounced in one- and two-dimensional systems, whereas effective equilibrium is 1 2 3 d established on large scales in three-dimensional systems. In 1D the canonical scaling towards strong coupling is balanced at an Figure 10. Equilibrium vs. non-equilibrium phase diagram for driven- attractive strong-coupling fixed point (SCFP) g by the loop cor- dissipative condensates (cf. Ref. [238]). The line g = 1, where g is ∗ defined in Eq. (187) and measures the deviation from equilibrium con- rection. This term vanishes at d = 3/2, and for d > 3/2 the one- ditions, separates the close-to-equilibrium regime for g < 1 from the loop equation does not have a stable SCFP, which, however, is strong-coupling, far-from-equilibrium regime at g > 1. Red dots indi- recovered in a non-perturbative FRG approach [238–240]. The cate the fixed-point values of g that are reached if the RG flow is initial- RG flow of g that is found within this approach is illustrated ized in the close-to-equilibrium regime. In dimensions one and two, the qualitatively in Fig. 10, which shows that also in 2D the flow is equilibrium fixed point at g = 0 is unstable, and the RG flow along the out of the shaded close-to-equilibrium regime with g < 1, and dashed lines is directed towards the blue line of strong-coupling fixed towards a strong-coupling, non-equilibrium fixed point. The sit- points. Thus, a system that is microscopically close to equilibrium will uation is quite different in 3D: in this case, if the microscopic exhibit strongly non-equilibrium behavior at large scales. On the other value of g is small, at large scales an effective equilibrium with a hand, in three spatial dimensions, an initially small value of g is dimin- renormalized value g → 0 is reached. However, for d > 2, there ished under renormalization, and the universal large-scale behavior is governed by the effective equilibrium fixed point at g = 0. The green exists a critical line of unstable fixed points gc, separating the line indicates the existence of a critical value gc in d > 2, corresponding basins of attraction of the equilibrium and non-equilibrium fixed to a transition between the effective equilibrium phase and a true non- points, for g < g and g > g respectively. Thus, in addition c c equilibrium phase that is realized for large microscopic values g > gc. to the effective equilibrium phase, a true non-equilibrium phase may be reached in systems that are far from equilibrium even at the microscopic level also in 3D [259]. The properties of this with the condensate phase as the single dominant gapless de- phase have not been explored so far. gree of freedom (cf. Sec. II D 6) has been formulated starting The rest of this section is organized as follows: in Sec.IVB, from the Langevin equation (80) for the complex order parame- we review the dynamical critical behavior at the driven- ter. Here we present an alternative and direct derivation within dissipative condensation transition in 3D [26, 172, 260], which, the Keldysh functional integral formalism [98, 169]. It is based according to the above discussion, is governed by an effective on the Keldysh action in terms of which the microscopic theory equilibrium fixed point. Signatures of the non-equilibrium na- of a bosonic many-body system with particle loss and gain is ture of the microscopic model are present in the asymptotic fade- formulated in Sec. II B 2. This differs from the derivation pre- out of the deviation from equilibrium at large scales. In contrast, sented in [19], which was based on the detour over the Langevin the universal scaling behavior of driven-dissipative condensates equation (80) that effectively captures the physics on a meso- in both 2D [19] and 1D [251–253] is quite distinct from the equi- scopic scale (cf. Fig.5 and the discussion in Sec.IIC). On this librium case and governed by the SCFP of the KPZ equation. level, amplitude fluctuations can be eliminated leading to the We review the resultant physical picture in Secs.IVC andIVD, KPZ equation for the phase. In this sense, we establish here a respectively. closer link between microscopic and mesoscopic theories, which is both appealing from a theoretical point of view and brings about a number of technical advantages. For example, physical A. Density-phase representation of the Keldysh action observables are usually represented by quantum mechanical op- erators or equivalently in terms of fields in a Keldysh functional Especially in reduced dimensions, at large scales the proper- integral description of the microscopic theory; here our approach ties of condensates are vitally influenced by fluctuations of the comes in handy as it yields the effective long-wavelength form Goldstone mode. To name an example, in 2D, both in [261] of generating functionals for expectation values and correlation and out of equilibrium [19] these fluctuations lead to a sup- functions of these observables which can then be evaluated fur- pression of long range correlations. In Ref. [19], an effective ther utilizing established approximation strategies. long-wavelength description of a driven-dissipative condensate Apart from specific applications, the derivation of the action 43 for the Goldstone mode presented here deepens our understand- where ζ1 and ζ2 are, respectively, real and imaginary parts of ing of general properties of the Keldysh formalism with regard ζ. Here, of all terms involving the products of fluctuations ζ1π to phase rotation symmetries. The crucial point is, that classical and ζ2π we only keep the dominant ones in the long-wavelength phase rotations introduced in Sec. II D 4 are a symmetry of the limit, i.e., we neglect contributions containing temporal deriva- 2 Keldysh action both in a closed system and in the presence of tives ζ2∂tπ, ζ1π∂tθ, and spatial derivatives ζ1∇ π, ζ2∇π · ∇θ, and 2 terms that describe incoherent pumping and losses, and as we ζ1π (∇θ) which are both small as compared to the mass-like have seen in Sec. II D 6, the spontaneous breaking of this sym- contributions ucζ1π and udζ2π in Eq. (192) for the Goldstone metry is sufficient to ensure the appearance of a Goldstone mode mode θ, in the regime ω → 0 for q → 0. Note that terms that corresponds to fluctuations of the phase of the order param- of higher order in π and ζ are contained in the original action eter [151, 181, 182, 262]. In the basis of classical and quantum Eq. (73) both in contributions involving derivatives and in the co- iα fields φc,q such phase rotations become φc,q 7→ φc,qe , showing herent and dissipative vertices. The validity of the saddle-point that the Goldstone mode corresponds to joint phase fluctuations approximation, therefore, is restricted to the low-frequency and of both the classical and the quantum fields. Therefore, in order low-momentum sector in a weakly interacting system. to derive the action for the Goldstone boson we represent the The action (192) resulting from this expansion is linear in π fields in the form and hence integration over this variable is trivial and yields a δ-functional, which in turn facilitates integration over the imag- √ iθ iθ φc = ρe , φq = ζe , (189) inary part ζ2 of ζ, Z Z where the density ρ is real whereas ζ is a complex variable. 0 iS ˜ iS KPZ The low-energy effective action for the Goldstone boson θ in a Z = D[θ, ζ1, ζ2] δ[ucζ1 − udζ2]e = D[θ, θ] e , (193) driven-dissipative system can be derived by integrating out fluc- tuations of the density ρ in Eq. (189) in the Keldysh partition where at each step a normalization factor is implicitly included function with action S given by Eq. (73), in the integration measure, ensuring Z = 1 [98, 169]. In the last ˜ √ Z Z equality we replaced ζ1 by the KPZ response field θ = i2 ρ0ζ1, ∗ ∗ iS ∗ iS and the KPZ action S is given by [5, 98] Z = D[φc, φc, φq, φq]e = D[ρ, θ, ζ, ζ ]e . (190) KPZ Z  λ  The equality of the integrals over complex classical and quan- S = θ˜ ∂ θ − D∇2θ − (∇θ)2 − ∆θ˜ , (194) KPZ t 2 tum fields and the variables introduced in the transforma- t,x tion Eq. (189) follows from the fact that this transforma- where the diffusion constant, non-linear coupling, and noise tion leaves the integration measure invariant, i.e., we have ∗ ∗ ∗ strength, respectively, are expressed in terms of the microscopic D[φc, φc, φq, φq] = D[ρ, θ, ζ, ζ ]. Note that this would not be the parameters in the original action Eq. (73) as case if instead of the density we introduced the amplitude of φc as a degree of freedom. Our goal is then to treat the integrals over  2  uc γ + 2udρ0  uc  ρ and ζ in Eq. (190) in a saddle-point approximation, which, as D = Kc , λ = −2Kc, ∆ = 1 +  . (195) u 2ρ  u2  we show below, is justified since fluctuations of the density are d 0 d gapped in the ordered phase with r < 0 and hence expected to d Along the lines of the derivation of the Langevin equation (80) be small. The first step is thus to find the saddle point, i.e., the from the action in the semiclassical limit in Eq. (79), the KPZ solutions to the classical field equations (cf. Eqs. (74)) action can be seen to be equivalent to the KPZ equation, which δS δS reads = 0, = 0. (191) δρ δζ λ ∂ θ = D∇2θ + (∇θ)2 + η, (196) t 2 For rd < 0, we recover the mean-field solution of Sec. II B 2, given by ρ = ρ0 = −rd/ud = −rc/uc (note that the last equality where the stochastic noise η has zero mean, hη(X)i = 0, and can always be satisfied by performing a gauge transformation is Gaussian with second moment hη(X)η(X0)i = 2∆δ(X − X0). to adjust the value of rc as described in Sec. II B 2) and ζ = Originally [2, 69], Eq. (196) was suggested by Kardar, Parisi, 0. We proceed to expand the action Eq. (73) to second order and Zhang as a model to describe the growth of a surface, e.g., in fluctuations of ρ and ζ around the saddle point. Note that due to the random deposition of atoms. In this context, h = the quantum vertex in the action Eq. (73) that is cubic in the θ is the height of the surface, and the origin of the non-linear quantum fields does not contribute at this order. Denoting the term is purely geometric [69]: the growth is assumed to occur density fluctuations as π = ρ − ρ0 we find in a direction that is locally normal to the surface and at a rate ds/dt = λ; If an increment ds = λdt is added along the normal, Z √  n h 2i 2 the corresponding change of the surface height is S = 2 ρ0 −ζ1 ∂tθ + Kc (∇θ) + Kcζ2∇ θ t,x o  q   2 2 2 λ 2 − (ucζ1 − udζ2) π + i (γ + 2udρ0) |ζ| , (192) dh = (λdt) + (λdt∇h) ≈ λ + (∇h) dt. (197) 2 44

Removing from dh/dt the average deposition rate λ by a trans- action has to be supplemented by infinitesimal regularization formation to a co-moving frame, h(t, x) 7→ h(t, x) + λt, and terms, discussed at the end of Sec. II B 1. adding a term D∇2h that describes surface tension, we obtain Finally, let us comment on the relation of the approach pre- the (deterministic part of the) growth equation (196). Intuitively, sented here to a Bogoliubov expansion in fluctuations of the it seems clear that the growth of a surface represents a genuine complex√ fields around the mean field average values, δφc = non-equilibrium process. More formally, this can be seen by not- φc − ρ0 and δφq = φq. The latter can be recovered formally ing that the non-linear term in the KPZ equation does in general by expanding the transformation Eq. (189) around the arbitrarily not satisfy a potential condition [5].12 chosen value θ = 0, which yields As pointed out above, an alternative derivation of the KPZ equation (196) as the effective long-wavelength description of √ π √ φc = ρ0 + √ + i ρ0θ, φq = ζ, (199) driven-dissipative condensates starts from the Langevin equa- 2 ρ0 tion (80)[19]. Then, the coefficients in the KPZ equation are slightly different, and they are given by Eqs. (185) and (186) and interpreting π and θ as the real and imaginary parts of instead of Eq. (195). To be specific, the differences are (i) the δφc. In the KPZ action in Eq. (194), such an expansion in θ absence in Eq. (186) of the tree-level shift ∝ udρ0 of the noise would amount to neglecting the non-linearity. Without the non- strength ∆ in Eq. (195), and (ii) the absence in Eq. (195) of the linearity, however, the KPZ action describes a free field cou- terms proportional to Kd in Eqs. (185) and (186). (i) is due to pled to a thermal bath. In other words, by performing a Bogoli- the fact that in the Langevin equation (80), which is valid in ubov approximation the non-equilibrium character that is intrin- the semiclassical limit (see Sec.IIC), the quantum vertex that is sic to the microscopic model with action (73) is lost in the long- ∗ ∗ proportional to φcφcφqφq in the action in Eq. (73) is neglected. wavelength limit. Formally, this failure is due to the fact that (ii) results from the absence of the diffusion term Kd in the mi- the saddle-point approximation is not valid for low-momentum croscopic model (73); on the other hand, in the Langevin equa- modes in the Goldstone direction since they are not gapped. tion (80) this term is included, as it is generated by integrating out fluctuations with wavelengths below the mesoscopic scale on which the Langevin equation is valid (cf. Fig.5). B. Critical dynamics in 3D Starting from the effective long-wavelength description of the condensate dynamics derived in this section, the universal scal- ing properties can be obtained from an RG analysis. For the KPZ In three spatial dimensions, the deviation from thermody- equation, this procedure, which leads at lowest order in a pertur- namic equilibrium conditions, which is quantified by the value bative expansion in λ to the one-loop flow equation (188),13 is of g defined in Eq. (187), is irrelevant in the RG sense (cf. described, e.g., in Refs. [5, 98]. Eq. (188)). Hence, at large scales, effective equilibrium is es- For completeness, we note that in the absence of drive and tablished. This immediately implies that the overall picture of Bose-Einstein condensation in thermal equilibrium in 3D re- dissipation, i.e., for Kd = rd = ud = 0, an analogous derivation for a weakly interacting Bose gas in thermal equilibrium leads mains valid also in the driven-dissipative context: in particu- to an effective action for the phase alone that is given by lar, the mean-field analysis of Sec. II B 2, which predicts a con- tinuous phase transition beyond which the classical phase ro- Z   tation symmetry (cf. Sec. II D 4) is spontaneously broken and ˜ 2 − 2∇2 S = θ ∂t c θ, (198) quanti- t,x off-diagonal long-range order is established, is modified tatively but not qualitatively once fluctuations around the mean- p where c = 2Kcucρ0 is the speed of sound, and describes the field condensate are taken into account. (Note, however, that in dissipationless propagation of sound waves with linear disper- equilibrium the condensation transition is induced by lowering sion ω = cq. To actually describe thermal equilibrium, this the temperature to a critical value, whereas driven-dissipative condensation is established by tuning the single-particle pump rate.) Yet, the non-equilibrium nature of the driven-dissipative system leaves its mark on the approach to the long-wavelength 12 The most general Langevin equation describing (near-) equilibrium dynamics thermalized regime in a fully universal way. contains both (i) relaxational and (ii) reversible contributions to the determin- istic dynamics [5]. The linear diffusion term in Eq. (196) is of type (i): it can An equilibrium system, fine-tuned to a critical point, beyond R − H H D ∇ 2 which spontaneous symmetry breaking occurs via a second or- be written as δ /δθ, where = 2 x ( θ) , and this term alone would cor- respond to relaxation to an equilibrium stationary distribution ∝ e−H/∆. The der phase transition, exhibits universal behavior. This is wit- non-linear term, on the other hand, can not be represented as the derivative nessed in non-analyticities in the free energy, and in the long- of a Hamiltonian functional and is hence of type (ii). However, for reversible range decay properties of correlation and response functions. contributions to be compatible with a thermal stationary state, they have to The decay properties are then governed by power laws, freed satisfy a potential condition. This is not the case for the non-linear term in the −r/ξ KPZ equation in dimensions d > 1. from the generic exponential cutoff ∼ e at large distances 13 Due to symmetries of the KPZ equation (for a comprehensive discussion see r due to the divergent correlation length ξ → ∞, defining the Ref. [263]), the RG flow is described by the single parameter g defined in critical point. While the concept of a free energy is not mean- Eq. (187). ingful in a non-equilibrium context, correlations and responses 45 can be considered also in the latter case. Then, universality en- dissipative fixed point still hosts information on the underlying tails that the algebraic long-wavelength, long-time decay of any quantum dynamics. The fadeout of this coherent dynamics is correlation or response function can be characterized in terms described by a new critical exponent, which can be shown to of a set of critical exponents, which do not depend on the mi- be independent of the static and dynamical exponents, i.e., it croscopic details of the problem but rather on symmetries and does not relate to the latter by scaling relations. Moreover, it dimensionality. The critical point is fully characterized by this can be seen that no more independent exponents could possibly set of critical exponents. be found [26, 260]. Finally, the value of this exponent distin- We emphasize a crucial difference between the critical be- guishes equilibrium from non-equilibrium conditions. This can havior in strictly non-interacting theories described by quadratic be understood from Fig.6: the exponent describes the fadeout Hamiltonians or quantum master equations, and the more of coherent couplings, i.e., the approach to the imaginary axis, −η generic – but also much more complex – case of interacting governed by a power law ∼ Λ r , where ηr is the new exponent. problems (for an example where a Gaussian fixed point is phys- In the equilibrium case, all couplings rotate uniformly, giving ically important, cf. Sec. III A). The non-interacting critical rise to ηr = −0.143 within the truncation of Refs. [26, 172]. In theories are described by a so-called Gaussian fixed point of contrast, in the non-equilibrium case, the slowest approach to the RG flow, while the interacting ones are associated to a so- the real axis is given by ηr = −0.101. The value of the non- called Wilson-Fisher fixed point. The structural difference be- equilibrium exponent can also be determined analytically from tween these fixed points is reflected by the fact that the values the field theoretic perturbative renormalization group in a dimen- of the critical exponents differ; in particular, at a Gaussian fixed sional expansion, yielding the result (to two-loop order) [260] point, all exponents are rational numbers reflecting the valid- ! (4 − d)2 4 ity of canonical power counting (cf. Sec.IIC), while one ob- ηr = − 4 ln − 1 . (200) tains non-rational numbers mirroring the importance of strong 50 3 long-wavelength fluctuation corrections at a Wilson-Fisher fixed (iii) Observability. — The drive exponent manifests itself, for point. The latter is more stable than the Gaussian one (where, in example, in the frequency and momentum resolved dynamical fact, all couplings are relevant in the sense of the RG), and thus single particle response as probed in homodyne detection, see the physically relevant one even if interactions are small on the Ref. [172] for details. It thus corresponds to a direct experi- microscopic scale. This is intuitive, taking into account that crit- mental observable, though its small value poses a challenge for ical behavior deals with the longest distances and timescales in experimental observation. a physical system. In summary, it is found that the correlation functions (both The impact of fluctuations on the values of the critical expo- static and dynamic) thermalize, and show identical univer- nents can be described quantitatively in the framework of the sal behavior to an equilibrium critical system with the same renormalization group, which provides a systematic way to deal symmetries. However, the dynamical response functions con- with the intricate long-wavelength, low-momentum divergences tain universal information distinguishing equilibrium from non- caused by the divergent correlation length. The critical behavior equilibrium systems. The microscopic drive conditions are thus of driven-dissipative systems in 3D was investigated in [26, 172] witnessed even at the largest macroscopic distances in a fully based on the functional renormalization group approach dis- universal way. cussed in Sec.IIE, and in [260] using the field theoretic per- In the following, we review how the results summarized turbative renormalization group [5]. These studies gave rise to above can be obtained from an open-system functional RG ap- the following picture of driven criticality: proach [26, 172]. The basic ingredients of this method are dis- The RG fixed point is purely dissipative (see Fig.6 (c)), i.e., cussed in Sec.IIE. all complex couplings rotate to the imaginary axis. However, it is the approach to this fixed point that contains universal in- formation, and this makes it possible to distinguish equilibrium 1. Effective action for driven-dissipative condensation from non-equilibrium critical behavior. The following key prop- erties are identified: The Wetterich equation (129) describes how the scale- (i) Asymptotic thermalization of correlation functions. — dependent effective action ΓΛ evolves from the microscopic ac- Both the static and dynamical critical exponents, governing, e.g., tion S at Λ = Λ0 to the full effective action Γ at Λ → 0, as the asymptotic decay of the spatial and temporal first order co- fluctuations with momenta larger than the cutoff Λ are integrated herence functions, are found to take the same values as in the out, and the latter is gradually lowered. It is an exact non-linear corresponding equilibrium problem. This can be made plausible differential equation for the functional ΓΛ. In the absence of an by the fact that the fixed point couplings indeed lie on a sin- exact solution, suitable schemes to approximate the functional gle ray in the complex plane — the imaginary axis. More for- differential equation have to be found. For the analysis of crit- mally, it is understood in terms of an emergent symmetry imply- ical behavior, progress can be made by choosing an ansatz for ing asymptotic emergence of detailed balance, or thermalization, ΓΛ that has a similar structure as the microscopic action S in at large spatial or temporal distances. Eq. (73). In this way, the effective action ΓΛ is parameterized (ii) Universal decoherence. — The approach to the purely in terms of the coupling constants appearing in the ansatz, and 46 consequently, the functional differential equation for ΓΛ can be 2. Non-equilibrium FRG flow equations cast in the form of a set of ordinary differential equations for the couplings, as we describe in detail below. The main idea of a projection prescription on a specific cou- Any such ansatz will contain only a finite number of cou- pling is to extract this coupling from the effective action ΓΛ by plings, and will hence truncate the most general structure of ΓΛ. taking appropriate derivatives of the latter with respect to the In other words, by choosing an ansatz of a particular form, one fields and coordinates, and subsequently√ setting the fields to makes an approximation, and the question arises, which cou- their stationary values φ¯c = φ¯0 = ρ¯0 and φ¯q = 0 (as noted plings should be included in the ansatz or truncation in order to in Sec. II D 6, choosing φ¯c to be real does not lead to a loss of obtain meaningful results. For the study of critical phenomena generality). Then, applying the very same projection descrip- at a second order phase transition, the power counting scheme tion to the Wetterich equation (129) yields the flow equation of Sec.IIC provides a guideline: following the arguments given for the corresponding coupling. For the actual evaluation of the there, we choose a truncation which includes all couplings that resulting flow equations, it is convenient to introduce rescaled are not irrelevant, and which therefore takes the form of the fields, which are related to the bare ones by absorbing the wave- semiclassical action in Eq. (79): function renormalization Z in the quantum field: φ = φ¯ , φ = Zφ¯ . (203) ( " # ) c c q q Z   ∂U¯ ∗ Γ = φ¯∗ iZ∗∂ + K¯ ∗∇2 φ¯ − + c.c. + iγ¯φ¯∗φ¯ . Λ q t c ¯∗ q q As a consequence of this transformation, and by rescaling all t,x ∂φc (201) couplings appropriately, it is possible to obtain a reduced set of (Recall that the variables of the effective action are the field flow equations, from which Z is eliminated. In a second step, the number of flow equations can be diminished further by in- expectation values Φ¯ ν, ν = c, q defined in Eq. (37).) Here, in addition to the complex prefactor K¯ = A¯ + iD¯ of the Lapla- troducing dimensionless renormalized variables, as detailed in cian, we are including a complex wave-function renormalization Sec. IV B 3 below. We start by deriving flow equations for the non-linear cou- Z = ZR + iZI. In the semiclassical limit, the homogeneous (i.e., not containing derivatives with respect to time or space) part of plings in the effective potential defined in Eq. (202). To see the action can be written in terms of an effective potential U¯ , how one can project the Wetterich equation onto flow equations ¯∗ ¯ for these couplings, consider the effective action, evaluated for which is a function ofρ ¯c = φcφc. Due to the invariance of the mi- croscopic action under classical phase rotations (cf. Sec. II D 4), homogeneous, i.e., space- and time-independent “background which is inherited by the effective action, only this combination fields:” of fields is allowed in the potential. The latter is given by  0 0∗  ΓΛ,cq = −Ω U¯ ρ¯cq + U¯ ρ¯qc − iγ¯ρ¯q . (204)

Here, the subscript cq in ΓΛ,cq indicates that both the classical 1 2 1 3 U¯ (¯ρc) = u¯2 (ρ¯c − ρ¯0) + u¯3 (ρ¯c − ρ¯0) . (202) and the quantum fields are set to constant but non-zero values; Ω 2 6 denotes the quantization volume, and we introduced the follow- ing products of fields, which are invariant under classical phase ¯ ¯∗ ¯ ∗ ¯∗ ¯ Here, both the two-body and three-body couplings,u ¯2 = λ + iκ¯ rotations (see Sec. II D 4):ρ ¯cq = φcφq = ρ¯qc andρ ¯q = φqφq. andu ¯3 = λ¯ 3+iκ¯3, respectively, are complex. The three-body term From the representation Eq. (204) it becomes immediately clear is marginal according to power counting, and therefore included how to project the flow equation (129) for ΓΛ onto a flow equa- in the truncation. In the FRG, it is advantageous to approach the tion for the derivative of the potential U¯ in Eq. (202) with respect transition from the ordered phase. Then, the form of the effective toρ ¯c: one has to (i) evaluate Eq. (129) for homogeneous fields, potential in Eq. (202) corresponds to an expansion around the (ii) take the derivative with respect toρ ¯cq, and finally (iii) set the stationary condensate densityρ ¯0. Indeed, this choice implies, quantum background fields to their stationary state value, ¯∗ ¯∗ that the field equations δΓΛ/δφc = 0, δΓΛ/δφq = 0 (Eqs. (40) in the absence of sources and evaluated with the scale-dependent 0 1 h i ∂`U¯ = − ∂ρ¯ ∂`ΓΛ,cq . (205) cq φ¯ =φ¯∗=0 effective action ΓΛ) are solved byρ ¯c = ρ¯0 and φ¯q = 0 on all Ω q q scales Λ. Here and in the following, we specify flow equations in terms of In the truncation Eq. (201), all couplings — including the con- the logarithmic cutoff scale ` = ln(Λ/Λ0). From the potential U¯ densate densityρ ¯0 — are scale dependent. As indicated above, in Eq. (202), the couplingsu ¯2,3 can be obtained by taking further by means of such an ansatz for the effective action ΓΛ, the func- derivatives with respect toρ ¯c. However, instead of projecting the tional differential equation (129) can be rewritten as a set of or- flow equation (205) in this way onto equations foru ¯2,3, it is more dinary differential equations for these running couplings, with convenient to introduce rescaled quantities as outlined above. initial conditions given by the microscopic action Eq. (79). This Inserting the representation Eq. (203) of the bare quantum field is achieved by applying projection prescriptions. In the follow- in the effective action (204) leads to appearance of a factor 1/Z∗ ing, we summarize this method for the problem at hand. For in front of the term involving the effective potential. This factor details we refer the reader to Ref. [172]. can be absorbed by introducing a rescaled potential via U¯ = ZU. 47

The flow equations of the bare and rescaled effective potential the lines of the derivation of Eqs. (211) and (212), with the result are related via (for details of the derivation see Ref. [172]; ηZR is the real part of ηZ) 0 0 0
∂`U¯ = Z −ηZU + ∂`U , (206) i h i ∂`γ = βγ = 2ηZRγ − ∂ρ ∂`ΓΛ,cq . (214) where ηZ denotes the anomalous dimension associated with Ω q ss the wave-function renormalization (∂`Z is specified below in Eq. (219)), Thus far we have specified how to project the Wetterich equa- tion onto flow equations for the couplings that parameterize the

ηZ = −∂`Z/Z. (207) homogeneous part of the effective action given in Eq. (204), where the classical and quantum fields are set to constant val-

Then, with ∂ρ¯cq = Z∂ρcq and inserting Eq. (205) on the RHS ues. In the following we review the derivation of flow equations of Eq. (208), the flow equation for the renormalized potential for the frequency- and momentum-dependent couplings, i.e., the becomes (here, primes denote derivatives with respect to ρc = wave-function renormalization Z and the coefficient K¯ of the ∗ φcφc) Laplacian in Eq. (201). This requires us to consider non-constant values of the fields. Moreover, we work in a basis of real fields 1 h i 1
0 0 0 0 which we introduced already in Sec. II D 6, φ¯ν = √ χ¯ ν,1 + iχ¯ ν,2 ∂tU = ηZU + ζ , ζ = − ∂ρcq ∂tΓk,cq ∗ . (208) 2 Ω φq=φq=0 for ν = c, q. Hence, the inverse propagator in this basis is given In analogy to Eq. (202), the renormalized effective potential can by the second variational derivative of the effective action with respect to the fieldsχ ¯ (cf. Eq. (41); to ease the notation, we be written as i   collect these fields in a vectorχ ¯ = χ¯ c,1, χ¯ c,2, χ¯ q,1, χ¯ q,2 ; the com- 1 2 1 3 U(ρc) = u2 (ρc − ρ0) + u3 (ρc − ρ0) , (209) ponents of this vector are labeled by i = 1,..., 4), and the flow 2 6 equation of the inverse propagator reads accordingly: with renormalized couplings defined as u = u¯ /Z = λ + iκ and " # 2 2 δ2∂ Γ u = u¯/Z = λ + iκ . To obtain flow equations for u , n = 2, 3, ¯ − 0 − 0 ` Λ 3 3 3 n ∂`Pi j(ω, q)δ(ω ω )δ(q q ) = 0 0 . (n) δχ¯ i(−ω, −q)δχ¯ j(ω , q ) one simply has to take derivatives of the relation un = U (ρ0) ss (215) with respect to the cutoff scale `, taking into account that also ρ0 is a running coupling: In particular, the inverse retarded propagator is given by 2 2 !  (n) (n+1) R −iZIω − Aq¯ − 2λ¯ρ¯0 iZRω − Dq¯ ∂`un = ∂`U (ρ0) + U (ρ0)∂`ρ0. (210) P¯ (ω, q) = 2 2 . (216) −iZRω + Dq¯ + 2¯κρ¯0 −iZIω − Aq¯ Inserting Eq. (208) on the RHS of this relation leads us to Note that the Goldstone theorem (i.e., the existence of a zero ¯ R ∂ u β η u u ∂ ρ ∂ ζ0 , eigenvalue of P (ω = 0, q = 0), see Sec. II D 6) is pre- ` 2 = u2 = Z 2 + 3 ` 0 + ρc ss (211) served during the flow. At the transition, whereρ ¯0 → 0, both ∂ u = β = η u + ∂2 ζ0 , (212) ` 3 u3 Z 3 ρc ss branches of the excitation spectrum that is encoded in the ze- ¯ R 0 ros of det P (ω, q) (cf. Eq. (77)) become gapless. As pointed where we evaluate ζ with ρc set to its stationary value ρc|ss = ρ0. out in Sec.IIE, in the FRG, the resulting infrared divergences Finally, flow equations for the real and imaginary parts of u2 and are regularized by introducing an additional contribution ∆S Λ, u3 can be obtained by taking the real and imaginary parts of given in Eq. (124), in the functional integral. In fact, by choosing Eq. (211) and (212), respectively. the following optimized form of the cutoff function [264, 265] The flow equation for the stationary density ρ0 cannot be (which obviously satisfies the requirements stated in Eqs. (127) specified without formal ambiguity: as we have already seen and (128)), in Sec. II B 2, both real and imaginary parts of the field equa- tion (75) yield conditions on ρ . However, we have seen as well 2  2 2 2 2 0 RΛ,K¯ (q ) = −K¯ Λ − q θ(Λ − q ), (217) that the condition stemming from the real part can always be satisfied by choosing the proper rotating frame. Therefore, the (2) in the regularized inverse propagator Γ + RΛ appearing in the physically correct choice is to assume that ρ is actually deter- Λ 0 Wetterich equation (129), the terms Aq¯ 2 in Eq. (216) are replaced mined by the imaginary part of the field equation, i.e., by the 0 by condition Im U (ρ0) = 0. Taking the derivative of this condition with respect to the cutoff `, we find  h   i A¯Λ2 for q2 < Λ2, ¯ 2 2 − 2 2 − 2  A q + Λ q θ(Λ q ) =  2 2 2 (218) ∂ ρ − ∂ U0
ρ / U00 ρ − ζ0 /κ, Aq¯ for q ≥ Λ , ` 0 = Im ` ( 0) Im ( 0) = Im ss (213) where ζ0 is the same as in Eq. (208). (and there is an analogous replacement for the terms Dq¯ 2). Having illustrated the main idea, the flow equation for the Hence, fluctuations with momenta below the cutoff scale Λ ac- rescaled noise strength γ = γ/¯ |Z|2 can easily be obtained along quire a mass ∼ Λ2, and the infrared divergences are lifted. 48

2 It remains to specify the flow equations for Z and K¯ . They can A∂`D/D , and expressing ∂`A as the real part of Eq. (221) etc. be obtained by choosing specific values of the indices i and j in In addition to r, we define another three scaling variables as the flow equation (215) for the inverse propagator, and by tak- 2 ! ing derivatives with respect to the frequency ω and the squared 2κρ0 γκ γ κ3 s = (w, κ,˜ κ˜3) = , , . (223) momentum q2, respectively: Λ2D 2ΛD2 4D3

1 h  R i The flow equations for the six dimensionless running couplings ∂`Z = − ∂ω tr 1 + σy ∂`P¯ (ω, q) , (219) 2 ω=0,q=0 collected in r and s form a closed set. Besides the Gaussian

 R R  fixed point at which the non-linear couplings vanish, these equa- ∂`K¯ = −∂q2 ∂`P¯ (ω, q) + i∂`P¯ (ω, q) . (220) 22 12 ω=0,q=0 tions have a non-trivial fixed point corresponding to the driven- dissipative condensation transition at Comparison with Eq. (216) shows, that these are indeed correct projection prescriptions. Note, however, that there is some ambi- r∗ = 0, s∗ = (0.475, 5.308, 51.383) . (224) guity in choosing these projection prescriptions: for example, A¯ ¯ R ¯ R ¯ This fixed point is reached in the RG flow, when the parame- appears both in P11 and P22. Our choice extracts K correspond- ing to the Goldstone direction, and mixes Goldstone and gapped ters in the microscopic action are chosen such that the system directions symmetrically in the projection on Z (see Ref. [172] is tuned precisely to the transition point. (Note that this point for details). Finally, the flow equation for the renormalized co- corresponds to the renormalized value of w ∝ ρ0 going to zero, efficient K = K¯ /Z is given by and not the bare one.) What are the physical implications of this fixed point? First, the value r∗ indicates, that the effective ∂`K = βK = ηZ K + ∂`K¯ /Z. (221) action at the fixed point is purely dissipative. As we have al- ready mentioned at the beginning of Sec.IV, for vanishing co- While Eqs. (211), (212), (213), (214), and (221) define a herent dynamics (or, in the terminology of equilibrium dynam- closed system of flow equations for the couplings u2, u3, γ, ρ0, ical models [1]: in the absence of reversible mode couplings), and K (as indicated above, the wave-function renormalization Z the driven-dissipative model reduces to the equilibrium model drops out of these equations), the explicit evaluation of the var- A. Thus, the values s∗ are the same as in model A, and there- ious projections is rather tedious. For details of this calculation fore the fixed point itself does not allow to distinguish whether we refer the reader to Ref. [172]. the microscopic starting point of the RG flow was in or out of equilibrium. However, the non-equilibrium nature of the driven- dissipative condensate is witnessed in the RG flow towards this 3. Scaling solutions and critical behavior effective equilibrium fixed point. In the following we consider the universal regime of the RG flow, which is reached in the At the critical point of a continuous phase transition, the cor- deep IR (i.e., for Λ/Λ0  1). In this regime, when the cou- relation length diverges, ξ → ∞. Then, instead of exponen- plings are close to their values at the fixed point, the RG flow −r/ξ tial decay according to ∼ e , correlation and response func- can be obtained from a linearization of the flow equations in tions depend on distance as power laws. This algebraic scaling δs = s − s∗, δr = r. The stability matrix governing the linearized behavior is reflected in the RG flow: indeed, the critical point flow takes block diagonal form, corresponds to a scaling solution to the RG flow equations. In ! ! ! practice, finding a scaling solution is facilitated by absorbing the δr N 0 δr ∂ = , (225) scaling factors ∼ Λθ (with some exponent θ for each coupling) ` δs 0 S δs in new variables, which thus take constant values at the critical point. Hence, the latter corresponds to a fixed point of the flow with 3 × 3 submatrices N and S . This block-diagonal structure equations for the rescaled couplings. Moreover, by means of a indicates, that the flow of r and s decouples in the IR. Therefore, suitable choice of rescaled variables it is often possible to further the flow of s close to the fixed point is the same as if we would reduce the number of flow equations, ending up with a minimal have set r = 0 from the very beginning. In other words, not set of independent equations. For the present case, the flow can only the values of the couplings s at the fixed point, but also the be specified in terms of just six real couplings. critical exponents encoded in the flow of s, which are the correla- At the beginning of this section, we introduced the quantity tion length exponent ν, the anomalous dimension η, and the dy- λ in Eq. (185) as a quantitative measure of the deviations from namical exponent z, see Ref. [172], are the same for both model thermal equilibrium conditions. In a similar spirit, the strength A and driven-dissipative condensates. (Note, however, that in of coherent relative to dissipative dynamics, which is encoded in model F [1], which describes condensation with particle num- the real and imaginary parts of the couplings in the microscopic ber conservation in equilibrium, the dynamical exponent takes a Keldysh action (73), is measured by the ratios different value than in model A, where particle number conser- vation is absent.) This confirms the asymptotic thermalization
r = rK, ru2 , ru3 = (A/D, λ/κ, λ3/κ3) . (222) of correlation functions mentioned in Sec.IVB (i). The values of the critical exponents we obtain from the truncation (201) are The flow equations for these ratios can be obtained straightfor- wardly by taking the RG scale derivatives, e.g., ∂`rK = ∂`A/D − ν = 0.716, η = 0.039, z = 2.121, (226) 49 and agree reasonably well with results from more sophisticated By definition, Eq. (230) is the location of the pole of the retarded calculations of ν to and η in the context of the static equilibrium Green’s function, and hence the coherent and diffusive parts en- problem [266]. code respectively the position and width of the peak of the spec- All the information on the universal properties of the driven- tral function defined in Eq. (61). The latter is probed, e.g., in dissipative transition, which are the same as in the equilibrium angle-resolved spectroscopy in exciton-polariton systems [267] model A, are encoded in s∗ and the block S of the stability ma- or radio-frequency spectroscopy in ultracold atoms [268]. How- trix in Eq. (225). The non-equilibrium nature of the microscopic ever, the small difference in the scaling of the position and width model, on the other hand, is betrayed by the block N, which de- of the peak predicted by Eq. (230) poses a challenge to its ex- scribes the flow of δr. This block has three positive eigenvalues, perimental observation.

n1 = 0.101, n2 = 0.143, n3 = 1.728, (227) C. Absence of algebraic order in 2D indicating that the ratios r are attracted to the fixed point value r∗ = 0. The general solution to the linearized flow equation for r reads Semiconductor microcavities hosting exciton-polaritons are effectively two-dimensional, and therefore this case has the 3 X greatest significance for current experiments. Even in thermal r = u c , (228) i i equilibrium, two-dimensional condensates are markedly differ- i=1 ent from their three-dimensional counterparts: first, according to where ui are the eigenvectors of N associated with the eigenval- the Mermin-Wagner theorem [261], in a two-dimensional con- ues ni in Eq. (227). The coefficients ci, which are referred to densate there cannot be true off-diagonal long-range order at any n ` n as scaling fields [169], take the scaling form ci ∼ e i ∼ Λ i . finite temperature. Instead, at low temperatures, spatial correla- Hence, for Λ → 0, the dominant contribution to r is given by tions decay algebraically with distance. Nevertheless, the sys- n1 −ηr r ∼ u1Λ = u1Λ , where we identified the drive exponent tem remains superfluid. Second, the algebraic or quasi-long- range order is established in an unusual transition, in which vor- − − ηr = n1 = 0.101. (229) tices, which proliferate at high temperatures, form bound pairs As anticipated in Sec.IVB (ii), this exponent governs the univer- as the temperature is tuned below the critical value. How is this sal fade-out of coherent dynamics ∝ r in the driven-dissipative scenario modified under non-equilibrium conditions? As a first system. Note that the existence of three distinct eigenval- step to answer this question, the issue of spatial correlations in ues (227) is due to the non-equilibrium character of the micro- two-dimensional driven-dissipative condensates is addressed in scopic model. Indeed, in model A with reversible mode cou- Ref. [19]. In this work, the results of which we describe in the plings, the equilibrium symmetry discussed in Sec. II D 1 allows following, the influence of non-topological phase fluctuations (spin waves) on the behavior of spatial correlations is analyzed, of only one ratio r = rK = ru2 = ru3 (cf. Eq. (98) and Fig.6). Then, the block N of the stability matrix in Eq. (225) has only which leads to the conclusion, that in driven-dissipative conden- sates algebraic decay is possible only on intermediate scales, and one single entry, which is given by the “middle” eigenvalue n2 in Eq. (227). This shows that also in the equilibrium setting the crosses over to stretched-exponential decay on the largest scales. dynamics becomes purely dissipative at the largest scales [254], The decay of correlations might be found to be even faster (i.e., however, the value of the critical exponent that governs universal exponential), once topological excitations (vortices) are taken decoherence is different. As pointed out above, this is due to the into account. absence of the equilibrium symmetry in the driven-dissipative In a weakly interacting Bose gas in thermal equilibrium, the case. The fact that the different values can be traced back to absence of true long-range order is caused by the vanishing en- a difference in symmetry supports the strength of the result, as ergy cost of long-wavelength phase fluctuations. These are gov- different symmetries are known to give rise to quantitatively dif- erned by the quadratic effective low-energy action (198), from ferent critical behavior [82]. which the behavior of spatial correlations at long distances can Decoherence at large scales has clear physical signatures be obtained straightforwardly, e.g., by introducing sources as de- which facilitate probing the drive exponent in experiments; to scribed in Sec.II and using the formulas for Gaussian functional wit, it implies that low-momentum excitations are diffusing integration collected in AppendixB. The result is rather than propagating (note that this is also predicted by mean- 0 ψ(x)ψ∗(x0) ≈ ρ hei(θ(x)−θ(x ))i field theory, cf. the discussion below Eq. (77)). A careful analy- 0 − 1 h(θ(x)−θ(x0))2i sis in the scaling regime reveals that the effective dispersion re- = ρ0e 2 (231) lation of single-particle excitations close to criticality takes the −α ∼ x − x0 , form [172] 2 z−ηr z 2.223 2.121 where α = m T/(2πρ0). One the other hand, the derivation in ω ∼ A0q − iD0q ∼ A0q − iD0q , (230) Sec.IVA shows that in the case of a driven-dissipative conden- where the diffusive contribution is supplemented by a subdomi- sate the phase-only action is non-linear and given by the KPZ nant (by the small difference of ηr in the exponent) coherent part. action (194). Hence, while the second equality in Eq. (231) still 50 applies to leading order in a cumulant expansion, the expecta- How do these findings compare to experimental results? tion value h(θ(x) − θ(x0))2i cannot be calculated directly.14 In Both in experiments on incoherently pumped polariton conden- the original context of the KPZ equation, where θ takes the role sates [281, 282] and simulations of parametrically pumped sys- of the height of a randomly growing surface [2, 69], the behav- tems [283], spatial correlations have been found to decay alge- ior of this correlation function is parameterized in terms of the braically within the confines of the system. This, however, is roughness exponent χ as not in contradiction to the present analysis based on the KPZ equation: indeed, if the microscopic value g0 of the rescaled 0
2 0 2χ h θ(x) − θ(x ) i ∼ x − x . (232) non-linearity (187) is small, which is actually the case in current experiments [19], a renormalized value of g = 1 is reached in The term roughness exponent is due to the fact that its value the RG flow only at the exponentially large scale distinguishes smooth from rough phases: for χ < 0, fluctuations L = ξ e8π/g0 , (234) of the surface height die out on large scales, and the surface is ∗ 0 smooth; on the other hand, if χ > 0, the interface is called rough. where ξ0 is a microscopic scale where the RG flow is initial- For any finite value of χ, the scaling behavior of the correlation ized, e.g., the healing length of the condensate. Indeed, to ob- function in Eq. (232) leads to stretched exponential decay of the tain this result, we have solved Eq. (188) with initial condition correlations of ψ, g0 at the scale ξ0. Then, in systems of a size L well below L∗, an effective equilibrium description is applicable, leading to the ∗ 0 −c|x−x0|2χ ψ(x)ψ (x ) ∼ e , (233) observed algebraic decay of correlations (below the equilibrium KT transition). In other words, even in an infinite system we where c is a non-universal constant. In the case that χ = 0 one should expect a smooth crossover from algebraic to exponential 0 2 usually expects logarithmic growth of h(θ(x) − θ(x )) i, which decay at the scale L∗. would lead to the equilibrium result in Eq. (231). However, as So far, our analysis has been based on the semiclassical discussed at the beginning of Sec.IV, in a 2D driven-dissipative Langevin equation (80) for the condensate dynamics, which condensate we should expect universal behavior that is quite dif- according to the arguments given at the beginning of Sec.IV ferent from the equilibrium case. Indeed, the FRG analysis re- correctly captures the universal scaling properties of driven- ported in Refs. [238–240, 263] and numerical simulations [269– dissipative condensates. However, in order to obtain an esti- 278] find χ ≈ 0.4 for the value of the roughness exponent, mate of L∗ for specific experimental parameters, a more micro- which implies that for |x − x0| → ∞ correlations in 2D driven- scopic model of exciton-polariton condensates is required. Start- dissipative condensates obey Eq. (233) and not Eq. (231)15, and ing from a widely used model, which has been introduced in decay stretched-exponentially. Ref. [151] and consists of a coupled system of equations for the A notable difference between the KPZ equation for randomly lower polariton field and the excitonic reservoir, the bare value growing interfaces, and the present context of the effective long- g0 and hence the scale L∗ can be seen to depend on the rate at wavelength description of a driven-dissipative condensate is that which the reservoir is replenished [19]. For high pump rates the analogue of the interface height in the latter case is a phase, θ, the KPZ scale L∗ grows rapidly, so that by pumping the system and as such it is compact, i.e., defined up to multiples of 2π. This strongly enough, algebraic correlations can be made to extend means that topological defects — vortices — in this field are over the entire system for any finite system size L. When the possible.16 Proliferation of vortices would lead to an even faster, pump rate is reduced, the system can loose its algebraic order in simple exponential decay of spatial correlations. The present two ways: either through the effect of the KPZ non-linearity if analysis does not take the possible presence of vortices into ac- L∗ drops below the system size, or — if L∗ is still much larger count. than the system size at the critical value of the pump strength for the equilibrium KT transition to occur — through the prolifera- tion of vortices. Note that as pointed out above, even when KPZ physics becomes relevant, vortices might still modify Eq. (233). 14 The non-linearity λ in the KPZ action (194) vanishes when the equilibrium These considerations lead to the finite-size phase diagram re- condition (98) is met, leading to algebraically decaying correlations also in ported in Ref. [19]. this case. This shows, that merely adding dissipation by coupling the system to The pump strengths at which the KT and KPZ crossovers oc- a bath in thermal equilibrium does not have an adverse effect on correlations. cur can be estimated based on the parameters given in Ref. [284]. (In the genuine case in which the condition (98) is met, the system is indeed coupled to a single bath. Otherwise, realizing this condition when the system It is convenient to introduce dimensionless pumping and loss is coupled to several baths would require a pathological fine-tuning of the rates as follows [19]: coupling parameters.) It is indeed the combination of independent drive and PR Rγ dissipation, which leads to the loss of algebraic coherence out of equilibrium. x − , γ l . 15 = 1 ¯ = (235) Note that as pointed out at the end of Sec.IVA, the KPZ non-linearity is γRγl γRuc neglected in Bogoliubov theory. Therefore, in 2D, this approach yields power- law decay of spatial correlations [182, 279]. Here, P is the rate at which the excitonic reservoir is replen- 16 This difference with the conventional KPZ equation also arises in “Active ished, while R is the amplification rate of the condensate due to Smectics” [280]. stimulated scattering of polaritons from the reservoir; γl and γR 51 are, respectively, the decay rates of lower polaritons and reser- equilibrium fixed point with g = 0 and Γ = −1. Then, algebraic voir excitons, and uc is the coherent polariton-polariton interac- correlations of the condensate field can survive if the effective tion [151]. For the parameters in [284], the KT transition should temperature at the fixed point, which is given by the renormal- p be expected at [19] xKT ≈ 0.02. Denoting by x∗ the pumping ized value of the dimensionless noise strength κ = ∆/ DxDy, strength at which the KPZ scale L∗ in Eq. (234) drops below the is below the critical value κc = π for the equilibrium KT transi- system size, we have [19] tion. Generalizing the microscopic model for exciton-polariton condensates mentioned above to account for spatial anisotropy, 2 x∗/xKT = γ¯ ln(L/ξ0) ≈ 0.04, (236) the dependence of the effective temperature on the strength of laser pumping can be obtained [19]. Remarkably, the transition where we tookγ ¯ ≈ 0.1, ξ0 ≈ 2 µm, and assumed a pump spot to the algebraically ordered phase is found to be reentrant: upon size of L ≈ 100 µm. Thus, approaching the transition from increasing the pump rate the ordered phase is first entered and above by lowering the pump power, the critical value xKT is then left again at even higher values of the pump rate. reached first, and the system loses algebraic order through un- binding of vortices [281–283]. On the other hand, the crossover to the disordered regime will be controlled by KPZ physics once x∗ ≥ xKT, which can be achieved by increasing the loss rate (i.e., D. KPZ scaling in 1D reducing the cavity Q) toγ ¯ ≈ 0.5. While the above analysis shows that algebraic order in 2D The marginality of g in two spatial dimensions is reflected in driven-dissipative condensates prevails only on intermediate the emergence of the exponentially large scale L in Eq. (234) scales, remarkably it can be restored on all scales in strongly ∗ beyond which KPZ scaling can be observed. In 1D, on the con- anisotropic systems [19]: consider a generalization of Eq. (80), trary, the KPZ non-linearity g is relevant (cf. the discussion at where the gradient terms are replaced by P Ki ∂2φ for α = i=x,y α i c the beginning of Sec.IV), which makes one-dimensional driven- c, d. Correspondingly, Eqs. (185) and (186) are replaced by dissipative condensates even more promising candidates to ob-  i   i  serve KPZ universality in finite-size systems. This possibility  K uc   K uc  D Ki  d  , λ − Ki  − d  , was explored numerically in Refs. [251–253], where the scaling i = c  i +  i = 2 c 1 i  (237) Kc ud Kcud properties of 1D driven-dissipative condensates were studied by simulations of the Langevin equation (80) for the condensate which for i = x, y are the diffusion constants and non-linearities field. appearing in the anisotropic KPZ equation. The RG flow of this equation has been analyzed in Refs. [280, 285]. It can be de- Experimentally, the most directly accessible signatures of KPZ universality are contained in the correlation function of the scribed in terms of the anisotropy parameter Γ = λyDx/λxDy (the system is anisotropic for Γ , 1), and the non-linearity condensate field (i.e., the Keldysh Green’s function defined in 2 2 p Eq. (34)), g = λx∆/(Dx DxDy). The flow equations are given by

dg g2   C(t − t0, x − x0) = hψ(t, x)ψ∗(t0, x0)i. (239) = − Γ2 + 4Γ − 1 , d` 32π (238) dΓ Γg   Indeed, in experiments with exciton-polaritons, both spatial cor- = − 1 − Γ2 . d` 32π relations C(0, x) and the autocorrelation function C(t, 0) can be obtained by performing interferometric measurements on the (Note that these equations differ from the RG equations in photoluminescence emitted from the semiconductor microcav- Ref. [19] by the sign on the RHS, which is due to the fact that ity [12, 284, 289]. Based on the mapping of the long-wavelength here we define the logarithmic scale as ` = ln(Λ/Λ0) instead condensate dynamics to the KPZ equation, one expects ex- of ` = ln(L/ξ0).) The line Γ = 0 divides the flow into two re- ponential decay of spatial correlations C(0, x) and stretched- gions with distinct fixed-point structure: for Γ > 0, which we exponential decay of the autocorrelation function according to denote as the regime of weak anisotropy, at large scales isotropy C(t, 0) ∼ exp(−ct2β), where β = 1/3 (in the original con- is restored, i.e., Γ → 1 for ` → ∞, and all the results discussed text of the KPZ equation, which is the stochastic growth of 17 above apply. In the regime of strong anisotropy corresponding driven interfaces, β is called the growth exponent [2]) and c is to Γ < 0, on the other hand, the flow is attracted to an effective a non-universal constant. This behavior was confirmed numer- ically [251–253]. In equilibrium, i.e., for g = λ = 0, the KPZ equation (196) reduces to a noisy diffusion equation, which in the surface growth context is known as the Edwards-Wilkinson 17 Current experiments with exciton-polaritons are in fact slightly anisotropic model [290]. Then, the behavior of spatial correlations is un- due to the interplay between polarization pinning to the crystal structure, and the splitting of transverse electric and transverse magnetic cavity modes [13, changed, whereas the exponent β governing the decay of tem- 286]. On the other hand, in experiments using the optical parametric oscillator poral correlations takes the value β = 1/4. The distinction be- regime pumping scheme [287, 288] (see also [79, 283]), strong anisotropy is tween one-dimensional condensates in equilibrium and driven- imprinted by the pump wavevector. dissipative condensates thus becomes manifest only in the dy- 52 namical properties.18 Moreover, in order to actually observe 10!1 KPZ scaling in the autocorrelation function, a large value of g, L#28 corresponding to a system far from equilibrium, is favorable. L#29 This can be achieved by making drive and dissipation the domi- !2 L#210 Α 10 nant contributions to the dynamics, as is the case in cavities with 2 11

L L#2

a reduced Q factor [251]. To be specific, for the parameters re- #! 12 t L#2 ported in Ref. [291], the Q factor would have to be reduced by , ! 13 L 3 L#2 a factor of ≈ 30 (corresponding to a polariton lifetime ≈ 1 ps " 10 instead of ≈ 30 ps achieved in the experiment) in order to make w KPZ scaling observable in a system of size ≈ 100 µm. !4 Another observable, which conveniently encodes the scaling 10 properties of the phase θ of the condensate, is defined as 10!3 10!2 10!1 100 z * !2+ t L 1 Z L 1 Z L w(L, t) = dx θ(t, x)2 − dx θ(t, x) . (240) L L 0 0 Figure 11. Finite-size scaling collapse of the roughness function w(L, t) defined in Eq. (240). The values of the roughness exponent α ≡ χ = 1/2 In the context of growing interfaces, were θ takes the role of ! the surface height, the quantity w(L, t) is known as the rough- and the dynamical exponent z = 3/2 are in the 1D KPZ universality class. Each curve corresponding to a specific system size L is an average ness function. It is a measure of the fluctuations of the sur- over 1000 noise realizations. The simulations were performed after face height over the linear extent of the system L. While the rescaling the Langevin equation (80) to bring it to dimensionless form. roughness function might not be easily accessible in experi- For details of the rescaling and the values of the parameters used in ments with driven-dissipative condensates, it allows a very com- the simulations, see Ref. [251]. (Copyright (2014) by The American pact demonstration of both static and dynamic KPZ scaling ex- Physical Society.) ponents if it is obtained numerically for a range of different system sizes [271]. Indeed, the finite-size scaling collapse of w(L, t) in Fig. 11 shows, that after a period of growth during → ∞ which w(L, t) ∼ t2β, the roughness function saturates at the time limit τ [42–44, 292–296]. An example which identifies z generic, universal features in the far from equilibrium dynamics Ts ∼ L ; the saturation value ws(L) scales with the system size as 2χ in a strongly interacting one-dimensional system is discussed in ws(L) ∼ L , where χ = 1/2 is the value of the roughness expo- nent in 1D [2,5, 98]. From the growth and roughness exponents, the present section. the usual dynamical exponent can be obtained as z = χ/β. The setting we consider here differs from the one in the pre- The numerical analysis reported in Ref. [251] was performed vious section not only in its focus on time evolution, but also in in the regime of weak noise, which is characterized by the ab- terms of underlying symmetries. Above we have studied sys- sence of phase slips in the spatiotemporal range covered by the tems that are open in the sense that both energy and particle simulations. Indeed, the mapping of the condensate dynamics number were not conserved, witnessed by the absence of the to the KPZ equation (196), in which the phase is regarded as a thermal symmetry (cf. Sec. II D 1) and the quantum phase rota- non-compact variable, does not take into account the possible tion symmetry (cf. Sec. II D 4), and leading to a breaking of de- occurrence of such defects. However, their presence at higher tailed balance and a low momentum diffusive Goldstone mode noise levels is expected to affect the scaling properties of driven- (cf. Sec. II D 6), respectively. Here we consider an open sys- dissipative condensates. tem, where only energy is not conserved, but particle number is. In fact, the absence of energy conservation here is reflected by a permanent inflow of energy into the system. The Lindblad V. UNIVERSAL HEATING DYNAMICS IN 1D operators are Hermitian in the present case, and this leads to continuous heating and ultimately to an entirely classical, infi- The notion of universality is not restricted to systems in ther- nite temperature stationary state, described by a density matrix mal equilibrium. As discussed in the previous sections, non- ρ ∼ 1, where the latter unit matrix is understood in the entire thermal steady states of driven-dissipative systems can show Fock space of the problem. This motivates us to study the time a large variety of universal features, such as scale invariance evolution of heating, with a focus on the short time dynamics and effective long-wavelength thermalization. However, in a following initialization in a pure zero temperature ground state, plethora of setups, aspects of universality can even be found in where quantum effects are still present. The presence of parti- the time evolution, which approaches a steady state only in the cle number conservation leads us to take a different strategy than in the previous sections. Here, accomodating number conserva- tion, which is at the heart of the strongly collective behavior of one dimensional systems, we first map the quantum master equa- 18 Concomitantly, also within Bogoliubov theory exponential decay of correla- tion in the operatorial formalism to an effective long-wavelength tions is found [181], cf. the discussion at the end of Sec.IVA and Footnote 15. description in terms of an open Luttinger liquid. In this way, 53 we can carefully account for the linear sound mode that is ex- of Luttinger liquid variables. In this regime, one can take the pected on the grounds of exact particle number conservation, cf. continuum limit bi → bx and express the bosonic operators in a Sec. II D 6. Only after this procedure, we perform the mapping phase and amplitude representation: to the Keldysh functional integral, similar to our strategy for the √ iθx spins in Sec. III. bx = ρxe , (243) ρx = ρ0 + ∂xφx/π,

A. Heating an interacting Luttinger liquid in terms of the Luttinger variables ∂xφx and θx, which represent smooth density and phase fluctuations and fulfill the commuta-   0 Consider a one-dimensional lattice of interacting bosons for tion relation ∂xφx, θx0 = iπδ(x − x ). The long-wavelength de- which the dynamics is described by the following master equa- scription of the Bose-Hubbard model is expressed by the Hamil- tion: tonian X Z   h n 2 oi 1 h i ∂tρ = −i H, ρ + γe 2niρni − ni , ρ . (241) 2 u 2 2 H = uK (∂xθx) − K (∂xφx) + κ (∂xφx)(∂xθx) , (244) i 2π x Here, H is a bosonic lattice Hamiltonian, whose long- which describes interacting Luttinger phonons on length scales wavelength physics is described by an interacting Luttinger liq- −1/2 x ≥ (ρ0Um) , above which a continuum representation of the uid. For concreteness, one can consider a Bose-Hubbard model Hamiltonian is appropriate. For weak interactions, the effective in one dimension away from integer filling, with Hamiltonian  1/2  1/2 parameters can be estimated to be u = ρ0U , K = π ρ0 . X X m 2 Um  † †  U The non-linearity in the Hamiltonian accounts for the leading H = −J bi bi+1 + bi+1bi + 2 ni(ni − 1), (242) i i order quasi-particle scattering term in the low energy regime (κ = 1/m), which is irrelevant in the sense of the renormaliza- which describes nearest neighbor hopping of particles with hop- tion group and does not modify static, equilibrium correlations. ping amplitude J and local interactions with interaction energies However, it is vital for the quantitative description of dynamic U. The dissipative contributions which drive the system out of correlation functions, and is non-negligible in a non-equilibrium † equilibrium, are the Hermitian jump operators ni = bi bi, mea- setting where the dynamics is affected by the non-linearity even suring the local particle number in terms of bosonic creation and on a qualitative level. The dissipative part of the master equation † annihilation operators bi , bi. For cold bosonic atoms in optical becomes quadratic in the Luttinger representation, such that the lattices, these jump operators represent the leading order contri- equation of motion reads bution of dissipation induced by spontaneous emission from the Z h n oi lattice drive laser [158] (see also the discussion in Sec.IC3), −   γ − 2 ∂tρ = i H, ρ + π2 2(∂xφx)ρ(∂xφx) (∂xφx) , ρ . (245) but this kind of dissipation can as well be realized by coupling x the bosonic particles to a phonon reservoir with a large effective This decoherence term is the leading order contribution of a temperature, which is equivalent to a locally fluctuating chemi- U(1)-symmetric (i.e., particle number conserving) decoherence cal potential mechanism in one-dimensional quantum wires, which features a linear increase of the energy in time. Furthermore, the U(1) µ(x, t) = µ0 + δµ(x, t), symmetry guarantees the existence of a linear sound mode and hδµ(x, t)δµ(x0, t0)i = γ δ(t − t0)δ(x − x0). e permits the transformation to the Luttinger framework in the co- It causes dephasing and leads to a linear increase of the energy herence dominated regime. From a microscopic perspective, it is evident that the Luttinger description has to break down for in the system, hHit ∝ Jγet [158, 159]. As a consequence of the linear energy increase, the system will never thermalize and is sufficiently strong decoherence, i.e., after the system has been constantly driven away from equilibrium, approaching the T → heated up sufficiently. This breakdown can be estimated by the ∞ state described by ρ ∝ 1 at infinite time t. We note that, for usual Luttinger criterion nq < Λ/|q|, and leads to a good esti- hermitean Lindblad operators, ρ ∝ 1 is always a solution to the mate for the relevant time scales up to which the dynamics of quantum master equation. the system is dominated by coherent sound modes and therefore In order to analyze the heating dynamics for short and tran- properly expressed in the Luttinger framework [40]. The corre- 2 −1 sient times, the master equation is transformed to the Luttinger sponding time regime is set by the condition t < u (κγ) . representation on the operatorial level, and only later on we per- The quadratic part of the equation of motion is diagonalized form the mapping to the Keldysh functional integral. The Lut- by the canonical Bogoliubov transformation tinger description is valid as long as the occupation of quasi- Z π 1/2 † −iqx particle modes does not exceed a critical value, which is deter- θx = θ0 + i ( 2|q|K ) (aq − a−q)e , mined by the Luttinger cutoff Λ [40]. Starting with a zero or q Z (246) low temperature initial state to which this applies, there exists a πK 1/2 † −iqx φx = φ0 − i ( 2|q| ) sgn(q)(aq + a−q)e , cutoff time tΛ, up to which the system can be described in terms q 54 which leads to the master equation The off-diagonal phonon density oscillates in the complex plane, thereby taking absolute values |ha†a† i | ≤ γ , which Z q −q t 2πuK h † i are negligibly small in the weak heating regime γ  uK, i.e., in ∂tρ = −i u|q|aqaq, ρ q the coherence dominated dynamics. It is therefore sufficient to Z h i Z consider only the diagonal elements in the phonon basis when γ|q| † − 1 { † } − (3) + πK AqρAq 2 AqAq, ρ [Hph , ρ]. (247) extending the analysis to the interacting model. q q The jump operators in the microscopic master equation (241) In this equation, the dissipative part is expressed via the opera- are the Hermitian, local density operators ni, which preserve the † † U(1) invariance of the dynamics even in the presence of dissi- tors Aq = aq + a−q = A−q in terms of bosonic phonon operators † pation. This leads to a decay of the off-diagonal elements of [aq, ap] = δ(q− p). The cubic Hamiltonian incorporates resonant the density matrix, i.e., to decoherence in the local number state phonon scattering processes representation, and an evolution of the density matrix towards q Z 0 p its diagonal ensemble. In the Luttinger representation, this de- (3) π | † Hph = 3κ 2K qp(p + q)(ap+qaqap + h.c.), (248) coherence expresses itself in a permanent production of photons q,p (251), i.e. a permanent heating of the Luttinger liquid, which which conserve momentum and the phonon energy. This is ex- features no compensation mechanism in the quadratic sector and R 0 pressed by , which signals to integrate only over configura- the consequent lack of a well defined steady state. The energy qp increases constantly, which will lead to an obvious breakdown tions {p, q} with |q| + |p| = |p + q|. The non-resonant processes of the Luttinger description as soon as the energy stored in the create only short-lived quantum states which do not contribute in long-wavelength modes exceeds a critical value. At this point, the long time dynamics due to dephasing, and which are there- the dynamics is no longer dominated by the coherences of ρ but fore not relevant for the forward dynamics of the system. by its diagonal elements, including the breakdown of superflu- In the Luttinger representation, the dissipative contribution is idity and quasi-long range order. quadratic and its effect is a constant population of the individual The way in which energy is distributed amongst the long- phonon modes. This can be seen most easily by computing the wavelength modes by the heating (249) is not typical for an in- time evolution of the phonon densities in the quadratic frame- teracting system at low energies, since it deviates strongly from a work, which yields Bose-Einstein distribution and the associated detailed balance of (3) | | energy. The phonon scattering terms in H favor detailed bal- ∂ ha†a i = γ q , (249) ph t q q t 2πK ance and strongly modify the actual distribution function com- † † γ|q| † † ∂thaqa−qit = − 2πK − 2iu|q|haqa−qit. (250) pared to (251), which makes them non-negligible in the present non-equilibrium setting. For a system prepared initially in the ground or finite temper- † ature state, the initial phonon densities are haqaqi0 = nB(u|q|) † † βu|q| −1 and ∂thaqa−qi0 = 0, where nB(u|q|) = (e − 1) is the Bose- B. Kinetic equation Einstein distribution evaluated on-shell. In this case, the solution of Eqs. (249), (250) is In order to determine the time evolution of the excitation den- sities in interacting systems out of equilibrium, a common and † γ|q|t haqaqit = nB(u|q|) + 2πK , (251) often successful strategy is the so-called kinetic equation ap- −iu|q|t ha†a† i = γe sin(u|q|t). (252) proach [98] (for an application to periodically driven Floquet q −q t 2πuK systems, cf. Ref. [297]), which determines the time evolution The first term describes an increase of the phonon density lin- of the distribution function of the excitations in terms of the sys- ear in time and momentum and leads to a linear increase of the tem’s self-energies. For the present setup, this approach has to be system energy, i.e., heats up the system according to modified in order to take into account the driven-dissipative na- ture of the system and the resonant character of the interactions. (2) (2) uγΛ2t The latter lead to a breakdown of perturbation theory and require ∆Et = hH it − hH i0 = . (253) 4πK non-perturbative techniques beyond one-loop corrections. A de- Equation (253) relates the effective long-wavelength Luttinger tailed derivation and discussion of the applicability and limita- parameters u, K, γ of the heating setup to the macroscopic heat- tions of such an approach can be found in [40, 41]. Other forms ing rate ∂t∆Et via the microscopic cutoff Λ. Since this heat- of kinetic equations for interacting Luttinger liquids, which fo- ing rate is not model specific but depends on the individual cus on a different set of non-equilibrium conditions, include a realization of the heating dynamics, it is not surprising that it perturbative treatment of phonon backscattering terms, resulting depends on macroscopic and microscopic parameters. In this from additional disordered or lattice potentials [298, 299], as sense, Eq. (253) should be viewed as the definition of the effec- well as a treatment of cubic phonon interactions in the presence tive heating parameter γ for a generic model in the presence of of a smooth background potential and a curved phonon disper- heating [40]. sion [300]. 55

The quantity of interest in this section is the time-dependent drops out, C(t, δ) ≡ C(0, δ), for all t. In Wigner coordinates, the † occupation of phonon modes nq,t = haqaqit in the presence parametrization of the Keldysh correlation function is of heating and phonon scattering. For bosonic modes and in K  R A G = G ◦ Fq − Fq ◦ G . (259) the steady state, the occupation of the modes is related to the q,t,δ q q t,δ Keldysh Green’s function via Here, we choose to neglect the subleading off-diagonal contribu- Z tions according to the above discussion, and consider only diag- K i Gq,ω = 2nq + 1. (254) onal modes in Nambu space. Eq. (259) contains the full Green’s ω functions of the system, which can be expressed via the Dyson relation in terms of the self-energies ΣR/A/K, For a system in thermal equilibrium, nq = nB(q, T) is the Bose distribution function, see Sec. II D 1, while for a general non- !−1 ! GK GR 0 GA − ΣA equilibrium steady state, n is a positive function, which has to = 0 . (260) q GA 0 GR − ΣR −ΣK be determined from the specific context. One can now introduce 0 the hermitian distribution function Fq,ω as in Eq. (72), but gen- The self-energies ΣR/A/K represent the correction to the bare eralized to a system with a continuum of momentum modes. In R/A −1 Green’s functions G0 = (i∂t − u|q|) due to phonon scatter- terms of the hermitian distribution function, the anti-hermitian ing and heating events. Inserting the Dyson representation into Keldysh Green’s function can be parameterized according to Eq. (259), it can be inverted and rearranged to read

K  R A K  R A G 0 = G ◦˜ Fq − Fq ◦˜ G . (255) ∂tFq,t,δ = iΣ − i Σ ◦ Fq − Fq ◦ G . (261) q,t,t q q t,t0 q,t,δ q q t,δ

† Here, the notion (...) expresses the fact that the whole ex- R A  R t,δ Here, Gq , Gq = Gq and Fq are two-time functions eval- pression in brackets should be transformed to Wigner coordi- ◦ R R R uated at equal momentum q and ˜ represents the convolution nates after performing the convolution Σ ◦ F ≡ Σ 0 F 0 q q t0 q,t1,t q,t ,t2 with respect to time and matrix multiplication according to the in ordinary time representation. The functionals ΣR/A/K are Nambu structure of the Green’s functions. For a system, which the self-energies in the retarded, advanced, and Keldysh sec- ◦ ◦ is diagonal in Nambu space, ˜ = reduces to a simple multi- tors, which incorporate the effect of interactions and the heat- plication. In the presence of off-diagonal occupations, i.e. for † † ing on the quadratic sector. The temporal derivative on the LHS ha−qaqi , 0, the operation◦ ˜ has to respect the symplectic struc- of (261) is the Wigner representation of the bare, non-interacting z ture of bosonic Nambu space and is promoted to◦ ˜ = σ ◦, as Green’s functions without heating. Taking the Fourier transform described in Ref. [41]. For a bosonic system in steady state, all of Eq. (261) with respect to the relative time coordinate δ yields terms in (255) are time-translational invariant and Fourier trans- the Wigner representation of the distribution function formation yields Z iωδ K R z z A Fq,t,ω = e Fq,t,δ, (262) Gq,ω = Gq,ωσ Fq,ω − Fq,ωσ Gq,ω. (256) δ In contrast, for a system out of equilibrium undergoing a non- for which   trivial time evolution, the mode occupations n remain well de- K R A q,t ∂tFq,t,ω = iΣq,t,ω − i Σ ◦ F − F ◦ Σ . (263) fined, but Eq. (255) is not time-translational invariant anymore. q,t,ω One then has to find a representation for the Green’s functions The corresponding transformation for the convolution inside the and F, which reveals the time-dependent occupations. This is parenthesis is done in the following, leading to the Wigner representation of ← → ← →    i ∂ ∂ −∂ ∂ the bosonic distribution function. R R 2 t ω ω t Σ ◦ F = Σq,t,ωe Fq,t,ω. (264) A convenient parameterization of non-equilibrium correlation q,t,ω and response functions is the so-called Wigner representation in Its explicit evaluation is nontrivial and in most cases simply im- time, which introduces a forward and a relative time coordinate possible. However, it is possible to approximate the complex (t, δ) for the phonon Green’s functions, according to exponential by the leading order expansion for many typical re- laxation dynamics [98]. In order to understand Eq. (264), one R † Gq,t,δ = −iθ(δ)h[aq,t+δ/2, aq,t−δ/2]i, (257) should take a closer look at its expansion up to first order in K † derivatives: Gq,t,δ = −ih{a , a − }i. (258) q,t+δ/2 q,t δ/2  R   R  R i ∂t Fq,t,ω ∂ωΣq,t,ω 2 Σ ◦ F = Σ Fq,t,ω 1 − R + (ω ↔ t) + O(∂ ). q,t,ω q,t,ω 2 Fq,t,ω Σ A two-time function C(t1, t2) can always be transformed to q,t,ω t1+t2 (265) Wigner coordinates, C(t1, t2) = C(t, δ), by defining t = 2 and The ratio κ f ≡ ∂t Fq,t,ω is the rate with which the distribution δ = t1 − t2. Here, the explicit dependence on t expresses the for- q Fq,t,ω ward time evolution of non-equilibrium systems, while for equi- F is changing in forward time, i.e., the forward time evolu- librium systems in the presence of time-translational invariance, tion rate, which is determined by the interplay between heat- the forward time dependence of generic two-time functions just ing and collective quasi-particle scattering. On the other hand, 56

 −1 ∂ ΣR r ω q,t,ω Here, p is the momentum variable and q labels the quantum κq ≡ R is identified with the inverse rate of the relative Σq,t,ω component of the Keldysh field variable. The dissipation thus time dynamics, which is dominated by fast single phonon prop- enters the action only in the quantum-quantum sector and does r f agation. As a consequence κq  κq and the correction terms not modify the spectrum in the quantum-classical sector. This in (265) can be safely neglected. This is a typical situation for is again a consequence of the Hermitian nature of the Lindblad many kinetic equation approaches and is termed the Wigner ap- operators, which lead to a continuously increasing occupation of proximation. For the present setup, a more careful analysis has phonons but do not introduce a compensating dissipative mecha- shown that the Wigner approximation is indeed satisfied as long nism in the retarded and advanced sector of the action 19. This is as the Luttinger representation of the problem is valid [40]. drastically different from the situation in the models of Secs. III In order to project Eq. (263) onto the quasi-particle densities, (cf. Eqs. (138) and (137)) andIV (cf. Eqs. (201) and (202)), R A it is multiplied by the spectral function Aq,t,ω = iGq,t,ω − iGq,t,ω, where the interplay of dissipation in the retarded/advanced sec- followed by a subsequent integration over frequencies ω. The tors and fluctuation or noise in the Keldysh sector of the action spectral function fulfills the sum rules lead to non-equilibrium fluctuation-dissipation relations describ- Z ing well-defined stationary states different from the trivial state A h a , a† i , q,t,ω = [ q,t,0 q,t,0] = 1 (266) ∝ ω ρ 1. K Z Z With the form of (270), the Keldysh self-energy isσ ˜ q,t = Wigner approx. K A A ◦ γ|p| q,t,ωFq,t,ω = ( F)q,t,ω = Gq,t,0 = 2nq,t + 1, (267) + σK , where the bare σK in this form is determined by ω ω 2πK q,t q,t the interactions alone. The resulting kinetic equation consists of h † i with nq,t = aq,taq,t being the phonon density. Applying this to three contributions Eq. (263) in the Wigner approximation yields ∂ n = γ|q| + σK − σR (2n + 1) . (271) Z   t q,t 2πK q,t q,t q,t ∂ n = i ΣK − ΣR F + ΣA F A .(268) |{z} |{z} | {z } t q,t 2 q,t,ω q,t,ω q,t,ω q,t,ω q,t,ω q,t,ω in-term heating in-term scattering out-term scattering ω For well defined quasi-particles, i.e., excitations with a well The first term represents the population of phonon modes due to defined energy-momentum relation, the spectral function re- the constant heating term, while the second and the third term flects the well-defined structure of the excitations and is sharply are effects of the elastic collisions redistributing energy. The peaked at the quasi-particle dispersion ω = u|q|, with a typical second term, proportional to the Keldysh self-energy, describes R scattering of phonons into the mode q due to the interactions, width σq,t  u|q|, which is the imaginary part of the self-energy R R while the third term, proportional to the retarded self-energy, evaluated on the mass shell σq,t = −Im(Σq,t,u|q|). If this is the case, the full self-energies in Eq. (268) multiplied with the spec- describes scattering of phonons out of the mode q, and is there- fore directly proportional to nq,t. Setting the interactions to zero, tral function can be approximated by their on-shell values, as R/K they are expected to vary only smoothly in the region where A both self-energies σ = 0 vanish and only the heating term remains, rendering the time evolution of the phonon density in is non-zero. The approximation ΣR/A/KA ≈ ΣR/A/KA is q,t,ω q,t,ω q,t,u|q| q,t,ω the absence of interactions into Eq. (249). called the quasi-particle approximation and in the present case, The kinetic equation (271) represents the foundation of the similar to the Wigner approximation, it is applicable in the en- analysis of the non-equilibrium dynamics in the presence of tire Luttinger regime [40, 41]. The latter is a consequence of heating and phonon scattering. In order to solve for the time- the subleading, RG-irrelevant nature of the interactions, which evolution of the phonon densities, one has to compute the self- lead to self-energies σR  u|q|. Performing the quasi-particle q,t energies σR/K for each momentum mode and at each time step. approximation, Eq. (268) obtains the simple form q,t The self-energies have to be determined by a non-perturbative K R ∂tnq,t = σ˜ q,t − σq,t(2nq,t + 1). (269) approach, which we discuss in the following. K In this equation, the anti-Hermitian Keldysh self-energy Σq,t,ω K K has been replaced by its on-shell value Σq,t,u|q| = −2iσ˜ q,t, with C. Self-consistent Born approximation K the real functionσ ˜ q,t. The kinetic equation (269) describes the time evolution of the For an interacting model of resonantly scattering phonons, the R K phonon density nq,t in terms of the on-shell self-energies σ , σ˜ , phonon self-energies are functionals of the phonon density, such which in turn are determined by both the interactions and the heating term. In order to identify the contribution from the heat- ing, one has to identify the impact of the dissipative contribution 19 The heating mechanism is operative for generic situations. For example, in Eq. (247) on the action S . Following the steps in Sec.IIA mean-field Mott initial states, which are the exact ground states of the Bose- carefully and setting the off-diagonal density contributions to Hubbard model for fine-tuned J = 0, are pure states which are not touched zero, according to Eq. (251), the dissipative contribution to the by the dissipator considered in this section. One point of view on this phe- microscopic action S is nomenology is that the infinite temperature state is an attractive fixed point Z of dynamics, but there are other unstable ones which need additional symme- γ|p| ∗ tries to be physically relevant (such as a spatially local gauge symmetry in the S D = i πK aq,p,taq,p,t. (270) J = 0 example). p,t 57 that the RHS of (271) contains an implicit, non-linear depen- quadratic self-energy is the second order expansion of the loga- dence on nq,t. In order to make this implicit dependence ex- rithm and its matrix elements are determined by the integrals plicit, the self-energies are typically evaluated perturbatively at Z 0 R K  2 R 2 A 2 R  one loop order and higher order corrections to the time evolu- ΣQ = 2i GP v (q, −p)GQ−P+ v (p, −q)GP−Q+ v (p, q)GP+Q , tion are neglected [98]. However, for the present scenario, the P interactions are resonant, i.e., describe scattering events inside a (277) Z 0 continuum of degenerate states, and therefore perturbative com- K h 2  K K R R A A  ΣQ = 2i v (q − p) GP GQ−P + GPGQ−P + GPGQ−P putations diverge at any order. This defines the need for non- P  i perturbative approaches to compute the phonon self-energies, +2v2(p, q) GK GK + GA GR + GR GR , (278) the simplest of which is the so-called self-consistent Born ap- P+Q P P+Q P P+Q P proximation, which we discuss in the following. where we used the collective indices Q = (q, ω), P = (p, ν) for The Keldysh action for interacting Luttinger liquids with heat- momentum and relative frequency. In Wigner approximation, ing is composed of a dissipative part S D, which has been dis- the Green’s functions are diagonal in forward time and therefore cussed in Eq. (270), and a Hamiltonian part S H, which results evaluated at equal forward time t. The integrals in (277), (278) from the Hamiltonian dynamics in Eqs. (247), (248). In the are performed only over resonant momentum configurations, see Keldysh representation, the action is the discussion around Eq. (248). In perturbation theory, the Green’s functions under the integral are the bare, non-interacting R −1 Z +! ! Green’s functions G = (ω − u|q| + i0+) , which diverge on the ∗ ∗ 0 i∂t − u|p| − i0 ac,p,t Q S = (a , a ) γ|p| mass-shell ω = u|q| and lead to a summation of infinities for c,p,t q,p,t i∂ − u|p| + i0+ i a p,t t πK q,p,t the self-energy. On the other hand, in self-consistent Born ap- Z 0 h ∗ proximation, the bare Green’s functions in Eqs. (277), (278) are + v(p, k) 2ac,k+p,tac,k,taq,p,t  −1 p,k,t R R replaced by the full Green’s functions GQ = ω − u|q| − ΣQ . ∗   i + aq,k+p,t ac,k,tac,p,t + aq,k,taq,p,t + h.c. (272) As a consequence, the on-shell Green’s function is regularized by the self-energy ΣR and takes the value p π | R  R −1  R−1 with the vertex function v(p, k) = 3κ 2K pk(p + k). One way of G = − Σ = −i σ . (279) computing the one-loop self-energy is to determine the one-loop q,u|q| q,u|q| q ∗ Inserting (279) in the definition for the retarded self-energy and correction to the effective action Γ[aα, aα] defined in Eq. (39) for general bosonic fields Φ. The effective action in the absence of evaluating the self-energy on-shell, one obtains   an external source is defined as Z    qp(q − p) pq(p + q)  R 2 ∂tnp,t   σq,t = v0 R + 2np,t + 1  +  Z σq,t  R R R R  ∗ ∗ ∗ 0

a scale in the self-energy equation (280). One important con- q→0 R sequence is that σq,t → 0 generically, i.e., the generation of a mass gap is forbidden by symmetry. A further consequence of Eq. (280) is that, whenever the term in brackets obeys a scaling law,

∂tnp ηn R + 2np + 1 ∼ γn|p| , (285) σp

R the solution for the self-energy will be a scaling function σq = η γR|p| R as well. Inserting this scaling ansatz in Eq. (280) yields

2 v γn ηR 0 4+ηn−ηR Figure 12. Illustration of the iterative process to compute the time evo- γR|p| = |p| Iηn,ηR (286) γR lution of the phonon density nq,t. For a specific forward time t, the on- R shell self-energy σq,t is determined via Eq. (280) and subsequently the and identifies result is inserted into the kinetic equation (281). In order to integrate q the phonon density, a Runge-Kutta solver for differential equations is ηn ηR = 2 + and γR = v0 γnIηn,ηR . (287) used, which determines nq,t numerically. 2

The dimensionless integral Iηn,ηR is defined as ! 1. Kinetics for small momenta Z x(1 − x) x(1 + x) I ηn ηn,ηR = x η η + η η , (288) 0 0 is continu- ously increased due to heating. The central result of the analysis Z ! R ηR γt is a scaling solution for the self-energies σq,t ∼ |q| with a new | | J 0 nq,t = nq,0 + q + t . (284) η / 2πK 0 xth(t). Here, xth(t) The momentum regime 0 < |q| < q2, defined as the regime where marks a thermal distance, below which the dynamics is domi- the non-linear contributions in |q| are negligible, depends on the nated by thermalized short distance modes and above which the actual value of the phonon density nq,t, and has to be determined phonon density increases linearly in momentum nq,t ∼ |q|, as it for each specific scenario individually. However, the existence was the case for the bare heating (249). The thermal distance 4/5 of q2 is guaranteed by the above general arguments, and the fact increases sub-ballistically xth(t) ∼ t in time, with a charac- that ∂tnq=0,t = 0 for all times t. The latter is a consequence of teristic heating exponent ηh = 4/5, while at the same time, the the U(1) symmetry of the present setup and the global particle effective temperature describing the distribution of the short dis- number conservation, as discussed in [41]. tance modes increases linearly in time T(t) ∼ t.

2. Scaling of the self-energy 1. Phonon densities

The fact that the present system is described by a U(1)- The results of a numerical simulation of the phonon densities invariant, massless field theory is reflected by the absence of are shown in Fig. 13. One can clearly identify the crossover 59

−1 achieved by the scaling of the inverse thermal length (xth(t)) to zero, instead of a direct filling of these modes.

2. Non-equilibrium scaling

−1 Away from the crossover scale q , xth , deep in the heating or thermal regimes, the phonon density exhibits scaling behavior and can be written as ( −1 c(t)|q| for |q|  xth nq,t = T(t) −1 , (289) u|q| for |q|  xth where the functions c(t), T(t) have to be determined numerically. In these regimes, according to Sec.VC2, one finds scaling be- havior of the phonon self-energy as well. The corresponding Figure 13. Time evolution of the phonon density nq,t in a sequence 10 scaling exponent is determined by (t1, t2, t3) = (2, 3, 4) · 2 in terms of q/Λ and for a heating rate γ = v0Λ 0.06v0Λ. In the heating regime, for small momenta, the distribution ∂tnq,t ≡ ∼ | |ηn increases linearly in momentum nq,t ∼ |q|, while it decreases as nq,t ∼ fq,t R + 2nq,t + 1 q . (290) −1 σq,t 1/|q| in the interaction dominated thermal regime. The crossover xth ∼ t−4/5 approaches zero as time evolves. The dashed line represents a In the heating dominated regime, n = c(t)|q| and therefore Bose-Einstein distribution corresponding to the phonon density at t = q,t t3. The dash-dotted line indicates nB(T(tc)) at the time tc, for which the 0 0 c (t) ηR>1 c (t) 1−ηR 1−ηR Luttinger description breaks down, i.e., T(tc) = uΛ. fq,t = |q| + c(t)|q| + 1 → |q| (291) γR γR

for small momenta, since ηR > 1 is guaranteed by the subleading T(t) nature of the interactions. This directly implies ηn = 1−ηR and a from an interaction dominated thermal regime with nq,t = u|q| super-diffusive exponent η = 5 in this regime, which lacks any at large momenta to a heating dominated regime with n ∼ |q|. R 3 q,t equilibrium counterpart. The crossover momentum between the two regimes represents −1 −4/5 On the other hand, in the large momentum regime fq,t is ob- the inverse thermal length (xth(t)) ∼ t . viously dominated by the term proportional to nq,t and ηn = −1, In the large momentum regime, the dominant contribution to which leads to the known thermal equilibrium exponent ηR = the kinetic equation is the collision term, which establishes an 3 2 [301] and a thermal scaling behavior for large momenta not approximate detailed balance between phonon absorption and only in the distribution function but also in the self-energy. The emission in the presence of the heating. This is expressed by the crossover between the two scaling regimes can be estimated by evolution of the phonon density towards a Bose-Einstein distri- equating the two relevant terms, which tend to dominate fq,t in bution, which is the fixed point of the collision term alone. In the corresponding regimes. This yields T(t) this regime nq,t = nB(u|q|, T(t)) ≈ u|q| with very good agreement ∂tnq and the only indicator of the permanent heating is the continu- th σR , = qth (292) ously increasing temperature T(t) ∼ t. This is contrasted by the 2nqth + 1 evolution of the phonon density in the low momentum regime, −1 which is dominated by strong phonon production. In this regime, for qth = xth and can be used to analytically estimate the scaling 4/5 the scattering of high-momentum phonons into low-momentum of xth(t) in time [40], resulting in xth(t) ∼ t , which agrees well modes enhances the effect of the heating and leads, due to the with the numerical findings. structure of the vertex, to an increase of the linear production rate according to Eq. (282). The pinning of the phonon occupation of the q = 0 mode is an exact result for the underlying U(1)- 3. Observability symmetric, i.e. particle number conserving dynamics. It can be shown that the phonon number fluctuations in the zero mo- The universal scaling of the phonon self-energies as well mentum mode are proportional to the fluctuations of the global as the scaling regimes of the phonon distribution function particle number in the system, see Ref. [40], and consequently can be observed in cold atom experiments via Bragg spec- they are integrals of motion for a U(1)-symmetry preserving dy- troscopy [40], which gives direct access to the characteristic namics. The pinning effect for the low momentum distribution universal features in the heating dynamics. In Bragg exper- at nq=0,t = nq=0,0 leads to a very slow, sub-ballistic thermaliza- iments, the detected Bragg signal is directly proportional to tion of the low momentum regime, since the formation of the the Fourier transform of the two-point density-density correla- typical, thermal Rayleigh-Jeans divergence nq,t ∼ 1/|q| is only tion function [302–305] or dynamical structure factor S q,t,ω = 60 R dδdxei(qx−ωδ)h{n(t + δ/2, x), n(t − δ/2, 0)}i. In the Luttinger atom setups could provide a physical platform to explore such framework in the Keldysh formalism this translates into into physics. S q,t,ω = −hρc(−q, t, −ω)ρc(q, t, ω)i. Explicit evaluation of the Even more ambitiously, such driven open quantum systems structure factor yields hold the potential for giving rise to truly new situations. One     example is presented by the question on the analogue of the (2nq,t+1)|q|K ω−q ˜ (ω→−ω) Kosterlitz-Thouless transition in two dimensional driven open S q,t,ω = πσR f σR + . (293) q,t q,t systems, which technically requires to analyze a KPZ equation Here, f˜(x) = 1/(1 + x2) is a dimensionless scaling function, with a compact variable, allowing for the presence of vortex de- which is centered at x = 0 and has unit width. As a consequence, fects. Keldysh field theory offers the flexibility to address such the dynamic structure factor is peaked at the mass shell ω = u|q| questions. R Quantum regime — The quantum dynamical field theory and has a typical width δω = σq,t, which reveals the scaling of the self-energies. framework developed here also allows us to address prob- The scaling of the distribution function and the time depen- lems where the limit of classical dynamical field theories (see Sec.IIC) is not applicable. Here the challenge is to identify dent crossover xth(t) does not necessitate dynamical (frequency resolved) information, and can be obtained from the static struc- traces of non-equilibrium quantum effects at macroscopic dis- ture factor alone. The latter represents the equal time density- tances. One instance of such a phenomenon has been established density correlation function. It is determined as the frequency recently in terms of a driven analogue of quantum critical be- integrated dynamic structure factor havior in a system with a dark state, which is decoupled from Z noise [27]. It remains to be seen whether a full classification of |q|K driven Markovian (quantum) criticality can be achieved, com- S q,t = S q,t,ω = π (2nq,t + 1) (294) ω plementing the seminal analysis of equilibrium classical criti- cality by Hohenberg and Halperin [1]. Beyond bosonic systems, and scales quadratically in the momentum in the heating regime this also includes fermionic systems, which have been shown to while approaching a constant in the thermalized regime, thereby exhibit scaling [57, 59, 60], but so far were analyzed at a Gaus- revealing the crossover between these two regimes. sian fixed point only. Certainly, these efforts will necessitate a more stringent understanding of conservation laws and dynami- VI. CONCLUSIONS AND OUTLOOK cal slow modes and their origin in the symmetries of the Keldysh functional integral. Another interesting direction in the context of collective quan- We have reviewed here recent progress in the theory of driven tum effects is provided by topologically non-trivial dark states open quantum systems, which are at the interface of quantum op- in atomic fermion systems [66, 308]. Here, fundamental ques- tics, many-body physics, and statistical mechanics. In particular, tions concern the possibility of topological responses for mixed we have developed a quantum field theoretical approach based states in systems, where the dynamics strongly breaks ergodic- on the Keldysh functional integral for open systems, which un- ity [205, 309]. The Keldysh approach is a proper tool to study derlies these advances. The formalism developed here paves the such problems systematically, via coupling the fermions to ar- way for many future applications and discoveries. We may struc- tificial gauge fields and the analysis of dissipative topological ture these into three groups of topics. field theories. Semi-classical regime — A key challenge here is to sharpen Dynamics — Addressing the time evolution of open sys- the contrast between equilibrium and genuine non-equilibrium tems adds a new twist to the question of thermalization [29, physics. One way of succeeding in this respect is to uncover 34, 310, 311] or, more generally, equilibration of quantum sys- new links of driven open quantum systems to paradigmatic sit- tems. This is also a necessary step to achieve a realistic de- uations in non-equilibrium (classical) statistical mechanics. We scription of broad classes of experiments, in particular, with ul- have discussed one such connection between the phase dynamics tracold atoms, as well as certain solid states systems in pump- of driven Bose condensates and surface growth in Sec.IV. Fur- probe setups [312, 313]. While first instances of universal be- ther instances have been pointed out in the literature recently: havior have been identified in low dimension in the short [40] for example, driven Rydberg gases can be brought into regimes and long [42, 44] dynamics, it is certainly fair to say that general where they connect to the physics of glasses [46] or the univer- principles so far remain elusive. sality class of directed percolation [47], and the late stages of heating of atoms in optical lattices show slow decoherence dy- namics described by non-linear diffusion, reminiscent of glasses as well [42, 43, 306]. Connections of such settings to field theo- VII. ACKNOWLEDGEMENTS retical non-linear reaction-diffusion models [102, 307] still await their exploration. A new kind of equilibrium to non-equilibrium The authors thank E. Altman, L. Chen, A. Chiocchetta, E. phase transition may be expected in three dimensional driven G. Dalla Torre, A. Gambassi, L. He, S. D. Huber, S. E. Huber, bosonic systems on the basis of the phase diagram for surface M. Lukin, J. Marino, S. Sachdev, P. Strack, U. Tauber,¨ and J. roughening, and it is intriguing to investigate whether ultracold Toner for collaboration on projects discussed in this review. We 61 are also grateful for inspiring and useful discussions with many A(x, x0) = ∇2δ(x− x0). By straightforward differentiation we find people at the KITP workshop “Many-Body Physics with Light.” δ2 Z δ Z In particular, we thank A. Altland, B. Altshuler, I. Carusotto, ∗ ∇2 ∗ ∇2 − 0 ∗ 0 φ (y) φ(y) = ∗ φ (y) δ(y x ) A. Daley, A. Gorshkov, M. Hafezi, M. Hartmann, R. Fazio, M. δφ (x)δφ(x ) y δφ (x) y Fleischhauer, A. Imamoglu, J. Koch, I. Lerner, I. Lesanovsky, P. δ = ∇2φ(x0) Rabl, A. Rosch, G. Shlyapnikov, M. Szymanska, J. Taylor and δφ∗(x) H. Tureci¨ for critical remarks and feedback on the manuscript. = ∇2δ(x − x0). We are indebted to J. Koch for proofreading the manuscript. We acknowledge support by the Austrian Science Fund (FWF) (A4) through the START grant Y 581-N16 and the German Research Finally, we note that the chain rule applies also in the case of Foundation (DFG) via ZUK 64 and through the Institutional functional differentiation. This last ingredient is required to per- Strategy of the University of Cologne within the German Ex- form the second variational derivatives in Eq. (34) in order to cellence Initiative (ZUK 81). S. D. also acknowledges support obtain the Green’s functions from the generating functional. by the Kavli Institute for Theoretical Physics, via the National Science Foundation under Grant No. NSF PHY11-25915, and L. M. S. support from the European Research Council through Appendix B: Gaussian functional integration Synergy Grant UQUAM. Here we summarize a number of useful formulas for Gaus- sian functional integration, which can be found in any textbook on field theory (see, e.g., Refs. [169, 171, 314]). The basic for- Appendix A: Functional differentiation d mula for real fields χ(x) = (χ1(x), . . . , χn(x)), x ∈ R (in the ap- plications discussed in the main text, the components of χ(x) are In this appendix, we give a brief account of functional or vari- fields on the closed time path or — after performing the Keldysh ational differentiation, following the presentation in Ref. [219]. rotation Eq. (32) — classical and quantum fields, and x collects The most transparent way to introduce the basic relations of temporal and spatial coordinates, x = (t, x)), is given by functional differentiation is by drawing an analogy to the famil- Z R R i χ(x)T D(x,x0)χ(x0)+i j(x)T χ(x) iar formulas of partial differentiation. Indeed, we can consider a D[χ] e 2 x,x0 x d field φ(x) with x ∈ R to be the continuum limit of a function φi R d −1/2 − i j(x)T D−1(x,x0) j(x0) defined on lattice points i ∈ Z . With this identification, relations = (det D) e 2 x,x0 , (B1) involving partial derivatives can be translated into corresponding R R d ones for functional derivatives simply by replacing discrete in- where = d x, j(x) = ( j1(x),..., jn(x)), and the inverse of P R R d x 0 dices i by continuous ones x, sums i by integrals x = d x, the integral kernel D(x, x ) is defined by means of the relation and partial derivatives ∂/∂φi by functional derivatives δ/δφ(x). Z Following this prescription, the basic formula ∂φi/∂φ j = δi j D(x, ξ)D−1(ξ, x0) = δ(x − x0)1. (B2) leads to ξ The above formula is valid for invertible symmetric kernels, δφ(x) − 0 D(x, x0) = D(x0, x)T , with positive semi-definite imaginary part. 0 = δ(x x ). (A1) δφ(x ) If D(x, x0) is not symmetric, on the LHS of Eq. (B1) it can be replaced by the symmetrized kernel When we are working with complex fields, φ and φ∗ are usually 1 h i treated as independent variables. Then, the generalization of D˜ (x, x0) = D(x, x0) + D(x0, x)T , (B3) 2   X  X leaving the value of the exponent invariant. Then, Eq. (B1) can ∂  ∗   φ Ai jφ j = Ak jφ j (A2) again be applied. ∂φ∗  i  k i, j j In case that the integral kernel is translaitionally invariant, i.e., it satisfies D(x, x0) = D(x − x0), it is advantageous to work in reads Fourier space. Then, Eq. (B1) can be written as Z Z Z i R R δ χ(−q)T D(q)χ(q)+i j(−q)T χ(q) ∗ 0 0 0 0 D[χ] e 2 q q ∗ φ (y)A(y, y )φ(y ) = A(x, y )φ(y ). (A3) δφ (x) 0 0 y,y y R − / − i j(−q)T D−1(q) j(q) = (det D) 1 2 e 2 q , (B4) 0 The expression A(x, x ) is the continuum limit of a matrix Ai j. R R ≡ dq In particular, it can be a differential operator. As an example, where we are using the shorthand notation q (2π)d . The we calculate the second functional derivative for the case that Fourier transformation turns the convolution in Eq. (B3) into a 62   multiplication, showing that D−1(q) can be obtained by inversion where we are assuming that −i D − D† is positive semi- of the matrix D(q), definite, but D(x, x0) does not have to be symmetric. The cor- responding formula for the case of a translationally invariant in- −1 D(q)D (q) = 1. (B5) tegral kernel reads (note the different signs of q in comparison to Eq. (B4)) As above, in Eq. (B4) we are assuming D(q) = D(−q)T . If this is not the case, D(q) has to be replaced by the symmetrized version 1 h i D˜ (q) = D(q) + D(−q)T , (B6) 2

Z R R before the formula can be applied. ∗ i ψ†(q)D(q)ψ(q)+i (φ†(q)ψ(q)+ψ†(q)χ(q)) D[ψ, ψ ] e q q For the case of complex fields ψ(x) = (ψ1(x), . . . , ψn(x)) (and R with corresponding definitions of φ(x) and χ(x)), Eq. (B1) is re- −1 −i φ†(q)D−1(q)χ(q) placed by = (det D) e q . (B8) Z R † 0 0 R † † ∗ i 0 ψ (x)D(x,x )ψ(x )+i (φ (x)ψ(x)+ψ (x)χ(x)) D[ψ, ψ ] e x,x x R −1 −i φ†(x)D−1(x,x0)χ(x) = (det D) e x,x0 , (B7)

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