Keldysh functional renormalization group for electronic properties of graphene

Keldysh functional renormalization group for electronic properties of graphene

Abstract

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

pacs:
Valid PACS appear here
{fmffile}

fgraphs

I Introduction

The band structure of graphene features two isolated points where valence and conduction bands touch.Castro Neto et al. (2009); Das Sarma et al. (2011); Peres (2010) At these touching points the electrons have a linear energy-momentum dispersion, similar to massless relativistic Dirac particles.Wallace (1947) This pseudorelativistic band structure is responsible for the appearance of phenomena usually related to the relativistic domain, such as Klein tunneling through potential barriers,Klein (1929); Cheianov and Falko (2006); Katsnelson et al. (2006); Beenakker (2008) the Zitterbewegung,Katsnelson (2006) or an anomalous quantized Hall effect.Gusynin and Sharapov (2005); Novoselov et al. (2005); Zhang et al. (2005); Novoselov et al. (2007)

For a description of realistic graphene samples, effects of disorder and electron-electron interactions have to be added to this idealized band structure. Disorder smears out the singularity at the nodal point, but preserves many of graphene’s remarkable electronic properties,Castro Neto et al. (2009); Das Sarma et al. (2011) and even leads to fundamentally new phenomena by itself, such as the absence of Anderson localization if disorder does not couple the nodal points.Bardarson et al. (2007); Nomura et al. (2007); San-Jose et al. (2007) The effect of interactions is most pronounced if the singularity in the density of states of the noninteracting theory is not smeared by disorder and the chemical potential is close to the nodal point.Kotov et al. (2012) The vanishing carrier density at the nodal point at zero temperature Hobson and Nierenberg (1953) implies the absence of screening, which leads to strongly enhanced interaction corrections. In particular, interactions are found to effectively renormalize the Fermi velocity at the nodal point, and the corrections to the velocity diverge logarithmically in the low-temperature limit.González et al. (1994); Vozmediano (2011) These logarithmic corrections have recently been verified experimentally, and good agreement with theoretical calculations was reported.Elias et al. (2011)

Although there is consensus about the way in which interactions affect the electronic structure of graphene,Kotov et al. (2012) a quantitative evaluation of the corrections proved to be problematic. The dimensionless interaction strength for the electrons in graphene is , which approximately equals  in the freestanding case in vacuum (). For such a large interaction strength a perturbative calculation of the renormalization effect cannot be reliable, and at first sight the reported agreement of one-loop perturbation theory with the experimentally observed increase of the Fermi velocity appears surprising. Indeed, a two-loop calculation leads a completely different result, a decrease of the Fermi velocity for small momenta.Mishchenko (2007); Vafek and Case (2008); Barnes et al. (2014) An alternative approach is to make use of the largeness of the number of fermion species (which is in graphene), and a perturbation theory in gives results largely consistent with the approach based on a perturbative treatment of the interaction strength.Foster and Aleiner (2008); González et al. (1999)

To address such a situation in which no small parameter, to organize a perturbative expansion, is available, nonperturbative methods have been applied to the problem of interacting Dirac fermions in two dimensions. One of those nonperturbative methods is the functional renormalization group (fRG), which shares some features with the celebrated Wilsonian renormalization group,Altland, A. and Simons, B. (2010); Bagnuls and Bervillier (2001) but rigorously extends the concept of flowing coupling constants to (one-particle irreducible vertex) functions. Initiated by Wetterich,Wetterich (1991); Note1 () this method has found widespread applications in high energy and in condensed matter physics.Metzner et al. (2012); Berges et al. (2002); Wetterich (2001); Gies and Wetterich (2002); Kopietz, P. and Bartosch, L. and Schütz, F. (2010) Of particular relevance to the present problem is the work of Bauer et al.,Bauer et al. (2015) who studied the Fermi velocity renormalization and the static dielectric function in graphene at zero temperature using the fRG framework and found excellent agreement with the experiment, surpassing the results of the conventional perturbative methods.

As powerful as the fRG is, it clearly has its limitations when used within its most commonly employed formulation in imaginary time. First and foremost, true nonequilibrium phenemena (beyond linear response) are out of reach of the Matsubara formalism. Second, even for linear response calculations the imaginary time formalism requires an analytical continuation from imaginary to real time at the end of a calculation, which may pose technical difficulties. The appropriate framework to describe true nonequilibrium dynamics is the Keldysh formalism.Chou et al. (1985); Kamenev (2011); Rammer (2007) The Keldysh formalism has the additional advantage that it erases the necessity of analytical continuations, which may also makes it a useful tool for equilibrium applications. Gezzi et al. implemented a Keldysh formulation of fRG for applications to impurity problems.Gezzi et al. (2007) Jakobs et al. further developed the theory, constructing a “Keldysh-compatible” cutoff scheme that respects causality, with applications to quantum dots and nanowires coupled to external baths.Jakobs et al. (2007, 2010) Keldysh formulations of fRG were also developed for various systems involving bosons.Berges and Mesterházy (2012); Berges and Hoffmeister (2009); Kloss and Kopietz (2011); Gasenzer and Pawlowski (2008); Gasenzer et al. (2010)

In the present article we construct a Keldysh fRG 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 velocity renormalization and the static dielectric function in graphene, finding full agreement with the zero-temperature Matsubara-formalism calculation of Bauer et al. Bauer et al. (2015) We also extend the calculation to finite temperatures, an extension that in principle is possible within the Matsubara formalism, too, but that comes at no additional calculational cost when done in the Keldysh formalism. We leave applications to true nonequilibrium properties of graphene for future work, but already notice that there is a vast body of perturbative (or in other ways approximate) true nonequilibrium theoretical results for graphene that such a theory can be compared with, see, e.g., Refs. Gorbar et al., 2002; Gusynin et al., 2007; Schütt et al., 2011; Narozhny et al., 2015. Although our theory focuses on graphene, a major part of the formalism we develop here is also applicable to conventional nonrelativistic fermions.

The extension of an imaginary-time fRG formulation to a Keldysh-based formulation involves quite a number of subtle steps and manipulations. One issue is the choice of a cut off scheme, which preferentially is compatible with the causality structure of the Keldysh formalism and, for equilibrium applications, with the fluctuation-dissipation theorem.Jakobs et al. (2007, 2010) Another issue is the possibility of an arbitrary nonequilibrium initial condition and the truncation of the (in principle) infinite hierarchy of flow equations in the fRG approach. To do justice to these issues, we have chosen to make this article self contained, although we tried to keep the discussion of standard issues as brief as possible.

The outline of the paper is as follows: In Sec. II we introduce the formal aspects of nonequilibrium quantum field theory, using the Keldysh technique applied to graphene. The originally purely fermionic problem is formulated as a coupled fermion-boson problem by means of a Hubbard-Stratonovich transformation, singling out the dominant interaction channel. The ideas of the functional renormalization group are reviewed in Sec. III, where we combine them with the nonequilibrium Keldysh formalism. We implement an infrared regularization and derive the exact spectral Dyson equations and quantum kinetic equations, as well as an exact flow equation, which incorporates all of the nonperturbative aspects of the theory. Finally, we perform a vertex expansion leading to an exact, infinite hierarchy of coupled integro-differential equations for the one-particle irreducible vertex functions. Section IV deals with a solution of our theory in thermal equilibrium. We discuss the necessary limitations for the construction of suitable regulator functions, which preserve causality and, at the same time, the fluctuation-dissipation theorem, allowing a solution of the quantum kinetic equations at all scales. We further present a simple truncation scheme for the calculation of the Fermi velocity and static dielectric function at finite temperature, extending the results of Bauer et al. Bauer et al. (2015)

Ii Nonequilibrium Quantum Field Theory

This section mainly serves as an introduction to the Fermi-Bose quantum field theory of interacting electrons in graphene in the nonequilibrium Keldysh formulation. The reader who is familiar with this formulation may skim through our notational conventions and continue reading at section III.

We consider interacting Dirac fermions in two dimensions, which are described by a grand canonical Hamiltonian in the Heisenberg picture

(1)

Here describes the low energy approximation of free electrons hopping on the honeycomb lattice, and contains the interaction effects. The first term reads Gusynin et al. (2007) ()

(2)

with the chemical potential and the external electromagnetic potentials and . The Dirac electrons are described by eight-dimensional spinors, where we choose the basis as , with

(3)

The indices denote the spin, the valley- and the sublattice degree of freedom. Further, is the two-dimensional unit matrix acting in spin space and , with the Pauli matrices and acting in valley and sublattice space, respectively. The interaction part is given by the instantaneous Coulomb interaction

(4)

where

(5)
(6)

and is the dielectric constant of the medium, being unity for freestanding graphene in vaccuum. Here the term is a background charge density, representing the charge accumulated on a nearby metal gate. Away from the charge neutrality point it essentially acts as a counterterm, which removes the zero wavenumber singularity of the Coulomb interaction at finite charge carrier density.

ii.1 Single-particle Green functions

Relevant physical observables can be expressed as correlation functions of the field operators, and the purpose of a field-theoretic treatment is to provide a formalism in which such correlation functions can be calculated efficiently. For an explicitly time-dependent Hamiltonian, such as the one above, one considers the evolution of the field operators along the “Schwinger-Keldysh contour”,Chou et al. (1985); Rammer (2007); Kamenev (2011) a closed time contour starting at a reference time , extending to , and eventually returning from to , see Fig. 1.

Figure 1: Schwinger-Keldysh time contour in the complex time plane with reference time as starting and end point. and are the forward and backward branch, respectively.

Consequently, the time arguments of the field operators are elevated to the “contour time”, and the building blocks of the theory are formed by the expectation values of “path ordered” products of the field operators. The concept of path ordering generalizes the concept of (imaginary) time ordering, such that field operators with a higher contour time appear to the right of operators with a lower contour time. In particular, the single-particle propagator reads

(7)

where the indices represent collectively the sublattice, valley and spin degrees of freedom. is the contour-time ordering operator and the expectation value is performed with respect to some initial density matrix given at a reference time

(8)

Since there are four possibilities where the two time variables can be located to each other with respect to the two time branches and , one can map the contour-ordered Green function to a matrix representation with time-arguments defined on the real axis

(9)

The constituents of this matrix are the time ordered, anti-time ordered, greater and lesser Green function, respectively,

(10a)
(10b)
(10c)
(10d)

By definition, these functions are linearly dependent and subject to the following constraint,Chou et al. (1985); Rammer (2007); Kamenev (2011)

(11)

which allows a basis transformation to three linearly independent propagators. This transformation is given by the involutional matrix , where is a Pauli matrix and is the orthogonal matrix

(12)

originally introduced by Keldysh.Keldysh (1964) Its application to Eq. (9) yields

(13)

with

(14a)
(14b)
(14c)

The functions are the retarded, advanced and Keldysh propagators, respectively. The latter one is also known as the statistical propagator. They obey the symmetry relations

(15)

as well as the causality relations Chou et al. (1985); Rammer (2007); Kamenev (2011)

(16a)
(16b)

Explicit expressions for the free propagators may easily be obtained in thermal equilibrium and in the absence of the electromagnetic potentials. To this end we send the reference time and Fourier transform the field operators following the conventions

(17)

with . After a short calculation one finds

(18a)
(18b)

where is the unit matrix in valley-sublattice space. Note that the entire statistical information of the system is contained in the Keldysh propagator. These expressions may be further simplified by expanding the propagators in the chiral basis

(19)

in which are the chiral projection operators

(20)

with . In the chiral basis the propagators then take the simple form

(21a)
(21b)
(21c)

The density of electrons in the system is given by

(22)

which is formally divergent. The charge carrier density, however, which is defined as Gusynin et al. (2007)

(23)

is finite. It is a function of the external doping and of the gauge invariant external electromagnetic fields. In the absence of such external fields, it vanishes at the charge neutrality point ().

ii.2 Contour-time generating functional

The entire physical content of the theory can be conveniently expressed by the partition function Chou et al. (1985); Negele, J.W. and Orland, H. (1998); Kamenev (2011); Berges and Mesterházy (2012); Calzetta and Hu (1988)

(24)

which is a generating functional for all -point correlation functions, including the single-particle propagators described above. Its arguments and , where only the former is shown on the left hand side for brevity, are eight component spinorial external source terms. Here and in the remainder of this article we employed a condensed vector notation

(25)

where labels space and (contour-) time coordinates, such that

(26)

The symbol indicates that the time integration has to 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 equal to zeroNote2 ()

(27)

In fact, this normalization is the very reason for the algebraic identity (11) and it leads to similar constraints for higher order correlation functions, see Ref. Chou et al., 1985. It further ensures that any correlation function computed from the partition function (24) does not contain disconnected bubble diagrams.

The partition function (24) can be represented in terms of a fermionic coherent state functional integral as Negele, J.W. and Orland, H. (1998); Kamenev (2011); Chou et al. (1985); Calzetta and Hu (1988)

(28)

Here is the contour-time action of the system and is the correlation functional, which incorporates the statistical information of the initial density matrix.Calzetta and Hu (1988); Chou et al. (1985); Berges and Cox (2001) Their dependence on the Grassmann-valued spinor fields and has been abbreviated by , as we did for the source field dependence of the partition function.

The action can be written as a contour-time integral over the Lagrangian

(29)

with

(30)

Similarly to the Hamiltonian (1), the action decomposes into free contribution and an interaction term,

(31)

expressions for which can be obtained immediately by substitution of Eqs. (2) and (4).

The functional describes the initial correlations of the system, corresponding to the density matrix . It may be expanded in powers of fields as

(32)

where the kernels are nonvanishing only, if all their respective contour-time arguments equal the initial time . The statistical information contained in the kernels , specifying the correlations present in the initial state, is in a one-to-one correspondence to the statistical information contained in the density matrix.Calzetta and Hu (1988); Berges and Cox (2001); Gasenzer et al. (2010) In practice, only a limited set of initial correlations is taken into account, either because of an implicit assumption that the initial state is a thermal equilibrium state for an effectively noninteracting system,Rammer (2007); Kamenev (2011) or as an expression of the finite knowledge that is available about an experimental setup.Berges and Cox (2001) In the remainder of this work we mainly focus on Gaussian density matrices, i.e., we truncate the series (32) after the first term, absorbing the statistical information of into the boundary conditions of the two-point function and simply write . Yet most of our results are not affected by this simplification and valid even in the general case. We come back to this issue in section III.5, where we comment on some questions regarding the possible implementation of correlated initial states.

Although it is possible to treat the theory presented so far within the formalism of the (fermionic) functional renormalization group,Tanizaki et al. (2014); Metzner et al. (2012) we here choose a formulation in which a bosonic field is introduced by means of a Hubbard-Stratonovich transformation, that decouples the Coulomb interaction.Negele, J.W. and Orland, H. (1998); Kamenev (2011); Kopietz, P. and Bartosch, L. and Schütz, F. (2010) It is well-known that bosonic degrees of freedom, such as Cooper pairs in the celebrated BCS-theory of superconductivity,Negele, J.W. and Orland, H. (1998) naturally emerge as collective, low energy degrees of freedom of composite fermions. Therefore, it is reasonable to introduce a collective bosonic field right from the beginning, which captures the dominant contributions of the interaction.

The Hubbard-Stratonovich transformation is an exact integral identity replacing the four-fermion interaction by a quadratic form of a real bosonic field and a Fermi-Bose interaction

(33)

The free bosonic part is given by

(34)

where is the inverse Coulomb interaction, understood in the distributional sense. The interaction term contains a trilinear Yukawa-type interaction and a linear term, describing the coupling of the Hubbard-Stratonovich boson to the background charge density 

(35)

Note that the fluctuating Bose field appears on the same footing as the external scalar potential , see Eq. (2).

We generalize the Hubbard-Stratonovich transformed partition function by introducing an additional source term , so that it gives access to bosonic as well as mixed Fermi-Bose correlators. The generalized Fermi-Bose partition function reads

(36)

with . It fulfills the same normalization condition, when the sources are set to zero, as the purely fermionic partition function

(37)

ii.3 Real-time representation

Although the contour-time representation allows for a compact and concise notation during any step of a 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 into forward () and backward () branch, thereby defining a doubled set of fields, and , allocated to the respective branch

(38)

In a next step, one performs a rotation from -field space to Keldysh space, using the involutional matrix , see Eq. (12), which was already employed for the rotation of the Green functions in Sec. II.1. Further, one defines the symmetric and antisymmetric linear combinations of the -fields as “classical” () and “quantum” () components, respectively, and combines these into vectors and as

(39)
(40)

The source fields are rotated and combined into vectors and likewise. Two remarks are in order. First, the mapping of the bosonic source term yields an additional factor of two, due to our choice of normalization in Eq. (40), which we choose to absorb into a redefinition of . The second remark is concerned about our definition of the Keldysh rotation for the fermionic field . Some authors prefer a different convention, which was originally proposed by Larkin and Ovchinnikov.Larkin, A. I. and Ovchinnikov, Yu. N. (1975) In a purely fermionic theory this is reasonable, since it leads to a certain technical simplification. However, this modified rotation is not possible for bosons. In the context of the coupled Fermi-Bose theory we are dealing with, the implementation of the Larkin-Ovchinnikov rotation would lead to an asymmetry in the arising Keldysh structures, which we want to avoid. Therefore, we define the Keldysh rotation as proposed in Eq. (39). Further, one has to keep in mind that the naming “classical” for the fermions is just terminology. For the bosons on the other hand this naming has a physical meaning.

We here summarize the main results of the real-time mapping and explain the structure of the theory obtained after the above Keldysh rotation. For the partition function we find

(41)

We have used here the short-hand notation

(42)

in which the Pauli matrix acts in Keldysh space, coupling a “classical” source to a “quantum” field and vice versa. Further, all the time integrations are defined from now on along the forward time branch only

(43)

The action is the sum of three contributions,

(44)

Its quadratic part in the fermionic sector is given by

(45)

The inverse free propagator has a trigonal matrix structure

(46)

with retarded/advanced and Keldysh blocks , which obey the symmetries Rammer (2007); Kamenev (2011)

(47)

The retarded/advanced blocks are the inverse free retarded/advanced propagators

(48)

where the gauge covariant derivative is given by

(49)

Note that the regularization term , which we have written here explicitly, enforces the retarded, respectively advanced, boundary condition. It has to be emphasized that the external gauge fields therein are understood as entirely classical

(50a)
(50b)

Since these fields are not quantized, their quantum components in Keldysh space vanish identically. Yet it is formally possible to keep them as source fields, which could be used to generate density-density or current-current correlation functions.Kamenev (2011) On the other hand this is not necessary, since we have the single-particle sources at our disposal. In contrast to the retarded and advanced blocks of Eq. (46), the Keldysh block does not take the form of a simple inverse propagator. It carries the statistical information of the theory and can be written as

(51)

with the noninteracting Keldysh Green function . Since the latter is an anti-hermitian matrix, see Eq. (15), it can be parametrized in terms of a hermitian matrix and the spectral functions as Kamenev (2011)

(52)

Substitution into Eq. (51) then yields that for noninteracting fermions the Keldysh block of the inverse matrix propagator is a pure regularization termNote3 ()

(53)

Only when interactions are considered the Keldysh block will acquire a finite value. We will come back to this issue in section III.3. The free propagator is obtained by inverting Eq. (46), where the Keldysh structure is given by Eq. (13).

The quadratic part of the action in the bosonic sector reads

(54)

The bosonic matrix has the same trigonal structure as the fermionic one

(55)

with the same symmetry relations as Eq. (47). Owing to the fact that the bosons are real, the above quantities fulfill the additional symmetries Kamenev (2011); Rammer (2007)

(56)

The retarded and advanced blocks are twice the inverse bare Coulomb interaction

(57)

The Keldysh component for bosons has the same structure as the fermionic one

(58)

Similarly to the fermionic case we can parametrize the bosonic Keldysh Green function in terms of a hermitian function  Kamenev (2011)

(59)

Since the bare Coulomb interaction is instantaneous, the above Keldysh propagator together with the Keldysh block (58) vanish identically. For that reason we may write

(60)

Again, the interaction with the fermions will eventually lead to a finite bosonic Keldysh self-energy and, hence, a nonvanishing Keldysh propagator as in the fermionic case.

Finally we discuss the Fermi-Bose interaction term. Its linear counterterm maps in the same way as the sources do, but with the important difference that the quantum component is identically zero. Nevertheless, we may still use the Keldysh vector notation for this term as well. The trilinear term maps to four interaction terms in real-time, which can be arranged in a matrix form similar to Eq. (45),

(61)

Note the factor of two in front of the linear term in comparison to the linear source term, which could not be absorbed into a redefinition of any of those fields as was the case for . Further observe that the classical components of the fluctuating Bose field appear in the same off-diagonal position as the external gauge field does in Eq. (45). The quantum components on the other hand are located in the diagonal.

Now that all of our notational conventions have been established we can move on to the central part of this work.

Iii Nonequilibrium Functional Renormalization Group

The idea of the functional renormalization group is to modify the bare action of the theory by introducing a dependence on a parameter , in such a way that the partition function can be easily (and exactly) calculated if is set equal to an initial value , whereas the true physical system corresponds to . Using the solution of the modified partition function at , one obtains the “physical” partition function at by tracking its changes upon lowering from to . In practice the parameter is chosen as an infrared regularization which effectively removes low-energy (or low-momentum) modes, determined by the cutoff , from the functional integration. In this case, the initial value is the ultraviolet cutoff of the action . For graphene, this ultraviolet cutoff is the momentum or energy at which the linear dispersion in Eq. (2) breaks down.

iii.1 Infrared regularization

We implement the idea of an infrared regularization by modifying the quadratic terms in the Fermi- and Bose-sectors of the contour-time action via additive regulator functions  Wetterich (1991); Berges et al. (2002)

(62a)
(62b)

It is also possible to regularize only one of the two sectors, by setting either or to zero. The regulators have to be analytic functions of . For they have to diverge, such that all infrared modes occuring in the functional integral are effectively frozen out, while for they have to vanish.Berges et al. (2002); Metzner et al. (2012); Kopietz, P. and Bartosch, L. and Schütz, F. (2010) In this way the partition function (36) becomes a cutoff dependent quantity, , where only the modes above contribute to the functional integral. In the limit it reduces to the original partition function of the previous section, see Eq. (41).

After mapping the contour-time regulator terms to a real-time repesentation and performing the Keldysh rotation as explained in Sec. II.3, the cutoff dependent quadratic parts of the action become

(63a)
(63b)

Note the absence of the factor 1/2 in front of the bosonic regulator term, which is due to our choice of normalization for the bosonic rotation (40). In principle, the most general choice for the contour-time regulators results in the following matrix structure for the real-time regulators

(64a)
(64b)

Although it is not strictly necessary if the evolution from to could be tracked exactly, for the correct implementation of approximate evolution schemes it is important that the regulators are chosen in such a way that they respect the symmetries and the causality structure of the theory. In particular, in order to ensure that the partition function is normalized to unity at any scale, and hence retain the algebraic identities among the correlation functions, cf. Eq. (11), we choose the regulators such that the “anomalous” components vanish. The remaining components are constructed in such a way that they are compatible with the symmetry and causality structure of the bare inverse propagators, see Eqs. (47) and (56). This choice of the regulator functions ensures that the partition function has the correct causality structure at any value of the cutoff , independent of eventual approximations made when solving the evolution equations.

In addition to the -dependencee of the action introduced via Eqs. (63) we allow the counterterm to be explicitly cutoff dependent, setting

(65)

The counterterm describes a flowing background charge density, which has to be tuned to remove potentially divergent contributions from the Coulomb interaction at finite charge carrier density.

iii.2 Connected functional and effective action

The evolution equation will not be derived for the partition function , but rather for the effective action , which is essentially the Legendre transformation of the cutoff dependent connected functional Negele, J.W. and Orland, H. (1998); Chou et al. (1985)

(66)

being a generating functional for connected correlation functions. Differentiation with respect to the sources yield the expectation values of the fields and ,

(67)
(68)

These expectation values, being complicated nonlinear functionals of the sources and , define “macroscopic” fields which inherit a -dependence from the regulators (and the counterterm). A macroscopic Fermi field can only exist when the sources are finite, otherwise it is strictly zero. The classical component of the macroscopic bosonic field , on the other hand, can very well acquire a finite value in the absence of source terms.Chou et al. (1985); Kloss and Kopietz (2011); Berges and Mesterházy (2012); Berges and Hoffmeister (2009) Such a macroscopic field expectation value may signal a spontaneous symmetry breaking, but in the theory we consider here this is not the case. The bosonic field is conjugate to the particle density and as such it reflects, e.g., a local deviation away from charge neutrality driven by an external potential . In the following we omit the brackets to denote the average of a single field, for brevity. Since we are always working with averages of fields, there can be no confusion.

The second derivatives of define the connected two-point correlators

(69)
(70)

where we introduced the connected average . Explicitly displaying the Keldysh structure, we have

(71)
(72)

The above propagators are source- and cutoff-dependent functionals, which do not obey the usual triangular structure. In particular the anomalous statistical propagators and are nonvanishing as long as the source terms are finite. By construction of the regulators, the familiar triangular structure together with the symmetry and causality relations arise once the single-particle sources are set to zero. All the other higher order connected correlation functions can be obtained by further differentiation as in the equilibrium Matsubara theory.Negele, J.W. and Orland, H. (1998)

The central object in the functional renormalization group is the effective action . It is the generating functional for one-particle irreducible vertex functions, and defined as the Legendre transform of the connected functional 

(73)

In the Legendre transform the single-particle sources must be understood as -dependent functionals of the field expectation values, obtained by inversion of the defining relations Eqs. (67) and (68). The Legendre transform is modified in such a way that the cutoff terms are subtracted on the right hand side. This ensures that the flowing action does not contain the cutoff terms at any scale, but spoils the convexity of an ordinary Legendre transform.

The properties and physical interpretation of this functional, mainly in the context of its equilibrium counterpart, have been discussed at length in the literature.Berges et al. (2002); Metzner et al. (2012); Kopietz, P. and Bartosch, L. and Schütz, F. (2010) Most importantly the flowing action has the nice property that it interpolates smoothly between the microscopic laws of physics, parametrized by an action , and the full effective action , where all thermal and quantum fluctuations are taken into account. In many cases the microscopic laws are simply governed by the bare action of the system . This latter statement, however, depends on the actual cutoff scheme. In certain situations it is preferable to devise a cutoff scheme where the initial effective action does not coincide with the bare action, and hence the initial conditions of the flow are nontrivial.Jakobs et al. (2010, 2007); Kopietz, P. and Bartosch, L. and Schütz, F. (2010); Schütz and Kopietz (2006); Sharma and Kopietz (2016) We will come back to this issue at the end of the next subsection.

Taking the first functional derivative of Eq. (73) with respect to the fields one finds that the effective action satisfies the “equations of motion”

(74)