Keldysh Functional Renormalization Group for Electronic Properties of Graphene

Keldysh functional renormalization group for electronic properties of graphene

Christian Fräßdorf Dahlem Center for Complex Quantum Systems and, Institut fürTheoretische Physik, Freie UniversitätBerlin, Arnimallee 14, 14195 Berlin, Germany

Johannes E. M. Mosig Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand (Dated: September 20, 2016) We construct a nonperturbative nonequilibrium theory for graphene electrons interacting via the instantaneous Coulomb interaction by combining the functional renormalization group method with the nonequilibrium Keldysh formalism. The Coulomb interaction is partially bosonized in the forward scattering channel resulting in a coupled Fermi-Bose theory. Quantum kinetic equations for the Dirac fermions and the Hubbard-Stratonovich boson are derived in Keldysh basis, together with the exact flow equation for the effective action and the hierarchy of one-particle irreducible vertex functions, taking into account a possible non-zero expectation value of the bosonic field. Eventually, the system of equations is solved approximately under thermal equilibrium conditions at finite temperature, providing results for the renormalized Fermi velocity and the static dielectric function, which extends the zero-temperature results of Bauer et al., Phys. Rev. B 92, 121409 (2015). PACS numbers: 11.10.Hi, 71.10.-w, 72.10.Bg, 72.80.Vp, 73.22.Pr, 73.61.-r, 81.05.ue

2 I. INTRODUCTION the electrons in graphene is α = e /0~vF , which approximately equals 2.2 in the freestanding case in vac- uum ( = 1). For such a large interaction strength The band structure of graphene features two isolated 0 a perturbative calculation of the renormalization effect points where valence and conduction bands touch.1–3 At cannot be reliable, and at first sight the reported agree- these touching points the electrons have a linear energy- ment of one-loop perturbation theory with the experi- momentum dispersion, similar to massless relativistic mentally observed increase of the Fermi velocity appears Dirac particles.4 This pseudorelativistic band structure surprising. Indeed, a two-loop calculation leads a com- is responsible for the appearance of phenomena usually pletely different result, a decrease of the Fermi veloc- related to the relativistic domain, such as Klein tunnel- ity for small momenta.22–24 An alternative approach is ing through potential barriers,5–8 the Zitterbewegung,9 to make use of the largeness of the number of fermion or an anomalous quantized Hall effect.10–13 species (which is Nf = 4 in graphene), and a perturba- For a description of realistic graphene samples, effects tion theory in 1/Nf gives results largely consistent with of disorder and electron-electron interactions have to be the approach based on a perturbative treatment of the added to this idealized band structure. Disorder smears interaction strength.25,26 out the singularity at the nodal point, but preserves To address such a situation in which no small pa- 1,2 many of graphene’s remarkable electronic properties, rameter, to organize a perturbative expansion, is avail- and even leads to fundamentally new phenomena by it- able, nonperturbative methods have been applied to the self, such as the absence of Anderson localization if dis- problem of interacting Dirac fermions in two dimen- 14–16 order does not couple the nodal points. The effect sions. One of those nonperturbative methods is the func- of interactions is most pronounced if the singularity in tional renormalization group (fRG), which shares some the density of states of the noninteracting theory is not features with the celebrated Wilsonian renormalization smeared by disorder and the chemical potential is close group,27,28 but rigorously extends the concept of flow- 17 to the nodal point. The vanishing carrier density at the ing coupling constants to (one-particle irreducible ver- 18 nodal point at zero temperature implies the absence of tex) functions. Initiated by Wetterich,29,30 this method screening, which leads to strongly enhanced interaction has found widespread applications in high energy and in corrections. In particular, interactions are found to effec- condensed matter physics.31–35 Of particular relevance to tively renormalize the Fermi velocity at the nodal point, 36

arXiv:1609.05679v1 [cond-mat.mes-hall] 19 Sep 2016 the present problem is the work of Bauer et al., who and the corrections to the velocity diverge logarithmi- studied the Fermi velocity renormalization and the static 19,20 cally in the low-temperature limit. These logarith- dielectric function in graphene at zero temperature using mic corrections have recently been verified experimen- the fRG framework and found excellent agreement with tally, and good agreement with theoretical calculations the experiment, surpassing the results of the conventional 21 was reported. perturbative methods. Although there is consensus about the way in which As powerful as the fRG is, it clearly has its limita- interactions affect the electronic structure of graphene,17 tions when used within its most commonly employed for- a quantitative evaluation of the corrections proved to be mulation in imaginary time. First and foremost, true problematic. The dimensionless interaction strength for nonequilibrium phenemena (beyond linear response) are 2 out of reach of the Matsubara formalism. Second, even Sec. III, where we combine them with the nonequilibrium for linear response calculations the imaginary time for- Keldysh formalism. We implement an infrared regular- malism requires an analytical continuation from imagi- ization and derive the exact spectral Dyson equations nary to real time at the end of a calculation, which may and quantum kinetic equations, as well as an exact flow pose technical difficulties. The appropriate framework equation, which incorporates all of the nonperturbative to describe true nonequilibrium dynamics is the Keldysh aspects of the theory. Finally, we perform a vertex ex- formalism.37–39 The Keldysh formalism has the addi- pansion leading to an exact, infinite hierarchy of cou- tional advantage that it erases the necessity of analyt- pled integro-differential equations for the one-particle ir- ical continuations, which may also makes it a useful tool reducible vertex functions. Section IV deals with a solu- for equilibrium applications. Gezzi et al. implemented a tion of our theory in thermal equilibrium. We discuss Keldysh formulation of fRG for applications to impurity the necessary limitations for the construction of suit- problems.40 Jakobs et al. further developed the theory, able regulator functions, which preserve causality and, constructing a “Keldysh-compatible” cutoff scheme that at the same time, the fluctuation-dissipation theorem, respects causality, with applications to quantum dots and allowing a solution of the quantum kinetic equations at nanowires coupled to external baths.41,42 Keldysh formu- all scales. We further present a simple truncation scheme lations of fRG were also developed for various systems for the calculation of the Fermi velocity and static dielec- involving bosons.43–47 tric function at finite temperature, extending the results In the present article we construct a Keldysh fRG of Bauer et al.36 theory for interacting Dirac fermions, as they occur at the nodal points in the graphene band structure. As a test of the formalism, we recalculate the Fermi II. NONEQUILIBRIUM QUANTUM FIELD velocity renormalization and the static dielectric func- THEORY tion in graphene, finding full agreement with the zero- temperature Matsubara-formalism calculation of Bauer This section mainly serves as an introduction to the et al.36 We also extend the calculation to finite temper- Fermi-Bose quantum field theory of interacting electrons atures, an extension that in principle is possible within in graphene in the nonequilibrium Keldysh formulation. the Matsubara formalism, too, but that comes at no ad- The reader who is familiar with this formulation may ditional calculational cost when done in the Keldysh for- skim through our notational conventions and continue malism. We leave applications to true nonequilibrium reading at section III. properties of graphene for future work, but already no- We consider interacting Dirac fermions in two dimen- tice that there is a vast body of perturbative (or in other sions, which are described by a grand canonical Hamil- ways approximate) true nonequilibrium theoretical re- tonian in the Heisenberg picture sults for graphene that such a theory can be compared with, see, e.g., Refs. 48–51. Although our theory fo- H(t) = Hf(t) + Hint(t) . (1) cuses on graphene, a major part of the formalism we develop here is also applicable to conventional nonrela- Here Hf describes the low energy approximation of free tivistic fermions. electrons hopping on the honeycomb lattice, and Hint The extension of an imaginary-time fRG formulation contains the interaction effects. The first term reads49 to a Keldysh-based formulation involves quite a number (~ = c = 1) of subtle steps and manipulations. One issue is the choice Z of a cut off scheme, which preferentially is compatible † Hf(t) = Ψ (~r, t) − µ + eϕ(~r, t) Ψ(~r, t) (2) with the causality structure of the Keldysh formalism ~r Z and, for equilibrium applications, with the fluctuation- † s ~ ~ dissipation theorem.41,42 Another issue is the possibility − ivF Ψ (~r, t)σ0 ⊗ Σ · ∇ + ieA(~r, t) Ψ(~r, t) , ~r of an arbitrary nonequilibrium initial condition and the truncation of the (in principle) infinite hierarchy of flow with the chemical potential µ and the external electro- equations in the fRG approach. To do justice to these is- magnetic potentials ϕ and A~. The Dirac electrons are sues, we have chosen to make this article self contained, described by eight-dimensional spinors, where we choose | although we tried to keep the discussion of standard is- the basis as Ψ ≡ Ψ↑ Ψ↓ , with sues as brief as possible. The outline of the paper is as follows: In Sec. II we ψ ψ ψ ψ | Ψσ ≡ AK+ BK+ BK− AK− σ . (3) introduce the formal aspects of nonequilibrium quantum field theory, using the Keldysh technique applied The indices σ =↑, ↓ denote the spin, K± the valley- and s to graphene. The originally purely fermionic problem A/B the sublattice degree of freedom. Further, σ0 is is formulated as a coupled fermion-boson problem by the two-dimensional unit matrix acting in spin space and means of a Hubbard-Stratonovich transformation, sin- Σx,y = τ3 ⊗ σx,y, with the Pauli matrices τ3 and σx,y gling out the dominant interaction channel. The ideas acting in valley and sublattice space, respectively. The of the functional renormalization group are reviewed in interaction part is given by the instantaneous Coulomb 3

Since there are four possibilities where the two time vari- ables can be located to each other with respect to the two time branches C+ and C−, one can map the contour- ordered Green function to a 2 × 2 matrix representation with time-arguments defined on the real axis FIG. 1. Schwinger-Keldysh time contour in the complex time ! plane with reference time t0 as starting and end point. C+ G++(~r, t; ~r0, t0) G+−(~r, t; ~r0, t0) and C are the forward and backward branch, respectively. 0 0 ij ij − Gij(~r, t; ~r , t ) = −+ 0 0 −− 0 0 Gij (~r, t; ~r , t ) Gij (~r, t; ~r , t ) GT (~r, t; ~r0, t0) G<(~r, t; ~r0, t0)! interaction ij ij = > 0 0 T¯ 0 0 . (9) Z Gij(~r, t; ~r , t ) Gij(~r, t; ~r , t ) 1 0 0 Hint(t) = δn(~r, t)V (~r − ~r )δn(~r , t) , (4) 2 ~r,~r0 The constituents of this matrix are the time ordered, anti-time ordered, greater and lesser Green function, re- where spectively, e2 V (~r − ~r0) = , (5) T 0 0 † 0 0 0 Gij(~r, t; ~r , t ) = −ihT ψi(~r, t)ψj (~r , t )i , (10a) 0|~r − ~r | ¯ † T 0 0 ¯ † 0 0 δn(~r, t) = Ψ (~r, t)Ψ(~r, t) − n˜(~r, t) , (6) Gij(~r, t; ~r , t ) = −ihT ψi(~r, t)ψj (~r , t )i , (10b) > 0 0 † 0 0 G (~r, t; ~r , t ) = −ihψi(~r, t)ψ (~r , t )i , (10c) and 0 is the dielectric constant of the medium, being ij j unity for freestanding graphene in vaccuum. Here the < 0 0 † 0 0 Gij(~r, t; ~r , t ) = +ihψj (~r , t )ψi(~r, t)i . (10d) termn ˜(~r, t) is a background charge density, representing the charge accumulated on a nearby metal gate. Away By definition, these functions are linearly dependent and from the charge neutrality point it essentially acts as a subject to the following constraint,37–39 counterterm, which removes the zero wavenumber singu- ˆT ˆ< ˆ> ˆT¯ larity of the Coulomb interaction at finite charge carrier G − G − G + G = 0 , (11) density. which allows a basis transformation to three linearly independent propagators. This transformation is given by the involutional matrix τ L, where τ is a Pauli matrix A. Single-particle Green functions 1 1 and L is the orthogonal matrix Relevant physical observables can be expressed as cor- 1 1 −1 L = √ , (12) relation functions of the field operators, and the purpose 2 1 1 of a field-theoretic treatment is to provide a formalism in which such correlation functions can be calculated originally introduced by Keldysh.52 Its application to efficiently. For an explicitly time-dependent Hamilto- Eq. (9) yields nian, such as the one above, one considers the evolu- ˆK ˆR −1 G G tion of the field operators along the “Schwinger-Keldysh τ1LGˆ(τ1L) = , (13) contour”,37–39 a closed time contour starting at a refer- GÂ 0 ence time t , extending to +∞, and eventually returning 0 with from +∞ to t0, see Fig. 1. Consequently, the time arguments t of the field operators are elevated to the “contour GˆR = 1 GˆT − Gˆ< + Gˆ> − GˆT¯ , (14a) time”, and the building blocks of the theory are formed 2 by the expectation values of “path ordered” products of GÂ = 1 GˆT + Gˆ< − Gˆ> − GˆT¯ , (14b) the field operators. The concept of path ordering gen- 2 eralizes the concept of (imaginary) time ordering, such ˆK 1 ˆT ˆ< ˆ> ˆT¯ G = 2 G + G + G + G . (14c) that field operators with a higher contour time appear to the right of operators with a lower contour time. In The functions GˆR/A/K are the retarded, advanced and particular, the single-particle propagator reads Keldysh propagators, respectively. The latter one is also known as the statistical propagator. They obey the sym- GTC (~r, t; ~r0, t0) = −ihT ψ (~r, t)ψ†(~r0, t0)i , (7) ij C i j metry relations where the indices i, j represent collectively the sublattice, (GˆR)† = GÂ , (GˆK )† = −GˆK , (15) valley and spin degrees of freedom. TC is the contour-time ordering operator and the expectation value is performed as well as the causality relations37–39 with respect to some initial density matrix given at a ˆR 0 0 0 reference time t0 G (~r, t; ~r , t ) = 0 , if t < t , (16a) GÂ(~r, t; ~r0, t0) = 0 , if t > t0 . (16b) h· · · i = Tr[ρ(t0) ··· ] . (8) 4

Explicit expressions for the free propagators may easily B. Contour-time generating functional be obtained in thermal equilibrium and in the absence of the electromagnetic potentials. To this end we send the The entire physical content of the theory can be con- reference time t0 → −∞ and Fourier transform the field veniently expressed by the partition function37,38,43,53,54 operators following the conventions D iη†Ψ+iΨ† E Z Z[η; ρ] = TCe , (24) +i~k·~r−iεt ~ ψi(~r, t) = e ψi(k, ε) , (17) ~k,ε which is a generating functional for all n-point correlation functions, including the single-particle propagators R R d2k dε described above. Its arguments η and η†, where only the with ~ ≡ 2 . After a short calculation one finds k,ε (2π) 2π former is shown on the left hand side for brevity, are eight component spinorial external source terms. Here and in ˆR/A ~ 1 G0 (k, ε) = , (18a) the remainder of this article we employed a condensed s ~ ~ σ0 ⊗ Σ0(ε + µ ± i0) − vF Σ · k vector notation ε Z Z GˆK (~k, ε) = tanh GˆR(~k, ε) − GÂ(~k, ε) , (18b) † † † † 0 2T 0 0 η Ψ ≡ η (x)Ψ(x) , Ψ η ≡ Ψ (x)η(x) , (25) C,x C,x where Σ0 = τ0 ⊗ σ0 is the 4 × 4 unit matrix in valley- where x = (~r, t) labels space and (contour-) time coordi- sublattice space. Note that the entire statistical infor- nates, such that mation of the system is contained in the Keldysh propagator. These expressions may be further simplified by Z Z Z ≡ dt d2r . (26) expanding the propagators in the chiral basis C,x C

R/A X R/A Gˆ (~k, ε) = Pˆ (kˆ)G (k, ε) , (19) The symbol C indicates that the time integration has to 0 ± ±,0 be performed along the Schwinger-Keldysh closed time ± contour. An important property of the partition function is that it is normalized to unity when the sources are set in which Pˆ (kˆ) are the chiral projection operators ± equal to zero55 ! Σ ± Σ~ · kˆ Z[0; ρ] = Tr ρ(t ) = 1 . (27) Pˆ (kˆ) = σs ⊗ 0 , (20) 0 ± 0 2 In fact, this normalization is the very reason for the algebraic identity (11) and it leads to similar constraints with kˆ = ~k/k. In the chiral basis the propagators then for higher order correlation functions, see Ref. 37. It fur- take the simple form ther ensures that any correlation function computed from the partition function (24) does not contain disconnected R 1 bubble diagrams. G±,0(k, ε) = , (21a) (ε + µ + i0) ∓ vF k The partition function (24) can be represented in

A 1 terms of a fermionic coherent state functional integral G±,0(k, ε) = , (21b) as37,38,53,54 (ε + µ − i0) ∓ vF k Z K ε † iS[ψ]+iK [ψ]+i~η†Ψ+iΨ~ †η G (k, ε) = −2πi tanh δ(ε + µ ∓ vF k) . (21c) Z[η; ρ] = DψDψ e ρ . (28) ±,0 2T

The density of electrons in the system is given by Here S[ψ] is the contour-time action of the system and Kρ[ψ] is the correlation functional, which incor- n(~r, t) = −i tr Gˆ<(~r, t, ~r, t) porates the statistical information of the initial density matrix.37,54,56 Their dependence on the Grassmann- i K R A = − tr Gˆ − Gˆ − Gˆ (~r, t, ~r, t) , (22) valued spinor fields Ψ and Ψ† has been abbreviated by ψ, 2 as we did for the source field dependence of the partition which is formally divergent. The charge carrier density, function. however, which is defined as49 The action can be written as a contour-time integral over the Lagrangian L(t) i n¯(~r, t) = − tr GˆK (~r, t, ~r, t) , (23) Z 2 S[ψ] = L(t) , (29) C,t is finite. It is a function of the external doping µ and of the gauge invariant external electromagnetic fields. In with Z the absence of such external fields, it vanishes at the † L(t) = Ψ (x)i∂tΨ(x) − H(t) . (30) charge neutrality point (µ = 0). ~r 5

Similarly to the Hamiltonian (1), the action decomposes expressions for which can be obtained immediately by into free contribution and an interaction term, substitution of Eqs. (2) and (4). The functional Kρ[ψ] describes the initial correlations of the system, corresponding to the density matrix ρ(t0). S[ψ] = Sf[ψ] + Sint[ψ] , (31) It may be expanded in powers of ﬁelds as

∞ m Z X (−1) X (2m) 0 0 0 0 Kρ[ψ] = 2 Kρ (x1i1, . . . , xmim; x1i1, . . . , xmim) (m!) 0 C,xmxm 0 m=0 im,im † † 0 0 ×ψ (x ) . . . ψ (x )ψ 0 (x ) . . . ψ 0 (x ) , (32) i1 1 im m im m i1 1

(2m) where the kernels Kρ are nonvanishing only, if all their The free bosonic part is given by respective contour-time arguments equal the initial time 1 Z t0. The statistical information contained in the kernels −1 (2m) Sb[φ] = φ(x)V (x − y)φ(y) , (34) Kρ , specifying the correlations present in the initial 2 C,xy state, is in a one-to-one correspondence to the statistical information contained in the density matrix.47,54,56 In where V −1 is the inverse Coulomb interaction, un- practice, only a limited set of initial correlations is taken derstood in the distributional sense. The interaction into account, either because of an implicit assumption term contains a trilinear Yukawa-type interaction and that the initial state is a thermal equilibrium state for an a linear term, describing the coupling of the Hubbard- effectively noninteracting system,38,39 or as an expression Stratonovich boson to the background charge den- of the finite knowledge that is available about an experi- sityn ˜(x) 56 mental setup. In the remainder of this work we mainly Z focus on Gaussian density matrices, i.e., we truncate the † Sint[ψ, φ] = − φ(x) Ψ (x)Ψ(x) − n˜(x) . (35) series (32) after the first term, absorbing the statistical C,x (2) information of Kρ into the boundary conditions of the two-point function and simply write Z[η, ρ] ≡ Z[η]. Yet Note that the fluctuating Bose field φ appears on the most of our results are not affected by this simplification same footing as the external scalar potential ϕ, see and valid even in the general case. We come back to this Eq. (2). issue in section III E, where we comment on some ques- We generalize the Hubbard-Stratonovich transformed tions regarding the possible implementation of correlated partition function by introducing an additional source initial states. term φ|J, so that it gives access to bosonic as well as mixed Fermi-Bose correlators. The generalized Fermi- Although it is possible to treat the theory presented Bose partition function reads so far within the formalism of the (fermionic) functional 31,57 Z renormalization group, we here choose a formula- † † | Z[η, J] = DψDψ†Dφ eiS[ψ,φ]+iη Ψ+iΨ η+iφ J , (36) tion in which a bosonic field is introduced by means of a Hubbard-Stratonovich transformation, that decou- 35,38,53 ples the Coulomb interaction. It is well-known that with S[ψ, φ] = Sf[ψ] + Sb[φ] + Sint[ψ, φ]. It fulfills the bosonic degrees of freedom, such as Cooper pairs in the same normalization condition, when the sources are set celebrated BCS-theory of superconductivity,53 naturally to zero, as the purely fermionic partition function emerge as collective, low energy degrees of freedom of composite fermions. Therefore, it is reasonable to intro- Z[0, 0] = 1 . (37) duce a collective bosonic field right from the beginning, which captures the dominant contributions of the interaction. C. Real-time representation The Hubbard-Stratonovich transformation is an exact integral identity replacing the four-fermion interaction Although the contour-time representation allows for a compact and concise notation during any step of a Sint[ψ] by a quadratic form of a real bosonic field and a Fermi-Bose interaction calculation, it is desirable to formulate the theory in a single-valued “physical” time which appeals to physical intuition and transparency. Hereto one splits the contour Z C into forward (C+) and backward (C−) branch, thereby eiSint[ψ] = Dφ eiSb[φ]+iSint[ψ,φ] . (33) defining a doubled set of fields, Ψ± and φ±, allocated to 6 the respective branch The action S[ψ, φ] is the sum of three contributions, Z S[ψ, φ] = L[ψ, φ] S[ψ, φ] = Sf[ψ] + Sb[φ] + Sint[ψ, φ] . (44) C,t Z Z Its quadratic part in the fermionic sector is given by = L[ψ+, φ+] + L[ψ−, φ−] . (38) Z Ψ (y) C+,t C−,t S [ψ] = Ψ†(x)Ψ†(x) Gˆ−1(x, y) c . (45) f c q 0 Ψ (y) In a next step, one performs a rotation from ±-field space xy q to Keldysh space, using the involutional matrix τ1L, see ˆ−1 Eq. (12), which was already employed for the rotation The inverse free propagator G0 has a trigonal matrix of the Green functions in Sec. II A. Further, one defines structure the symmetric and antisymmetric linear combinations of 0 (GÂ)−1 Gˆ−1 = 0 , (46) the ±-fields as “classical” (c) and “quantum” (q) com- 0 (GˆR)−1 (Gˆ−1)K ponents, respectively, and combines these into vectors 0 0 † Ψ, Ψ and φ as ˆR/A −1 with retarded/advanced (G0 ) and Keldysh blocks Ψ Ψ (Gˆ−1)K , which obey the symmetries38,39 Ψ ≡ c ≡ τ L + , Ψ† = (Ψ)† , (39) 0 Ψ 1 Ψ q − † † (GˆR)−1 = (GÂ)−1 , (Gˆ−1)K = −(Gˆ−1)K . φc 1 φ+ 0 0 0 0 φ ≡ ≡ √ τ1L . (40) φq 2 φ− (47) The retarded/advanced blocks are the inverse free re- The source fields are rotated and combined into vectors tarded/advanced propagators η, η† and J likewise. Two remarks are in order. First, the mapping of the bosonic source term yields an addi- (GˆR/A)−1(x, y) = δ(x − y)σs ⊗ Σ iD + iv Σ~ ·D , tional factor of two, due to our choice of normalization 0 0 0 y0 F ~y in Eq. (40), which we choose to absorb into a redefini- (48) tion of J. The second remark is concerned about our where the gauge covariant derivative is given by definition of the Keldysh rotation for the fermionic field Ψ†. Some authors prefer a different convention, which ~ iDx = i∂x ± i0 + µ − eϕ(x) , D~x = ∂~x + ieA(x) . (49) was originally proposed by Larkin and Ovchinnikov.58 In 0 0 a purely fermionic theory this is reasonable, since it leads Note that the regularization term ±i0, which we have to a certain technical simplification. However, this mod- written here explicitly, enforces the retarded, respectively ified rotation is not possible for bosons. In the context advanced, boundary condition. It has to be emphasized of the coupled Fermi-Bose theory we are dealing with, that the external gauge fields therein are understood as the implementation of the Larkin-Ovchinnikov rotation entirely classical would lead to an asymmetry in the arising Keldysh struc- 1 tures, which we want to avoid. Therefore, we define the ϕ(x) ≡ ϕc(x) = 2 ϕ+(x) + ϕ−(x) , (50a) Keldysh rotation as proposed in Eq. (39). Further, one ~ ~ 1 ~ ~ has to keep in mind that the naming “classical” for the A(x) ≡ Ac(x) = 2 A+(x) + A−(x) . (50b) fermions is just terminology. For the bosons on the other hand this naming has a physical meaning. Since these fields are not quantized, their quantum com- We here summarize the main results of the real-time ponents in Keldysh space vanish identically. Yet it is for- mapping and explain the structure of the theory obtained mally possible to keep them as source fields, which could after the above Keldysh rotation. For the partition func- be used to generate density-density or current-current tion Z[η, J] we find correlation functions.38 On the other hand this is not necessary, since we have the single-particle sources η at Z † † Z[η, J] = DψDψ†Dφ eiS[ψ,φ]+iη τ1Ψ+iΨ τ1η+iφ|τ1J . our disposal. In contrast to the retarded and advanced ˆ−1 K blocks of Eq. (46), the Keldysh block (G0 ) does not (41) take the form of a simple inverse propagator. It carries We have used here the short-hand notation the statistical information of the theory and can be written as Z Ψ (x) η†τ Ψ ≡ η†(x) η†(x) τ c , (42) 1 c q 1 Ψ (x) ˆ−1 K ˆR −1 ˆK Â −1 x q (G0 ) = −(G0 ) G0 (G0 ) , (51) in which the Pauli matrix τ acts in Keldysh space, cou- 1 with the noninteracting Keldysh Green function GˆK . pling a “classical” source to a “quantum” field and vice 0 Since the latter is an anti-hermitian matrix, see Eq. (15), versa. Further, all the time integrations are defined from it can be parametrized in terms of a hermitian matrix Fˆ0 now on along the forward time branch C+ only and the spectral functions GˆR/A as38 Z Z Z ∞ Z 0 2 ≡ = dt d r . (43) ˆK ˆR ˆ ˆ Â x C+,x t0 G0 = G0 F0 − F0G0 . (52) 7

Substitution into Eq. (51) then yields that for noninter- well. The trilinear term maps to four interaction terms in acting fermions the Keldysh block of the inverse matrix real-time, which can be arranged in a matrix form similar propagator is a pure regularization term59 to Eq. (45), ˆ−1 K ˆ Z (G0 ) = 2i0F0 . (53) † φq(x) φc(x) Sint[ψ, φ] = − Ψ (x) Ψ(x) x φc(x) φq(x) Only when interactions are considered the Keldysh block Z will acquire a finite value. We will come back to this issue + 2 φ|(x)τ1n˜(x) . (61) in section III C. The free propagator Gˆ0 is obtained by x inverting Eq. (46), where the Keldysh structure is given Note the factor of two in front of the linear term in com- by Eq. (13). parison to the linear source term, which could not be The quadratic part of the action in the bosonic sector absorbed into a redefinition of any of those fields as was reads the case for J. Further observe that the classical com- 1 Z ponents of the fluctuating Bose field appear in the same | −1 Sb[φ] = φ (x)D0 (x, y)φ(y) . (54) 2 xy off-diagonal position as the external gauge field ϕ does in Eq. (45). The quantum components on the other hand −1 The bosonic matrix D0 has the same trigonal structure are located in the diagonal. as the fermionic one Now that all of our notational conventions have been established we can move on to the central part of this A −1 −1 0 (D0 ) work. D0 = R −1 −1 K , (55) (D0 ) (D0 ) with the same symmetry relations as Eq. (47). Owing III. NONEQUILIBRIUM FUNCTIONAL to the fact that the bosons are real, the above quantities RENORMALIZATION GROUP fulfill the additional symmetries38,39

R −1| A −1 −1 K | −1 K The idea of the functional renormalization group is to (D0 ) = (D0 ) , (D0 ) = (D0 ) . (56) modify the bare action of the theory by introducing a dependence on a parameter Λ, in such a way that the The retarded and advanced blocks are twice the inverse partition function can be easily (and exactly) calculated bare Coulomb interaction if Λ is set equal to an initial value Λ0, whereas the true R/A −1 −1 physical system corresponds to Λ = 0. Using the solution (D0 ) (x, y) = 2V (x − y) . (57) of the modified partition function at Λ = Λ0, one obtains The Keldysh component for bosons has the same struc- the “physical” partition function at Λ = 0 by tracking ture as the fermionic one its changes upon lowering Λ from Λ0 to 0. In practice the parameter Λ is chosen as an infrared regularization −1 K R −1 K A −1 which effectively removes low-energy (or low-momentum) (D0 ) = −(D0 ) D0 (D0 ) . (58) modes, determined by the cutoff Λ, from the functional Similarly to the fermionic case we can parametrize the integration. In this case, the initial value Λ0 is the ul- bosonic Keldysh Green function in terms of a hermitian traviolet cutoff of the action S[ψ, φ]. For graphene, this 38 function B0 ultraviolet cutoff is the momentum or energy at which the linear dispersion in Eq. (2) breaks down. K R A D0 = D0 B0 − B0D0 . (59)

Since the bare Coulomb interaction is instantaneous, the A. Infrared regularization above Keldysh propagator together with the Keldysh block (58) vanish identically. For that reason we may write We implement the idea of an infrared regularization by modifying the quadratic terms in the Fermi- and Bose- −1 −1 −1 D = 2V ≡ 2V τ1 . (60) sectors of the contour-time action via additive regulator 0 29,32 functions Rˆf,Λ,Rb,Λ Again, the interaction with the fermions will eventually † lead to a finite bosonic Keldysh self-energy and, hence, Sf[ψ] → Sf,Λ[ψ] = Sf[ψ] + Ψ Rˆf,ΛΨ , (62a) a nonvanishing Keldysh propagator as in the fermionic 1 S [φ] → S [φ] = S [φ] + φ|R φ . (62b) case. b b,Λ b 2 b,Λ Finally we discuss the Fermi-Bose interaction term. Its linear counterterm maps in the same way as the sources It is also possible to regularize only one of the two sec- do, but with the important difference that the quantum tors, by setting either Rˆf,Λ or Rb,Λ to zero. The regula- componentn ˜q(x) is identically zero. Nevertheless, we tors have to be analytic functions of Λ. For Λ → Λ0 they may still use the Keldysh vector notation for this term as have to diverge, such that all infrared modes occuring in 8 the functional integral are effectively frozen out, while for B. Connected functional and effective action Λ → 0 they have to vanish.31,32,35 In this way the partition function (36) becomes a cutoff dependent quantity, The evolution equation will not be derived for the par- Z[η, J] → Z [η, J], where only the modes above Λ con- Λ tition function ZΛ[η, J], but rather for the effective ac- tribute to the functional integral. In the limit Λ → 0 it tion ΓΛ[ψ, φ], which is essentially the Legendre transfor- reduces to the original partition function of the previous mation of the cutoff dependent connected functional37,53 section, see Eq. (41). After mapping the contour-time regulator terms to a WΛ[η, J] = −ilnZΛ[η, J] , (66) real-time repesentation and performing the Keldysh rotation as explained in Sec. II C, the cutoff dependent being a generating functional for connected correlation quadratic parts of the action become functions. Differentiation with respect to the sources yield the expectation values of the fields Ψ(x) and φ(x), † ˆ Sf,Λ[ψ] = Sf[ψ] + Ψ Rf,ΛΨ , (63a) δWΛ δWΛ † † = +τ1 Ψ(x) , = − Ψ (x) τ1 , (67) Sb,Λ[φ] = Sb[φ] + φ|Rb,Λφ . (63b) δη (x) δη(x) δWΛ = φ|(x) τ . (68) Note the absence of the factor 1/2 in front of the bosonic δJ(x) 1 regulator term, which is due to our choice of normalization for the bosonic rotation (40). In principle, the most These expectation values, being complicated nonlinear general choice for the contour-time regulators results in functionals of the sources η and J, define “macroscopic” the following 2×2 matrix structure for the real-time reg- fields which inherit a Λ-dependence from the regulators ulators (and the counterterm). A macroscopic Fermi field can only exist when the sources are finite, otherwise it is ˆZ Â ! strictly zero. The classical component of the macroscopic Rf,Λ(x, y) Rf,Λ(x, y) Rˆf,Λ(x, y) = , (64a) bosonic field hφc(x)i, on the other hand, can very well ac- ˆR ˆK 37,43–45 Rf,Λ(x, y) Rf,Λ(x, y) quire a finite value in the absence of source terms. RZ (x, y) RA (x, y) Such a macroscopic field expectation value may signal a R (x, y) = b,Λ b,Λ . (64b) b,Λ RR (x, y) RK (x, y) spontaneous symmetry breaking, but in the theory we b,Λ b,Λ consider here this is not the case. The bosonic field φ(x) is conjugate to the particle density n(x) and as such it re- Although it is not strictly necessary if the evolution flects, e.g., a local deviation away from charge neutrality from Λ = Λ0 to Λ = 0 could be tracked exactly, driven by an external potential ϕ(x). In the following we for the correct implementation of approximate evolution omit the brackets to denote the average of a single field, schemes it is important that the regulators are chosen for brevity. Since we are always working with averages in such a way that they respect the symmetries and the of fields, there can be no confusion. causality structure of the theory. In particular, in or- The second derivatives of WΛ define the connected two- der to ensure that the partition function is normalized to point correlators unity at any scale, and hence retain the algebraic iden- δ2W tities among the correlation functions, cf. Eq. (11), we Λ = −iτ Ψ(x)Ψ†(y) τ , (69) choose the regulators such that the “anomalous” compo- δη†(x)δη(y) 1 c 1 nents RˆZ ,RZ vanish. The remaining components are 2 f,Λ b,Λ δ WΛ constructed in such a way that they are compatible with = +iτ1 φ(x)φ|(y) τ1 , (70) δJ |(x)δJ(y) c the symmetry and causality structure of the bare inverse propagators, see Eqs. (47) and (56). This choice of the where we introduced the connected average hABic ≡ regulator functions ensures that the partition function hABi − hAihBi. Explicitly displaying the 2 × 2 Keldysh has the correct causality structure at any value of the structure, we have cutoff Λ, independent of eventual approximations made † when solving the evolution equations. hΨ(x)Ψ (y)ic ≡ iGˆΛ(x, y|η, J) (71) In addition to the Λ-dependencee of the action intro- GˆK (x, y|η, J) GˆR(x, y|η, J) = i Λ Λ , duced via Eqs. (63) we allow the counterterm to be ex- GÂ(x, y|η, J) GˆZ (x, y|η, J) plicitly cutoff dependent, setting Λ Λ

hφ(x)φ|(y)ic ≡ iDΛ(x, y|η, J) (72) n˜ → n˜Λ (65) K R DΛ (x, y|η, J) DΛ (x, y|η, J) = i A Z . DΛ (x, y|η, J) DΛ (x, y|η, J) The countertermn ˜Λ describes a flowing background charge density, which has to be tuned to remove poten- The above propagators are source- and cutoff-dependent tially divergent contributions from the Coulomb interac- functionals, which do not obey the usual triangular struc- tion at finite charge carrier density. ture. In particular the anomalous statistical propagators 9

ˆZ Z GΛ and DΛ are nonvanishing as long as the source terms Here we have defined the matrices are finite. By construction of the regulators, the famil- ˆ ˆ ˆ| iar triangular structure together with the symmetry and RΛ ≡ diag −Rf,Λ, Rf,Λ, 2Rb,Λ , (78) causality relations arise once the single-particle sources τ ≡ diag(τ , τ , τ ) , (79) are set to zero. All the other higher order connected 1 1 1 1 correlation functions can be obtained by further differen- ˆ (2) ˆ(2) 53 and the Hesse matrices WΛ and ΓΛ of second func- tiation as in the equilibrium Matsubara theory. tional derivatives The central object in the functional renormalization  |  group is the effective action ΓΛ[ψ, φ]. It is the generating δη† δη −δη† δη† −δη† δJ (2) | | | | functional for one-particle irreducible vertex functions, Wˆ = −δ δη δ δ † δ δJ  W , (80) Λ  η η η~ η~  Λ and defined as the Legendre transform of the connected | | | | −δJ δη δJ δη† δJ δJ functional WΛ  |  δΨ† δΨ δΨ† δ † δΨ† δφ † † Ψ Γ [ψ, φ] =W [η , J ] − η τ Ψ − Ψ τ η − φ|τ J (2) Λ Λ Λ Λ Λ 1 1 Λ 1 Λ Γˆ =  δ| δ δ| δ| δ| δ  Γ . (81) Λ Ψ Ψ Ψ Ψ~ † Ψ~ φ Λ †   − Ψ~ Rˆf,ΛΨ~ − φ|Rb,Λφ . (73) | | | | δφδΨ δφδΨ† δφδφ In the Legendre transform the single-particle sources The inversion relation (77) generalizes the standard must be understood as Λ-dependent functionals of the Dyson equations for single-particle propagators to field expectation values, obtained by inversion of the the case of source-dependent functional propagators, defining relations Eqs. (67) and (68). The Legendre Eqs. (71), (72). If the sources are finite, Eq. (77) also transform is modified in such a way that the cutoff terms includes mixed Fermi-Bose correlators, which disappear are subtracted on the right hand side. This ensures that upon setting the sources to zero. Applying further func- the flowing action does not contain the cutoff terms at tional derivatives to this equation yields a tree expansion any scale, but spoils the convexity of an ordinary Legen- of a connected n-particle correlation function in terms m- dre transform. particle vertex functions (m ≤ n) and full propagators; The properties and physical interpretation of this see Refs. 31, 35, and 53. functional, mainly in the context of its equilibrium counterpart, have been discussed at length in the literature.31,32,35 Most importantly the flowing action has C. Dyson and quantum kinetic equations in the the nice property that it interpolates smoothly between functional renormalization group the microscopic laws of physics, parametrized by an action ΓΛ0 , and the full effective action ΓΛ=0, where all Evaluating the generalized Dyson equation at vanish- thermal and quantum fluctuations are taken into ac- ing sources we obtain the scale dependent nonequilibrium count. In many cases the microscopic laws are simply Dyson equations for Fermions and Bosons governed by the bare action of the system Γ = S. This Λ0 latter statement, however, depends on the actual cutoff ˆ−1 ˆ ˆ ˆ ˆ G0 − ΣΛ + Rf,Λ GΛ = 1 , (82) scheme. In certain situations it is preferable to devise a −1 cutoff scheme where the initial effective action does not 2 V + ΠΛ + Rb,Λ DΛ = 1 , (83) coincide with the bare action, and hence the initial conditions of the flow are nontrivial.35,41,42,60,61 We will come where we employed the definition of the unregularized back to this issue at the end of the next subsection. inverse full propagators Taking the first functional derivative of Eq. (73) with 2 respect to the fields one finds that the effective action δ ΓΛ ˆ−1 ˆ = − G − ΣΛ (x, y) , (84) δΨ†(x)δΨ(y) 0 satisfies the “equations of motion” φc=φ¯c Z δ2Γ δΓΛ Λ = 2 V −1 + Π (x, y) . (85) = −τ1ηΛ(x) − Rˆf,Λ(x, y)Ψ(y) , (74) Λ † δφ|(x)δφ(y) ¯ δΨ (x) y φc=φc Z δΓΛ † † ˆ Here Eqs. (84) and (85) define the (fermionic) self-energy = +ηΛ(x)τ1 + Ψ (y)Rf,Λ(y, x) , (75) δΨ(x) y Σˆ Λ and the (bosonic) polarization function ΠΛ, respec- Z δΓΛ tively. The latter is also known as bosonic self-energy, = −τ1JΛ(x) − 2 Rb,Λ(x, y)φ(y) . (76) which will be used synonymously in the remainder of this δφ|(x) y work.62 By construction of the regulators, the fermionic The second functional derivatives of the connected func- and bosonic self-energies have the same trigonal structure tional WΛ[η, J] and the second functional derivatives of in Keldysh space as the inverse free propagators and the the effective action ΓΛ[ψ, φ] are subject to an inversion regulators relation,31,35,53 which can be written in the compact form 0 Σˆ A 0 ΠA Σˆ = Λ , Π = Λ . (86) ˆ(2) ˆ ˆ (2) ˆ Λ ˆ R ˆ K Λ ΠR ΠK − ΓΛ + RΛ τ1WΛ τ1 = 1 . (77) ΣΛ ΣΛ Λ Λ 10

Besides, they inherit their causality and symmetry rela- Eqs. (51) and (58) to the interacting case and the pres- tions, see Eqs. (47) and (56). ence of infrared regulators. The diagonal components of Eqs. (82) and (83) contain Continuing the parallels with the noninteracting case, ˆK K the respective retarded and advanced Dyson equations the flowing full Keldysh propagators GΛ and DΛ can be paramterized in terms of cutoff dependent hermitian ˆ ˆR/A−1 ˆ R/A ˆR/A ˆR/A ˆ matrices FΛ and BΛ, respectively, G0 − ΣΛ + Rf,Λ GΛ = 1 , (87) ˆK ˆR ˆ ˆ Â −1 R/A R/A R/A GΛ = GΛ FΛ − FΛGΛ , (91) 2 V + ΠΛ + R D = 1 , (88) b,Λ K R A DΛ = DΛ BΛ − BΛDΛ . (92) whereas their off-diagonal yield the Keldysh Green func- This paramterization can be used to derive an equation tions of motion for each of the distribution functions FˆΛ and B . Such equations of motion are known as the quan- ˆK ˆR ˆ K ˆK Â Λ GΛ = −GΛ −ΣΛ + Rf,Λ GΛ , (89) tum kinetic equations.38 To this end we insert the above

K R K K A parametrization into Eq. (89), respectively Eq. (90). Ap- DΛ = −2DΛ ΠΛ + Rb,Λ DΛ . (90) plying the retarded inverse full propagator from the left and the advanced one from the right, we obtain the two These relations are a straightforward generalization of kinetic equations

h ˆ ˆ−1i ˆK ˆR ˆ ˆ ˆA ˆ K ˆ R ˆ ˆ ˆ A FΛ, G0 + Rf,Λ − Rf,ΛFΛ − FΛRf,Λ = ΣΛ − ΣΛ FΛ − FΛΣΛ , (93)

h −1i K R A K R A BΛ,V + Rb,Λ − Rb,ΛBΛ − BΛRb,Λ = −ΠΛ + ΠΛ BΛ − BΛΠΛ , (94)

where [·, ·] denotes the commutator. The left hand side the Keldysh propagators of these equations is the kinetic term, while their right ˆK ˆR ˆ ˆ Â hand side is known as the collision integral. Note that the Rf,Λ = Rf,ΛFΛ − FΛRf,Λ , (95) K R A commutator for the bosonic distribution function BΛ does Rb,Λ = Rb,ΛBΛ − BΛRb,Λ . (96) not involve any time derivatives: The dynamics of BΛ is entirely driven by the bosonic collision integral, and thus As a consequence the regulators on the left hand side of induced by the dynamics of the fermions. In a general the above kinetic equations drop out and we are left with nonequilibrium situation the kinetic terms do not vanish the kinetic equations in their standard form, as if no reg- and, hence, the Keldysh self-energies do not admit the ulators were present, see Ref. 38. Especially in the treat- same decomposition as the Keldysh propagators, leading ment of equilibrium problems this fact has a great advan- to a finite collision integral. tage. Namely, it is possible to solve the kinetic equations at all scales simultaneously with the well-known equilib- We want to stress that the Λ-dependence of the dis- rium distribution functions. In this way the results of tribution functions, since it is a parametric one, poses the Matsubara formalism are reproduced directly in real a serious complication. The kinetic equations have to time, avoiding the necessity of cumbersome analytic con- be solved at each scale, together with the flow equa- tinuations. We will come back to the equilibrium problem tions for the various self-energies and higher order vertex in the final section of this article, Sec. IV. The drawback functions, selfconsistently. The latter set of flow equa- of these schemes, however, is that the initial conditions tions will be derived in the next subsections. Therefore of the flow equations, become nontrivial as pointed out further approximations are inevitable, if one hopes to in the Refs. 41–44, meaning that ΓΛ in the limit Λ → Λ0 obtain numerical solutions for a specific nonequilibrium does not coincide with the bare action S. On the other problem. For example, if the external fields are taken hand this is a rather small price to pay. to be slowly varying functions of time and/or space, one could use a Wigner transformation and perform a gradient expansion to some low order.38,63 Often this approx- D. Exact flow equation imation is combined with the so-called quasiparticle approximation, which reduces the phase space of the distri- The implementation of the infrared regulators de- bution functions and eventually leads to the Boltzmann scribed above enables us to derive an exact evolution transport equation. An important technical simplifica- equation for the effective action ΓΛ, which describes its tion is achieved by the class of cutoff schemes where the flow in the infinite dimensional space of all possible ac- Keldysh regulators are parametrized in the same way as tions as a function of the flowing cutoff Λ. The flow 11

31,32 equation for ΓΛ follows upon taking the Λ-derivative of other. We thus find the defining relation, Eq. (73), at a fixed field configu- ~ † ˆ ~ ration. To this end recall that the connected functional ∂ΛΓΛ = ∂ΛWΛ − Ψ ∂ΛRf,ΛΨ − φ|∂ΛRb,Λφ , (97) WΛ[ηΛ, JΛ] therein has an explicit and an implicit Λ- where the scale-derivative of the first term on the right dependence. The flow of the sources ηΛ and JΛ, viewed as functionals of the fields Ψ and φ, does not contribute hand side, ∂ΛWΛ, has to be performed for fixed source fields η and J. It obeys an exact flow equation as well, to the flow of ΓΛ as the respective terms cancel each which is readily derived from the definition (66)

Here the trace Tr encompasses an integration over posi- E. Vertex expansion tion and time, as well as a summation over the Keldysh components c and q and, for fermions, a summation over In practice, the exact flow equation (101) is too com- the spin, valley and sublattice indices. Note the oc- plex to be solved directly. Instead, one has to resort to curence of the flowing counterterm n˜Λ on the right hand approximation schemes. side, and recall that it possesses a classical component A particularly crude approximation scheme is to ne- only. Upon insertion of Eq. (98) into Eq. (97) the addi- ˆ(2) tional regulator terms cancel, such that the flow equation glect the Λ-dependence of ΓΛ on the right hand side contains connected functional propagators and the coun- of the flow equation (101) and replace it by its initial terterm only. Making use of Eqs. (69), (70) and (80), we value at the scale Λ = Λ0. In this approximation the can write our intermediate result in the compact form single-scale derivative ∂/Λ turns into an ordinary one and the flow equation can be integrated exactly. For certain i (2) cutoff schemes (see Ref. 32) this approximation then im- ∂ Γ = − STr (∂ Rˆ )τ Wˆ τ + 2φ|τ ∂ n˜ . Λ Λ 2 Λ Λ 1 Λ 1 1 Λ Λ mediately yields the effective action to one-loop order in (99) perturbation theory We recognize here the well-known one-loop structure of the flow equation with the cutoff-insertion ∂ Rˆ . The i Λ Λ Γ [ψ, φ] = S[ψ, φ] + STr ln Sˆ(2)[ψ, φ] . (102) usual minus sign for a closed fermion loop has been ab- 1-loop 2 sorbed into the definition    Other approximations, such as the random phase approx- −1 0 0 imation, can be obtained by similar considerations. STr(··· ) ≡ Tr  0 −1 0 ···  . (100) In recent years there have been many proposals 0 0 1 for systematic approximations of the effective action, which are capable of describing truly nonperturbative In order to close Eq. (99) we make use of the generalized phenomena.32 We here pursue an expansion into powers Dyson equation (77) and write of fields Ψ and φ, following Refs. 31, 35, and 60. As- suming the effective action to be an analytic functional of −1 i ˆ ˆ(2) ˆ the fields, we can perform a formally exact Taylor expan- ∂ΛΓΛ = STr (∂ΛRΛ) ΓΛ + RΛ 2 sion, known as “vertex expansion”. It can be employed + 2φ|τ1∂Λn˜Λ to replace the single functional integro-differential equation by an equivalent infinite hierarchy of coupled ordi- i ˆ(2) ˆ = ∂/ΛSTr ln Γ + RΛ + 2φ|τ1∂Λn˜Λ , (101) nary integro-differential equations for the one-particle ir- 2 Λ reducible vertex functions. Clearly, to solve the complete where we have defined the “single-scale derivative” ∂/Λ hierarchy exactly is still an impossible task. However, a in the third line, which acts on the regulator only. This truncation of the infinite hierarchy at a certain finite or- equation is the desired exact flow equation for the ef- der is still nonperturbative in essence and does not nec- fective action of a Fermi-Bose theory in the nonequilib- essarily rely on the presence of a smallness parameter in rium Keldysh formalism. Despite its apparent simplicity the interaction Sint. it is a highly complicated nonlinear functional integro- Taking into account that the bosonic field may develop ¯ differential equation, which captures all of the nonper- a finite expectation value φc(x), e.g. due to a finite exter- turbative features of the theory. nal scalar potential, we should expand the bosonic field 12 around this macroscopic field, rather than around zero, perfields”, a condensed notation collecting fermionic and bosonic degrees of freedom into a single field, in thermal ¯ 35,60 φc(x) = φc(x) + ∆φc(x) , φq(x) = ∆φq(x) (103) equilibrium by Schützand Kopietz. In our case the vertex expansion reads The general vertex expansion in the presence of bosonic field expectation values has been worked out for “su-

∞ ∞ X X (−1)m 1 Z X X Z X ΓΛ[ψ, φ] = 2 (m!) n! 0 xm,xm 0 0 yn m=0 n=0 im,im αm,αm βn (2m,n) 0 0 0 0 0 0 × ΓΛ (x1i1α1, . . . , xmimαm; x1i1α1, . . . , xmimαm; y1β1, . . . , ynβn) † † 0 0 × ψ (x ) ··· ψ (x )ψ 0 0 (x ) ··· ψ 0 0 (x )∆φ (y ) ··· ∆φ (y ) . (104) i1α1 1 imαm m imαm m i1α1 1 β1 1 βn n

Here, latin indices in collectively denote the discrete fermionic degrees of freedom, sublattice, valley and spin, whereas greek indices αn, βn are reserved for the degrees of freedom in Keldysh space, the classical and quantum components. (2m,n) The coeﬃcient-functions ΓΛ in this expansion deﬁne the one-particle irreducible vertex functions

(2m,n) 0 0 0 0 0 0 ΓΛ (x1i1α1, . . . , xmimαm; x1i1α1, . . . , xmimαm; y1β1, . . . , ynβn) =

δ(2m+n)Γ Λ . (105) † † 0 0 0 0 0 0 δψ (x1) ··· δψ (xm)δψi α (xm) ··· δψi α (x )δφβ1 (y1) ··· δφβn (yn) i1α1 imαm m m 1 1 1 φc=φ¯c

In the above definition it is understood that after performing the (2m + n)-fold derivative, all fields have to be set to ¯ zero except the classical component of the bosonic field, which is set to its possibly nonzero expectation value φc(x). This notation has already been employed in Eqs. (84) and (85). Further, the normalization of the partition function implies that vertex functions which possess classical indices only vanish identically.37 To obtain the hierarchy of flow equations for the vertex functions, we have to insert the expansion (104) into the exact flow equation (101) and compare coefficients. It is important to emphasize that both the vertex functions, as well as the bosonic expectation value are functions of the flowing cutoff Λ. Thus, we obtain two contributions on the left hand side of (101)

∞ ∞ X X (−1)m 1 Z X X Z X ∂ΛΓΛ = 2 (m!) n! 0 xm,xm 0 0 yn m=0 n=0 im,im αm,αm βn Z (2m,n) (2m,n+1) ¯ × ∂ΛΓΛ (... ; y1β1, . . . , ynβn) − ΓΛ (... ; y1β1, . . . , ynβn, yc)∂Λφc(y) y † † 0 0 × ψ (x ) ··· ψ (x )ψ 0 0 (x ) ··· ψ 0 0 (x )∆φ (y ) ··· ∆φ (y ) . (106) i1α1 1 imαm m imαm m i1α1 1 β1 1 βn n

(2m,n) (2m,n+1) In the second line we suppressed the fermionic arguments of the vertex functions ΓΛ and ΓΛ for clarity. ˆ(2) For the right hand side it is beneficial to separate the field-independent part from the field-dependent part of ΓΛ . Recalling the generalized Dyson equation (77), we write31

ˆ(2) ˆ−1 ˆ ΓΛ [ψ, φ] = GΛ − ΣΛ[ψ, φ] , (107) with

ˆ−1 ˆ(2) ˆ ˆ(2) ˆ(2) GΛ = ΓΛ |φc=φ¯c , ΣΛ[ψ, φ] = ΓΛ |φc=φ¯c − ΓΛ . (108)

ˆ−1 Here GΛ is a 3 × 3 matrix in field space, which contains the unregularized inverse full propagators, see Eqs. (84) and (85), whereas Σˆ Λ[ψ, φ] is the field dependent self-energy, which must not be confused with the (field independent) self-energy in the inverse full propagators. Now we can expand the logarithm on the right hand side of Eq. (101) in 13

−1 ˆ−1 ˆ terms of (regularized) full propagators GΛ + RΛ as follows

−1 ˆ(2) ˆ ˆ−1 ˆ ˆ ln ΓΛ + RΛ = ln 1 − GΛ + RΛ ΣΛ[ψ, φ] ∞ n X 1 −1 −1 = − Gˆ + Rˆ Σˆ [ψ, φ] . (109) n Λ Λ Λ n=1 The desired hierarchy of flow equations is given by comparing coefficients in the expansions of (106) and (109). This can be done in a systematic way, because, by construction, the field dependent self-energy Σˆ Λ[ψ, φ] only contains terms which are at least linear in one field variable. In the following we present a truncated set of equations, with the further approximation that only vertices with one bosonic and two fermionic legs have been kept. The motivation for this approximation is, that these structures are already present in the bare action. Accounting for the purely bosonic three-vertex and other higher order vertices, which are inevitably generated by the flow, is possible by using the strategy explained above. The equation for the bosonic field expectation value reads Z −1 R R 0 ¯ 0 R 0 ¯ 0 (V + ΠΛ + Rb,Λ)(x1, x1)∂Λφc(x1) + ∂ΛRb,Λ(x1, x1)φc(x1) 0 x1 Z i X X αβ 0 0 (2,1) 0 0 = − ∂/Λ GΛ,kl(x1, x2)ΓΛ,lk (x2β, x1α; x1q) − ∂Λn˜Λ(x1) , (110) 2 0 0 α,β k,l x1,x2

The derivation of this equation makes use of the equation given in App. A, of motion (76) at its extremal value Ψ = 0, ∆φ = 0, replacing the ﬂow of the one-point function. ∂ΛΣˆ Λ = i∂/Λ + ,

In the presence of a bosonic regulator a graphical representation of the above equation is rather exceptional and not very helpful. However, for purely fermionic cutoff (111) schemes, we recognize the typical tadpole structure, also i −1 R −1 ∂ΛΠΛ = ∂/Λ , (112) known as Hartree diagrams, by applying (V + ΠΛ ) 2 on both sides of the equation. Further note the counterterm flow on the right hand side. It is the only location where the background charge densityn ˜(x) enters the flow equations explicitly. We can understand its presence here by considering exemplarily the space-time translation in- (2,1) ∂ΛΓ = i∂/Λ . (113) variant system at finite density. In that case the first Λ term on the right hand side is finite and closely related to the charge carrier density. (In fact, in the simple truncation scheme where the three-vertex flow is neglected it In these diagrams, the straight line corresponds to a is identical to the charge carrier density.) In turn, this fermionic full propagator, the wiggly line to a bosonic ¯ would imply that the expectation value φc has to be fi- full propagator, the triangle to a vertex and the crossed nite. The counterterm, however, cancels the finite con- circle to the bosonic field expectation value. The dot tribution on the right hand side at any scale, such that above the crossed circle denotes the scale derivative act- the expectation value is consistently removed from the ing on the expectation value. Summation over discrete theory and all tadpole diagrams with it. In other physi- degrees of freedom (including Keldysh space) and inte- cal situations, depending on the experimental setup, the gration over continuous ones is implied. The above flow counterterm flow has to be constructed by further phys- equations closely resemble one-loop perturbation theory, ical considerations. a fact which is not surprising, since the exact flow equation (101) has a one-loop structure. By construction, the single-scale derivative ∂/Λ appear- We here show the flow equations for the fermionic self- ing in the above expressions does not act on the vertex energy, bosonic polarization and the three-vertex in their functions, but only on the regulator occuring in the ex- graphical form only. Their explicit analytical form is pressions for the internal full propagators (such as the 14

αβ factor GΛ,kl in Eq. (110)). In other words, ∂/Λ is a scale- IV. THERMAL EQUILIBRIUM derivative at constant self-energy, which yield what is 31,35 known in the literature as single-scale propagators As a first application and a test of the methods developed in the previous section, we now apply the general ∂/ΛGˆΛ = −GˆΛ∂ΛRˆf,ΛGˆΛ ≡ Sˆf,Λ , (114) nonequilibrium formalism to the equilibrium case and show how the results of the Matsubara imaginary-time ∂/ΛDΛ = −DΛ∂Λ2Rb,ΛDΛ ≡ Sb,Λ . (115) formalism are recovered. In thermal equilibrium physical observables do not depend on time. In particular, the Graphically, the single-scale propagators are often de- reference time t0 drops out in any calculation, so that picted as a (straight or wiggly) line with a slash. They the limit t0 → −∞ may be taken at the beginning of the have the same trigonal structure as the flowing propaga- calculation and a Fourier transform to frequency space tors can be performed. In contrast to the Matsubara formalism, the frequencies in the Keldysh formulation are real ! SˆK SˆR SK SR and continuous, which removes the need for an analytical Sˆ = f,Λ f,Λ , S = b,Λ b,Λ . (116) f,Λ SÂ 0 b,Λ SA 0 continuation at the end of a calculation. The tempera- f,Λ b,Λ ture dependence enters through the solution of the kinetic equations and the fluctuation-dissipation theorem, which The advantage of using the single-scale derivative is, that will be discussed below. the computational effort to arrive at the vertex flow equa- In the following we further restrict ourselves to spa- tions as well as their analysis is greatly reduced. The tially translation invariant systems, setting the external reason being, in particular, that ∂/Λ obeys the product electromagnetic potentials to zero. Since the propagators rule for differentiation, according to which, at the graph- and each vertex now conserve energy and momentum, ical level, for each internal line on the right hand side the flow equations simplify considerably. We also limit of Eqs. (111)–(113) the single-scale derivative produces ourselves to intrinsic, freestanding graphene, setting the an additional equivalent term, where the corresponding chemical potential µ and the background charge density line has been substituted by a single-scale propagator. n˜ to zero, and the dielectric constant of the medium 0 Therefore, one may perform all analytical manipulations to unity. As a consequence the bosonic field expectation within the integrals first and apply the scale derivative value and the counterterm vanish. After discussing some afterwards. general aspects, we present a simple truncation scheme We close this section by discussing the role of cor- for the flow equations, and solve the resulting system of related initial states in the above set of exact flow equations numerically for finite temperatures. equations. Recall from section II B that correlated initial states manifest themselves as higher order terms in the expansion of the correlation functional K [ψ], see ρ A. Fluctuation-dissipation theorem and cutoff Eq. (32). The kernels of this expansion would appear schemes within the effective action ΓΛ as a contribution to the respective higher order vertex function in the expansion (104) already at the initial scale Λ .56 Since a com- The equilibrium state is uniquely specified by the 0 ˆ mon truncation strategy of the infinite hierarchy of flow Boltzmann statistical operatorρ ˆ = exp(−βH). This equations is to keep only those vertices which are already particular density matrix leads to a periodicity of the present in the bare action, the number of flow equations, fermionic and bosonic field operators along the imag- which should be considered for correlated initial states, inary time axis, which can be expressed by the KMS 53 grows rapidly. Even for the simplest possible nongaus- boundary conditions. Eventually, these boundary con- sian extension, which is a quartic term in the fermionic ditions manifest themselves as constraining relations be- correlation functional, the analysis is considerably im- tween the various n-point correlation functions, which is peded. First, one would have to keep the four-vertex known as the fluctuation-dissipation theorem. Demand- contribution to the fermionic self-energy flow, and sec- ing its validity at any scale greatly reduces the numer- ond it should be revised, if it is justifiable to neglect the ical effort, since the flow equations themselves have to four-vertex flow entirely or if at least the flow of some preserve these constraints. Thus, the number of inde- dominant interaction channel has to be taken into ac- pendent flow equations is diminished. We here concen- count. Owing to the complicated structure of the flow trate on the fluctuation-dissipation relation for the con- equations, it becomes clear that the study of nongaussian nected two-point correlators and self-energies. We refer initial correlations is practically limited to a low order.64 to Refs. 65 and 66 for further reading. On the other hand the field is vastly unexplored and may In the Keldysh formalism the fluctuation-dissipation lead to interesting new physical effects. In any case the theorem can be very elegantly formulated. The neces- nonequilibrium functional renormalization group as we sary condition for thermal equilibrium is the vanishing of presented above is an excellent framework for such an the kinetic term in the quantum kinetic equations (93) ~ undertaking. and (94). Assuming that the hermitian matrix FˆΛ(k, ε) 15 is proportional to the unit matrix we thus have even further by eliminating one of the integrations in- volved on their right hand side. ˆ K ~ ~ ˆ R ~ ˆ A ~ ΣΛ (k, ε) = FΛ(k, ε) ΣΛ (k, ε) − ΣΛ (k, ε) , (117) Several aspects of this choice of the regulator function are worthwhile discussing. The first issue is the role of K R A ΠΛ (~q, ω) = BΛ(~q, ω) ΠΛ (~q, ω) − ΠΛ (~q, ω) . (118) the Fermi surface. At charge neutrality the Fermi surface consists of the points located at the K+ and K− points, The fluctuation-dissipation theorem states that the dis- a fact that is not altered by the interaction. This is a tribution functions FΛ and BΛ take the simple, scale in- major simplification, because there is no need to adapt dependent form the regulators to a continuously changing Fermi surface. ε Since this simplification is special to the charge neutral- F (~k, ε) = tanh , (119) Λ 2T ity point, other regularization schemes may be preferable ω away from it, see our discussion below. B (~q, ω) = coth . (120) Λ 2T Second, the above choice of regularization function transforms the additive regularization into a multiplica- Since the equilibrium solution is unique, their indepen- tive one. Such multiplicative regularizations are also dence of the scale Λ is crucial. Using the above solu- common in the literature, see, e.g., Refs. 31 and 35. The tion, we can immediately write down the corresponding Keldysh regulator has been set to zero in order to guar- Keldysh propagators antee the trivial initial conditions ΓΛ0 = S. Although now the kinetic terms in the kinetic equations contain ˆK ~ ε ˆR ~ Â ~ GΛ (k, ε) = tanh GΛ (k, ε) − GΛ (k, ε) , (121) explicitly the regulators, it is still possible to obtain the 2T scale independent equilibrium solutions of the previous K ω R A DΛ (~q, ω) = coth DΛ (~q, ω) − DΛ (~q, ω) . (122) section. This fact is a simple consequence of the scalar 2T multiplicative cutoff. Whereas the fluctuation-dissipation theorem is gener- At the end of section III C we discussed that a ˆK K ally valid in thermal equilibrium for the physical limit parametrization of the Keldysh regulators Rf,Λ and Rb,Λ Λ → 0, its validity at all scales Λ is not automatic. Re- in terms of the distribution functions FˆΛ and BΛ, requiring Eqs. (120) for arbitrary cutoff Λ puts strong con- spectively, in principle leads to a simplification of the ki- straints on the choice of the infrared regulators. As dis- netic terms in the kinetic equations, see Eqs. (93)–(96). cussed in the previous section, these constraints have to In this parametrization the kinetic terms no longer ex- be implemented together with the restrictions that ensure plicitly contain the regularization functions. As a re- that the cutoff scheme preserves causality and respects all sult the kinetic equations can be solved immediately by the symmetries of the model. the above scale independent distribution functions. This Following Ref. 36, we now describe a regularization fact applies to regulator functions that act in the mo- scheme that meets these conditions. We have adopted mentum and/or frequency domain. The possibility to this regularization scheme for our numerical calculations, use cutoffs in the frequency domain that manifestly pre- in order to facilitate the comparison of our results and serve causality is a major technical advantage of the those of Ref. 36. In this scheme, regularization is applied Keldysh formulation and does not exist for frequency in the fermionic sector only, cutoffs in the imaginary-time formulation, where the causality structure is usually destroyed. Of course, in R/A/K Rb,Λ = 0. (123) frequency-independent regularization schemes, such as the one of Eqs. (124), causality issues are avoided for As a consequence, the bosonic single-scale propagators both approaches. An additional advantage of a frequency vanish identically. For the fermionic degrees of free- cutoff in the fermionic sector is that no explicit reference dom, we consider a regulator with momentum depen- to a Fermi surface needs to be made. dence only, An example for a cutoff scheme, which incorporates all of the above mentioned properties, is the “hybridiza- R A −1 −1 ˆ ~ ˆ ~ ˆ ~ ˆ ~ 41,42 Rf,Λ(k, ε) = Rf,Λ(k, ε) = G0,Λ(k, ε) − G0 (k, ε) , tion cutoff” of Jakobs et al. In this scheme the in- RˆK (~k, ε) = 0 (124) finitesimal regulators ±i0 in the inverse bare propagators b,Λ and the Keldysh blocks are elevated to cutoff dependent with quantities ±iΛ. Being essentially a frequency cutoff, the hybridization scheme is particularly useful in those cases ˆ−1 ~ ˆ−1 ~ −1 where a momentum cutoff is not appropriate, such as G0,Λ(k, ε) = G0 (k, ε)(Θ(k − Λ)) . (125) graphene away from the charge neutrality point or the The absence of a frequency dependence of the regulator presence of a finite magnetic field. In both cases, the function implies that the frequency structure of the prop- Fermi surface (if it can be defined at all) will be subject to agators is untouched by the regularization procedure and change during the renormalization group flow, requiring causality is manifestly preserved. The sharp Θ-function a continuous adjustment of the momentum cutoff. The cutoff in momentum space simplifies the flow equations frequency cutoff of Refs. 41 and 42, on the other hand, is 16 insensitive to a changing Fermi surface and compatible The retarded and advanced propagators in the bosonic with spatially varying external fields. Another example sector are given by of a frequency cutoff is the “outscattering rate cutoff” employed by Kloss and Kopietz.45 It is similar to the hy- R/A 1 1 DΛ (q, ω) = , (130) bridization cutoff, but has the important difference that 2 −1 R/A V (q) + ΠΛ (q, ω) the Keldysh blocks of the inverse free propagators are not regularized. In this case the distribution functions where V (q) is the two-dimensional Fourier transform of become explicitly scale dependent and the fluctuation the Coulomb interaction, dissipation theorem is manifestly violated, making the 2πe2 outscattering rate cutoff not suitable for an equilibrium V (q) = . (131) setting. q The Λ dependence of the bosonic propagators is entirely determined by the flowing polarization function. By in- B. Dressed flowing propagators troducing the dielectric function

After having discussed the regularization scheme, we R/A(q, ω) ≡ 1 + V (q)ΠR/A(q, ω) , (132) can now give explicit expressions for the dressed flowing Λ Λ propagators, which are central to the flow equations of the propagators can be written in the convenient form the functional renormalization group. The temperature arguments of the fermionic and bosonic self-energies are 1 V (q) DR/A(q, ω) = . (133) suppressed in the following. Λ R/A 2 (q, ω) As discussed in Sec. II A, the expressions for the Λ fermionic propagators take their simplest form in the chiral basis. Since by construction of the regulators the C. Fermi velocity and static dielectric function at exact flow equation preserves chirality at all scales, the finite temperature same holds true for the fermionic self-energy and the flowing propagators We now proceed to solve the truncated flow equations R/A X R/A Σˆ (~k, ε) = Pˆ (kˆ)Σ (k, ε) , (126) using a finite-temperature real-time analogue of the trun- Λ ± ±,Λ cation scheme employed by Bauer et al.36 We consider ± intrinsic graphene, so that the bosonic field expectation ˆR/A ~ X ˆ ˆ R/A GΛ (k, ε) = P±(k)G±,Λ (k, ε) , (127) value φc and the counterterm are absent. The system of ± equations (111)–(113), see also App. A, is further simplified by neglecting the flow of the three-vertex functions ˆ ˆ where the P±(k) are the chiral projection operators, see entirely, keeping these at their initial values at Λ = Λ0. Eq. (20). Thus, the retarded and advanced chiral flowing We also neglect the Λ dependence of the scalar self energy propagators can be written in the compact form R/A Σε,Λ of Eq. (129a), as well as the frequency dependence Θ(k − Λ) of the scalar self energy ΣR/A of Eq. (129b). These ap- GR/A(k, ε) = , (128) v,Λ ±,Λ R/A proximations lead to well-defined Λ-dependent poles of ε ∓ v k − Σ (k, ε) F ±,Λ the single-particle propagators (128) at Θ(k − Λ) = , R/A R/A ξΛ(k) = vΛ(k)k , (134) ε − Σε,Λ (k, ε) ∓ vF + Σv,Λ (k, ε) k where the renormalized Fermi velocity is given by where we have defined 1 vΛ(k) = vF + Σv,Λ(k) . (135) ΣR/A(k, ε) = ΣR/A + ΣR/A (k, ε) , (129a) ε,Λ 2 +,Λ −,Λ Finally, we neglect the frequency dependence of the di- R/A 1 R/A R/A Σ (k, ε) = Σ − Σ (k, ε) . (129b) electric function R/A(q, ω) = (q). As a consequence v,Λ 2k +,Λ −,Λ Λ Λ the bosonic Keldysh propagator remains identically zero Recall that the single-scale derivative only acts on the during the flow. Θ-function, such that the sharp momentum cutoff yields The complete truncation scheme can be conveniently a particularly simple single-scale propagator. expressed if we parameterize the effective action as 17

Z 0 Σ (ε − i0) + v (k)Σ~ · ~k Γ [Ψ, φ] = Ψ†(~k, ε)σs ⊗ 0 Λ Ψ(~k, ε) Λ 0 ~ ~ ε ~k,ε Σ0(ε + i0) + vΛ(k)Σ · k 2i0 tanh 2T Σ0 −1! Z 0 V (q)/ (q) | Λ + φ (−~q, −ω) −1 φ(~q, ω) ~q,ω V (q)/Λ(q) 0 Z φ (~q, ω) φ (~q, ω) − Ψ†(~k + ~q, ε + ω) q c Ψ(~k, ε) . (136) ~k,ε,~q,ω φc(~q, ω) φq(~q, ω)

We note that if one wishes to go beyond the static approximation of Eq. (136), and include the dynamical effects of plasmons and quasiparticle wavefunction renormalization, one should not neglect the three-vertex flow entirely. A R/A R/A R/A naive extension, where only the renormalization of Σε,Λ and the frequency dependences of Σv,Λ and Λ are taken into account, is not sufficient. As Bauer et al. have shown,36 one should at least include the marginal part of the three-vertex in the analysis. In that case the vertex flow reduces to a differential form of a Ward identity, leading to a partial cancellation of fermionic self-energy- and vertex-corrections. Neglecting the vertex flow would violate the Ward identity and lead to an inconsistency in the flow of the quasiparticle wavefunction renormalization. The sequence of approximations described above results in two coupled flow equations, one for the Fermi velocity vΛ(k) and one for the static dielectric function Λ(q). The approximations are self consistent in the sense that neither a quasipaticle wavefunction renormalization nor a frequency dependence of the dielectric function are generated during the flow. Within the truncation of the effective action given above, we obtain the flow equation for the Fermi velocity

2 Z π e Λ ξΛ(Λ) cosϕ 1 Λ∂ΛvΛ(k) = − dϕ tanh , (137) 2π k 2T q 2 q 2 0 1 + k − 2 k cosϕ k k Λ Λ Λ Λ 1 + Λ − 2 Λ cosϕ whereas the ﬂow equation for the static dielectric function takes the form

2e2 Z π/2 2Λ 1 Λ∂ΛΛ(q) = − q dϕ Θ cosϕ + − 1 q (138) π 0 q q 2 q 2 1 + 2Λ cosϕ − 2Λ " ξ (Λ) ξ (Λ + qcosϕ) sin2ϕ × tanh Λ + tanh Λ 2T 2T ξΛ(Λ) + ξΛ(Λ + qcosϕ) # ξ (Λ) ξ (Λ + qcosϕ) (2Λ/q + cosϕ)2 − 1 + tanh Λ − tanh Λ . 2T 2T ξΛ(Λ) − ξΛ(Λ + qcosϕ)

The derivation of Eq. (138) requires the use of elliptic Pauli blockade by opening the intra-band phase space for coordinates. At the initial scale Λ = Λ0 the fermionic momenta of the order T , leading to the additional term and bosonic self-energies vanish, which translates to the in the third line. initial conditions vΛ0 (k) = vF , Λ0 (k) = 1. In the limit T → 0 our equations reduce to the expressions given in The above equations have been solved numerically for Ref. 36. The temperature dependence enters the Fermi different temperatures with the dimensionless coupling 2 velocity flow equation only as a simple factor in the in- constant α = e /vF = 2.2. Specifically, they have tegrand, due to the absence of plasmonic effects. The been rewritten as pure Volterra integral equations of the temperature dependence of the dielectric function flow second kind by integration over the scale variable Λ, equation, on the other hand, is more complicated. The see Ref. 67, and using the initial conditions. We dis- two contributions in the second and third line of Eq. (138) cretized the parameter spaces by nonuniform, adaptive can be traced back to inter- and intra-band transitions, grids, which were interpolated linearly when intermedi- repectively. At T = 0 the valence band is fully occupied, ate values were required. The case k = 0 could not be while the conduction band is empty. Thus, the fermionic included in the grids due to divergent terms. Therefore, −5 phase space for intra-band transitions is Pauli blocked we built the grids down to k/Λ0 = 10 and extrapolated and only inter-band transitions contribute to the polar- for lower momenta if necessary. The coupled system of ization function. A finite temperature, however, lifts this integral equations has been solved iteratively, starting from the initial values vΛ(k) = vF and Λ(q) = 1 for the 18

7 F

v 5 )/

6 k ( v 4 k → 0 5 lim

F 3 0.00025 0.001 0.005 4 T / vF Λ0 v ( k )/ v 3

1 10-5 10-4 10-3 0.01 0.1 1

k/Λ 0

FIG. 3. (Color online) Fermi velocity versus momentum k, for −4 −4 −3 temperatures T/vF Λ0 = 0; 5.0 × 10 ; 7.5 × 10 ; 1.0 × 10 ; 2.5 × 10−3; 5.0 × 10−3; and 7.5 × 10−3 (top to bottom data sets). The inset shows the logarithmic temperature dependence of the Fermi velocity at k = 0. The single data point −5 at v(10 )/vF = 6 shows a non-physical deviation from the logarithmic divergence at zero temperature, indicating that FIG. 2. (Color online) Cutoff dependent Fermi velocity v (k) Λ our numerical algorithm breaks down there. This behaviour at temperature T/v Λ = 10−3. The physical limit corre- F 0 could be expected, since the grids have only been built down sponds to Λ = 0. Note that the renormalized Fermi velocity to k/Λ = 10−5. At finite temperatures similar conver- is finite at Λ = k = 0. Further observe that the figure is 0 gence problems occur upon approaching the lower grid cutoff almost symmetric around k = Λ, suggesting that the momen- T/v Λ ≈ 10−5. tum k acts as an infrared cutoff for the Fermi velocity viewed F as a function of Λ in the same way as Λ acts as a cutoff for the Fermi velocity as a function of k. This fact can be readily explained by the presence of thermally excited charge carriers, which can screen the zero-temperature calculation and continuing the iteration bare Coulomb interaction at long wavelengths. Thus, the until a self-consistent solution was obtained. During the effective Coulomb interaction becomes short ranged, cut- iterative procedure the grids were occasionally refined acting off the divergence at small momenta. The larger the cording to a gradient criterion. For finite temperature temperature the more charge carriers are excited, lead- we used previously computed and converged results at a ing to an enhancement in the suppression of the diver- nearby temperature as an initial value in order to mini- gence. Indeed our numerics show that this suppression is mize the computation time. a logarithmic function of the temperature, which could The results of the numerical integration for the Fermi be fitted by velocity vΛ(k) in its full parameter space is shown exem- −3 lim v(k) = C + D ln(vF Λ0/T ) , (140) plarily for the reduced temperature T/vF Λ0 = 10 in k→0 the figure 2, whereas figure 3 summarizes our result in the physical limit Λ = 0 for all temperatures we considered. with C = 0.84(33) and D = 0.57(6). For momenta k T/v the long-wavelength screening of the Coulomb The corresponding results for the dielectric function Λ(q) F are shown in the figures 4 and 5, respectively. interaction becomes irrelevant and the Fermi velocity At zero temperature the Fermi velocity shows the well- asymptotically approaches the zero-temperature value. known logarithmic renormalization, which has been re- A well known issue in the comparison with experimen- ported previously by many authors within one-loop per- tal data is the value of the ultraviolet cutoff Λ0. Since turbation theory.1,17,20 Our numerical result could be fit- we already fixed the numerical value of the bare Fermi ted by velocity by setting α = 2.2, the cut-off Λ0 can be used as a fit parameter. Alternatively, one could take the ul- v(k) = A + B ln(Λ0/k) , (139) traviolet cutoff to be fixed (given by the inverse lattice spacing), and instead use α, i.e. vF , as a fit parameter. with A = 1.34(4) and B = 0.52(1), which coincides with The drawback of the latter method, however, is that the the result of Bauer et al.36 within numerical accuracy. At dimension of the free parameter space would be enlarged. nonzero temperature we find that v is finite for k → 0, One would have to solve the flow equations for different while for large momenta the Fermi velocity merges into temperatures and couplings α, which would increase the the logarithmic behaviour found at zero temperature. numerical effort even further. 19

deviation to slightly larger values at momenta of order unity. This fact may be explained by differences in the numerical implementation of the flow equations. At finite temperature, however, a strong temperature dependence, proportional to 1/q, sets in for momenta q . T/vF . The emergence of the power law divergence for small momenta can be easily understood from perturbation theory, already at the one-loop level.50,68,69 In the regime q T/vF the static polarization function becomes momentum independent, scaling linearly with temperature, which results in the one-loop dielectric function

Λ (q) = 1 + a(T ) 0 , v q T, (141) 1-loop q F

with T a(T ) = 8 ln2 α . (142) vF Λ0 The divergence at zero momentum is a consequence of the presence of thermally excited charge carriers, screening the bare Coulomb interaction. Our numerical calculations qualitatively confirm this one-loop picture as they FIG. 4. (Color online) Cutoff dependent dielectric function −3 reproduce the 1/q dependence as well as the linear tem- Λ(q) at temperature T/vFΛ0 = 10 . Note the sharp feature at Λ = 0 for momenta q T/v . perature dependence of the prefactor a(T ). On the quan- . F titative level, however, we find a considerable deviation in the numerical value of the proportionality constant: 1000 The numerics could be fitted by a(T ) = 0.98(5)T/vF Λ0, 0.01 which is about one order of magnitude lower than the one-loop prediction. This discrepancy can be understood )

T 0.005 100 ( by considering the fact that a one-loop calculation em- a ploys only noninteracting propagators, while the fRG re- 0. sult is obtained by a selfconsistent calculation, using fully ϵ ( q ) interacting propagators, such that a strong renormaliza- 10 0. 0.0025 0.005 0.0075 0.01 tion of the former result is to be expected. T / vF Λ0 In the high-temperature limit we expect a strong screening of the Coulomb interaction, implying the ab- 1 sence of velocity renormalization, due to its logarith- -5 -4 mic suppression with increasing temperatures, and hence 10 10 0.001 0.010 0.100 1 the emergence of a free field fix point. However, such q/Λ 0 an asymptotically free fix point has little practical relevance, since in that regime the electron-phonon interac- FIG. 5. (Color online) Dielectric function as a function of tion should be taken into account in a realistic model, −4 70 momentum q for temperatures T/vFΛ0 = 0, 5.0 × 10 , 7.5 × which would drive the system into a crumpled phase 10−4, 1.0 × 10−3, 2.5 × 10−3, 5.0 × 10−3, and 7.5 × 10−3 and eventually lead to an instability of the underlying (bottom to top data sets). For momenta q/Λ0 below the honeycomb lattice. In other words, graphene would have ˜ reduced temperature T = T/vFΛ0 our data are consistent melted long before the free field fix point would have been with the 1/q dependence predicted by perturbation theory. reached. The temperature dependence of the prefactor could be fitted by (q) = 1+a(T˜)Λ0/q, indicated by dashed lines, where a(T˜) is a linear function as shown in the inset. V. CONCLUSIONS

The zero-temperature result for the dielectic function In this article we formulated a nonperturbative (q) is only very weakly momentum dependent for large nonequilibrium theory for Dirac electrons interacting via momenta, while for q → 0 it logarithmically approaches the Coulomb interaction, which is based on the Keldysh unity, in contrast to the momentum independence of the functional renormalization group. Our theory should one-loop prediction. This behaviour is in accord with the be a good description of the low-energy properties of result of Bauer et al.,36 although we observe a systematic graphene. 20

The essential parts of the theoretical description are the (integer) quantum Hall regime, since then momen- the exact Dyson equations for the real-time Fermi-Bose tum is not a well-defined quantum number, and hence theory, from which the quantum kinetic equations follow, cannot be employed as a flow parameter. as well as an exact flow equation for the effective action. Whereas the use of the Keldysh formulation is techni- The functional flow equation has been transformed into cally convenient (but not essential) for equilibrium prob- a hierarchy of ordinary coupled integro-differential equa- lems, because it avoids the necessity of an analytical con- tions, describing the flow of the one-particle irreducible tinuation, for nonequilibrium problems the Keldysh for- vertex functions, by means of a vertex expansion. This malism is essential. Possible applications of the formal- hierarchy has to be solved approximately using a self- ism developed here are nonthermal fixed points, thermal- consistent truncation scheme. As a test of our formal- ization, and quantum transport in linear or even beyond ism, we reproduced the results for the Fermi velocity linear response. Another issue of interest is the topic renormalization and the dielectric function at zero tem- of nongaussian initial correlations, for which we outlined perature that were previously obtained by Bauer et al.36 their implementation within our theoretical framework, using the imaginary-time Matsubara formalism, and we although an actual application is beyond the scope of the extended these results to finite temperature. present article. The research provided in this article can be extended For applications to realistic graphene samples, not only into several different ways. For equilibrium problems one interactions, but also disorder has to be taken into ac- may take into account dynamical effects, yielding the count. This applies to quantum transport problems in dynamical polarization function and quasiparticle wave- particular, see Refs. 38, 72, and 73. The Keldysh for- function renormalization. This extension would go hand mulation we presented here is perfectly suited for such a in hand with a nonperturbative study of collective plas- research programme. As is well-known the normalization mon modes. A purely bosonic cutoff combined with ex- of the partition function can be exploited to perform the act Schwinger-Dyson equations, as recently proposed by impurity average directly on the level of the partition Sharma and Kopietz in Ref. 61, would be highly advan- function. There is no need for supersymmetry or the tageous for such an undertaking. replica trick in the Matsubara formalism. For gaussian Another interesting extension is to investigate modifi- correlated disorder the averaging procedure leads to a cations of the isotropic Dirac spectrum, such as trigonal quartic fermionic pseudo-interaction term. Especially at warping,1 or anisotropies in strained graphene.71 Both the charge neutrality point the deviations from the usual phenomena would require the modification of the nonin- Fermi-liquid behaviour should be strongly pronounced. teracting Hamiltonian Hf, see Eq. (2), but the general Similarly to the Coulomb interaction treated here, the structure of the calculation is not modified. Further- theory at this point lacks a smallness parameter and more, it would be interesting to study the fate of gaps, conventional approximation strategies, such as the self- or masses, in the spectrum under the renormalization consistent Born approximation, break down. Since the group flow. A particularly exciting scenario is the possi- common truncation strategies of the infinite hierarchy of bility of a spontaneous mass generation,17 for which one flow equations do not rely on the existence of a smallness starts from an infinitesimal mass term at the initial scale parameter, the Keldysh fRG offers the necessary tools to Λ = Λ0, which may be elevated to a finite value at the go beyond these approximations in consistent manner. end of the flow. This extension, too, requires no modifi- As a closing remark we want to point out that the cations of the general formalism, as the vertex expansion good agreement between the functional forms of the mo- in Sec. III E is sufficiently general enough to cope with mentum and temperature dependences of the one-loop such situations. perturbation theory and the functional renormalization The application of our formalism to extrinsic graphene group results presented here may come as a surprise, requires a different cutoff scheme than the one we used since there is no small parameter justifying a perturbative here, since the presence of a finite Fermi surface is in- approach. Indeed, a two-loop calculation for the Fermi compatible with the use of a simple “static” momentum velocity already shows the lack of convergence of the per- cutoff in the fermionic sector. One possibility would be turbative approach, as it predicts a logarithmic decrease to modify the momentum cutoff to “dynamically” adapt for small momenta.23 Nevertheless, the exact flow equa- to a continuously changing Fermi surface at each scale tion (101) has a one-loop structure, so it becomes clear Λ. However, this modification would complicate the flow that some features of one-loop perturbation theory are equations considerably and is therefore not convenient.31 qualitatively reproduced. For the future applications dis- An alternative cutoff scheme, circumventing this dif- cussed above it is, therefore, reasonable to expect that ficulty, is the causality preserving frequency cutoff of the results derived from a perturbative one-loop calcula- Jakobs et al.41,42, which may be used either within the tion at least hint into the right direction, although not all simple rotation invariant conical Dirac spectrum consid- features of the exact theory are reproduced quantitatively ered in this work or within one of the modifications of correctly. In the end, quantitatively accurate results can the bare spectrum mentioned above. Moreover, as ex- be expected only by more sophisticated nonperturbative plained at the end of Sec. IV A, frequency cutoffs are approaches, such as the functional renormalization group advantageous for the study of external magnetic fields in developed here. 21

ACKNOWLEDGMENTS bosonic expectation value here again. We emphasize once more that the purely bosonic three-vertex, as well as all We want to thank Piet Brouwer for support in the higher order vertices have already been neglected. preparation of the manuscript and for discussions, as well We employ here a condensed notation, where the nu- as Severin Jakobs, Peter Kopietz, Johannes Reuter and merical indices such as 1 and 10 represent space-time 0 0 0 Georg Schwiete for helpful discussions. This work is sup- coordinates x1 = (~r1, t1) and x1 = (~r1, t1), respec- ported by the German Research Foundation (DFG) in tively, and the integration sign with a prime denotes the framework of the Priority Program 1459 “Graphene”. integration over all primed space-time coordinates. Be- sides, the three-vertices are written in the compact form (2,1) αβγ ΓΛ (1iα, 2jβ; 3γ) = ΓΛ,ij (1, 2; 3). As explained in sec- Appendix A: Analytical form of the vertex flow tion III E, latin indices denote the internal degrees of equations freedom of the fermions (sublattice, valley, spin), and greek letters denote the Keldysh degrees of freedom (clas- In this appendix we give the explicit analytical form of sical and quantum). In the following we also omit the the flow equations for the fermionic self-energy, bosonic Λ-indices for brevity, since all quantities appearing here polarization and the three-legged Fermi-Bose vertex. For are scale dependent (except for the bare Coulomb inter- completeness sake we also state the flow equation for the action V ), and thus there can be no confusion.

a. Field expectation value Z 0 −1 R R 0 ¯ 0 R 0 ¯ 0 V + Π + Rb (1, 1 )∂Λφc(1 ) + ∂ΛRb (1, 1 )φc(1 )

i X X Z 0 = − ∂/ Gαβ(10, 20)Γβαq(20, 10; 1) − ∂ n˜(1) (A1) 2 Λ kl lk Λ α,β k,l b. Self-energy Z 0 αβ αβc 0 ¯ 0 ∂ΛΣij (1, 2) = Γij (1, 2; 1 )∂Λφc(1 ) Z 0 X X αγ1γ4 0 0 γ1γ2 0 0 γ2βγ3 0 0 γ3γ4 0 0 + i∂/Λ Γik (1, 1 ; 4 )Gkl (1 , 2 )Γlj (2 , 2; 3 )D (3 , 4 ) (A2) γ1,γ2,γ3,γ4 k,l c. Polarization i X X Z 0 ∂ Παβ(1, 2) = ∂/ Gγ1,γ2 (10, 20)Γγ2γ3α(20, 30; 1)Gγ3,γ4 (30, 40)Γγ4,γ2β(40, 10; 2) (A3) Λ 2 Λ kl lm mn nk γ1,γ2,γ3,γ4 k,l,m,n d. 3-vertex Z 0 αβγ X X αγ1γ6 0 0 γ4γ2γ5 0 0 γ2γ3γ3 0 0 ∂ΛΓij (1, 2; 3) = i∂/Λ Γik (1, 1 ; 6 )Γnj (4 , 2; 5 )Γlm (2 , 3 ; 3) γi k,l,m,n i=1,...,6

γ1γ2 0 0 γ3γ4 0 0 γ5γ6 0 0 × Gkl (1 , 2 )Gmn (3 , 4 )D (5 , 6 ) (A4) Recall that in the bare action only four of the above 3-vertices are present, namely the ones with the Keldysh indices cqc, qcc, ccq, qqq being all equal to unity. The remaining three ones, with the Keldysh indices qqc, cqq, qcq, are generated during the ﬂow, while the ccc-vertex is constrained to vanish at all scales. Therefore, we state in the following a further truncated set of the above equations, where only the four 3-vertices present in the bare action have been kept. These equations were the starting point for our analysis of thermal equilibrium in section IV.

e. Field expectation value Z 0 −1 R R 0 ¯ 0 R 0 ¯ 0 V + Π + Rb (1, 1 )∂Λφc(1 ) + ∂ΛRb (1, 1 )φc(1 )

i X Z 0 = − ∂/ GK (10, 20)Γccq(20, 10; 1) − ∂ n˜(1) (A5) 2 Λ kl lk Λ k,l 22 f. Self-energy Z 0 Z 0 R qcc 0 ¯ 0 X qcc 0 0 K 0 0 ccq 0 0 A 0 0 ∂ΛΣij(1, 2) = Γij (1, 2; 1 )∂Λφc(1 ) + i∂/Λ Γik (1, 1 ; 4 )Gkl(1 , 2 )Γlj (2 , 2; 3 )D (3 , 4 ) k,l qcc 0 0 R 0 0 qcc 0 0 K 0 0 +Γik (1, 1 ; 4 )Gkl(1 , 2 )Γlj (2 , 2; 3 )D (3 , 4 ) (A6)

Z 0 Z 0 A cqc 0 ¯ 0 X ccq 0 0 K 0 0 cqc 0 0 R 0 0 ∂ΛΣij(1, 2) = Γij (1, 2; 1 )∂Λφc(1 ) + i∂/Λ Γik (1, 1 ; 4 )Gkl(1 , 2 )Γlj (2 , 2; 3 )D (3 , 4 ) k,l cqc 0 0 A 0 0 cqc 0 0 K 0 0 +Γik (1, 1 ; 4 )Gkl(1 , 2 )Γlj (2 , 2; 3 )D (3 , 4 ) (A7)

Z 0 K X qcc 0 0 K 0 0 cqc 0 0 K 0 0 ∂ΛΣij (1, 2) = i∂/Λ Γik (1, 1 ; 4 )Gkl(1 , 2 )Γlj (2 , 2; 3 )D (3 , 4 ) k,l qcc 0 0 R 0 0 qqq 0 0 A 0 0 +Γik (1, 1 ; 4 )Gkl(1 , 2 )Γlj (2 , 2; 3 )D (3 , 4 ) qqq 0 0 A 0 0 cqc 0 0 R 0 0 +Γik (1, 1 ; 4 )Gkl(1 , 2 )Γlj (2 , 2; 3 )D (3 , 4 ) (A8)

Z 0 ! Z X ccq 0 0 R 0 0 qcc 0 0 R 0 0 0 = ∂ΛΣij(1, 2) = i∂/Λ Γik (1, 1 ; 4 )Gkl(1 , 2 )Γlj (2 , 2; 3 )D (3 , 4 ) k,l cqc 0 0 A 0 0 ccq 0 0 A 0 0 +Γik (1, 1 ; 4 )Gkl(1 , 2 )Γlj (2 , 2; 3 )D (3 , 4 ) (A9) g. Polarization i X Z 0 ∂ ΠR(1, 2) = ∂/ GK (10, 20)Γccq(20, 30; 1)GR (30, 40)Γqcc(40, 10; 2) Λ 2 Λ kl lm mn nk k,l,m,n A 0 0 ccq 0 0 K 0 0 cqc 0 0 +Gkl(1 , 2 )Γlm (2 , 3 ; 1)Gmn(3 , 4 )Γnk (4 , 1 ; 2) (A10)

i X Z 0 ∂ ΠA(1, 2) = ∂/ GK (10, 20)Γcqc(20, 30; 1)GA (30, 40)Γccq(40, 10; 2) Λ 2 Λ kl lm mn nk k,l,m,n R 0 0 qcc 0 0 K 0 0 ccq 0 0 +Gkl(1 , 2 )Γlm (2 , 3 ; 1)Gmn(3 , 4 )Γnk (4 , 1 ; 2) (A11)

i X Z 0 ∂ ΠK (1, 2) = ∂/ GK (10, 20)Γccq(20, 30; 1)GK (30, 40)Γccq(40, 10; 2) Λ 2 Λ kl lm mn nk k,l,m,n R 0 0 qqq 0 0 A 0 0 ccq 0 0 +Gkl(1 , 2 )Γlm (2 , 3 ; 1)Gmn(3 , 4 )Γnk (4 , 1 ; 2) A 0 0 ccq 0 0 R 0 0 qqq 0 0 +Gkl(1 , 2 )Γlm (2 , 3 ; 1)Gmn(3 , 4 )Γnk (4 , 1 ; 2) (A12)

Z 0 ! i X 0 = ∂ ΠZ (1, 2) = ∂/ GR (10, 20)Γqcc(20, 30; 1)GR (30, 40)Γqcc(40, 10; 2) Λ 2 Λ kl lm mn nk k,l,m,n A 0 0 cqc 0 0 A 0 0 cqc 0 0 +Gkl(1 , 2 )Γlm (2 , 3 ; 1)Gmn(3 , 4 )Γnk (4 , 1 ; 2) (A13) h. 3-vertex Z 0 ccq X K 0 0 cqc 0 0 A 0 0 ccq 0 0 R 0 0 qcc 0 0 ∂ΛΓij (1, 2; 3) = i∂/Λ D (5 , 6 )Γik (1, 1 ; 6 )Gkl(1 , 2 )Γlm (2 , 3 ; 3)Gmn(3 , 4 )Γnj (4 , 2; 5 ) k,l,m,n R 0 0 ccq 0 0 K 0 0 ccq 0 0 R 0 0 qcc 0 0 +D (5 , 6 )Γik (1, 1 ; 6 )Gkl(1 , 2 )Γlm (2 , 3 ; 3)Gmn(3 , 4 )Γnj (4 , 2; 5 ) A 0 0 cqc 0 0 A 0 0 ccq 0 0 K 0 0 ccq 0 0 +D (5 , 6 )Γik (1, 1 ; 6 )Gkl(1 , 2 )Γlm (2 , 3 ; 3)Gmn(3 , 4 )Γnj (4 , 2; 5 ) (A14) 23

Z 0 cqc X K 0 0 cqc 0 0 A 0 0 cqc 0 0 A 0 0 cqc 0 0 ∂ΛΓij (1, 2; 3) = i∂/Λ D (5 , 6 )Γik (1, 1 ; 6 )Gkl(1 , 2 )Γlm (2 , 3 ; 3)Gmn(3 , 4 )Γnj (4 , 2; 5 ) k,l,m,n R 0 0 ccq 0 0 K 0 0 cqc 0 0 A 0 0 cqc 0 0 +D (5 , 6 )Γik (1, 1 ; 6 )Gkl(1 , 2 )Γlm (2 , 3 ; 3)Gmn(3 , 4 )Γnj (4 , 2; 5 ) R 0 0 ccq 0 0 R 0 0 qcc 0 0 K 0 0 cqc 0 0 +D (5 , 6 )Γik (1, 1 ; 6 )Gkl(1 , 2 )Γlm (2 , 3 ; 3)Gmn(3 , 4 )Γnj (4 , 2; 5 ) (A15)

Z 0 qcc X K 0 0 qcc 0 0 R 0 0 qcc 0 0 R 0 0 qcc 0 0 ∂ΛΓij (1, 2; 3) = i∂/Λ D (5 , 6 )Γik (1, 1 ; 6 )Gkl(1 , 2 )Γlm (2 , 3 ; 3)Gmn(3 , 4 )Γnj (4 , 2; 5 ) k,l,m,n A 0 0 qcc 0 0 R 0 0 qcc 0 0 K 0 0 ccq 0 0 +D (5 , 6 )Γik (1, 1 ; 6 )Gkl(1 , 2 )Γlm (2 , 3 ; 3)Gmn(3 , 4 )Γnj (4 , 2; 5 ) A 0 0 qcc 0 0 K 0 0 cqc 0 0 A 0 0 ccq 0 0 +D (5 , 6 )Γik (1, 1 ; 6 )Gkl(1 , 2 )Γlm (2 , 3 ; 3)Gmn(3 , 4 )Γnj (4 , 2; 5 ) (A16)

Z 0 qqq X K 0 0 qcc 0 0 K 0 0 ccq 0 0 K 0 0 cqc 0 0 ∂ΛΓij (1, 2; 3) = i∂/Λ D (5 , 6 )Γik (1, 1 ; 6 )Gkl(1 , 2 )Γlm (2 , 3 ; 3)Gmn(3 , 4 )Γnj (4 , 2; 5 ) k,l,m,n K 0 0 qcc 0 0 R 0 0 qqq 0 0 A 0 0 cqc 0 0 +D (5 , 6 )Γik (1, 1 ; 6 )Gkl(1 , 2 )Γlm (2 , 3 ; 3)Gmn(3 , 4 )Γnj (4 , 2; 5 ) R 0 0 qqq 0 0 A 0 0 ccq 0 0 K 0 0 cqc 0 0 +D (5 , 6 )Γik (1, 1 ; 6 )Gkl(1 , 2 )Γlm (2 , 3 ; 3)Gmn(3 , 4 )Γnj (4 , 2; 5 ) A 0 0 qcc 0 0 K 0 0 ccq 0 0 R 0 0 qqq 0 0 +D (5 , 6 )Γik (1, 1 ; 6 )Gkl(1 , 2 )Γlm (2 , 3 ; 3)Gmn(3 , 4 )Γnj (4 , 2; 5 ) (A17)

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