PHYSICAL REVIEW X 5, 041005 (2015)

Far-from-Equilibrium Field Theory of Many-Body Quantum Spin Systems: Prethermalization and Relaxation of Spin Spiral States in Three Dimensions

Mehrtash Babadi,1 Eugene Demler,2 and Michael Knap2,3,4 1Institute for Quantum Information and Matter, Caltech, Pasadena, California 91125, USA 2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 3ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA 4Physik Department, Walter Schottky Institut, and Institute for Advanced Study, Technische Universität München, 85748 Garching, Germany (Received 26 April 2015; published 12 October 2015) We study theoretically the far-from-equilibrium relaxation dynamics of spin spiral states in the three- dimensional isotropic Heisenberg model. The investigated problem serves as an archetype for under- standing quantum dynamics of isolated many-body systems in the vicinity of a spontaneously broken continuous symmetry. We present a field-theoretical formalism that systematically improves on the mean field for describing the real-time quantum dynamics of generic spin-1=2 systems. This is achieved by mapping spins to Majorana fermions followed by a 1=N expansion of the resulting two-particle-irreducible effective action. Our analysis reveals rich fluctuation-induced relaxation dynamics in the unitary evolution of spin spiral states. In particular, we find the sudden appearance of long-lived prethermalized plateaus with diverging lifetimes as the spiral winding is tuned toward the thermodynamically stable ferro- or antiferromagnetic phases. The emerging prethermalized states are characterized by different bosonic modes being thermally populated at different effective temperatures and by a hierarchical relaxation process reminiscent of glassy systems. Spin-spin correlators found by solving the nonequilibrium Bethe- Salpeter equation provide further insight into the dynamic formation of correlations, the fate of unstable collective modes, and the emergence of fluctuation-dissipation relations. Our predictions can be verified experimentally using recent realizations of spin spiral states with ultracold atoms in a quantum microscope [S. Hild et al., Phys. Rev. Lett. 113, 147205 (2014)].

DOI: 10.1103/PhysRevX.5.041005 Subject Areas: Atomic and Molecular Physics, , Nonlinear Dynamics

I. INTRODUCTION Recent theoretical and experimental investigations of The equilibration of isolated quantum many-body sys- strongly correlated systems, however, suggest significant tems is a fundamental and ubiquitous question in physics. It refinements to this dichotomy. For instance, certain systems “ plays a central role in understanding a broad range of can be trapped for long times in quasistationary prether- ” phenomena, including the dynamics of the early Universe malized states with properties strikingly different from [1], the evolution of neutron stars [2], pump-probe experi- true thermal equilibrium [10]. Examples include nearly – ments in condensed-matter systems [3], and the operation integrable one-dimensional systems [11 15] and systems of devices [4]. The simplest perspective on with vastly different microscopic energy scales in which the problem is to recognize a dichotomy between ergodic slow dynamics results from the slow modes providing and nonergodic systems. The former exhibit fast relaxation configurational disorder and thereby localizing the fast – to local equilibrium states occurring at microscopic time modes [16 18]. Even subtler examples of slow dynamics scales, followed by a slower relaxation process to global include the Griffiths of interacting disordered sys- thermal equilibrium described by classical hydrodynamics tems [19,20] and translationally invariant systems in higher of a few conserved quantities [5–7]. In contrast, nonergodic dimensions with emergent slow degrees of freedom – systems possess an extensive set of conservation laws that [10,21 29]. prevent their thermalization [8,9]. In this work, we discuss the emergence of slow dynamics and prethermalization in translationally invariant spin systems that possess continuous symmetries. In higher dimensions, these systems can exhibit thermodynamically Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- stable symmetry-broken phases along with gapless bution of this work must maintain attribution to the author(s) and Goldstone modes. Here, we show that the relaxation the published article’s title, journal citation, and DOI. dynamics of low-energy initial states that allow symmetry

2160-3308=15=5(4)=041005(18) 041005-1 Published by the American Physical Society MEHRTASH BABADI, EUGENE DEMLER, AND MICHAEL KNAP PHYS. REV. X 5, 041005 (2015) breaking upon thermalization is remarkably different from different mean-field solutions of the classical Heisenberg the relaxation of high-energy states, thereby establishing a model, all of which are thermodynamically unstable with connection between aspects of equilibrium and nonequili- the exception of Q ¼ 0; π. The fluctuation-induced destruc- brium phenomena in these systems. In particular, the slow tion of the initial order and the emergence of thermody- Goldstone modes in the former case result in the emergence namically stable ordered or disordered phases at longer of long-lived nonthermal states with a hierarchical relax- times is another question we address here. Last, spin spiral ation dynamics that closely resembles aging in systems states have been recently realized in one and two dimen- with quenched disorder. A nonperturbative treatment and sions using ultracold atoms in a quantum gas microscope beyond mean-field corrections are both found to be crucial [33,34]. An extension of these experiments to three for describing the relaxation process. dimensions makes a direct experimental scrutiny of our We specifically study the dynamics of the three-dimen- predictions possible. sional (3D) isotropic Heisenberg model initially prepared in Our results indicate that spiral states tuned toward FM or a spiral state; see Fig. 1(a). Our choice of spin spiral states AFM states exhibit a slow hierarchical relaxation and can is motivated by the following considerations. First, the come arbitrarily close to a dynamical arrest; see Figs. 1(b) winding of a spiral Q serves as a tuning parameter for and 1(c). Surprisingly, the relaxation dynamics is neither the energy density of the state. The full spectrum of the compatible with the trivial relaxation to local thermal Heisenberg model is traversed from ferromagnetic (FM) to equilibrium and slow hydrodynamic evolution, since the Néel antiferromagnetic (AFM) states upon sweeping Q spin current is not conserved, nor with the linearized from 0 to π, respectively. In light of the eigenstate thermal- dynamics of the collective modes, which predicts expo- ization hypothesis (ETH) [30–32], the energy density of the nentially growing out-of-plane instabilities. In fact, we find initial state fully determines the fate of all local observables the instabilities to self-regulate and slow down signifi- at late times in ergodic systems. One of the main objectives cantly. As we elaborate in the following sections, the of this work is to understand the route toward thermal- physical phenomena discussed here are expected to gen- ization of these states. Second, spiral states represent eralize to a broad range of models that exhibit a finite- temperature between a disordered and a symmetry-broken phase. The relaxation of the Néel spin spiral state with Q ¼ π has been previously studied in the 1D Heisenberg model [33,35–37]. In contrast to the 3D case studied in the present work, the 1D Heisenberg model does not exhibit a symmetry-broken thermal phase and, in turn, cannot exhibit the type of prethermalization we discuss here. More recently, the dynamics of the Néel state in the Fermi-Hubbard model on an infinite-dimensional Bethe lattice has been investigated [38]; however, the approach to the steady state could not be studied due to the small effective exchange interaction. From a technical perspective, our investigation of the nonequilibrium dynamics of spiral states has been enabled by developing a nonperturbative field-theoretic formalism applicable to generic spin-1=2 systems for arbitrary initial states and geometries, which we refer to as the “spin-2PI” formalism. This is achieved using a Majorana fermion representation of spin-1=2 operators [39,40], enlargement of the spin-coordination number by a replica-symmetric extension, and ultimately a systematic 1=N fluctuation FIG. 1. Relaxation of spin spiral states in the 3D isotropic expansion of the real-time two-particle-irreducible (2PI) Heisenberg model. (a) The system is prepared in a spin spiral effective action [41,42]. xy Q Q; Q; Q state in the plane with the winding ¼ð Þ as tuning The recent rapid progress in the phenomenology of parameter. The figure illustrates the case Q ¼ π=2. (b) The real- far-from-equilibrium quantum dynamics and its broad of the transverse magnetization M⊥ for three different Q as indicated in the plot. For Q ¼ 7π=8, a hierarchical applications has been enabled by similar nonperturbative relaxation process emerges with a nonthermal plateau at inter- functional techniques. Examples include extensive studies mediate times. The time scale is switched to logarithmic at tJ ¼ 5 of the OðNÞ model in nonequilibrium [43–45]; thermal- for better visibility. (c) A global view of the spiral dynamics. ization, prethermalization, and nonthermal fixed points Nonthermal plateaus appear near Q ∼ 0; π. [10,22,46–48]; particle production; reheating and defect

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dS X generation in inflationary universe models [49–53]; and j α αβ β φj × Sj; φ V S : 2 dynamics of ultracold fermionic and bosonic dt ¼ j ¼ jk k ð Þ k∈L [54–56]. The present work is the first to utilize this powerful technique to study the far-from-equilibrium In case of quantum spins, the Bloch equation only describes dynamics of interacting quantum spin systems. ˆ the evolution of the spin expectation values hSi to the extent Understanding the emergence of slow dynamics near Sˆ Sˆ ≈ Sˆ Sˆ thermodynamic phase transitions has implications reaching of which the mean-field ansatz h j ki h jih ki is valid. far beyond the domain of condensed-matter physics. For The latter, however, is only justified for lattices with large instance, studies of nonequilibrium quantum fields in the coordination number, high spin particles, or in the presence context of inflation and early Universe dynamics have of a high-temperature bath. The crucial role of quantum 1=2 suggested that the slowing down of quantum evolution near fluctuations in the dynamics of isolated spin- systems in phase transitions is a plausible explanation for the large finite-dimensional lattices is beyond the reach of semi- number of light particles and broken symmetries in the classical methods and demands a more careful treatment. observable Universe [57]. Given that the experimental Here, we propose a formalism for transcending verification of theories about early Universe phenomena the mean-field approximation for spin evolution by a are typically rather indirect, experiments with synthetic systematic inclusion of quantum corrections. This is many-body systems that allow precise monitoring of real- achieved using functional methods and a variant of the N time dynamics close to phase transitions could play an large- expansion technique. As a first step, we con- important role in elucidating the emergence of slow struct an auxiliary model in which each spin is replicated N N2 evolution. The dynamics of various interacting spin sys- times, and each bond is promoted to bonds between tems have been already investigated in experiments with the replicas, with equal weight but with an overall scale 1=N synthetic quantum matter, including domain formation in factor of . The Hamiltonian of the auxiliary model is spinor condensates [58,59], the precise measurement of the written as evolution of spin flips in the ground state of 1D lattice spin 1 X X 1 XN XN systems [60–63], quantum coherences in long-range mod- ˆ αβ ˆ β;σ0 ˆ α;σ HN ¼ V S S : ð3Þ els [64], and the relaxation dynamics of spin spiral states in 2 jk N k j j∈L k∈L σ0¼1 σ¼1 1D and 2D Heisenberg models [33,34]. The experimental observation of the dynamical phenomena discussed here The initial state jΨ0i is also subsequently promoted to an are thus expected to be within close reach. N This paper is organized as follows: In Sec. II,we uncorrelated product in the replica space ⨂ jΨ0iσ. The σ¼1 introduce the spin-2PI formalism, a technique we develop N 1 to study the dynamics of interacting spin systems. original problem is recovered by setting ¼ . We refer to the sum appearing in the parentheses in Eq. (3) as the Complementary technical details are presented in the φˆ Appendix. We discuss the relaxation of spin spiral states exchange-field operator j, which plays the role of an in Sec. III. The phenomenon of dynamical slowing down effective fluctuating with which the spins N and arrest will be presented Sec. III A, the long-time interact. The described large- construction effectively z Nz thermalization in Sec. III B, and the dynamic formation increases the coordination number of each spin to , thereby suppressing the fluctuations of φˆ j according to the of correlations and instabilities in Sec. III C. We conclude pffiffiffiffiffiffi our findings in Sec. IV. law of large numbers φˆ j ¼ φc;j þ Oð1= NzÞ, where φc;j ≡ hφji is the mean exchange field. In the limit of N II. THE SPIN-2PI FORMALISM infinite , the exchange-field operator becomes effectively classical such that mean-field dynamics of the original Consider a generic Hamiltonian describing the pairwise model Hˆ emerges as the asymptotically exact description of interaction between localized spin degrees of freedom on a ˆ the dynamics in limN→∞HN. For large but finite N, the given lattice L: fluctuations of φˆ are small but not negligible and can be 1 X systematically incorporated into the dynamics order by Hˆ VαβSˆ αSˆ β; 1=N ¼ 2 jk j k ð1Þ order in . This program can be carried out within the j;k∈L functional method of 2PI effective actions. Crucially, truncating the expansion at a finite order in 1=N and V j k where is an arbitrary interaction, and denote lattice taking the limit N → 1 yields nonperturbative and con- ˆ α sites, and fS g are spin-1=2 operators. Summation over the serving approximations for the spin dynamics. We refer to repeated spin indices is assumed. Had Sˆ been classical this method as the spin-2PI formalism, which is illustrated angular momentum variables, the Hamiltonian dynamics of schematically in Fig. 2. In brief, spins precess about a self- the system would be governed by the (nonlinear) Bloch consistently determined exchange mean field, and quantum equation: spin fluctuations are mediated by the local and nonlocal

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1 X XN L η ηα;σi∂ ηα;σ ½ ¼2 j t j j∈L σ¼1 X XN 1 αβ V η η α;σ1 η η β;σ2 : þ 8N jk ð j × jÞ ð k × kÞ j;k∈L σ1;σ2¼1 ð5Þ

The exchange field is introduced by a Hubbard- Stratonovich decoupling of the quartic term using a real- vector boson φj on each lattice site. The nonequilibrium exchange mean field φc, exchange-field-fluctuation propa- gator D, and Majorana propagator G are introduced as

φcð1Þ¼hφˆ ð1Þi; FIG. 2. The spin-2PI formalism illustrated. The spins (green iDð1; 2Þ¼hTC½φˆ ð1Þφˆ ð2Þi − φcð1Þφcð2Þ; arrows) precess about a fluctuating exchange field (uncertain blue arrows). The quantum fluctuations of the exchange field are iGð1; 2Þ¼hTC½ηð1Þηð2Þi: ð6Þ mediated by real-vector bosons (wiggly lines) and are suppressed 1=N by a factor of , permitting a systematic expansion. The integer variables are shorthand for the bundle of lattice site, contour time, spin, and replica index. According to exchange of a real-vector boson whose propagator is Eq. (4), the local spin expectation value is proportional to suppressed by a factor of 1=N. the fermion tadpole: In the remainder of this section, we briefly outline the field-theoretical developments that underlie the spin-2PI 1 Sˆ α t ε Gβγ tþ;t : formalism. Complementary technical details are given in h j ð Þi ¼ 2 αβγ jj ð Þ ð7Þ the Appendix. A path integral for the spin-1=2 operators is constructed using a representation invoking Majorana We obtain the real-time evolution equations for G, D, and fermions [39,40]: φc using the 2PI effective action formalism [41]. The effective action Γ½G; D; φc is found by sourcing G, D, and φc and performing Legendre transformations: ˆ i Sj ¼ − ηj × ηj: ð4Þ 2 1 1 1 Γ G; D; φ G−1 G−1G − D−1 ½ c¼2 tr ln þ 2 tr½ 0 2 tr ln η1; η2; η3 1 The Majorana operators at each site f j j j g satisfy the − D−1D Γ G; D; φ ; μ ν μν tr½ 0 þ int½ c ð8aÞ Clifford algebra fηj ; ηkg¼δjkδ , from which the SUð2Þ 2 ˆ α ˆ β ˆ γ algebra for spins ½Sj ; S ¼iδjkεαβγS and the Casimir k j 1 i condition S2 ¼ 3=4 follow. The latter ensures a faithful Γ G; D; φ − M φ G φ D−1φ Γ G; D : j int½ c¼ 2 tr½ ½ c þ2 c 0 c þ 2½ spin-1=2 representation without introducing any unphys- ical states and obviates the necessity of using constraint ð8bÞ gauge fields in contrast to the Schwinger slave-particle approach [65]; see the Appendices of Ref. [66] for a The bare Majorana and exchange propagators are given as G−1 1; 2 i∂ δ 1; 2 D−1 1; 2 N V−1 δ t ;t detailed treatment of the Majorana representation for spin- 0 ð Þ¼ t1 ð Þ and 0 ð Þ¼ ð Þ ð 1 2Þ, 1=2 operators. respectively. M½φc is the leading-order (LO) self-energy Replacing the spin operators in Hˆ using Eq. (4), the [see Eq. (A9)]. The evolution equations follow from Hamiltonian is mapped to that of a many-body system of making Γ stationary with respect to G, D, and φc; see Majorana fermions with quartic interactions. The large-N Eqs. (A7a)–(A7c). program can be identically followed by replicating the slave Save for Γ2½G; D, the rest of the terms appearing in Majorana particles and assigning a replica index to each. Γ½φc; G; D scale as OðNÞ and together comprise the LO We proceed by constructing a path integral for the approximation. The next-to-leading-order (NLO) correc- Majorana fermions using fermionic coherent states on tions and beyond are represented by Γ2½G; D, which the closed-time-path (CTP) Schwinger-Keldysh contour. formally corresponds to the sum of 2PI vacuum diagrams The Lagrangian is given as arising from the cubic interaction vertex:

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and functional methods. As discussed before, these tools ð9Þ significantly simplify and streamline the calculation of higher-order corrections. Furthermore, there is no preferred axis for spin quantization in the spin-2PI formalism, 1=N Γ allowing us to study magnetically ordered and disordered The expansion of int to the next-to-next-to-leading order (NNLO) is diagrammatically given as states in a unified way.

Γ III. RELAXATION OF SPIN SPIRAL STATES IN THE 3D HEISENBERG MODEL In this section, we investigate the unitary evolution of the ð10Þ spin spiral state on a 3D cubic lattice P ˆ z −i Q·RjSj We have used the stationarity condition, Eq. (A7c),toomit jspðQÞi ¼ e j ⨂ j →ij; ð11Þ 3 φc in favor of G in the LO interaction terms. The Feynman j∈Z diagram rules are given in Appendix A2. Truncating the 1=N expansion of Γ at a finite order and under the isotropic Heisenberg Hamiltonian Hˆ ¼ N 1 P ˆ ˆ setting ¼ yields systematic improvements of the mean- −J ij Si · Sj using the spin-2PI formalism developed field spin dynamics. The ensuing approximate theories are h i in the previous section. Here, j →ij denotes the x-polarized self-consistent and nonperturbative by construction and state on lattice site j. The spiral is prepared in the xy plane respect the conservation laws associated with the global with a winding wave vector Q. We assume ferromagnetic symmetries of the microscopic action, such as magnetiza- couplings J>0 for concreteness, even though the sign of tion and energy. The latter is crucial for the long-time the J does not affect the unitary evolution due to the time- stability of the nonequilibrium dynamics. reversal symmetry of the Heisenberg model. The Bloch equation is recovered upon truncating Γ at the The spiral state jspðQÞi is a simultaneous eigenstate of LO level; see Appendix A3. Truncations at NLO and Sˆ Q ≡ Tˆ Rˆ Q a x; y; z Tˆ Rˆ Q beyond give rise to memory effects due to the dynamical að Þ a zð aÞ, ¼ , where a and zð aÞ a fluctuations of the exchange field and result in a two-time denote the translation by one lattice site along the axis Q z Kadanoff-Baym integro-differential equation instead of the and rotation by angle a about the axis, respectively. The mean-field Bloch equation; see Eqs. (A13a)–(A14b). translation and rotation symmetries of the isotropic ˆ ˆ Finally, higher-order correlators, in particular, the spin- Heisenberg model imply ½H; Sa¼0, so that the spiral ˆ ˆ ˆ ˆ Q Sˆ Q spin correlator iχð1; 2Þ ≡ hTC½Sð1ÞSð2Þi − hSð1ÞihSð2Þi, state jspð Þi remains a simultaneous eigenstate of að Þ can be reconstructed with the knowledge of G and D by at all times in the course of unitary evolution. As a result, ˆ z solving the nonequilibrium Bethe-Salpeter integral equa- the out-of-plane magnetization hSjðtÞi vanishes identically, tion on the Schwinger-Keldysh contour; see Appendix A3. and the spiral magnetic order with the initial winding Q We remark that in systems with large spin-coordination persists at all times. The transverse magnetization number z, fluctuations of the exchange field are inherently X 1 x y suppressed and the expansion parameter is more accurately M Q;t ≡ e−iQ·Rj Sˆ t i Sˆ t 1= zN N ⊥ð Þ L3 ½h j ð Þi þ h j ð Þi ð12Þ identified with ð Þ. Therefore, the large- expansion of j∈L the spin-2PI effective action in models with z ≳ 1 is expected to be controlled and rapidly converging, even is the only degree of freedom at the level of single spin after taking the limit N → 1. Studies of the O N model ð Þ observables. Also, M⊥ðk;tÞ¼0 for k ≠ Q. We remark show that the most important correction to the mean-field that even though the magnetization dynamics is signifi- “ (LO) approximation is captured by the NLO fluctuation- cantly constrained at the level of single spin observables by ” exchange diagram, along with negligible quantitative symmetries, arbitrary spin correlations are allowed to form corrections from the subleading terms [67,68]. in the course of evolution, including both in- and out-of- 1=N The replica-based expansion proposed here differs plane spin correlations at arbitrary wave vectors. 1=S from the usual semiclassical expansion in significant A simplifying aspect of the present problem is that ways even though they improve upon the same mean-field the apparently broken translation symmetry of the limit. For instance, the replicated Fock space of a single spiral state can be restoredP using an “unwinding” unitary ˆ z spin is reducible and has many more states compared to a i Q·RjS Uˆ ≡ e j j pure spin-N=2 representation. A technical advantage of our transformation Q under which the x approach is that it preserves the underlying spin-1=2 spiral state transforms into a uniform -polarized product ~ ˆ degrees of freedom, which in conjunction to the state jΨ0i¼UQjspðQÞi ¼⊗j j →ij. The unwinding Majorana representation leads to the familiar diagrammatic transformation, however, transforms the Hamiltonian

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ˆ ~ ˆ ˆ ˆ † M H → H ¼ UQHUQ to an anisotropic Heisenberg model The evolution of ⊥ is shown in Fig. 1(b) for several Q Q ∈ 0; π with a Dzyaloshinskii-Moriya term: choices of , along with a global surface plot for ½ tJ ∈ 0; 30 X and ½ in Fig. 1(c). The stationarity of the FM ~ ˆ z ˆ z ˆ x ˆ x ˆ y ˆ y state (Q ¼ 0) and the rapid settlement of the Néel state H ¼ −J ½S S þ cos Q · ðRj − RkÞðSj Sk þ S S Þ j k j k Q π hj;ki ( ¼ ) to a steady state with finite staggered magnetiza- tion is observed. − Q R − R Sˆ xSˆ y − Sˆ ySˆ x : sin · ð j kÞð j k j kÞ ð13Þ Short-time dephasing dynamics.—For all Q, the first stage of dynamics is a short-time relaxation of the form 2 The translation invariance of the initial state in the spiral M⊥ ≈ 1=2 − νQt arising from the dephasing between frame significantly simplifies the structure of the spin-2PI the eigenstates that overlap with the spiral. A straight- equations: G and Σ become local in the real space while D ˆ forward calculation using the short-time expansion hSðtÞi ¼ depends only on the distance between the sites. These Sˆ it H;ˆ Sˆ it 2=2 H;ˆ H;ˆ Sˆ simplifications hold for arbitrary truncations of Γ . h i0 þ h½ i0 þ½ð Þ h½ ½ i0 þ gives int ν 3 J2 Q − 1 2 ν Additionally, the bosonic self-energy Π becomes local in Q ¼ 8 ðcos Þ .Thevaluesof Q extracted from the real space at the NLO truncation. The magnetization is the numerically obtained M⊥ are in agreement with the exact nonvanishing only along the x direction in the spiral frame result; see Fig. 6. The second stage of relaxation dynamics due to the symmetry considerations mentioned earlier. The depends on the winding of spiral and is either directly quantities calculated in the spiral frame can be readily thermalizing or exhibits long-lived prethermalized states transformed to the lab frame using appropriate rotations. In preceding the true thermalization. We discuss these cases 23;> particular, Eq. (7) gives M⊥ðQ;tÞ¼ð1=2ÞG ðt; tÞ with separately. G calculated in the spiral frame. We choose the winding to Spiral states with Q ∼ π=2.—Spin spiral states with Q ∼ π=2 be along the diagonal direction Q ¼ðQ; Q; QÞ hereafter have a high-energy density with respect to both the FM and refer to the spiral winding with the single scalar and the AFM states. Thus, Q ∼ π=2 spiral states overlap Q ∈ ½0; π. with a large number of eigenstates of the Heisenberg At the LO level, the spin dynamics is governed by the model. Such a broad superposition of states lead to fast Bloch equation, Eq. (2). The exchange mean field φc is dephasing, which is found to be within a few exchange times. Our results indicate a rapid onset of exponential parallel to the local magnetization at all lattice sites in a −γ t spiral state, implying the absence of any dynamics. In other decay M⊥ ∼ e Q with the fastest rate occurring at words, the spiral states are fixed points of the mean-field Q ¼ 0.55ð1Þπ ∼ π=2. dynamics for all windings Q. Spiral states with Q ∼ 0 and Q ∼ π.—A complex multi- Going beyond the LO dynamics and including the scale relaxation scenario emerges for spirals with windings exchange-field fluctuations by taking into account the tuned to Q ∼ 0 and Q ∼ π, lying close to FM and AFM NLO corrections, the spiral state exhibits an intriguing magnetic orders, respectively. The transverse magnetiza- fluctuation-induced relaxation dynamics. States with differ- tion exhibits an intermediate plateau for these initial states ent windings have different energy densities, along with which appears continuously upon tuning Q; see Fig. 1. The different strengths of in-plane and out-of-plane spin fluc- plot of M⊥ shown in Fig. 1(b) for Q ¼ 7π=8 displays the tuations, and are found to relax in strikingly different ways. intermediate plateau followed by relaxation at later times. As we discuss below, these factors conspire to give rise to a As Q is tuned closer toward 0 or π, the lifetime of the nontrivial hierarchical relaxation scenario for spiral states plateau increases abruptly and the magnetization comes to a lying close to thermodynamically stable orders, exhibiting dynamical arrest. We investigate the nature of such long- prethermalization [10], and dynamical arrest resembling lived plateaus in more detail in the following sections. glassy systems [69]. B. Prethermalization vs thermalization A. Relaxation and dynamical arrest Because of the nonintegrability of the 3D Heisenberg of the transverse magnetization model, the energy distribution of spin fluctuations is The spiral state for Q ¼ 0 is a fully polarized FM expected to approach a thermal population in the long- eigenstate of the Heisenberg model and is therefore sta- time limit, according to the ETH [30–32]. We investigate tionary. The Q ¼ π spiral, on the other hand, corresponds the nature of steady states emerging in the dynamics by to an uncorrelated Néel state which in three dimensions has calculating the spin-spin correlation and response func- a large overlap with the correlated AFM state lying at the tions, corresponding to the Keldysh (K) and retarded (R) upper end of the spectrum of the FM Heisenberg model. As components of the CTP spin-spin correlator χðt; t0Þ,by a result, the system is expected to achieve a steady state solving the nonequilibrium Bethe-Salpeter equation (see marked with a finite staggered magnetization after a short Appendix A3). At thermal equilibrium, these quantities are course of dephasing dynamics, provided that the generated related via the bosonic fluctuation-dissipation relation effective temperature is below the ordering temperature. (FDR):

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K R iχ ðωÞ¼−2 cothðω=2kBTÞIm½χ ðωÞ; ð14Þ Thermalization of spiral states with Q ∼ π=2.—For a range of spiral wave vectors π=4 ≲ Q ≲ 3π=4, the steady- where T is the effective temperature. Here, ω refers to the state temperatures obtained from all bosonic modes, i.e., Fourier frequency in the time difference t − t0 in the steady local and nonlocal in- and out-of-plane spin and exchange- state achieved at long times. Likewise, one can define an field fluctuations, agree with each other, suggesting the effective temperature for the exchange-field fluctuations complete thermalization of the system and in accordance K R using the bosonic FDR between D and D . We refer to the with the ETH. temperatures obtained from local spin χ and exchange Spirals with Q ¼ π=2 flow to an infinite-temperature D T T fluctuations as spin and fluct, respectively. thermal state, which is understood from the duality The effective temperatures obtained from the FDR in the Q → π − Q, J → −J present in the classical Heisenberg steady state are shown in Fig. 3(a). For all spiral windings model. This classical duality extends to the quantum Q , we find that FDR is satisfied to an excellent degree for Heisenberg model in the high-temperature regime. The both local spin- and exchange-fluctuation correlators once duality point Q ¼ π=2 further marks the resonance from the steady state is reached; see Fig. 3(b).However,aswe positive temperatures for Q<π=2 to negative temperatures discuss below, the effective temperature obtained from spin for Q>π=2; see the inset of Fig. 3(a). The T<0 thermal and exchange fluctuations may disagree with each other. states of the FM Heisenberg model with coupling −jJj This allows us to distinguish prethermalization from true correspond to T>0 states of the AFM Heisenberg model thermalization. with coupling þjJj, and vice versa. Negative-temperature states naturally arise in isolated systems with bounded energy spectra as legitimate thermal states and occur when the initial energy density lies closer to the upper edge of the energy spectrum. Prethermalization of spiral states with Q ∼ 0; π.—For spiral states with Q ∼ 0; π, where the system develops a prethermal plateau, the effective spin- and exchange- field-fluctuation temperatures disagree, even though the FDR is satisfied well for each mode individually. This finding supports the prethermalized nature of such steady states. It is understood that the temperatures calculated within the prethermal plateau [shown as shaded regions in Fig. 3(b)] correspond to the effective temperature of individual modes and not the true thermodynamical tem- perature. We expect the two temperatures to approach each other at longer times once the system exits the prether- malized plateau and progresses toward a fully thermalized state. The spiral state with Q ¼ 0 is an exact ground state of the system, and FDR yields T ¼ 0 as expected. In contrast, the Q ¼ π state approaches a finite temperature, which is understood by the fact that the uncorrelated Néel state must “ ” FIG. 3. Thermalization of the spin spiral state. (a) The effective be dressed with spin correlations before the steady state is inverse temperature of local spin T and local exchange-field reached; see the inset of Fig. 3(a). The disparity between spin Q 0 Q π fluctuations Tfluct obtained from the fluctuation-dissipation rela- the evolution of ¼ and ¼ states reveals the tions in the steady state. The two temperatures are in agreement quantum mechanical nature of spins and the breakdown for spiral windings near Q ∼ π=2, supporting the true thermal- of the classical duality Q → π − Q in the low-temperature ization of the system. The temperatures calculated in the regime. Q ∼ 0; π prethermalized plateaus (shaded regions) disagree with The 3D Heisenberg model exhibits a finite-temperature each other and generically differ from the temperature of the true equilibrium phase transition from the disordered para- thermal states that emerge at later times. Inset: The temperature k T Q magnetic phase to the ordered FM or AFM phase, depend- B as a function of (same data as in the main panel) displays a J resonance from positive infinite temperature to negative infinite ing on the sign of the exchange coupling . For the spiral at Q π temperature at the classical duality point Q ¼ π=2. (b) The ¼ , the FDRs of the spin fluctuations are not well approach of Tfluct to the steady state (light to dark) as obtained fulfilled at accessible times while those for exchange-field from fluctuation-dissipation relations for Q ¼ π=4 (left) and Q ¼ fluctuations are. The temperature extracted from the latter π T Q π 0 82J (right). The steady-state temperatures are shown on the plots. j fluctð ¼ Þj ¼ . lies below the AFM ordering

041005-7 MEHRTASH BABADI, EUGENE DEMLER, AND MICHAEL KNAP PHYS. REV. X 5, 041005 (2015)

AFM temperature Tc ¼ 0.946ð1ÞJ. The latter has been to form out-of-plane textures as the energy of spiral states obtained from quantum Monte Carlo simulations [70]. can be reduced by an appropriate out-of-plane tilt. Crucially, the near-thermal distribution of fluctuations in Therefore, in a thermodynamical ensemble where arbitrary the prethermalized plateaus, the stability of FM and AFM out-of-plane fluctuations are allowed, these saddle points ordered phases at finite energy densities in the 3D fail to give rise to symmetry-broken states, leaving Q ¼ 0 Heisenberg model, and the proximity of Q ∼ 0; π spiral and Q ¼ π as the only thermodynamically stable orders in states to these stable orders allow us to draw a connection the Heisenberg model. Therefore, the present situation must between the long-time stability of such spiral states and be regarded from the perspective of quantum dynamics, spontaneous symmetry breaking at equilibrium: The spiral i.e., the unitary evolution of a pure state jspðQÞi rather than winding Q sets the energy density of the system and the statistical fluctuations in a mixed thermodynamical subsequently the effective temperature TðQÞ in the pre- ensemble. Here, the system remains in a pure state at all thermal state. TðQÞ approximately dictates the magnitude times and the magnetic order is confined to the xy spiral of spin fluctuations on the top of the spiral states which plane due to the symmetries discussed at the beginning of locally resemble either FM or AFM for Q ∼ 0; π. Sec. III. It is therefore conceivable that symmetry-protected Depending on Q, TðQÞ can either lie below or above dynamical constraints allow thermodynamically unstable FM AFM the critical transition temperature Tc or Tc , thereby saddle points to become long-lived states in the course of providing an approximate condition for the local stability of unitary dynamics. the spiral order. We will study the global instability of the Out-of-plane instability.—The out-of-plane instability of spiral states and their destruction at longer times in the next the spiral state in the Heisenberg model can be studied section. either by performing a linear response analysis of the Bloch According to the above discussions, the connection equations, or similarly from the Holstein-Primakoff spin- made between the emergence of slow dynamics and wave analysis. Either way, the dispersion of out-of-plane symmetry breaking at equilibrium essentially hinges on spin wavespffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi forming on the top of the spiral is found as 2 2 the eigenstate thermalization hypothesis and the Mermin- ωk ¼ ϵk − Δk [33,71], where Wagner theorem. Therefore, this connection is expected to reach beyond the present discussion and to generalize to a X3 broader range of initial states and models that exhibit ϵk ¼ −JS ½ð1 þ cos Q · eˆ dÞ cos k · eˆ d − 2 cos Q · eˆ d; spontaneous continuous symmetry breaking. d¼1 X3 Δk ¼ −JS ½ð1 − cos Q · eˆ dÞ cos k · eˆ d: ð15Þ C. Instabilities and correlations d¼1 The dynamical stabilization of spirals near the FM and AFM orders, and consequently the appearance of prether- Unstable modes arise when ωk assumes imaginary mal plateaus, were understood on the basis of thermody- values. Except for Q ¼ 0; π=2; π, one always finds such namical arguments in the previous section. However, even unstable modes: for Q<π=2, the fastest-growing mode is though the spiral states are fixed points of the mean-field k ¼ðk; k; kÞ with k ¼ cos−1½cos2ðQ=2Þ along with a dynamical equations, they are unstable and have a tendency sharp cutoff jkj ≤ Q; for Q>π=2, unstable modes occur

FIG. 4. The evolution of spin correlations. Top panels: Growth rate of out-of-plane instable modes obtained from a linear response ˆ z ˆ z zz;K analysis. Bottom panels: Numerically calculated correlation function hSkS−kiðtÞ¼iχk ðt; tÞ as a function of the lattice wave vector k ¼ðk; k; kÞ within the spin-2PI formalism including NLO corrections. (a) Q ¼ 3π=8, (b) Q ¼ π=2, and (c) Q ¼ 3π=4. The inset in ˆ þ ˆ − (c) shows the connected part of the in-plane correlations hSk S−kiðtÞ.

041005-8 FAR-FROM-EQUILIBRIUM FIELD THEORY OF MANY- … PHYS. REV. X 5, 041005 (2015) for jk − πj ≤ Q, with the fastest mode always being the embedded in our self-consistent approach. The time scale staggered k ¼ π mode, independent of Q. for the formation of correlations is found to be on the order A simple estimate for the lifetime of the prethermal of the dephasing time, reflecting the fact that the short-time plateaus is obtained by calculating the time it takes for the dephasing dynamics and formation of correlations are typical unstable out-of-plane collective mode to grow to manifestations of the same phenomenon. Oð1Þ. The rationale behind this estimate is that the in-plane For spiral states that thermalize within the numerically order can not coexist with strong enough out-of-plane achievable time scales, we observe a smooth shift in both fluctuations. Expanding around Q ¼ 0; π, we obtain a in-plane and out-of-plane spin correlations from the initial scaling t ∼ 1=Q2 for the FM-like and t ∼ 1=ðπ − QÞ for Q-dependent enhanced modes to either k ¼ 0 or k ¼ π, the AFM-like spirals, up to logarithmic corrections. depending on whether Q<π=2 or Q>π=2, respectively However, the lifetime of plateaus as found from the [see Figs. 4(a), 4(c), and the inset]. Even though the linear spin-2PI formalism exceeds the above estimates; in par- response analysis correctly indicates the wave vector of the ticular, as Q is tuned closer toward 0 or π, we observe a fastest-growing out-of-plane mode, the spin correlations faster increase of the lifetime of prethermal plateaus. As we rapidly saturate to their maximum values, as opposed to an discuss below, the increased lifetime can be explained on unbounded exponential growth. A similar rapid dynamical the basis of the self-regulation of out-of-plane spin regulation of the growth of unstable modes was previously fluctuations. reported in Refs. [22,49] in the context of parametric The top panels in Fig. 4 show Im½ωk for several values resonance in the OðNÞ model. In the present context, this of Q, along with the evolution of equal-time out-of-plane phenomenon explains why the lifetime of the plateaus zz;K ˆ z ˆ z spin correlations iχk ðt; tÞ¼hSkðtÞS−kðtÞi obtained by exceeds the estimate obtained from the linear response solving the nonequilibrium Bethe-Salpeter equation in the analysis and indicates the important role of mode coupling NLO approximation; see the bottom panels of Fig. 4. between spin waves and the necessity of nonperturbative Further, the connected part of in-plane correlations treatments. þ−;K ˆ þ ˆ − ˆ þ ˆ − iχ ðt; tÞ¼hS ðtÞS−kðtÞi − hS ðtÞihS−kðtÞi is shown For spiral states that exhibit long-lived prethermal k k k D in the inset of Fig. 4(c). plateaus, we study the exchange-field correlations ,a χ At t ¼ 0, spin correlations are 0 in accordance with the quantity that is closely related to but can be calculated for initial spiral state jspðQÞi being an uncorrelated product much longer times with fewer computational resources. zz;K þ−;K state. The out-of-plane correlations form at times t ∼ J. The The evolution of Dk ðt; tÞ and Dk ðt; tÞ for Q ¼ π=4 most enhanced correlations coincide with the wave vector and Q ¼ 7π=8 are shown in Fig. 5(a). The former corre- predicted by the linear response analysis to a good degree. sponds to a spiral state that thermalizes at about 20J−1, The sharp cutoffs predicted by this analysis are found to be while the latter exhibits a magnetization plateau up to −1 smeared, which is expected due to the mode coupling τM ∼ 20J , as shown in Fig. 1.Fort ≲ τM, the most

20 60 10

15 40 8 10 20 6 5 0 0 4 0 0 10 10 2 2200 0 30 30 1

20 60 0.8

15 40 0.6 10 20 5 0.4 2 0 0 0.2 0 0 10 1010 0 0 0 1 2 3 4 5 10 20 30 0 1 2 3 4 5 10 20 30 20 20 30 30

FIG. 5. Dynamics of exchange-field fluctuations. (a) The out-of-plane (top) and in-plane (bottom) exchange-field fluctuations as a function of time and momentum k ¼ðk; k; kÞ for Q ¼ π=4 (left) and Q ¼ 7π=8 (right). The red lines indicate the most enhanced mode in the long-time limit; the dashed blue line in the lower right plot corresponds to the k ¼ Q in-plane mode, which initially exhibits the strongest enhancement of correlations. (b) The evolution of the late-time most enhanced mode for Q ¼ π=4; π=2 (left) and Q ¼ 7π=8; π (right). In cases where the system thermalizes, left column, SUð2Þ symmetry emerges in the long-time limit, while it is broken in the prethermal case Q ¼ 7π=8 and for Q ¼ π, right column. In the latter case, the system can exhibit true long-range order, provided its effective temperature is below the critical temperature of the equilibrium phase transition and can thus be thermal and simultaneously break SUð2Þ symmetry.

041005-9 MEHRTASH BABADI, EUGENE DEMLER, AND MICHAEL KNAP PHYS. REV. X 5, 041005 (2015) enhanced in-plane mode occurs at k ¼ Q, which upon demagnetization smoothly switches to k ¼ π for t ≳ τM. The most unstable out-of-plane mode is always at k ¼ π. As a further check for thermalizing behavior, we study the restoration of the SUð2Þ symmetry in the exchange- zz;K þ−;K field fluctuations Dk ðt; tÞ → ð1=2ÞDk ðt; tÞ; see Fig. 5(b).ForQ ¼ π=4 and Q ¼ π=2, we find that the SUð2Þ symmetry is restored at longer times (left column), while for Q ¼ 7π=8 and the Néel initial state Q ¼ π, the SUð2Þ symmetry remains broken at all accessible times (right column). Notably, the out-of-plane fluctuations for Q ¼ 7π=8 are found to be an order of magnitude stronger than the Q ¼ π, in agreement with the previously men- tioned existence of an unstable out-of-plane mode for the former state and its absence in the latter. FIG. 6. Comparison between spin-2PI and semiclassical dy- The magnitude of in-plane fluctuations remain essen- namics. The spin-2PI results ( black lines) are compared to tially constant in the plateau for Q ¼ 7π=8 (top right), the semiclassical dynamics obtained from the dTWA (dashed blue while the out-of-plane fluctuations monotonically increase lines). The short-time analytic result from Sec. III A is also shown −1 and reach a maximum at t ∼ 20J ∼ τM, precisely when for reference (thick red lines). The long-time dynamics of the two the prethermal magnetization decays. This finding connects methods are significantly different. In particular, the dTWA is not the decay of the the spiral to the growth of out-of-plane capable of describing the long-lived prethermal plateaus in contrast to the spin-2PI formalism. The scale of the time axis fluctuations. The time τM also marks a reversal in the trend is switched from linear to logarithmic at tJ ¼ 5 for better of out-of-plane and in-plane correlations. Even though this visibility. change indicates a first step toward establishing SUð2Þ- symmetric correlations, the condition is far from being Q π t ∼ τ ¼ . We note that the latter is supported by exact satisfied at M and is bound to occur on much longer quantum Monte Carlo calculations. time scales, indicating a hierarchical relaxation scenario with the relaxation of magnetization preceding the relax- ation of correlations. IV. CONCLUSIONS AND OUTLOOK The appearance of long-lived prethermal states and the We formulated a nonperturbative and conserving field- hierarchical relaxation is reminiscent of aging dynamics in theoretic technique for describing the far-from-equilibrium classical structural glass models with quenched disorder quantum dynamics of strongly interacting spin-1=2 sys- [69] and kinematically constrained models [72]. Similar tems for arbitrary lattices and initial states. Referred to as multiscale glassy relaxation dynamics has been recently the spin-2PI formalism, this method systematically reported in the quench dynamics of fermions in a nearly improves upon the mean-field description by including integrable 1D model using a different method [15]. quantum fluctuations by means of an asymptotic 1=N Comparison to semiclassical methods.—According to expansion, which is controlled in models with intrinsically the discussions presented so far, the relaxation dynamics of large lattice-coordination number. the spiral state accompanies the formation of in- and out-of- We utilized the spin-2PI technique to study far-from- plane quantum correlations in the system. In order to study equilibrium phenomena in spin systems with continuous the role of correlations further, we compare our predictions symmetries. Specifically, we explored the relaxation with the results obtained from the discrete truncated Wigner dynamics of spin spiral states in the 3D Heisenberg model, approximation (dTWA) [73,74], a variant of the semi- treating the spiral winding Q as a tuning parameter. Going classical TWA method [75] that relies on mean-field beyond the trivial mean-field (LO) dynamics by including trajectories. The magnetizations obtained from spin-2PI the NLO correction, we found the spiral states with (solid black lines) and dTWA (dashed blue lines) are different windings to relax in remarkably different ways. compared in Fig. 6. The two methods generically agree In particular, spiral states resembling FM and AFM ordered with the analytic short-time expansion (thick red lines), states, corresponding to Q ∼ 0 and π, respectively, get with the exception that dTWA does not reproduce the trapped for long times in nonthermal states, i.e., “false correct short-time dynamics for small Q; see Q ¼ π=4 in vacuums” whose lifetimes diverge as the windings are Fig. 6 [76]. The two methods, however, predict strikingly tuned to Q ¼ 0 or π. In contrast, the spiral states far from different long-time dynamics. Even though dTWA exhibits Q ¼ 0; π relax rapidly. some degree of dynamical slowing down for FM-like and We calculated the effective temperature of spin and AFM-like spirals, it produces neither the prethermal plateau exchange-field fluctuations from the fluctuation-dissipation for Q ¼ 7π=8 nor the finite steady-state magnetization for relation. For Q ∼ π=2 spiral states, all modes reach a single

041005-10 FAR-FROM-EQUILIBRIUM FIELD THEORY OF MANY- … PHYS. REV. X 5, 041005 (2015) temperature, supporting full thermalization in accordance Matter, an NSF Physics Frontiers Center with support of with the eigenstate thermalization hypothesis. In contrast, the Gordon and Betty Moore Foundation, the Austrian the different bosonic modes of prethermalizing spirals settle Science Fund (FWF) Project No. J 3361-N20, as well as at different temperatures. Technische Universität München–Institute for Advanced We investigated the dynamical formation of correlations Study, funded by the German Excellence Initiative and the and found that the collective modes predicted to be unstable European Union FP7 under grant agreement No. 291763. from a linear response analysis are self-regularized at rather E. D. acknowledges support from Dr. Max Rössler, the short time scales, demonstrating the importance of the Walter Haefner Foundation, and the ETH Foundation. nonlinear effects and nonperturbative treatments. The growth of out-of-plane fluctuations causes the eventual decay of the prethermal states. The restoration of SUð2Þ APPENDIX: SUMMARY OF THE TRUNCATED symmetry occurs much later after the decay of magneti- SPIN-2PI FORMALISM AT THE NLO LEVEL zation, suggesting a hierarchical relaxation reminiscent of In this Appendix, we provide supplemental material for coarsening and aging in classical glassy systems. Our the spin-2PI formalism along with a brief account of the results can be tested readily in ultracold-atom experiments numerical methods. The covered material includes the with two-component Mott insulators in 3D optical lattices, explicit derivation of the approximate dynamical equations such as a 3D extension of the experiments in Refs. [33,34]. from the NLO truncated 2PI effective action and the This work can be extended in several directions. A reconstruction of real-time spin-spin correlators from the straightforward extension is to investigate the relaxation of Bethe-Salpeter equation. spiral states in anisotropic models, or in lower dimensions. Another immediately accessible direction is to study spin 1. Correlation functions on the Schwinger-Keldysh systems with long-range interactions, as realized, for time contour instance, with Rydberg atoms, polar molecules, or trapped ions, and their instability toward dynamic crystallization. In the Schwinger-Keldysh formalism, the nonequili- The effect of small deviations from the initial spiral state on brium dynamics of quantum fields is most elegantly the quantum evolution could be studied as well. We expect derived from a path integral defined on the round-trip C Cþ∪C− our predictions to carry over to the case of weakly contour ¼ : disordered initial states, provided that the deviations from the pristine spiral state remain small by the time the prethermalization plateau is reached. A conservative bound t = t0 t =+∞ for the allowed degree of disorder can be estimated from the presented linear response analysis; however, a more real- istic calculation must take into account self-regulation and slowing down of the unstable modes in the presence of The Majorana operators η and real-vector boson φ are disorder, which is a computationally challenging task. On replaced by Grassmann and real-vector-valued variables in related grounds, it is desirable to study the formation of the path integral, along with antiperiodic and periodic topological defects in quenches to the ordered phase, boundary conditions at the contour end points, respectively. corresponding to the instantaneous limit of the quantum The correlation functions defined on the C contour can be Kibble-Zurek mechanism [77,78]. For the 3D Heisenberg thought of 2 × 2 matrices in the two-dimensional space of model with SUð2Þ symmetry, topologically stable hedge- the contour branch index. For example, the Majorana two- hogs [79] are expected to form with universal scaling laws. point correlator G can be explicitly written as Other possible research directions include extension to þþ þ− open spin systems, studying the role of NNLO corrections G ðt1;t2Þ G ðt1;t2Þ G t1;t2 ; A1 ð Þ¼ −þ −− ð Þ to assess the robustness of the NLO results, and comparison G ðt1;t2Þ G ðt1;t2Þ with other systematic expansions, such as 1=D expansion in D-dimensional lattices, and the semiclassical 1=S where the times appearing in the matrix are ordinary times. expansion. We have dropped the discrete indices for brevity. The off- diagonal matrix elements are identified with the “lesser” ACKNOWLEDGMENTS and “greater” explicitly ordered correlators:

We thank S. Gopalakrishnan, B. I. Halperin, and Gþ− t ;t ≡ G< t ;t i η t η t ; S. Sachdev for useful discussions. We acknowledge support ð 1 2Þ ð 1 2Þ¼þh ð 2Þ ð 1Þi −þ > from Harvard-MIT CUA, NSF Grant No. DMR-1308435, G ðt1;t2Þ ≡ G ðt1;t2Þ¼−ihηðt1Þηðt2Þi: ðA2Þ AFOSR Quantum Simulation MURI, the ARO-MURI on Atomtronics, ARO MURI Quism program, Humboldt The diagonal matrix elements are related to each other by Foundation, the Institute for Quantum Information and the virtue of the unitarity of evolution:

041005-11 MEHRTASH BABADI, EUGENE DEMLER, AND MICHAEL KNAP PHYS. REV. X 5, 041005 (2015)

þþ > < G ðt1;t2Þ¼þθðt1 − t2Þ½G ðt1;t2Þ − G ðt1;t2Þ; The vacuum diagrams accompany symmetry factors that −− > < must be worked out case by case. The self-energy Σ; Π and G ðt1;t2Þ¼−θðt2 − t1Þ½G ðt1;t2Þ − G ðt1;t2Þ; ðA3Þ four-point vertex Λð2Þ diagrams have an extra factor of i and i2 which are identified with the usual retarded and advanced , respectively. response functions Gþþ ≡ GR and G−− ≡ GA. While the lesser and greater correlation functions are independent 3. Evolution of correlation functions functions for Dirac (complex) fermions, they are related to in the spin-2PI formalism each other for Majorana fermions by transposition and The transition from the path integral to the 2PI effective negation, as it can be seen from Eq. (A3): action Γ½G; D; φ was briefly outlined in the main text and is a straightforward generalization of the results of G>ð1; 2Þ¼−G<ð2; 1Þ: ðA4Þ Cornwall, Jackiw, and Tomboulis [41]. Within this formal- In summary, the two-point correlator of Majorana fermions ism, the evolution equations follow from a variational on the contour is fully specified by a single real-time principle, reminiscent of Lagrangian dynamics of classical correlator, e.g., G>ð1; 2Þ. It is easily shown that the same particles, with the quantum correlators playing the role of decomposition and relations hold for the Majorana self- generalized coordinates. Going back to Eqs. (8a) and (8b) energy Σ. The correlator of real bosons D and the bosonic and making Γ stationary with respect to G, D, and φ,we self-energy Π admit a similar decomposition, except for the obtain absence of the relative minus sign in the definition of D> < −1 −1 and D : G ¼ G0 − M½φc − Σ½G; D; ðA7aÞ

þ− < −1 −1 D ðt1;t2Þ ≡ D ðt1;t2Þ¼−ihφðt2Þφðt1Þi; D ¼ D0 − Π½G; D; ðA7bÞ −þ > D ðt1;t2Þ ≡ D ðt1;t2Þ¼−ihφðt1Þφðt2Þi; ðA5Þ 1 XN φμ t Vμνϵ Gγ;σ;λ;σ tþ;t : which imply c;jð Þ¼2N jk νγλ jj ð Þ ðA7cÞ σ¼1 D>ð1; 2Þ¼D<ð2; 1Þ: ðA6Þ With the spin, replica, time, and space indices laid out Similar to Majorana correlators, the two-point correlator of explicitly, the “bare” fermion and boson propagators are real bosons on the contour is fully specified by a single real- written as time correlator, e.g., D>. The same result holds for the bosonic self-energy Π. G−1 1; 2 δ δ δ i∂ δ t ;t ; 0 ð Þ¼ σ1σ2 α1α2 j1j2 t1 Cð 1 2Þ

−1 −1 α1α2 D ð1; 2Þ¼NðV Þ δCðt1;t2Þ; ðA8Þ 2. Feynman rules for the spin-2PI formalism 0 j1j2 The conventional rules are used for respectively. The contour Dirac δ function is defined as interpreting the diagrams appearing throughout this work: δCðt1;t2Þ¼δðt1 − t2Þ with the sign corresponding to t1;t2 ∈ C , respectively. In Eq. (A7a), M½φcð1; 2Þ¼ −iδ δ t ;tþ δ φμ t ε σ1σ2 Cð 1 2 Þ j1j2 c;j1 ð 1Þ μα1α2 the LO interaction effect and describes the coupling of Majorana fermions with the classical spin mean field φc. According to Eq. (A7c), the latter is instantaneously determined by the Majorana tadpole contracted with a bare interaction line. Thus, we find

The integer indices refer to the bundle of lattice site and contour time in the diagrams above. Since the Majorana fermion propagators possess no charge flow direction, one may arbitrarily assign a direction to each line. The overall sign of each diagram, however, must be determined at the ðA9Þ end by counting the number of fermion permutations. The power counting of the large-N extension is per- which resembles the familiar Hartree self-energy that formed as follows: (1) each Majorana fermion loop describes the mean-field effects. We emphasize that the introduces a factor of N resulting from the replica summa- Majorana tadpole is identified with the magnetization in þ tion and (2) each interaction and boson line introduce a our formalism. Note that the Mð1; 2Þ ∝ δCðt1;t2 Þ is instan- factor of 1=N. taneous and carries no memory effect. We will later show

041005-12 FAR-FROM-EQUILIBRIUM FIELD THEORY OF MANY- … PHYS. REV. X 5, 041005 (2015) that truncating the approximation at this level and neglect- results from the replica summation in the Majorana bubble, NLO ing the self-energies indeed yields the Bloch equation. we find Γ2 ∼ Oð1Þ. This must be compared to the LO term Σ G; D Γ O N Going beyond the LO approximation, ½ in int which is ð Þ. The resulting NLO self-energies are and Π½G; D describe memory effects associated from given as the spatiotemporal fluctuations of the exchange field. By definition, these self-energies are obtained from the variations of Γ2½G; D:

δΓ2½G; D Σ½G; Dð1; 2Þ ≡ 2 ; δGð1; 2Þ δΓ2½G; D Π½G; Dð1; 2Þ ≡ 2 : ðA10Þ δDð1; 2Þ ðA12Þ We recall that Γ2½G; D is formally equivalent to the sum of 2PI vacuum diagrams constructed from the interaction vertex L η; φ i ε φαηβ;σηγ;σ Having derived the explicit expressions for the self- int½ ¼2 αβγ j j j and admits a systematic expan- sion in 1=N. Here, we truncate the series at the NLO level: energies, we discuss the derivation of evolution equations as the next step. Our starting points are the coupled Dyson’s equations given in Eqs. (A7a) and (A7b). Strictly speaking, Γ ðA11Þ Dyson’s equations are differential identities on the contour functions. They can be cast into a more useful form by P acting them from the left- and right-hand sides by G and D, Πμν 1;2 iε ε N Gα;σ;γ;σ t ;t Gβ;σ;λ;σ t ;t where 0 ð Þ¼ μαβ νγλ σ¼1 j1j2 ð 1 2Þ j1j2 ð 1 2Þ respectively, resulting in a set of contour integro- is the Majorana bubble. Since D ∼ 1=N and the factor of N differential equations:

Z iδ δ ∂ iφν t ε Gμα2 t ;t δ 1; 2 dτΣα1μ t ; τ Gμα2 τ;t ; ½ α1μ j1k t1 þ c;kð 1Þ να1μ kj ð 1 2Þ¼ ð Þþ j k ð 1 Þ kj ð 2Þ ðA13aÞ 2 C 1 2 Z − iδ δ ∂ − iφν t ε Gα1μ t ;t δ 1; 2 dτGα1μ t ; τ Σμα2 τ;t ; ½ μα2 kj2 t2 c;kð 1Þ νμα2 j k ð 1 2Þ¼ ð Þþ j k ð 1 Þ kj ð 2Þ ðA13bÞ 1 C 1 2

Z 1 1 α1β1 α1α2 α1μ μν να2 Dj j ðt1;t2Þ¼ Vj j δCðt1;t2Þþ Vj k dτΠkl ðt1; τÞDlj ðτ;t2Þ; ðA14aÞ 1 2 N 1 2 N 1 C 2

Z 1 1 α1β1 α1α2 α1μ μν να2 Dj j ðt1;t2Þ¼ Vj j δCðt1;t2Þþ dτDj k ðt1; τÞΠkl ðτ;t2ÞVlj : ðA14bÞ 1 2 N 1 2 N C 1 2

δ 1; 2 ≡ δ δ δ t ;t G t ;t ∝ δ t ≤ t ;t ≤ T We have defined shorthand ðR Þ j1j2 α1α2 Cð 1 2Þ, that the assumption j1j2 ð 1 2Þ j1j2 for 0 1 2 dtA t Σ t ;t ∝ δ andR the contour integralR C ð Þ is interpreted implies j1j2 ð 1 2Þ j1j2 in the same domain. The causal ∞ dtA t ∈ Cþ − ∞ dtA t ∈ C− as t0 ð Þ t0 ð Þ. Equation (A13a) structure of Eqs. (A13a) and (A13b) subsequently extends and its adjoint Eq. (A13b) are referred to as Kadanoff- this property to infinitesimally larger domains and even- Baym (KB) equations. The convolution integrals of tually to all times by induction. Therefore, we can always self-energies and correlators manifestly show memory make the following simplifying substitution in the KB effects, which is a shared feature of beyond mean-field equation: approximations.

The spatial structure of Eqs. (A13a)–(A14b) can be α1α2 α1α2 G ðt1;t2Þ → δj j G ðt1;t2Þ; simplified by noting that physical initial states imply initial j1j2 1 2 j1j1 Σα1α2 t ;t → δ Σα1α2 t ;t : correlations between pairs of Majorana operators on the j1j2 ð 1 2Þ j1j2 j1j1 ð 1 2Þ ðA15Þ Gα1α2 t ;t ∝ δ same site, i.e., j1j2 ð 0 0Þ j1j2 . Crucially, this property extends to all times in the KB dynamics, independent of the The LO approximation.—The KB equations reduce to order of truncation in 1=N. To see this, one first establishes the mean-field Bloch equation upon truncation at the LO

041005-13 MEHRTASH BABADI, EUGENE DEMLER, AND MICHAEL KNAP PHYS. REV. X 5, 041005 (2015) level, which amounts to neglecting fluctuation self-energy [see Eq. (13)]. The two-point correlator of Majorana corrections Σ → 0. In this limit, the KB equations fermions at t ¼ t0 is easily found as imply 0 1 −i=20 0−i=2 B 0 −i=21=20C α1α2;> ν μα2 α1α2;> B C i∂ G t ;t iφ t ε G t ;t 0; Gj j ðt0;t0Þ¼δj j : t1 jj ð 1 2Þþ c;jð 1Þ να1μ jj ð 1 2Þ¼ 1 2 1 2 @ 0 −1=2 −i=20A −i∂ Gα1α2;> t ;t iφν t ε Gα1μ t ;t 0: −i=20 0−i=2 t2 jj ð 1 2Þþ c;jð 2Þ νμα2 jj ð 1 2Þ¼ ðA16Þ ðA18Þ Subtracting the equations from one another, setting t2 ¼ t1 ¼ t, and using Eq. (7), we finally obtain The exchange-field correlator at t ¼ t0 is not an indepen- dent degree of freedom and is determined by Gðt0;t0Þ; see ˆ ˆ ∂thSjðtÞi ¼ φc;j × hSjðtÞi; ðA17Þ Eq. (A12). For translationally invariant initial states as such, G and Σ further become independent of the lattice site. Furthermore, Dj j depends only on the distance, and at the which is the Bloch equation as anticipated. 1 2 NLO level, Πj j is local as well. The simplified structure of The NLO approximation.—Including self-energy cor- 1 2 rections, the time convolutions appearing in the KB the correlators and self-energies is summarized as follows: equations prohibit us from arriving at a closed equation αβ Gð1; 2Þ → δR R G ðt1;t2Þ; for the equal-time Green’s functions and we inevitably need 1 2 ’ Σ 1; 2 → δ Σαβ t ;t ; to solve for the complete unequal-time Green s function. ð Þ R1R2 ð 1 2Þ For concreteness, we consider the case of spin spirals D 1; 2 → Dαβ t ;t ⟶F:T: Dαβ t ;t ; hereafter. The spatial structure of the KB equations can be ð Þ R1−R2 ð 1 2Þ k ð 1 2Þ significantly simplified by applying the unwinding unitary αβ Πð1; 2Þ → δr r Π ðt1;t2Þ: transformation, either directly on Eqs. (A14a) and (A14b) 1 2 or on the spin Hamiltonian. Either way, the initial spiral The KB equations can be written explicitly in terms of state transforms into an uncorrelated x-polarized FM state G> and D> using the Langreth rules [80]. We quote the ~ jΨ0i ≡ ⨂j →ij at the expense of an anisotropic interaction final result, setting N ¼ 1: j Z t α μ 1 i∂ Gα1α2 t ;t iφ 1 t Gμα2;> t ;t dτ Σα1μ;> t ; τ Σμα1;> τ;t Gμα2;> τ;t t1 ð 1 2Þþ c ð 1Þ ð 1 2Þ¼ ½ ð 1 Þþ ð Þ ð 2Þ t0Z t 2 α μ;> μα ;> α μ;> − dτΣ 1 ðt1; τÞ½G 2 ðτ;t2ÞþG 2 ðt2; τÞ; ðA19aÞ t0 Z t μα 1 −i∂ Gα1α2 t ;t iGα1μ;> t ;t φ 2 t dτ Gα1μ;> t ; τ Gμα1;> τ;t Σμα2;> τ;t t2 ð 1 2Þþ ð 1 2Þ c ð 2Þ¼ ½ ð 1 Þþ ð Þ ð 2Þ t0Z t 2 α μ;> μα ;> α μ;> − dτG 1 ðt1; τÞ½Σ 2 ðτ;t2ÞþΣ 2 ðt2; τÞ; ðA19bÞ t0

Z t1 α1α2;> α1μ μν;> να2 α1μ μν;> νμ;> να2;> Dk ðt1;t2Þ¼Vk Π ðt1;t2ÞVk þ Vk dτ½Π ðt1; τÞ − Π ðτ;t1ÞDk ðτ;t2Þ Z t0 t2 α1μ μν;> να2;> α2ν;> − Vk dτΠ ðt1; τÞ½Dk ðτ;t2Þ − Dk ðt2; τÞ; ðA20aÞ t0 Z t1 α1α2;> α1μ μν;> να2 α1μ;> μα1;> μν;> να2 Dk ðt1;t2Þ¼Vk Π ðt1;t2ÞVk þ dτ½Dk ðt1; τÞ − Dk ðτ;t1ÞΠ ðτ;t2ÞVk Z t0 t2 α1μ;> μν;> νμ;> να2 − dτDk ðt1; τÞ½Π ðτ;t2Þ − Π ðt2; τÞVk : ðA20bÞ t0

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The self-energies Σ> and Π> are read from Eq. (A12). These vertex corrections are structurally similar to the The last explicit equations are suitable for devising a random-phase approximation (RPA), Maki-Thompson numerical forward propagation scheme. Starting from (MT), and Aslamazov-Larkin (AL) vertex corrections G0ðt0;t0Þ, we calculate Σðt0;t0Þ and Πðt0;t0Þ from accounting for superconducting fluctuations in metals Eq. (A12) and Dðt0;t0Þ from Eq. (A20a). The casual [81]. Explicitly, these vertex parts are given as structure of Eqs. (A19a)–(A20b) allows us to propagate 1 1 fG; Σ; D; Πg in ðt1;t2Þ in discrete time steps of size Δt. ð2Þ ¯ ¯ μν Λ ð33; 44Þ¼ εα α μiV ενα α δj j δj j This is achieved using a robust predictor-corrector method RPA 2 3 3¯ j3j4 2 4 4¯ 3 3¯ 4 4¯ O Δt3 with guaranteed accuracy to ð Þ. × δCðt3;t3¯ ÞδCðt4;t4¯ ÞδCðt3;t4Þ; ðA26aÞ Calculating spin-spin correlators in the spin-2PI for- malism.—In the framework of 2PI effective actions, higher- 1 1 Λð2Þ 33;¯ 44¯ ε iDμν t ;t ε δ δ ð Þ¼ α α μ j j ð 3 3¯ Þ να¯ α¯ j j j¯ j¯ order correlators are “reconstructed” from the history of MT 2 3 4 3 3¯ 2 3 4 3 4 3 4 two-point correlators. Here, we are interested in the × δCðt3;t4ÞδCðt3¯ ;t4¯ Þ; ðA26bÞ connected spin-spin correlator:

α α α α α α ð2Þ β3β3¯ β4β4¯ 1 2 ˆ 1 ˆ 2 ˆ 1 ˆ 2 Λ 33;¯ 44¯ iG t ;t¯ iG t ;t¯ iχ t ;t T S t S t − S t S t ; ð Þ¼ j j¯ ð 3 3Þ j j¯ ð 4 4Þ j1j2 ð 1 2Þ¼h C½ j1 ð 1Þ j2 ð 2Þi h j1 ð 1Þih j2 ð 2Þi AL 3 3 4 4 1 1 ðA21Þ μν × εα β μiD ðt3;t4Þ ενα β 2 3 3 j3j4 2 4 4 where the spin operators are shorthand notations for 1 1 ε iDμ¯ ν¯ t ;t ε : × α¯ β¯ μ¯ j j ð 3¯ 4¯ Þ να¯ ¯ β¯ ðA26cÞ Eq. (4). The spin-spin correlator is found from the 2 3 3 3¯ 4¯ 2 4 4 Majorana L function, defined as 2 2 ¯ ¯ ¯ ¯ ¯ ¯ It is easily noticed that Λð Þ ∼ Oð1Þ, Λð Þ ∼ Oð1=NÞ, and Lð11; 22Þ ≡ hTC½ηð1Þηð1Þηð2Þηð2Þi − iGð1; 1ÞiGð2; 2Þ; RPA MT Λð2Þ ∼ O 1=N2 ðA22Þ AL ð Þ. Therefore, we may drop the latter if accuracy at the NLO order is desired. by contracting ε symbols with its left and right pairs of For translation-invariant states, it can be shown that α α¯ ;α α¯ L 11;¯ 22¯ ∼ L 1 1 2 2 t t¯ ; t t¯ δ δ fermion lines: ð Þ R1−R2 ð 1 1 2 2Þ R1R1¯ R2R2¯ . Taking a Fourier transform in R1 − R2 yields decoupled integral i α1α2 β1γ1;β2γ2 þ þ equations for each momentum transfer q. The temporal χj j ðt1;t2Þ¼ εα β γ Lj j ;j j ðt1 ;t1;t2 ;t2Þεα β γ : ðA23Þ 1 2 4 1 1 1 1 1 2 2 2 2 2 structure of the Bethe-Salpeter equation remains formi- dable. Performing the contour integral and discrete sum- The L function, in turn, satisfies a nonequilibrium Bethe- mations over 3; 3¯ variables in Eq. (A24) and contracting the Salpeter equation on the C contour: right legs according to Eq. (A23), we reach to an integral Z α1α1¯ ;μ equation for the three-time object Γq t1;t1¯ ; t2 ≡ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ð2Þ ð Þ Lð11; 22Þ¼Π2ð11; 22Þþ d3d3d4d4Π2ð11; 33ÞΛ α1α1¯ ;α2α2¯ −i=2 L t ;t¯ ; tþ;t ε C ð Þ q ð 1 1 2 2Þ μα2α2¯ in two contour times t ;t t × ð33;¯ 44¯ÞLð44;¯ 22¯Þ; ðA24Þ 1 1¯ (with fixed external time 2). The first step in solving the Bethe-Salpeter equation is to ¯ ¯ ¯ ¯ ¯ ¯ where Π2ð11; 22Þ¼Gð12ÞGð2 1Þ − Gð12ÞGð12Þ and the recast it in terms of functions of ordinary times. In Λð2Þ 33;¯ 44¯ δ2Γ G =δG 33¯ comparison to the KB equation, this step is significantly 2PI irreducible vertex ð Þ¼ int½ ð Þ × δG 44¯ Γ G φ D more involved here due to the complex real-time structure ð Þ; here, int½ is given in Eq. (8b) with c and substituted in terms of G from the stationarity condition of three-time and four-time CTP functions and multiple Eqs. (A7b) and (A7c). contour integrals. We leave the cumbersome details for a The NLO effective action yields three contributions to separate publication and solely outline the procedure here. 2 We showed earlier in Appendix A1that the four real-time Λð Þ: matrix elements of two-time functions such as G and D can be fully specified using a single real-time function, e.g., G>. α1α1¯ ;μ A similar analysis of Γq ðt1;t1¯ ; t2Þ, taking into account symmetries and unitarity of evolution, reveals that the eight real-time components of a three-time function as such can be fully specified by three independent functions. ðA25Þ Accordingly, the contour Bethe-Salpeter equation can be explicitly written as three coupled two-dimensional integral The last symbol stands for the three permutations of the first equations in ordinary times; the latter is numerically solved ¯ ¯ three diagrams obtained by ð3 ↔ 3Þ, ð4 ↔ 4Þ, and by discretizing the integrals using approximate quadratures ð3 ↔ 3¯; 4 ↔ 4¯Þ with signs −, −, and þ, respectively. and solving the resulting linear system.

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