Keldysh approach to the renormalization group analysis of the disordered electron liquid

G. Schwiete1, ∗ and A. M. Finkel’stein2, 3 1Dahlem Center for Complex Quantum Systems and Institut f¨urTheoretische Physik, Freie Universit¨atBerlin, 14195 Berlin, Germany 2Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843-4242, USA 3Department of Condensed Matter Physics, The Weizmann Institute of Science, 76100 Rehovot, Israel (Dated: September 9, 2018) We present a Keldysh nonlinear sigma-model approach to the renormalization group analysis of the disordered electron liquid. We include both the Coulomb interaction and Fermi-liquid type in- teractions in the singlet and triplet channels into the formalism. Based on this model, we reproduce the coupled renormalization group equations for the diffusion coefficient, the frequency and inter- action constants previously derived with the replica model in the imaginary time technique. With the help of source fields coupling to the particle-number and spin densities we study the density- density and spin density-spin density correlation functions in the diffusive regime. This allows us to obtain results for the electric conductivity and the spin susceptibility and thereby to re-derive the main results of the one-loop renormalization group analysis of the disordered electron liquid in the Keldysh formalism.

PACS numbers: 71.10.Ay, 72.10.-d, 72.15.Eb, 73.23.-b

I. INTRODUCTION range of applicability of the NLσM can be broader than the conditions of its derivation. In disordered conductors, perturbations of charge and The original formulations of the NLσM for non- spin relax diffusively at low frequencies and large dis- interacting17–19 as well as for interacting systems7 were tances. In a system obeying time-reversal symmetry, the based on the replica method,20 in combination with the low-energy modes in the Cooper channel also have a dif- imaginary time technique.21 In this scheme, the partition fusive form. These modes, Diffusons and Cooperons, de- function is replicated n times before the averaging over scribe the low-energy dynamics of disordered electrons. disorder-configurations is performed; at the end of the The electron-electron (e-e) interaction causes a scattering calculation, the limit n 0 needs to be taken in order to → of the diffusion modes. As a result, the diffusion constant, remove certain unphysical terms that are present in the frequency, and interaction constants acquire corrections, theory for finite n. As the main object of study is the which in two dimensions are logarithmically divergent at equilibrium partition function, the theory can serve as a low temperatures.1–6 The procedure that handles these platform for studying thermodynamic quantities as well mutually coupled corrections corresponds to a renormal- as the response to weak perturbations through the calcu- ization group (RG) analysis.7–9 The derivation of the lation of equilibrium correlation functions. The replica coupled RG equations is conveniently based on a gener- sigma model is very convenient for perturbative RG cal- alized nonlinear sigma model (NLσM) that includes the culations, which are at the heart of the mentioned suc- effects of electron-electron interactions.7 The structure cesses of this approach. of the theory remains intact during the course of renor- Despite these successes, the theory in its original for- malization, albeit with effective temperature-dependent mulation has certain limitations. The study of equilib- parameters. Among other things, the RG analysis re- rium correlation functions may be obscured by the re- veals the importance of spin3 (as well as valley10) fluc- quired analytical continuation from imaginary frequen- tuations for establishing the strange metallic phase at cies to real ones, which can be very involved. Most no- low temperatures, which does not exist in two dimen- tably, however, true non-equilibrium phenomena are be- sions in the absence of e-e interactions.11–13 Based on yond the scope of this theory as it is constructed with this theory, both quantitative and qualitative statements the help of the equilibrium imaginary time technique. arXiv:1311.1371v2 [cond-mat.mes-hall] 19 Nov 2014 about transport and thermodynamic quantities close to An alternative approach to interacting many-body sys- the metal-insulator transition in two-dimensional elec- tems, which is free of these limitations, is the so-called tron systems can be obtained for the case when it is Keldysh technique.22–25 It is closely related to real-time driven by disorder and interactions.6,14–16 techniques developed for classical systems.25–29 In these By its essence, the NLσM is a minimal microscopic approaches, correlation functions are calculated directly theory, which incorporates all symmetry constraints and in real time, thereby rendering the analytical contin- conservation laws relevant for the low-energy dynamics of uation unnecessary. The range of applicability of the electrons in disordered conductors. Phenomenologically, Keldysh approach includes systems in thermodynamic such a theory may be considered as an analog of the equilibrium as well as non-equilibrium problems. In this Fermi liquid theory for the diffusion modes. As such, the context, the intimate connection to quantum kinetics is 2 of particular advantage. An additional property is very NLσM and cast it into a form that is convenient for the convenient when treating quenched disordered systems: RG analysis. Due to the complex structure of the ap- the normalization of the Keldysh partition function is in- pearing fields and matrices in spin, Keldysh, time (fre- dependent of the disorder potential. The disorder aver- quency) and coordinate (momentum) spaces, the nota- aging can therefore be performed straightforwardly with- tion can at times be involved. We therefore include, from out introducing replicated fields as was already noted in the very beginning, a compact summary of our notations Refs. 27 and 30. as a reference point in Sec. II A. Section III is concerned In this work, we analyze a Keldysh NLσM for e-e in- with the general structure of correlation functions for teractions in disordered electron systems. The Keldysh particle-number densities and spin densities in the diffu- NLσM was first employed for non-interacting electrons sive regime. We perform their calculation in the Keldysh in Ref. 31. [A combination of replicas and the Keldysh formalism emphasizing the important role of conserva- approach was already used in Ref. 32.] For disor- tion laws. In Sec. IV we present the RG analysis of the dered fermions with short-range interactions a Keldysh model. After introducing the general formalism, we dis- sigma model was constructed in Ref. 33, and the RG cuss in detail the renormalization of the parameters (RG- equations7,8 were re-derived for this case. A sigma model charges) appearing in the model, and derive the set of for electrons with long-range Coulomb interaction was coupled RG equations. In Sec. V we return to the analy- introduced in Ref. 34, and generalized to include the in- sis of the correlation functions and calculate corrections teraction in the Cooper channel in a subsequent work.35 to the static parts as well as vertex corrections that arise Our study differs from previous related works33–35 in sev- in connection with the source fields for particle-number eral aspects. In contrast to Ref. 34, we account for both and spin densities. This allows us to obtain the temper- the Coulomb interaction and Fermi liquid-type interac- ature dependence of the spin susceptibility in Sec. V A, tions in the singlet and the triplet channels in order to and the electric conductivity in Sec. V C. Finally, we con- find the Keldysh analog of the original model of Ref. 7. clude in Sec. VI. The obtained model allows us to perform the full RG analysis in the presence of a perturbation that violates the time-reversal symmetry, i.e., when the Cooperons can II. KELDYSH SIGMA MODEL FOR be neglected. Unlike Ref. 33, we implement the proce- INTERACTING ELECTRON SYSTEMS dure directly in the Larkin-Ovchinnikov representation, (for a review, see Ref. 25), which is very convenient for In this section, we present a derivation of the Keldysh the calculation of retarded correlation functions. We also NLσM for the interacting electron liquid. We include the introduce source fields coupled to the particle and spin Coulomb interaction and Fermi-liquid type interactions densities. They allow us to derive the density-density and in the singlet and triplet channels as well as source fields spin-density spin-density correlation functions. This re- coupling to density and spin, see Sec. II B. The resulting quires an analysis of the static and dynamic parts of the sigma model, which contains the Fermi liquid renormal- correlation functions, including vertex corrections, and izations, is presented in Sec. II C. In Sec. II D we rewrite enables us, in particular, to obtain the low-temperature the sigma-model in a form that is convenient for the RG behavior of the electric conductivity and the spin suscep- procedure that will be presented later in Sec. IV. For the tibility. In this way, we re-derive the main results of the convenience of the reader, we first summarize our nota- RG theory of the disordered electron liquid with the help tions in Sec. II A. of the Keldysh sigma model. Whenever possible, we try to highlight those aspects of the analysis that are specific for the Keldysh approach. We conclude, that despite the A. Notations differences related to working with Keldysh matrices in- stead of replicas, the RG-procedure in both schemes is In the approach we use, the original Keldysh contour24 rather similar. disappears from the explicit formulation of the theory The relevance of this study goes beyond a mere con- which, instead, is reformulated in terms of matrices.25 firmation of previously obtained results. We consider it The 2 2 matrices in Keldysh space are decorated with × as a step towards tackling problems that are sensitive to a hat and labeled by a lower index, e.g.,γ ˆ2 orσ ˆ3. For the kinetics of the electronic system at energy scales of the Hubbard-Stratonovich (H-S) fields generating the the order of the temperature or below. Such problems are electron-electron interactions the lower index is also re- transparently treated within the Keldysh formalism. The lated to the Keldysh space. We write, e.g., θk, where renormalized Keldysh NLσM allows to analyze the sub- k = 1, 2 indicates the so-called classical or quantum temperature regime with effective parameters encoding fields. the physics originating from the RG interval, i.e., from The Pauli matrices written without hats and labeled energies exceeding temperature. An important problem by the upper indices are used to describe interactions in 36 of this kind is the calculation of thermal transport. the density/spin-density channels. They can be unified This paper is organized as follows. In Sec. II we de- into the four-component vector ~σ = (σ0, σ1, σ2, σ3)T or scribe the main steps of the derivation of the Keldysh the three-component vector σ = (σ1, σ2, σ3)T . For the 3

H-S fields, e.g. for θl, where l = (0, 1 3), the upper in- Here, symbolizes the Keldysh contour,22–25 which con- dex indicates whether the field acts in− the density chan- sists ofC the forward (+) and backward ( ) paths; x = T † ∗− ∗ nel (component 0) or spin-density channels (components (r, t) and ψx = (ψ↑(x), ψ↓(x)) , ψx = (ψ↑(x), ψ↓(x)) are 1 3). Vector fields combine all four components, θ~ = vectors of Grassmann fields comprising the two spin com- (θ−0, θ1, θ2, θ3)T , or three components, θ = (θ1, θ2, θ3)T . ponents. K is the grand canonical hamiltonian Usually, each of the components of these vectors itself is l K = H µN, H = H0 + Hint. (4) a two-component vector in the Keldysh space, e.g., θk. In − total, the vectors θ~ and θ acquire eight or six components, The non-interacting part of the Hamiltonian is respectively. Besides the H-S fields, the auxiliary poten- tials (fields) ~ϕ = (ϕ, ϕ1, ϕ2, ϕ3)T , ϕ = (ϕ1, ϕ2, ϕ3)T are † introduced to generate the correlation functions describ- H0 = ψxh0ψx, (5) ing the density (singlet) and spin-density (triplet) chan- Zr nels. 2 ∗ where h0 = /2m +udis. Here, udis(r) is the disorder We will use the symbols tr and Tr for traces. The sym- potential and−∇m∗ is the (renormalized) mass. The inter- bol tr includes a trace in Keldysh space, an integration action Hamiltonian Hint can be subdivided into singlet over frequencies, and a summation over spin degrees of and triplet parts, Hint = Hint,ρ + Hint,σ, where freedom. The symbol Tr, in addition to all above, in- cludes an integration over the spatial coordinates of all 1 0 0 the functions appearing under the trace. Hint,ρ = n(r, t) Vρ(r r ) n(r , t) (6) 2 0 − Underscoring of matrices and fields denote multipli- Zr,r 0 0 cation by the matricesu ˆ from the left and right, e.g., Hint,σ = 2 s(r, t) Vσ(r r ) s(r , t). (7) ˆ ˆ 0 − Q =u ˆ Q uˆ; here the convolution is in the time do- Zr,r main. After◦ ◦ the Fourier transform, the convolution con- We introduced the particle-number density and spin den- verts into an algebraic product, Qˆ =u ˆεQˆε,ε0 uˆε0 . The ε,ε0 sities definition of the matrixu ˆε is given in Eq. (33); these ma- trices carry the information on the fermionic equilibrium † 0 1 † n(x) = ψxσ ψx, s(x) = ψxσψx. (8) distribution function ε = tanh(ε/2T ). 2 Finally, in order toF lighten the notation we will in the ∞ The interactions in the singlet and triplet channels are following often write t = −∞ dt and x = r,t. When- ever the frequency integration is made explicit, we use described in terms of the amplitudes R R R R the symbol ε = dε/2π. Furthermore,ε ˆ acts trivially ρ σ F0 F0 on a matrix in the frequency space asε ˆQˆ 0 = εQˆ 0 . Vρ(q) = V0(q) + ,Vσ = . (9) R R εε εε 2ν 2ν The term irreducible correlation function in this paper means that only those diagrams should be considered, ρ σ Here, F0 and F0 are the Fermi liquid parameters known which cannot be separated into two disconnected parts from the phenomenological Fermi liquid theory21,37 and by cutting a single Coulomb interaction line. In order ν is the single-particle density of states per spin direc- to find the irreducible correlation function in the singlet tion. In Vρ(q) the bare long-range part of the Coulomb channel, the long-range Coulomb interaction V0(q) has to interaction is separated from the short-range part. The be separated from the rest of the interaction amplitudes. latter determines the Fermi liquid renormalization of the The argument q in any amplitude of the electron-electron polarization operator. interaction indicates that this amplitude is reducible with Next, we introduce fields on the forward and backward respect to the Coulomb interaction. paths of the Keldysh contour, ψ±, and group them into the vector B. Derivation of the model ψ ψ~ = + . (10) ψ−   Starting point for the derivation is the Keldysh par- tition function for the interacting electron liquid in the The corresponding action reads coherent state representation ∞ S[ψ~†, ψ~] = dt [ψ† , ψ ] [ψ† , ψ ] . (11) † † + + − − = D[ψ , ψ] exp(iS[ψ , ψ]), (1) −∞ L − L Z Z   Z where the action S is defined as The interaction part can be decoupled with the help of a four-component H-S field for each of the paths, ϑl , ± ± S[ψ†, ψ] = dt [ψ†, ψ] (2) organized into a matrix C L Z l † † † ˆl ϑ+ 0 [ψ , ψ] = ψxi∂tψx K[ψ , ψ]. (3) ϑ = l , l = (0, 1 3). (12) L − 0 ϑ− − Zr   4

As a result, the partition function can be written as Working with classical and quantum fields is useful for the calculation of physical quantities like correlation 25 = D[ϑ~]D[ψ~†, ψ~]exp(iS[ψ~†, ψ,~ ϑ~]), (13) functions. Z Z The first step in the derivation of the NLσM is the where averaging of the partition function over disorder config- urations. For the sake of simplicity, we will work with † † l l a delta-correlated impurity potential. This choice corre- S[ψ~ , ψ,~ ϑ~] = ψ~ i∂t h0 + µ + ϑˆ σ σˆ3ψ~x x − sponds to the statistical weight Zx 1   ~T −1 0 ~ 0 R 2 + ϑ (r, t)V (r r )ˆσ3ϑ(r , t). (14) −πντ drudis(r) 2 0 − ... = D[udis](... ) e . (23) Zr,r ,t h idis N Z In the last formula, the sum over the repeated index l The normalization factor is chosen so that 1 dis = 1. from 0 to 3 is implied, whileσ ˆ3 is the third Pauli matrix Averaging of the disorder-dependentN part of theh partitioni in the space of forward and backward fields. As we have function gives already noted in Sec. II A, ϑ~ has eight components: each −i R Ψ~ † u (r)Ψ~ iS of the l-components has two components in the Keldysh e x x dis x = e dis , (24) space. (The same will hold for θ~ and ~ϕ introduced below.) dis D E We also introduced a matrix V comprising the interaction where potentials for the singlet and triplet channels i ~ † ~ ~ † ~ Sdis = (Ψr,tΨr,t)(Ψr,t0 Ψr,t0 ). (25) V = diag(Vρ,Vσ,Vσ,Vσ). (15) 4πντ 0 Zr,t,t It is convenient to change the basis and perform the Following further the standard route for the derivation 25,38 39 Keldysh rotation defined by of the NLσM, the four fermion term Sdis is decoupled ˆ † † −1 with a H-S field Q as Ψ~ = ψ~ Lˆ , Ψ~ = Lˆσˆ3ψ,~ (16) 1 R † iS − Ψ~ Qˆ 0 (r)Ψ~ 0 where the rotation matrix L is given by e dis = D[Qˆ]e 2τ r,t,t0 r,t t,t r,t

Z πν R − tr[Qˆ 0 (r)Qˆ 0 (r)] 1 1 1 −1 T e 4τ r,t,t0 t,t t ,t . (26) Lˆ = − , Lˆ = Lˆ =σ ˆ3Lˆσˆ3. (17) √2 1 1 ×   The matrix Qˆ is Hermitian (note that the transposition Under the rotation Lˆ, the field ϑˆ transforms into θˆ (the involves the interchange of the time arguments). upper index l is not shown), To summarize, the Keldysh partition function has been presented in the form θ θ θˆ LˆϑˆLˆ−1 = cl q . (18) ≡ θq θcl † †   = D[Q]D[Ψ~ , Ψ]~ D[θ~] exp(iS[Ψ~ , Ψ~ , θ,~ Qˆ]), (27) Z As a result, we come to a description in terms of the Z classical (cl) and quantum (q) components of the bosonic where fields S[Ψ~ †, Ψ~ , θ,~ Q] (28) θi = (ϑi ϑi )/2. (19) cl/q + ± − ~ † ˆ−1 0 i ˆ ~ = Ψx G0 (x x ) + δr−r0 Qt,t0 (r) Ψx0 0 − 2τ With the help of two matrices in Keldysh space, Zx,x   † l l iπν ~ ˆ ~ ˆ 0 r ˆ 0 r γˆ1 =σ ˆ0, γˆ2 =σ ˆ1, (20) + Ψxθ (x)σ Ψx + tr[Qt,t ( )Qt ,t( )] 4τ 0 Zx Zr,t,t one may write T −1 0 0 + θ~ (r, t)ˆγ2V (r r )θ~(r , t). r,r0,t − ˆl l Z θ = θkγˆk, (21) k=1,2 After the averaging, the matrix Green’s function X ˆ 0 ~ ~ † G(x, x ) = i ΨxΨx0 (averaging is with respect to Ψ, where k = 1 denotes the classical component, while Q and θ) acquires− h thei typical triangular structure k = 2 denotes the quantum component. As a result, the Keldysh action in the rotated basis reads GR GK Gˆ = , (29) 0 GA ~ † ~ ~ ~ † ˆl l ~   S[Ψ , Ψ, θ] = Ψx i∂t h0 + µ + θ σ Ψx x − where GR, GA, and GK are the retarded, advanced and Z   T −1 0 0 Keldysh components, respectively. Needless to say, the + θ~ (r, t)ˆγ2V (r r )θ~(r , t). (22) 0 − free Green’s function Gˆ0 has the same structure. Zr,r ,t 5

At this point it is convenient to introduce the auxiliary the matrix field Qˆ in the absence of the e-e interaction potentials ϕl (x) into the theory. To this end we replace cl,q −1 i −1 i ˆ 0 r ˆ ˆ † l l † l l l Q0;t,t ( ) = G0 + Q0 . (31) Ψ~ θˆ σ Ψ~ Ψ~ θˆ ϕˆ σ Ψ~ . (30) πν 2τ r,r,t,t0 → −     ˆ Here, ϕcl(x) can be interpreted as a classical external po- In equilibrium, it can be solved by the ansatz Q0;t,t0 (r) = i ˆ tential, while ϕcl (i = 1, 2, 3) describes a magnetic cou- Λt−t0 , where pling to the spin degrees of freedom. The corresponding quantum components do not have an immediate physical 1 2 ε Λˆ = , (32) interpretation. They merely play the role of source fields ε 0 F1 used to generate correlation functions, see Sec. III.  −  The main purpose of the manipulations presented in and ε = tanh(ε/2T ) is the fermionic equilibrium distri- F this section so far was to perform the disorder average bution function. It is sometimes important to remember and to cast the Keldysh partition function into a form that the saddle point Λˆ inherits the analytical structure that is convenient for further analysis. No approxima- of the Keldysh Green’s function. In particular, the uni- tions have been introduced. The resulting functional ties in the 11 and 22 components should be interpreted as with action (28) is still very complicated. On the other retarded and advanced elements, i.e., slightly displaced in hand, as is well known, perturbations of charge and spin the time domain in accordance with the analytical prop- relax diffusively at low temperatures. One may therefore erties of the Green’s function. It is instructive to present seek to find a low-energy theory of the disordered system Λˆ in the form Λˆ =u ˆ σˆ3 uˆ (here, symbolizes a convo- ◦ ◦ ◦ by integrating out the fast electronic degrees of freedom lution), where and focus on diffusion modes only. As described below, this eventually yields the low-energy field theory that de- −1 1 ε uˆε =u ˆ F . (33) scribes the diffusion modes including effects of their re- ε ≡ 0 1  −  scattering, the so-called NLσM. For non-interacting elec- trons, the NLσM was first introduced by Wegner.17 In order to discuss slow (in space and time) fluctuations In a system with time-reversal symmetry, the modes around this saddle point, we parametrize the matrix Qˆ in the particle-particle channel (i.e. the Cooper channel) as also have a diffusive form. Therefore, the two mentioned ˆ types of diffusion modes, known as Diffusons and Cooper- Qˆ = Uˆ σˆ3 U, (34) ◦ ◦ ons, should both be included in the effective description. Initially, the generalization of the sigma model descrip- where Uˆ = Uˆ −1. We will also often use the matrix Qˆ tion to the interacting electron liquid with the help of the defined as replica approach concentrated on the charge and spin de- grees of freedom.7 Subsequently, both the electron inter- Qˆ =u ˆ Qˆ u.ˆ (35) action in the Cooper channel and the Cooperon modes ◦ ◦ 40 ˆ ˆ were also included into the RG analysis. Compared to Recall in this connection that Λ =σ ˆ3. The so defined Q the model presented in (28), this generalization requires and Qˆ fulfill the constraint a further doubling of the size of vectors Ψ and matri- 39 ces Q as to include the so-called time-reversal sector. Qˆ Qˆ = Qˆ Qˆ = 1ˆ. (36) For the sake of clarity, Cooperons and the interaction in ◦ ◦ the Cooper channel will be ignored in the present work. ˆ Physically, this corresponds to the effect of a weak per- The frequency representation of the matrix Q is formed pendicular magnetic field. according to the prescription The next important step in the derivation of the NLσM iεt−iε0t0 is to find a saddle point for the field Qˆ. In the presence Qˆεε0 (r) = Qˆtt0 (r) e . (37) 0 of the e-e interaction, this is a highly nontrivial task. Zt,t One possible route to deal with this problem is to use ˆ ˆ ˆ the saddle point of the non interacting theory (i.e., in The matrices Q and U transform as Q does, following the same prescription. Naturally, we will consider the the absence of θ) as a first approximation, and then to ˆ ˆ analyze deviations with respect to this reference point. Fourier transformed quantities Qεε0 , Uεε0 , etc., as ma- This is the strategy chosen by Finkel’stein7 in its orig- trices in frequency space and write the parametrization ˆ ˆ inal work and we also will follow this route here. (An presented in Eq. (34) as Qˆ = Uˆσˆ3U, UˆU = 1,ˆ so that alternative course was chosen in Ref. 34. There, a part Qˆ2 = 1.ˆ When choosing this parametrization, we imme- of the effects of the electron interaction was accounted for diately restrict ourselves to the so-called ”massless” man- by a modification of the equation determining the saddle ifold. Fluctuations that violate the constraint (36) are point.) massive and their dynamics is beyond our interest.17,18,39 Let us, therefore, write the saddle point equation for The parametrization of Eq. (35) is very convenient for the 6

RG procedure. For frequencies exceeding the tempera- Notation S0 indicates that the corresponding terms in the ture, matricesu ˆ are almost frequency-independent. One action are not yet written in the final form suitable for may therefore integrate out Uˆ until the moment whenu ˆ the RG analysis, and will be treated further in Sec. II D. introduces the information about temperature. As a result of the integration in θ~, the source fields After integrating the fermionic fields Ψ, Ψ†, one can ~ϕ = (ϕ, ϕT )T acquire static vertex corrections describing perform a gradient expansion in the slow fields Uˆ and the FL renormalizations and screening. Namely, we get Uˆ and also expand in the fields θ~ and sources ~ϕ (which for the singlet component are slowly varying by definition). The relevant steps have ϕ(q, t) been described many times in the literature and we refer, ϕFL(q, t) = ρ , (44) e.g., to Refs. 39 and 25 for details. The result is the 1 + F0 + 2νV0(q) nonlinear sigma model in the form and for the triplet components πνi S = Tr D( Qˆ)2 + 4i εˆ+ (θˆl ϕˆl)σl Qˆ ϕi 4 ∇ − ϕi = , i = 1, 2, 3. (45) FL 1 + F σ h T −1 0 0  i 0 + θ~ (r, t)γ ˆ2V (r r ) θ~(r , t) 0 − Zr,r ,t Furthermore, the interaction amplitudes in the singlet ˜ T and triplet channels, symbolized by Γρ and Γσ respec- +2ν (θ~ ~ϕ) (x)γ ˆ2 (θ~ ~ϕ)(x), (38) − − tively, acquire the desired form Zx ρ σ whereε ˆ acts trivially on a matrix in the frequency space 2νV0(q) + F0 F0 Γ˜ρ(q) = , Γσ = . (46) ˆ 0 ˆ 0 ρ σ asε ˆQεε = εQεε . Note that for non-interacting elec- 1 + (2νV0(q) + F0 ) 1 + F0 trons (in contrast to the case of e-e interactions), owing to the trace operation Tr(ˆεQˆ) = Tr(ˆεQˆ), only the source For future purposes it will be convenient to decompose 3,7 term prevents one from removing the distribution func- the interaction in the singlet channel into two parts. tion from the action. The last term in Eq. (38) arises One of them is the statically screened Coulomb interac- as a result of integrating out fast electronic degrees of tion Γ0(q), while the other one is the short-range interac- freedom with energies exceeding 1/τ. The interval of en- tion Γρ which acts within the polarization operator along ergies below 1/τ down to temperature T is dominated by with Γσ, diffusion modes, and it will be studied later in Secs. III Γ˜ (q) = 2Γ (q) + Γ , (47) and IV on the basis of the NLσM. ρ 0 ρ where ρ C. NLσM after Fermi liquid renormalizations ν 1 F0 Γ0(q) = ρ 2 −1 ∂n , Γρ = ρ . (48) (1 + F0 ) V0 (q) + ∂µ 1 + F0 The last term in Eq. (38) allows us to obtain the Fermi ∂n liquid (FL) renormalizations in the NLσM in a systematic We also obtained the FL renormalization for ∂µ , the way, including the renormalization of the source fields. A quantity that determines the value of the polarization similar treatment of the Fermi liquid corrections in the operator in the static limit Keldysh formalism can be found, e.g., in Refs. 41 and 42. Upon integration in θ, one finds the action of the Keldysh ∂n 2ν = ρ . (49) sigma model for interacting electrons in the form ∂µ 1 + F0 0 0 S = S0 + Sint + Sϕ, (39) This concludes the derivation of the Keldysh sigma model which, in principle, can be used as a starting point where for the RG analysis of the disordered electron liquid. In 0 πνi ˆ 2 ˆ the next section, we will nevertheless cast the NLσM in S0 = tr D( Q) + 4iεˆQ , (40) 4 r ∇ an equivalent form that will turn out to be more suitable Z h i π2ν for the renormalization group analysis. ˆ r ij ˜ r r0 ˆ r0 Sint = tr[ˆγiQtt( )]ˆγ2 Γρ( )tr[ˆγjQtt( )] − 8 rr0t − 2 Z π ν ij D. NLσM: Preparation for the RG-procedure tr[ˆγiσQˆ (r)]ˆγ Γσtr[ˆγjσQˆ (r)], (41) − 8 tt 2 tt Zrt As a preparation for the RG analysis, we will now and S0 = S0 + S0 with ϕ ϕQ ϕϕ present the model in a slightly modified form. We write the action as 0 l l ˆ SϕQ = πν tr ϕˆFL(r)σ Q(r) , (42) r Z h i S = S0 + Sint + Sϕ, (50) S0 = 2ν ~ϕT (r, t)ˆγ ~ϕ(r, t). (43) ϕϕ FL 2 and comment on the individual terms next. Zrt 7

The second (i.e., the frequency) term in the expres- acterize by their correlations 0 sion for S0, Eq. (40), acquires logarithmic corrections at low temperatures in the presence of the electron interac- i j 0 i 0 0 ij φ0(x)φ0(x ) = Γ0(r r )δ(t t )γ2 , (57) tions. In other words, not only D, but also the dynamics h i 2ν − − of the diffusion modes is modified in the course of the i j 0 i 0 ij φ (x)φ (x ) = Γ1δ(x x )γ , (58) renormalization of the NLσM. Following Refs. 7,9,43 we h 1 1 i 2ν − 2 will introduce the parameter z into the model in order to i j 0 i 0 ij φ2,αβ(x)φ2,γδ(x ) = Γ2δαδδβγ δ(x x )γ2 . (59) account for these changes. As a result, S0 takes the form h i −2ν −

This definition allows us to cast Sint in a compact form πνi 2 S0 = tr D( Qˆ) + 4izεˆQˆ . (51) 4 r ∇ 2 2 Z h i i(πν) 0 0 Sint = tr[φˆn(r)Qˆ(r)]tr[φˆn(r )Qˆ(r )] .(60) For technical reasons, it is convenient to rewrite the 2 rr0 h i n=0 Z interaction term, Eq. (41), in a different form. Instead of X organizing the short-range part of the interaction ampli- Here, the frequency representation of the fields φn has tudes into the singlet and triplet channel amplitudes, Γρ been introduced in the matrix form, φˆn;εε0 , according to and Γσ, we will pass to a representation that separates the convention: small-angle and large-angle scattering, described by Γ1 i(ε−ε0)t and Γ2, respectively. The RG-analysis takes a simpler φˆn;εε0 (r) = φˆn(r, t) e . (61) 3,7 form in this representation. To this end we rewrite the Zt interaction terms with the help of the identity We will sometimes use the notation φˆ =u ˆ φˆ uˆ in analogy ◦ ◦ Γσσ~ αβσ~ γδ = 2Γσδαδδβγ Γσδαβδγδ, (52) to Eq. (35), so that tr[φˆn(r)Qˆ(r)] = tr[φˆn(r)Qˆ(r)]. − We had to split the interaction in the singlet channel where α, β, γ and δ are spin indices. The interaction into φ0 and φ1, because for the calculation of the irre- amplitudes Γ1 and Γ2 are defined as ducible density-density correlation function (i.e., the po- larization operator) one needs to consider the Coulomb 1 and the short-range parts of the interaction separately. Γ1 = (Γρ Γσ) , Γ2 = Γσ. (53) 2 − − (Recall that the term irreducible in this context means that only those contributions should be considered, which The amplitude Γ2 describes large angle scattering, while cannot be separated into two disconnected parts by cut- Γ1 describes small angle scattering. It is therefore con- ting a single Coulomb interaction line.) We encounter venient to define a new amplitude Γ(q), which comprises this problem considering the source terms associated with both Γ1 and the screened Coulomb interaction Γ0(q): the singlet channel, see Eq. (44). Source fields were in- troduced because they allow generating correlation func- Γ(q) = Γ0(q) + Γ1. (54) tions by functional differentiation of the Keldysh parti- tion function, for details see Sec. III below. The potential In terms of the new amplitudes one finds the relation related to the singlet channel, ϕ, can be used to obtain Γ˜ρ(q) = 2Γ(q) Γ2, cf. Eq. (47). Note that in the limit − the density-density correlation function which, in turn, of small q, the effective amplitude in the ρ channel can is related to electric conductivity through the Einstein be expressed in terms of Γ1 and Γ2 as follows relation, see Sec. V C. It is important to note, however, that only the knowledge of the irreducible density-density 1 Γ˜ρ(q 0) = ρ + 2Γ1 Γ2. (55) correlation function is required for that purpose [for a de- → 1 + F0 − tailed discussion of this point we refer to Ref. 44]. For 0 this reason we will not work with the source term Sϕ, Returning to the action, the interaction term can be but with a slightly modified one, Sϕ, for which the de- (identically) rewritten as pendence on V0(q) is removed. Note that the triplet part is unaffected by this change. Sint = Finally, we write Sϕ = SϕQ + Sϕϕ, where 2 π ν ˆ ij 0 ˆ 0 tr[ˆγiQαα;tt(r)]γ2 Γ(r r )tr[ˆγjQββ;tt(r )] − 4 0 − ρ σ ˆ Zrr t SϕQ = πν tr (γ/ ϕˆ(r) + γ/ ϕˆ(r)σ) Q(r) (62) π2ν r ˆ r ij ˆ r Z h i + tr[ˆγiQαβ;tt( )]γ2 Γ2tr[ˆγjQβα;tt( )]. (56) T ρ σ σ σ 4 rt r r Z Sϕϕ = 2ν ~ϕ ( , t)ˆγ2diag(γ• , γ• , γ• , γ• )~ϕ( , t). (63) Zrt In order to obtain a more tractable form for the inter- ρ/σ ρ/σ action part of the action, let us introduce a set of H-S Here, the constants γ/ and γ• have been introduced. ρ/σ ρ/σ fields: real φ0(x), φ1(x) and Hermitian φ2,αβ(x), each γ/ characterize the (triangular) vertices and γ• the with classical and quantum components, which we char- static part of the correlation function. By comparison 8 with Eqs. (42)-(45) and keeping in mind the previous In Eq. (67), θ(t t0) is the Heaviside function and thermal remarks one finds that the initial values for the renor- averaging is with− respect to the grand canonical ensem- malization procedure read ble,

1 1 ˆ ρ ρ σ σ e−K/T γ/ = γ• = ρ , γ/ = γ• = σ . (64) 1 + F 1 + F ... T = tr [ˆρ . . . ] , ρˆ = , (70) 0 0 h i tr(e−K/Tˆ ) ρ/σ ρ/σ As one can see, γ/ = γ• initially coincide. It is a pri- ˆ ˆ ˆ ˆ ˆ ori not obvious, however, whether this important relation where K = H µN, H is the Hamiltonian, and N the − ˆ ˆ† remains true under renormalization, and this is why the number operator. The field operators ψ and ψ are writ- ˆ different constants have been introduced. ten in the Heisenberg representation with respect to K. Using the time ordered product T [ˆo(t1)ˆo(t2)] = θ(t1 To summarize, the nonlinear sigma model contains sev- − t2)ˆo(t1)ˆo(t2) + θ(t2 t1)ˆo(t2)ˆo(t1) and anti-time ordered eral parameters (”charges”) that may in principle acquire − product T˜[ˆo(t2)ˆo(t1)] = θ(t1 t2)ˆo(t2)ˆo(t1) + θ(t2 logarithmic corrections at low temperatures, D, z,Γ1 and − − ρ/σ ρ/σ t1)ˆo(t1)ˆo(t2), one may present the correlation function Γ2, γ/ and γ• . Let us state the initial values, which follow directly from the derivation presented in Sec. II, as namely R i ˜ χoo(x1 x2) = T [ˆo(x1)ˆo(x2)] T [ˆo(x2)ˆo(x1)] 2 − −2 − D = vF τ/2, z = 1; (65) D +ˆo(x1)ˆo(x2) oˆ(x2)ˆo(x1) . (71) − T ρ σ σ 1 F0 F0 F0 E Γ1 = , Γ2 = , (66) In the Keldysh formalism, this expression can conve- 2 1 + F ρ − 1 + F σ −1 + F σ  0 0  0 niently be represented with the help of the functional ρ/σ ρ/σ integral, namely and the values for γ• , γ/ are written in Eq. (64).

R i χ (x1 x2) = o+(x1)o+(x2) o−(x1)o−(x2) oo − −2 − III. CORRELATION FUNCTIONS + o−(x1)o+(x2) o−(x2)o+(x1) − i In this section we first recall how retarded correlation = [o+ + o−](x1)[o+ o−](x2) ,(72) functions can be generated from the Keldysh partition −2 h − i function by taking derivatives with respect to the so- where o± are now the corresponding (bosonic) fields called quantum and classical components of suitably cho- on forward and backward paths of the Keldysh contour sen source fields. Next, we discuss the general structure and averaging is with respect to the action S (compare of the correlation functions for particle-number densities Eq. (11)). Introducing the classical and quantum com- and spin densities in the diffusive regime. The conserva- 1 ponents of the densities o as ocl/q = 2 (o+ o−), one may tion laws for the total number of particles and for spin write the correlation function in the form± impose important constraints on the structure of these R correlation functions. χ (x1 x2) = 2i ocl(x1)oq(x2) . (73) oo − − h i The source term that has been introduced into the ac- A. Generalities tion in Eq. (30) can be re-written as follows:

† i i We are interested in the retarded correlation functions, Ssource = Ψ~ ϕˆ σ Ψ~ (74) which are defined as a commutator of operators: − = 2(ϕ2ncl + ϕ1nq + 2ϕ2scl + 2ϕ1sq) − R χ (x1 x2) = iθ(t1 t2) [ˆo(x1), oˆ(x2)] . (67) oo − − − h iT Therefore, the correlation functions for the density n and i In order to be in line with common notation, we use the spin components s can conveniently be written as hats to denote operators in this section up to Eq. (71). 2 R i δ Afterwards, the hat symbol will again be reserved for χnn(x1 x2) = Z , (75) − 2 δϕ2(x1)δϕ1(x2) Keldysh matrices only. In Eq. (67),o ˆ can be either the ϕ1=ϕ2=0 operator of the densityn ˆ or of a component of the spin 2 R i δ s χ i j (x x ) = . (76) density operator ˆ, s s 1 2 i Z j − 8 δϕ2(x1)δϕ1(x2) ϕ1=ϕ2=0 ˆ† 0 ˆ nˆ(x) = ψα(x)σαβψβ(x), (68) αβ This is rather intuitive, as X 1 ˆ† ˆ ˆs(x) = ψα(x)σαβψβ(x). (69) i δZ i i δZ 2 ncl(x) = , scl(x) = i (77) αβ h i 2 δϕ (x) h i 4 δϕ (x) X 2 2 9 are the average particle-number and spin densities in the avoid overcounting of the relevant degrees of freedom. presence of the external (classical) potentials and, hence, The chosen parametrization is not the only possible one. the correlation functions describe the corresponding re- In fact, it gives rise to a non-trivial Jacobian, which, how- sponses. ever, does not become relevant for the one-loop calcula- A very important observation can be made directly tion discussed in this manuscript. Other parametriza- from the definition of the correlation function, Eq. (67). tions exist; for an instructive discussion within the con- To this end, let us first define the Fourier transform text of the Keldysh NLσM we refer to Ref. 45. Returning R of the retarded correlation functions as χoo(x1 x2) = to the exponential parametrization, Eq. (78), note that R − χoo(q, t1 t2) exp(iq(r1 r2)). Since for any given Qˆ =σ ˆ exp(Pˆ). Further, the matrices Pˆ can be written q − − 3 time the operators of the total density and spin nˆ(x) as R r and ˆs(x) commute with Kˆ , as the total number of r R ˆ 0 dcl;εε0 (r) particles and the total spin are conserved, the correla- Pεε0 (r) = , (79) R dq;εε0 (r) 0 tion function χ(q, t1 t2) vanishes in the limit q 0.   − → (This fact imposes an important constraint on the RG where dcl/q are hermitian matrices both in the frequency flow of the various charges in the model.) When further αβ ∗ βα domain and in spin space, [d 0 ] = d 0 . Expand- introducing the Fourier transform with respect to time, cl/q;εε cl/q;ε ε χR (q, t t ) = χR (q, ω) exp( iω(t t )), it should ing S0 + Sint (see Eqs. (51) and (60)) up to second order oo 1 2 ω oo 1 2 ˆ be appreciated− that the limits q− 0− and ω 0 do in the generators P , one obtains not commute withR each other. In particular,→ if the→ limit iπν 2 2 ω 0 is taken first, the correlation functions do not S = tr[D( Pˆ) 2izεˆσˆ3Pˆ ] (80) vanish,→ but their values are related to the corresponding − 4 ∇ − 2Z thermodynamic susceptibilities. i 2 ˆ r ˆ r ˆ r0 ˆ r0 + (πν) tr[φn( )ˆσ3P ( )]tr[φn( )ˆσ3P ( )] . 2 0 n=0 rr h i X Z B. Correlation functions from the sigma model Recall that for the frequency representation of the fields φn, the matrix form φˆn;εε0 has been introduced. We will now discuss the density-density and spin-spin By inverting the corresponding quadratic form, i.e., correlation functions in the framework of the NLσM. The in the Gaussian approximation, this action gives rise to discussion will be restricted to the so-called ladder ap- certain correlations for the components of Pˆ. The result proximation, i.e., to an approximation, for which no in- is most easily obtained after separation into singlet and ternal momentum and frequency integrations over diffu- triplet channels. Defining sion modes are carried out. In fact, those integrations give rise to logarithmic corrections (arising from the in- l 1 l αβ d 0 = σ d 0 , l = (0, 1 3), (81) terval T < ε < 1/τ), which is the essence of the RG- cl/q;εε 2 βα cl/q;εε − αβ scheme. The logarithmic corrections may be absorbed X into the various charges of the model, while the form of one obtains for the correlation functions describing dif- the model is unchanged. The results for the correlation fusion of the particle-hole pairs in the singlet (indicated functions obtained in the ladder approximation are there- by 0) and triplet (indicated by i, j 1, 2, 3 ) channels, fore applicable at different scales (or temperatures) once respectively: ∈ { } the appearing charges are replaced by their renormalized 1 values. As already mentioned before, the conservation d0 (q)d0 ( q) = (q, ω) (82) laws for the number of particles and the total spin im- cl;ε1ε2 q;ε3ε4 − −πν D × pose certain constraints on the relation between different δε ,ε δε ,ε δω,ε −ε iπ∆ε ε Γ˜ρ(q) ˜1(q, ω) , RG charges which must be obeyed at each step of the 1 4 2 3 − 4 3 1 2 D 5,6,40   renormalization procedure. This observation serves and as an important check for the correctness of the obtained RG equations. 1 di (q)dj ( q) = δij (q, ω) (83) In short, we now find the correlation functions for den- cl;ε1ε2 q;ε3ε4 − −πν D × sity and spin in the Gaussian approximation with respect δε ,ε δε ,ε δω,ε −ε iπ∆ε ε Γσ 2(q, ω) , to fluctuations, i.e., with respect to diffusion modes. We 1 4 2 3 − 4 3 1 2 D   thereby assume that all non-Gaussian integrations that 0 where ω = ε ε , ∆ 0 = 0 and δ 0 = 2πδ(ε ε ). lead to RG-corrections have been already performed. As 1 2 ε,ε ε ε ε,ε Obviously, on− the level ofF the−F Gaussian fluctuations,− the a preparation, let us start with the parametrization of the singlet and triplet channels do not interfere with each matrix Uˆ. A convenient choice of the parametrization is other. Note that three types of diffusons have been 3,7 ˆ ˆ introduced in the above correlation functions: Uˆ = e−P/2, Uˆ = eP/2, (78) 1 ˆ (q, ω) = (84) with the additional constraint P, σˆ3 = 0 in order to D Dq2 izω { } − 10

1 ˜ + + + ... 1(q, ω) = 2 (85) D Dq iz˜1ω ρ ρ ρ ρ ρ ρ − γ￿ γ￿ γ￿ Γρ γ￿ γ￿ Γρ Γρ γ￿ 1 2(q, ω) = 2 , (86) D Dq iz2ω − + + + ... wherez ˜1(q) = z 2Γ(q) + Γ2 = z Γ˜ρ(q), and z2 = σ σ σ σ σ σ − − γ￿ γ￿ γ￿ Γσ γ￿ γ￿ Γσ Γσ γ￿ z +Γ2 = z Γσ. We will see soon that actuallyz ˜1(q) 0 − ≈ and, therefore, ˜ does not depend on ω.7,9 dyn,R 1 FIG. 1: Dynamical correlation functionsχ ¯nn (top) and TransformingD back to the original representation in dyn,R χsksk (bottom). terms of spin projections, one finds

αβ γδ d (q)dq;ε ε ( q) (87) A R cl;ε1ε2 3 4 − where χoo(ω) = χoo( ω), and D E − 2 K R A = [δαδδβγ δε1,ε4 δε2,ε3 (q, ω) χ (ω) = ω χ (ω) χ (ω) . (93) −πν D oo B oo − oo

+δαδδβγ δω,ε4−ε3 iπ∆ε1,ε2 (q, ω)Γ2 2(q, ω)
D D Furthermore, the two terms in Eq. (90) for Sϕϕ;d give ˜ rise to the static (st) and dynamical (dyn) parts of the δαβδγδδω,ε4−ε3 iπ∆ε1,ε2 2(q, ω)Γ(q) 1(q, ω) . − D D correlation functions, respectively. As can directly be i In order to demonstrate the general structure of the cor- read off from Eq (63), the contribution from Sϕϕ is relation functions for conserved quantities, we will be st,R ρ st,R σ interested in the irreducible correlation function in the χ¯ = 2νγ , χ i i = 2νγ , (94) nn − • s s − • singlet channel, χ¯ˆnn χˆnn irr. For that, the ladder which is irreducible with≡ respect| to the Coulomb inter- while for the dynamical part one finds action is required. It can be found simply by exclud- i q 2 ing Γ0( ), so that the expression for the irreducible av- SϕQ irr (95) 0 0 2 erage d (q)d ( q) coincides with the one 2 cl;ε1ε2 q;ε3ε4 2 − irr i(πν) ρ σ = tr (γ ϕˆ(r) + γ ϕˆ(r)σ)ˆσ3Pˆ(r) statedD in Eq. (82) up to the replacementE Γ(˜ q) Γρ and / / 2 r → **Z  ++irr ˜1 1, where h i D → D T dyn 0 0 1 = ~ϕ (x)Xˆ (x x )~ϕ(x ), − xx0 − 1(q, ω) = 2 (88) D Dq iz1ω Z − ˆ dyn dyn dyn dyn dyn with where X = diag(ˆχnn , 4ˆχsxsx , 4ˆχsy sy , 4ˆχsz sz ). The components ofχ ˆdyn have again the structure indicated z1 = z 2Γ1 + Γ2 = z Γρ. (89) in Eq. (92), and − − With this preparation, the correlation functions in dyn,R / 2 χ¯ (q, ω) = 2ν(γ ) iω 1(q, ω), (96) the ladder approximation can be calculated. In view of nn − ρ D ˆ dyn,R / 2 Eqs. (75) and (76), we may integrate out the P modes χsisi (q, ω) = 2ν(γσ) iω 2(q, ω). (97) and keep resulting terms only up to quadratic order in − D ϕ. Therefore, we calculate the dressed term For a diagrammatic illustration see Fig. 1. In order to obtain this result, the following relation has i 2 Sϕϕ;d = Sϕϕ + S , (90) been used 2 ϕQ irr 1 ε+ ω ε− ω = ω ε+ ω ε− ω , (98) where for the second term both appearing matrices Qˆ are − F 2 F 2 B F 2 − F 2 ˆ replaced byσ ˆ3P , and the averaging is with respect to the where
action (80) for which the contraction rules obtained above can be used. In Eq. (90), ... denotes the connected ω hh ii ω = coth (99) average. One may anticipate that Sϕϕ;d has the following B 2T   form, is the bosonic equilibrium correlation function. A second T 0 0 important identity is Sϕϕ;d = ~ϕ (x)Xˆ(x x )~ϕ(x ), (91) − 0 − Zxx ω ω π ε+ 2 ε− 2 = ω. (100) where Xˆ = diag(χ¯ˆnn, 4ˆχsxsx , 4ˆχsy sy , 4ˆχsz sz ), and the F − F Zε 2 2 blocksχ ˆoo have a structure that is typical for cor-
relation× functions in the Keldysh formalism. Indeed, The total correlation function is then found by adding the static and the dynamical parts, 0 χA χˆ = oo , (92) oo χR χK χR (q, ω) = χst,R + χdyn,R(q, ω), (101)  oo oo  oo oo oo 11 with the result Finally, a comment is in order. The vanishing of the correlation function χ (q, ω) in the limit q 0 does (γρ)2 nn 2 / → Dq iω z1 ρ not request it to be irreducible. However, the obtained R ρ − − γ• χ¯nn(q, ω) = 2νγ• 2 , (102) universal form for the diffusive correlation functions will − Dq  iz1ω  − σ 2 be lost for the reducible correlation function because of 2 (γ/ ) Dq iω z2 γσ plasmons. Recall that the irreducible correlation function R σ − − • χsisi (q, ω) = 2νγ• 2 .(103) χ¯nn is, in fact, the polarization operator. Furthermore, − Dq  iz2ω  − we need to know only the irreducible functionχ ¯nn(q, ω) As discussed in Sec. III A, conservation of charge and in order to extract the conductivity using the Einstein spin demands that relation.6,7

R χoo(q = 0, ω 0) = 0. (104) → IV. RENORMALIZATION In order to fulfill these conditions, the following relations must hold in view of Eqs. (102) and (103), The renormalization group approach for the problem ρ 2 σ 2 at hand follows a general philosophy that is common to (γ/ ) (γ/ ) z1 = ρ , z2 = , (105) many problems in condensed matter physics. For the RG γ γσ • • procedure, the fields in the action are separated into fast where the first relation is related to charge and slow modes. Subsequently, the fast modes are inte- conservation3,7 and the second one to the conser- grated out with logarithmic accuracy, leading to an effec- vation of spin.6,40 One may readily check that for the tive action for the slow modes with scale-dependent pa- ρ/σ ρ/σ rameters, i.e., RG charges. A remark about the RG pro- bare values of z1, γ• and γ/ , these relations are fulfilled. Below, we will discuss the renormalization of cedure in the Keldysh technique is in order: for any the- the NLσM for interacting electrons. In the RG scheme, ory in which a quenched disorder average is performed, diagrams that can be cut into separate parts by cutting the parameters z,Γ1,Γ2 (which determine z1 and z2) as σ σ only impurity lines should not appear. In the original well as γ• and γ/ acquire logarithmic corrections and thereby become scale-dependent. It will be an important model of Ref. 7 the so-called replica trick was used in or- check of the theory that the two conditions displayed in der to make sure that such contributions vanish. When Eq. (105) still hold after renormalization. Indeed, we using the Keldysh approach, the vanishing is effected in will find that a somewhat different way. Generally speaking, the most important observation about the vanishing of unphysi- ρ ρ 1 cal terms in the Keldysh technique is that the frequency γ/ = γ• = ρ (106) 1 + F0 integral over a product of several retarded or advanced functions (but not a mixture of them) vanishes. This ar- are not renormalized, and that the relation z1 = 1/(1 + ρ gument will frequently be used later on. The argument, F0 ) holds under the RG flow. Therefore, the first relation however, does not carry over to the case when a single in Eq. (105) is fulfilled. As a byproduct, it follows from retarded or advanced function is connected to the rest q ρ these relations that 2Γ0( ) = 1/(1+F0 ) for small enough by impurity lines only. This special case is discussed in −1 q q q when V0 ( )∂µ/∂n 1. Therefore,z ˜1( ) = 0 in this connection with Fig. 7 in Sec. IV C. [An alternative to  2 limit and, hence, ˜1(q, ω) = 1/Dq . D the replica and Keldysh approaches exists, the so-called Further, we will find that supersymmetry technique.39 It is a very powerful tool for noninteracting systems. Its application to interacting γσ = γσ = z , (107) / • 2 systems, however, is a formidable challenge, and progress 46 and the relation for the conservation of spin also holds, in this direction is so far limited. ] so that In order to lighten notations, starting from Sec. IV C we will leave out the hats symbolizing matrices in 2 R ∂n Dq Keldysh space. χ¯nn(q, ω) = 2 ω (108) −∂µ Dq i 1+F ρ − 0 2 R Dq ij A. Generalities χsisj (q, ω) = 2νz2 2 δ . (109) − Dq iz2ω − For the NLσM the separation into fast and slow modes The correlation functionsχ ¯R (q, ω) and χR (q, ω) have a nn ss should be done in such a way that the nonlinear con- universal form, which is typical for diffusive correlation straint Qˆ2 = 1 is preserved47 functions of the densities of a conserved quantity. In a ˆ ˆ ˆ ˆ separate publication, we show that the same structure, Qˆ = UˆQˆ0U, Qˆ0 = Uˆ0σˆ3U 0, Uˆ0U 0 = UˆU = 1ˆ. (110) compare Eqs. (108) and (109), also holds for the heat ˆ density - heat density correlation function reflecting en- Here, Qˆ0 contains the fast variables, Uˆ and U represent ergy conservation.36 the slow degrees of freedom. It is also convenient to in- 12 troduce the slow field Qˆs as For the frequency term in the action, one should explic- itly distinguish fast and slow frequencies, i.e.,ε ˆf andε ˆs. ˆ Qˆs = Uˆσˆ3U. (111) Then

2 ˆ 2 ˆ 2 When inserting Qˆ in the form specified in Eq. (110) into Tr zεˆUˆσˆ3Pˆ U = Tr zεˆsUˆσˆ3Pˆ U + Tr zεˆf σˆ3Pˆ . the action S0, one obtains h i h i h (116)i πνi 2 2 S0 = Tr D( Qˆ0) + D[Qˆ0, Φ]ˆ 4 ∇ We will now present a list of all the terms that are rele- h ˆ vant for the one-loop RG-analysis. The following terms +2DΦ[ˆ Qˆ0, Qˆ0] + 4izεˆUˆQˆ0U , (112) ∇ contain only slow modes i ˆ ˆ where Φˆ = U Uˆ = UUˆ. Using this notation, the iπνD 2 SD = Tr ( Qˆs) (117) interaction reads∇ −∇ 4 ∇ h ˆ i 2 2 Sz = πνzTr εˆsQs (118) i(πν) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ − Sint = Tr φnUQ0U Tr φnUQ0U . (113) h i 2 i 2 n=0 SΓ = (πν) Tr φˆnQˆs Tr φˆnQˆs (119) X D h i h iE 2 For the RG-procedure, a particular parametrization for Dρ h σ i ˆh iE Sγ/ = πνTr γ/ ϕˆ + γ/ ϕˆσ Qs (120) the fast degrees of freedom needs to be chosen. In accord h i with the previous Section, we will work with the exponen- T
ρ σ σ σ Sγ• = 2ν ~ϕ (x)ˆγ2diag(γ• , γ• , γ• , γ• )~ϕ(x).(121) tial parametrization Uˆ0 = exp( P/ˆ 2), Qˆ0 =σ ˆ3 exp(Pˆ), x − Z σˆ3, Pˆ = 0. It turns out to be sufficient to expand up to { } Terms S and S arise from the source term S . In second order in Pˆ. We left out terms linear in Pˆ. Such γ/ γ• ϕ fact, Sγ• is identical to Sϕϕ; the present notation is used terms describe the decay (or fusion) of a fast mode into ρ/σ slow modes. These processes do not not play any role in to emphasize the dependence on the parameters γ• . the RG-analysis. Then the result of the expansion reads Next, we come to the terms containing fast modes. The terms originating from S0 read πνi ˆ 2 ˆ S0 = Tr D( Qs) + 4izεˆQs (114) iπν 4 ∇ S = Tr D( Pˆ)2 2izεˆ σˆ Pˆ2 (122) h i f,0 f 3 πνi ˆ 2 ˆ2 ˆ 2 − 4 ∇ − + Tr D(ˆσ3P Φ) + DP (Φˆσ3) h i 2 πνi S1 = Tr DΦ[ˆ P,ˆ Pˆ] (123) h 2 ˆ − 2 ∇ +DΦ[ˆ P,ˆ Pˆ] + izεˆUˆσˆ3Pˆ U ∇ πνi h 2 2 i 2 S2 = Tr DPˆ (Φˆˆσ3) + D(ˆσ3PˆΦ)ˆ (124) πνi 2 i 2 Tr D( Pˆ) , − 4 ∇ πν h 2 ˆ i Sε = Tr zεˆsUˆσˆ3Pˆ U . (125) h i − 2 h i 2 i 2 Here, S2 has two parts, which we label as S2a and S2b in Sint = (πν) Tr φˆnQˆs Tr φˆnQˆs (115) 2 the order of appearance. n=0 D h i h iE The interaction part of the action Sint gives rise to the X2 i 2 2 ˆ following terms + (πν) Tr φˆnQˆs tr φˆnUˆσˆ3Pˆ U 2 n=0 2 X D h i h iE i 2 ˆ ˆ 2 Sint,1 = (πν) Tr φˆnUˆσˆ3PˆU Tr φˆnUˆσˆ3PˆU i 2 ˆ ˆ 2 + (πν) Tr φˆnUˆσˆ3PˆU φˆnUˆσˆ3PˆU . n=0 2 X D h i h iE n=0 2 X D h i h iE i 2 2 ˆ Sint,2 = (πν) Tr φˆnQs Tr φˆnUˆσˆ3Pˆ U . So far, the separation into fast and slow degrees was 2 n=0 purely formal. Let us now qualify this distinction: X D h i h iE (126) 1. Frequencies in the interval λτ −1 < ε < τ −1, 0 < λ < 1 and momenta in the shell λτ −| 1| < Dk2/z < Note that the labeling of these two terms refers to their τ −1 are referred to as fast. different structure with respect to Pˆ, and is not related to the fields φ1 and φ2. 0 2. If at least one of the frequencies ε or ε for the slow Finally, the source term SϕQ, see (62), generates a term field Uˆεε0 is fast, it has to be set equal to the unit matrix. πν ρ σ ˆ ˆ2 ˆ Sϕ,2 = Tr γ/ ϕˆ + γ/ ϕˆσ Uσ3P U , (127) ˆ 2 3. In the fast variables Pεε0 at least one of the frequen- h i 0
cies ε, ε or the momentum should be fast. where the labeling is chosen in analogy to Sint,2. 13

The terms containing fast modes are conveniently Sf,0 represented in a diagrammatic language as depicted in Figs. 2, 3 and 4. We want to integrate out fast modes Pˆ in the Gaus- S1 sian approximation, and in this way generate a new effec- tive action. Besides the slow part of the action, compare Eqs. (117) to (121), corrections arise from the term

S2,a ∆S = i ln D[Pˆ] eiS1+iS2+iSε+iSint+iSϕ,2 eiSf,0 .(128) − Z  In general, if there are N different parts in the action S2,b in which slow and fast modes couple to each other, one finds

i PN S iS ∆S = i ln D[Pˆ] e i=1 i e f,0 (129) − Z  S N N  N  " i 1 = Si + SiSj SiSjSk + ... h i 2 hh ii− 6 hh ii i=1 ij=1 ijk=1 FIG. 2: The elements of the RG-procedure originating from X X X the noninteracting part of the action. Open ends imply P . Here, the connected average means that contractions be- Closed sleeves correspond to U or U. When separated by an tween different terms must be taken as angle, a gradient acts on one of them. A slow frequency εs stands in the vertex marked by a dot. AB = AB A B , (130) hh ii h i − h i h i and so on. When integrating out fast modes, two cases should be distinguished. If at least one of the frequencies of the Pˆ- Sint,1 matrix is slow, then the contractions should be performed using Sf,0 alone. One can formulate two contraction rules for this case. Rule (i) applies when the two contracted Pˆs stand under different traces S int,1;d ˆ ˆ ˆ ˆ tr APε1ε2 (r1) tr BPε3ε4 (r2) (131)

D 2h ⊥ i h ⊥ iE = tr Aˆ Πˆ ε ε (r1 r2)Bˆ δε ,ε δε ,ε , −πν 1 2 − 1 4 2 3 Sint,2 h ⊥ 1 i where we denote Aˆ = (Aˆ σˆ3Aˆσˆ3), and 2 − (q, ω) 0 Πˆ ε+ ω ε− ω (q) = D (132) 2 2 0 (q, ω)  D  Sint,2;d contains a retarded diffuson and an advanced one, (ω) = ( ω). A second contractionD rule (ii) applies D D − when two contracted Pˆs appear within one trace. It reads as follows FIG. 3: The elements of the RG-procedure originating from the interaction part of the action. A shaded square implies tr [APε1ε2 (r1)BPε3ε4 (r2)] (133) one of the interaction amplitudes. A ladder means that the h i 1 ˆ interaction was dressed by ladder diagrams. Such terms are = tr[AΠε1ε2 (r1 r2)]tr[B] indicated by the subscript ”d”. −πν −  tr[Aσˆ3Πˆ ε ε (r1 r2)]tr[Bσˆ3] δε ε δε ,ε . − 1 2 − 1 4 2 3  In the second case, when both frequencies of the Pˆ matrix are fast, the free Gaussian action of the fast modes Sϕ,2 besides Sf,0 also contains a part originating from Sint,1. In the case in question, it takes the form Sint,1 Sf,int, where → 2 FIG. 4: Source term i 2 Sf,int = (πν) Tr φˆnσˆ3Pˆ Tr φˆnσˆ3Pˆ . (134) 2 n=0 X D h i h iE 14

Correspondingly, one should take the contraction with = + + + ... the full quadratic form Γρ;d Γ˜ρ Γ˜ρ Γ˜ρ Γ˜ρ Γ˜ρ Γ˜ρ

Sf = Sf,0 + Sf,int. (135) = + + + ... The relevant contraction rule for the components of Pˆ has already been stated in Eqs. (82), (83) and (87). Γσ;d Γσ Γσ Γσ Γσ Γσ Γσ As is clear from the discussion presented in connection with these formulas in Sec. III B, the extension of the FIG. 5: Dressed interactions. quadratic form corresponds to ”dressed” diffusons, which include not only impurity scattering but also a rescatter- Keldysh space structure (compare with Eq. (92)). As a ing in the singlet and triplet channels as described by result, one gets the ampitudes Γρ and Γσ. An example when this exten- K R sion becomes important is the dressing of the interaction Γµ;d(q, ω)Γµ;d(q, ω) Γˆµ;d(q, ω) = , µ = ρ, σ , which will be discussed next. ΓA (q, ω) 0 { }  µ;d  (139) B. Dressed interaction where ΓA (q, ω) = ΓR (q, ω), (140) Suppose that a certain average contains the interaction µ;d µ;d − K R A part of the action, Sint. Besides Sint, one may as well in- Γµ;d(q, ω) = ω(Γµ;d(q, ω) Γµ;d(q, ω)), (141) sert in its place the second cumulant i S2 , where for B − 2 hh intii and each of the interaction terms one Qˆεε0 will be replaced by ˆ 0 R ˜ ˜ ˜ the fastσ ˆ3Pεε with both frequencies fast, so that adja- Γρ;d(q, ω) = Γρ(q) 1 iωΓρ(q) 1(q, ω) (142) ˆ ˆ − D cent U, U should be substituted by 1. The contraction of R   ˆ Γ (q, ω) = Γσ 1 iωΓσ 2(q, ω) . (143) such fast P s has to be taken with respect to Sf . This case σ;d − D may occur because the interaction fixes only the differ-   ence of frequencies ε ε0 rather than the two frequencies Obviously, the difference between the dressed and bare − amplitudes is in the dynamic properties; in the static individually. It means that Sint should be replaced by its dressed (extended) counterpart limit the amplitudes are equal. A diagrammatic illustra- tion of dressing is shown in Fig. 5. R i 2 Clearly, Γρ/σ;d describe rescattering in the singlet and Sint;d = Sint + S , (136) 2 int triplet channels with intermediate sections composed of a pair of retarded and advanced Green’s functions (some- where specifically times referred to as RA sections). Each RA section gives rise to a window function ∆ε+ω/2,ε−ω/2 which, when in- i 2 i 2 ˆ ˆ 2 Sint = (πν) Tr[φnQ]Tr[φnσˆ3P ] φ (137) tegrated in ε, produces a factor of ω (compare relation 2 −2 h i Sf (100)). This is why the coefficients of the frequencies of DD EE and we indicated by the labels φ and Sf which kind of the diffusion modes ˜1 and 2 are modified by the in- average should be used. For the calculation of this object teraction amplitudes,D see EqsD (84)-(86). An important a separation into singlet and triplet channel is useful, in difference to the calculation of the correlation function is close analogy to the calculation of the correlation func- that in the present case the interaction may be reducible tions demonstrated before, see Fig. 1. The calculation with respect to the Coulomb interaction, and Γ˜ρ(q) and gives ˜1 appear in the singlet channel. D A somewhat simplified way to express the same result Sint;d (138) is 2 π ν 0 0 0 ˆ r ˆ r ˜1 2 = tr[ˆγiσ Qε1ε2 ( )]tr[ˆγjσ Qε3ε4 ( )] R ˜ R − 8 0 Γρ;d(q, ω) = Γρ(q)D , Γσ;d(q, ω) = Γσ D . (144) Zrr ,εi ij 0 D D Γˆ (r r , ε1 ε2)δε −ε ,ε −ε × ρ;d − − 1 2 4 3 In order to obtain Γd and Γ2;d, one may use the relations π2ν R 1 R R R R ˆ ˆ 0 Γd = 2 Γρ;d Γσ;d and Γ2;d = Γσ;d to find tr[ˆγiσQε1ε2 (r)]tr[ˆγjσQε3ε4 (r )] − − − 8 0 rr ,εi   Z ˜ ˆij 0 R 1 2 R 2 Γσ;d(r r , ε1 ε2)δε1−ε2,ε4−ε3 . Γ (q, ω) = Γ(q)D D , Γ (q, ω) = Γ D . (145) × − − d 2 2;d 2 D D The dressed (d) interaction can be obtained by the Needless to say, ΓR and ΓR are components of interac- ij ˆij ij d 2;d substitutionsγ ˆ2 Γρ(q) Γρ;d(q, ω) andγ ˆ2 Γσ(q) ˆ ˆ → → tion matrices Γd and Γ2;d with a structure as indicated ˆij ˆij Γσ;d(q, ω), where the interaction matrices Γρ/σ;d have a in Eq. (139). 15

If a model for a disordered Fermi liquid with short range interactions is considered, one may use the replace- ment Γ(q) Γ1, ˜1 1 in the final expressions. For the Coulomb→ case,D it→ is useful D to single out the screened Coulomb interaction explicitly. To this end, one may use the identity (a) S (b) i S S ˜ ˜ ￿ int,1￿ ￿￿ 1 int,1￿￿ R 1 2 1 1 1 2 Γd = Γ(q)D D2 = Γ0(q)D D2 + Γ1 D D2 . (146) D D D After defining

˜ ˜R 1 Γ0;d = Γ0(q)D , (147) 1 D (c) i S S (d) 1 S2S one may single out the Coulomb interaction ￿￿ 2 int,1￿￿ − 2 1 int,1 ￿￿ ￿￿ R R R Γd = Γ0;d + Γ1;d, (148) FIG. 6: The four different terms contributing to ∆SD. For the terms (a) and (b) a gradient expansion is needed. where

2 1 2 R ˜ 1 R means that we can use S1 at most twice or S2 once. Addi- Γ0;d = Γ0;d D , Γ1;d = Γ1 D D . (149) 2 2 tionally, gradients can be generated by Taylor expansion D0 D0 of the slow fields U, U¯. As a result, one should consider Note that the entire dependence on the Coulomb interac- ˜ tion is delegated to Γ0;d. Furthermore, with the use of the ∆SD = Sint + i S1Sint + i S2Sint (154) ρ ρ h i hh ii hh ii identities ∂µn = 2ν/(1 + F ) as well as z1 = 1/(1 + F ), 0 0 1 2 ˜ S Sint . one can obtain Γ0;d in the form −2 1

2 −1 We will discuss these terms one by one and use the oppor- ˜R ν −1 ∂n Dq Γ0;d(q, ω) = ρ 2 V0 (q) + 2 (150). tunity to highlight some aspects that are specific for the (1 + F ) ∂µ Dq iz1ω 0  −  RG procedure in the Keldysh formalism. For a diagram- matic illustration of the four terms, see Fig. 6. Recall ˜R We observe that Γ0;d is the dynamically screened that for notational simplicity, we will from now on leave Coulomb interaction. Following this decomposition of out hats for matrices in Keldysh space. the dressed interaction, we elevate relations (57), (58) and (59) to

1. hSinti i φi (x)φj (x0) = Γˆij (x x0), (151) h 0 0 i 2ν 0;d − Sint consists of two parts, Sint,1 and Sint,2. First con- i j 0 i ˆij 0 φ1(x)φ1(x ) = Γ1;d(x x ), (152) sider Sint,2 . The corresponding expression contains the h i 2ν − followingh averagei i j 0 i ij 0 φ (x)φ (x ) = Γˆ (x x )δαδδβγ , (153) h 2,αβ 2,γδ i −2ν 2;d − (ω) 0 Pε ε Pε ε D (155) h 1 2 2 3 i ∝ 0 (ω) whenever the dressed interaction is used. We remind Zε2  D  that Γ and Γ are defined in Eq. (149) and Γ 0;d 1;d 2;d The diagram for S is displayed in Fig. 7. It is imme- in Eq. (145). The appearing interaction matrices have int,2 diately obvious thath thisi diagram can be cut into separate the typical Keldysh structure, compare Eq. (139). When parts by cutting only impurity lines. As is well known, dressing is not needed (such as for external vertices de- R such diagrams should not appear for any theory in which fined below), the static limit may be taken and Γ Γn n;d → a quenched disorder average is performed. The so-called for n = 0 2. 20 − replica method was invented to eliminate such contri- butions. Indeed, the internal Green’s function allows for a free summation over the replica index, and therefore the C. Renormalization of the diffusion coefficient diagram vanishes in the zero-replica limit. In the Keldysh technique, the vanishing of unphysical terms mostly oc- In this section, we discuss the renormalization of the curs because the frequency integral over a product of sev- diffusive term SD in the one loop approximation. This eral retarded or advanced functions (but not a mixture of term contains two slow momenta (spatial gradients). It them) vanishes. This argument, however, does not carry 16

εf

ε2 FIG. 7: Diagram for hS i. This term vanishes as discussed int,2 ε in the text. 1

ε2 over to the case of a single retarded or advanced func- ε1 tion as is relevant for the discussed term. In this case one needs to argue that the contribution of the unphys- εf ical diagram to the calculation of any physical quantity will always contain the frequency integral of the sum of FIG. 8: This figure illustrates the two choices of εf for the one retarded and one advanced function, and it is simple average hSint,1i, where εf symbolizes the fast frequency. In to see that their sum vanishes. In the example at hand, fact, both choices are equivalent and in this way one comes the retarded and advanced diffuson appear as separate from Eq. (156) to Eq. (157). elements of the matrix Mε ε = Pε ε Pε ε . When- 1 3 ε2 h 1 2 2 3 i ever physical quantities are calculated, all modes have R to be integrated out, which implies that eventually the 2. i hhS1Sintii sum of retarded and advanced functions will appear. An- ticipating this fact, diagrams as encountered for Sint,2 The relevant contribution comes from Sint,1 only. One may safely be dropped; Fig. 7 illustrates this importanth i finds point. 2 For the other term, Sint,1 , see Fig. 6(a), one finds h i i S1Sint,1 = 2iπν D( r00 r0 ) (158) hh ii − ∇ 3 − ∇ 3 Sint,1 (156) n=0 Zri,εi h i X⊥ ⊥ 2 ¯ r r r ¯ r r r ⊥ tr U( 1)φn( 1)U( 1) ε ε U( 2)φn( 2)U( 2) ε ε = iπν tr U¯(r1)φn(r1)U(r1) 1 2 2 3 ε1ε2 n=0 r1r2,ε1ε2 D hk ˆ
0 ˆ 00
Z D h Φε ε (r3)Πε1ε2 (r1 r3)Πε3ε2 (r3 r2) , X
3 1 r00=r0 =r ¯ ⊥ × − − 3 3 3 U(r2)φn(r2)U(r2) Πε1ε2 (r1 r2) . iE ε2ε1 −

⊥ iE where ε1 and ε3 are necessarily slow, because of Φε1ε3 . Here and in the following we denote M = (M Recall that Φ = U U = UU, and it is clear that it σ Mσ )/2, and M k = (M + σ Mσ )/2, so that M =− ∇ −∇ 3 3 3 3 can only have two slow indices or vanish. Therefore ε2 M k + M ⊥. M k is the diagonal part of M in Keldysh ⊥ k needs to be fast and space, and M the off-diagonal part; [M , σ3] = 0, ⊥ M , σ3 = 0. We will mostly work in such a way that 2 { } contractions in P are performed first, while the choice i S1Sint,1 = 2iπν D( r00 r0 ) (159) hh ii − ∇ 3 − ∇ 3 of fast and slow frequencies for the P -matrices is made n=0 ri,εi X Z a posteriori. This is a straightforward procedure since ⊥ ⊥ tr U¯(r1)φn(r1) φn(r2)U(r2) the frequency arguments of P always reappear explic- ε1εf εf ε3 itly as arguments of the diffusion propagators Π. For D hk 0 00 Φ (r3)Πε ε (
r1 r )Πε ε (r
r2) . ε3ε1 1 f 3 3 f 3 00 0 the renormalization of the diffusion coefficient, in the × − − r3 =r3=r3 iE discussed contribution precisely one frequency argument of the P -matrices is fast. This might be either ε1 or Fig. 6(b) illustrates the structure of this term. One may ε2, see Fig. 8 for an illustration. Due to the identity already notice the structural similarity to Eq. (157); the ⊥ ˆ ˆ ⊥ X Πε1ε2 = Πε2ε1 X for any matrix X in Keldysh space same observation also holds for the remaining contribu- both possibilities are equivalent. For definiteness, we tions to ∆SD. This is why the further evaluation is post- choose here ε2 as fast and write ε2 = εf . This leads poned until all four terms have been discussed. to the intermediate result 2 ¯ r r ⊥ Sint,1 = 2iπν tr (U( 1)φn( 1))ε1εf 3. i hhS2Sintii h i n=0 r1r2,ε1εf X Z D h r r ⊥ r r S contains two terms, S and S . In S , all fre- (φn( 2)U( 2))εf ε1 Πε1εf ( 1 2) .(157) 2 2a 2b 2b × − quencies of the P matrices are forced to be slow due iE We do not evaluate this term right now, but first proceed to the presence of Φ and this does not lead to an RG- with the other terms. contribution to the diffusion coefficient. The relevant 17 contribution comes from a combination of S2a and Sint,1: other contributions to ∆SD.

2

i S2aSint,1 = 2iπνD (160) hh ii n=0 Zri,εi ⊥ X ⊥ tr U¯(r1)φn(r1) φn(r2)U(r2) 1 2 ε1εf εf ε3 4. − 2 S1 Sint D h k (Φ(r3)ΛΦ(r3)Λ)
Πε ε (r1 r3)Π
ε ε (r3 r2) . × ε3ε1 1 f − 3 f − iE Similarly to the previously discussed terms, the dom- For an illustration of this contribution see Fig. 6(c). The inant contribution comes from Sint,1. The contractions expression will be evaluated further together with the can be performed in several ways, as indicated below

2 4 ↔ ↔ 1 2 i(πν) 2 S Sint,1 = D 2Tr[ΦP P ]Tr[ΦP P ] Tr[φnUσ3P U¯]Tr[φnUσ3P U¯] (161) − 2 1 16 ∇ ∇ h i n=0  X ↔ ↔ +4Tr[ΦP P ]Tr[ΦP P ] Tr[φnUσ3P U¯]Tr[φnUσ3P U¯] ∇ ∇ h i ↔ ↔ +2Tr[ΦP P ]Tr[ΦP P ] Tr[φnUσ3P U¯]Tr[φnUσ3P U¯] . ∇ ∇ h i 

5. The correction ∆D

In the previous sections, expressions were obtained for the four different contributions to the RG-corrections to SD. They can be found in Eqs. (157), (159), (160) and (162). As is obvious from these formulas, and also from FIG. 9: Terms of the kind displayed in this figure arise when 1 2 the diagrammatic representation in Fig. 6, the following evaluating the average − 2 S1 Sint,1 , compare the first and third terms in Eq. (161). All frequencies involved are bound to block is common to all four terms be small. This makes the contributions of this type irrelevant. 2 ⊥ ⊥ U¯(r1)φn(r1) φn(r2)U(r2) (163) ε1εf εf ε3 n=0 For the first and last terms, all frequencies of P are fixed X D

E to be slow by the presence of two Φ-fields. This is why i ˆij = ε ε (r1 r2) terms of this kind are irrelevant for the RG; see Fig. 9 for 2ν V 5 f − Zε5 an illustration. Out of the three terms, the relevant one ⊥ ⊥ ε ε (r1)γiuε uε γj ε ε (r2) , is the second which reduces to the contribution displayed × U 1 5 f f U 5 3 in Fig. 6(d). It gives

where = uU, ¯ = Uu¯ and = Γd 2Γ2;d. U U V − 1 2 ¯ S Sint = (162) The gradient expansion of U and U mentioned at the 2 1 − beginning of the calculation is necessary for Sint and 2 hh ii 2 i S1Sint only, since the expressions for i S2Sint and 2iπνD ( r00 r0 )( r00 r0 ) hh1 2 ii hh ii ∇ 3 − ∇ 2 ∇ 4 − ∇ 4 2 S1 Sint already contain two slow gradients (via Φ). n=0 ri,εi − hh ii X Z Since εf is fast and all other frequencies are slow, we can ⊥ ⊥ 00 tr (U¯(r1)φn(r1)) (φn(r2)U(r2)) Πε ε (r1, r ) neglect, with the logarithmic accuracy, the slow frequen- × ε1εf εf ε3 3 f 3 cies εi compared to εf in the RG-integrals. Putting these h k 0 00 k 0 Φ (z)Πε ε (r , r )Φ (r4)Πε ε (r , r2) . remarks into effect, one finds × ε3ε4 4 f 3 4 ε4ε1 1 f 4 i 18

1 2 ∆SD = Sint + i S1Sint + i S2Sint S Sint (164) h i hh ii hh ii− 2 1 k k k 0 00 r 00 r 0 r r = π tr δε3ε1 D r r Φε3ε1 ( )D r + Φε3ε1 ( )D r + D(Φε3ε4 ( )σ3Φε4ε1 ( )σ3) − r,p,ε − ∇ ∇ − ∇ ∇ Z i hn o ¯ 0 ⊥ ˆij 2 00 ⊥ ε1ε5 (r )γiuεf −ε (p)Πε (p) uεf γj ε5ε3 (r ) U V f f U r00=r0=r

4
k k
i k k 0 00 r 00 r 0 r r π tr δε3ε1 D r r + Φε3ε1 ( )D r Φε3ε1 ( )D r DΦε3ε4 ( )Φε4ε1 ( ) −d r,p,ε ∇ ∇ ∇ − ∇ − Z i hn o ¯ 0 ⊥ ˆij 2 3 00 ⊥ ε1ε5 (r )γiuεf −ε (p)Dp Πε (p) uεf γj ε5ε3 (r ) , U V f f U r00=r0=r i

where d is the dimension. An additional term, which does Employing again the approximations of Eq. (168), we see not contain any gradients, was left out here. Fortunately, that both components reduce to integrals over a product such terms need to cancel once all corrections are consid- of only retarded or only advanced functions. A similar ered, as they would make the diffuson massive. (We have structure, obviously, holds for ˜m (p). In perturbative R⊥k checked this cancellation by a perturbative calculation.) calculations such terms vanish after integration in fre- In order to further evaluate this expression, we study the quency a discussed earlier. In the RG procedure it is a quantity little bit more complicated. After integration in momen- tum, such terms are odd functions in frequency. Thus, m a ij m b ˜ (p) = [γiuε ] (p)Π (p)[uε γj] , (165) although the integration over the fast frequency is per- Rab f V−εf εf f Zεf formed within limited intervals, the sum over the positive where a, b , , m = 2, 3 is the power with which the and negative frequency-intervals vanishes. It is useful in ∈ {k ⊥} this connection to compare the expressions for the diag- diffusons enter the expressions, and ˜m (p) is a matrix in Rab onal and off-diagonal matrices ˜. The diagonal ones, see Keldysh space. For example, ˜m (p) is a diagonal matrix R Rkk Eq. (170), contain an additional factor σf which makes with entries the εf -integrals finite. m R A m Therefore, we need to keep only the and com- ˜ (p)11 = ε + (Fε ε ) , (166) kk ⊥⊥ kk f εf f f εf εf ponents. Coming back to ∆SD as given in Eq. (164), one R ε B V − B V D Z f   obtains m A R m ˜ (p)22 = ε + ( ε Fε ) .(167) kk f εf f f εf εf ⊥ ⊥ R ε −B V B − V D ij n f ¯ε ε γiuε Π uε γj ε ε Z   U 1 5 f V−εf εf 5 U f 3 Zε5,εf For the RG calculation in 2d, these integrals need to be

¯a ˜m b found with logarithmic accuracy only. To this end note = a0b0 U R U ε1ε3 that for the purpose of the RG analysis, we may set a,b=⊥k X h i ¯k k ¯⊥ ⊥ m R εf εf sign(εf ). (168) = + σf ε ε , (172) F ≈ B ≈ ε ε f f U U U U 1 3 εf D V Due to the frequent occurrence of the sign-factor, let us   Z introduce the notation where in the second line we denoted 0 = , 0= for a0 and b0, and used the obvious fact⊥ thatk thek off-⊥ σ = sign(ε ). (169) f f diagonal part of the product C = AB is given by ⊥ a As a consequence C = a=⊥,k A Ba0 . As only the parallel component of the total matrix considered in Eq. (172) enters the m m R P ˜ (p) σf . (170) trace in Eq. (164), we may effectively replace Rkk ≈ Dεf Vεf Zεf ¯k(r0) k(r00) + ¯⊥(r0) ⊥(r00) U¯(r0)U(r00). (173) In a similar way one finds ˜m (p) = ˜m (p). U U U U → R⊥⊥ Rkk Next, consider the off-diagonal matrix ˜m with en- It was used that the matrices u cancel. Let us further Rk⊥ tries introduce the notation

˜m p R A m ˜2 p 2 p R p k⊥( )12 = εf εf εf + εf (1 εf εf ) εf , 2 = kk( ) = σf εf ( ) εf ( ), (174) R ε F B V V − F B D R p R p,ε D V Z f   Z Z f m n A 2 3 2 3 R ˜ (p)21 = . (171) 3 = Dp ˜ (p) = σf Dp (p) (p). Rk⊥ Dεf Vεf R Rkk Dεf Vεf Zεf Zp Zp,εf 19

The expression for the renormalization of the diffusion constant reads

k ∆SD = π 2Tr D U U D UΦ U − R − ∇ ∇ − ∇ h +DUΦk U + D[(Φk)2 (Φ⊥)2] ∇ − 4π k i 3Tr D U U + D UΦ U − d R ∇ ∇ ∇ FIG. 10: Diagrammatic representation for i hhSεSint,1ii. This h term contributes to ∆z. DUΦk U DΦkΦk − ∇ − 4π ⊥ 2 i = 3Tr D(Φ ) . (175) Below we present some details of the calculation. As it d R turns out, the dominant contributions arise from those   We see that the two-diffuson contributions cancel out (as terms for which Sint is replaced by Sint,1. it may be expected from general arguments3,6), and the remaining term comes from the three-diffuson term only. ⊥ 2 1 2 1. i hhSεSint,1ii Using Tr[(Φ ) ] = Tr[( Qs) ], one finds − 4 ∇ π 2 After evaluating the relevant contractions in the P - ∆SD = tr[D( Qs) ] (176) − d ∇ × matrices, one obtains the expression Z 2 3 R R 2 σf Dp (p) Γ (p, εf ) 2Γ (p, εf ) , Dεf d − 2,d Zp,εf i SεSint,1 = 2πν (180)   hh ii ri,εi This leads to the following result for the correction to the n=0 Z X ⊥ tr (U(r1)φn(r1)U(r1)σ3) Πε ε (r3 r2) diffusion coefficient ε1ε2 2 3 − Π (r r )(U(r )φ (r )U(r ))⊥ 4iD 2 3  ε2ε1 3 1 2 n 2 2 ε2ε3 ∆D = σf Dp ε (p) × − dν f k p,εf D r r Z (U( 3)zεsU( 3))ε3ε1 ] . R R × Γd (p, εf ) 2Γ2,d(p, εf ) . (177) E × − The frequencies ε1 and ε3 are bound to be slow due to The factor d in the denominator results from an aver- presence of εs, while ε2 is fast. This observation directly aging over the direction of momentum. The logarithmic leads to the result integral will be evaluated in Sec. IV F below. i SεSint,1 = πi 2Tr [zεsQs] . (181) Finally, the situation with the abandoned terms, where hh ii − R all frequencies were forced to be slow, is worth comment- The corresponding diagram is displayed in Fig. 10. ing. See Fig. 9 as an example. Such terms have a hybrid structure, as they resemble at the same time the SD-term and the interaction term of the action: they contain gra- 2. hSint,1i dients and mix frequencies. The remaining momentum integrals are not logarithmic, and are determined by the lower cutoff λτ −1 of the RG-interval. Compared to the This term is somewhat special, as it contains a contri- electron-electron interaction terms, the discussed terms bution from the boundaries of the frequency integration contain a small parameter ρDk2/(λτ −1), which is not interval. Starting point is formula (157), see also Fig. 11, r compensated by a large logarithm. Here, the small pa- where (unlike previously) 2 may directly be set equal to r rameter ρ is the only small parameter introduced for the 1, but an expansion in slow frequencies is performed. In RG analysis: order to see how it works, it is convenient to first perform the average in φ 1 ρ = . (178) ⊥ ij 2 Sint,1 πTr ( ε ε γiuε ) Πˆ ε −ε (p) (p) (2π) νD h i ≈ − U 1 5 f f 1 Vε5−εf h ⊥ It has the meaning of the sheet resistance measured in (uεf γj ε5ε1 ) . (182) × U dimensional units; note an extra factor π as compared to  the quantum resistance. An expansion in slow frequencies could be either in ε5 or in ε1. When expanding in ε1, the matrices , ¯ cancel following the previous arguments. Therefore,U oneU should D. Renormalization of z consider an expansion in ε5 and study ˜1 a ij ab(p; ε5) = [γiuεf ] ε5−εf (p) −εf (p) There are two corrections to Sz, R V − V Zεf  b  ∆Sz = i SεSint + Sint . (179) Πε (p)[uε γj] , a, b , . (183) hh ii h i × f f ∈ {⊥ k} 20

by the spin structure from the other two and, therefore, corrections to either of these two classes are easily iden- tified. The amplitudes Γ0(q) and Γ1 have the same spin structure, but they differ in another aspect. Recall that Γ0(q) is the statically screened long-range Coulomb inter- action, while Γ1 is short-range as it is directly related to the Fermi liquid amplitudes. A correction to Γ0(q) could FIG. 11: The average hSint,1i as relevant for the calculation arise only from diagrams, for which the Coulomb inter- of ∆z. Expansion in the slow frequency is needed to be per- action is not part of the logarithmic integration. Such formed. type of diagrams can be generated with the help of Sint,2 and closely resemble vertex corrections for a scalar ver- tex. Importantly, such corrections, although they arise Only the and components give a logarithmic con- ⊥⊥ kk from individual diagrams, eventually cancel, once all con- tribution. Further, it should be noted that an expansion tributions are summed up. Indeed, it turns out that the of the distribution function in ε5 is not necessary since cancellation occurs between certain pairs of diagrams. such terms would be exponentially suppressed in the RG The calculation will, therefore, be organized in such a regime. Defining way that these pair diagrams are treated together. As 1 already indicated, the cancellation of the corrections to p R p 1 = σf εf ( )∂εf Vεf ( ), (184) Γ0(q) also reflects itself in the fact that the scalar tri- R z εf ,p D ρ Z angular vertex γ/ remains unrenormalized. This will be one obtains demonstrated explicitly below in Secs. V A and V B. In contrast, the correction to the amplitude Γ1, which is kk ⊥⊥ ˜ ˜ zε5 1σ3, (185) short-range in character, is finite. R1,ε5 ≈ −R1,ε5 ≈ − R and further on Generally, the RG-equations at the one-loop level sum the series of logarithmic corrections of the kind Sint,1 π 1Tr [zεQs] . (186) n h i ≈ − R (ρ ln 1/T τ) , where ρ, the small parameter of the RG ex- pansion, has been introduced in Eq. (178). Corrections The integral may be rearranged with the use of a 1 to the interaction amplitude may contain a product of partial integrationR in ε : f several interaction amplitudes, with some of them being 1 dressed. Even on the level of the one-loop approximation, p R p 1 = σf ( ) εf ( ) i 2, it is a priori not clear whether the number of diagrams R 2πz p D V bound − R Z that needs to be considered in order to derive such a sys- (187) tem of equations is finite. As has first been demonstrated by Finkel’stein in Ref. 7, it is fortunately the case and the where the index bound indicates that expression should product of at most four (dressed and undressed) interac- be evaluated at the boundaries of the frequency integra- tion amplitudes is involved in the calculation. The main tion interval. guiding rule here is that the order of the RG-equation is determined by the number of momentum integrations: 3. The correction ∆z each integration generates the small parameter ρ. There cannot be too many dressed amplitudes, because other- wise it is impossible to arrange them without an addi- When combining the two contributions, Eqs. (181) and tional momentum integration. (186), a partial cancellation occurs and only the bound- ary terms remain. For the total correction to z one reads In order to structure the calculation, we will present off the correction to SΓ as the sum of 6 individual contri- 1 butions. Apart from the first one, all of them consist p R p ∆z = σf εf ( ) εf ( ) . (188) of pairs of diagrams. These pairs arise as a result of a 2πν p D V bound Z different choice of the fast frequency for the logarithmic

It is important to note that once the integrand is eval- integration. The above mentioned cancellation of cor- uated at the two boundaries, i.e., the upper and lower rections to Γ0(q) takes place between the two partner limits of the frequency integral, the momentum integral diagrams forming a pair [whenever such correction ap- is convergent and yields a logarithmic correction. pears]. For the corrections to Γ1 and Γ2 the cancellation is not complete, and these corrections remain finite. We write E. Renormalization of the interaction amplitudes

5

The interaction term SΓ contains three interaction am- ∆SΓ = (∆SΓ)i, (189) plitudes, Γ0(q), Γ1 and Γ2. The amplitude Γ2 differs i=0 X 21 where

(∆SΓ)0 = Sint,1 (190) h i i 2 (∆SΓ)1 = S (191) 2 int,1 (∆SΓ)2 = i Sint,1S int,2 (192) hh ii FIG. 12: hS i as relevant for the renormalization of the 1 2 int,1 (∆SΓ)3 = Sint,1Sint,2 (193) interaction amplitudes. In this case, all frequencies involved −2 are slow, the logarithmic correction arises from an integration 1 2 (∆SΓ)4 = Sint,1S (194) over fast momenta. −2 int,2 i 2 2 (∆SΓ)5 = S S . (195) −4 int,1 int,2 obtains π Sint,1 = Tr Qs,αβ;ε2ε1 γiQs,βα;ε4,ε3 γj (197) We will present details of the calculation of the first two h i 4 εi Z h i contributions, the other ones can be considered in a sim- ij Γˆ (p) (p)δε −ε ,ε −ε ilar way, but we will only state the results and display × D 1 4 2 3 Zr the corresponding diagrams. As already mentioned, the π calculation of vertex corrections presented in Secs. V A tr Qs,αα;ε2ε1 γiQs,ββ;ε4,ε3 γj − 4 and V B have a close similarity to some of the diagrams Z h i ij that are important here. The interested reader may find Γˆ (p)δε −ε ,ε −ε . × 2 D 1 4 2 3 additional information there, in particular about the can- Zr celations for pair diagrams. As the frequency arguments of Γ, Γ2 are slow (while the momenta are fast), no dressing of the interaction line was included, and the static amplitudes can be used. As was already noted before, in such a case Γˆ and Γˆ2 are off- diagonal matrices in Keldysh space and take the simple 1. hS i int,1 form

0 Γ 0 Γ2 Γˆ = , Γˆ2 = . (198) This term has been considered before and we may use Γ 0 Γ2 0 formula (156) for Sint,1 as our starting point. In the     h i present context, we consider the case that the two fre- We can use the relation (recall that γ1 is the unit matrix) quency arguments ε and ε are slow, while the mo- 1 2 Tr[Q Q γ ] + Tr[Q γ Q ] mentum entering Π is fast, see Fig. 12. Therefore we 1 2 2 1 2 2 can approximate it by just Π(p, 0), i.e., for the range = Tr[Q1]Tr[γ2Q2] + Tr[γ2Q1]Tr[Q2], (199) of momenta p that are of interest, the frequency de- where all appearing Q-matrices have fixed frequency ar- pendence may be neglected. In this approximation. guments and spin indices. The result is Π(p, 0) (p, 0) (p) becomes proportional to the ≈ D ≡ D unit matrix in Keldysh space and additionally the sum- π ij Sint,1 = tr γiQs,αβ;ε2ε1 (r) γ2 (200) mation in ε1 and ε2 may be performed. As no expansion h i 4 × Zr,εi in slow momenta is required, we may put r2 r1 for the h i → arguments of the slow modes: tr γjQs,βα;ε4,ε3 (r) δε1−ε4,ε2−ε3 Γ(p) (p) p D h i Z π ij 2 tr γiQs,αα;ε2ε1 (r) γ2 ⊥ − 4 r,εi × Sint,1 = iπν tr[ U¯(r)φn(r)U(r) (196) Z h i h i 0 n=0 Zrr tr γ Q (r) δ Γ (p). X D
j s,ββ;ε4ε3 ε1−ε4,ε2−ε3 2 0 0 ⊥ 0 × p D U¯(r)φn(r )U(r ) ] (r r ) h i Z × D − Comparing to the original interaction term, Eq. (56), one 2
E iπν 0 finds that the structure of the Γ1 and Γ2 terms are repro- = tr[φn(r)φn(r ) 2 0 duced, leading to the resulting corrections from (∆SΓ)0: n=0 Zr,r X 0 0 Qs(r)φn(r)Qs(r)φn(r )] (r r ). 1 (∆Γ1)0 = Γ2 (p), (201) − D − πν D Zp 1 The first term in the last equation is just a constant and (∆Γ2)0 = Γ(p) (p). (202) πν D can be dropped. After performing the average in φ one Zp 22

2. Pairs of diagrams 1(a) 1(b)

As we have already mentioned, pairs of diagrams arise as a result of a different choice of the fast frequency εf for the logarithmic integration. These pairs of diagrams are displayed as two columns in Fig. 13. As an illustration, 2(a) 2(b) we discuss in detail one pair of diagrams, labeled as 1(a) and 1(b). This pair gives rise to the correction (∆SΓ)1, and originates from

i 2 Sint,1 = (203) 2 3(a) 3(b) i (πν)4 Tr[φ2Uσ3P U]Tr[φ2Uσ3P U] − 8 φ2 0 DD 0 Tr[φ Uσ3P U]Tr[φ Uσ3P U] φ0 . ×h 2 2 i 2 0 EE Note that φ2 has the same correlation as φ2 (As it will 4(a) 4(b) become clear later, only the φ2-contractions have to be considered in all diagrams presented in Fig. 13. Other- wise, the contributions are canceled out within each of the pairs.) We perform the contractions, and introduce a symme- try factor two: 5(a) 5(b) i S2 = i(πν)2 (204) 2 int,1 − Zri,εi ⊥ 0 ⊥ tr[(Uφ2U) (r1)Πε ε (r1 r4)(Uφ U) (r4)] h ε1ε2 2 1 − 2 ε2ε1 0 ⊥ ⊥ 0 tr[(Uφ2U)ε ε (r3)Πε4ε3 (r3 r2)(Uφ2U)ε ε (r2)] φ2φ . 3 4 − 4 3 i 2 FIG. 13: The pairs of diagrams related to (∆SΓ)i, i = 1 − 5. The different ways in which the occurring frequencies can Diagrams labeled as (a) give rise to the corrections (∆Γ1)i and be chosen as being fast are as follows: diagrams labeled as (b) to the corrections (∆Γ2)i. Only those contributions remain, for which all interaction amplitudes are (a) (ε2, ε3) fast or equivalently (ε1, ε4) fast (