Keldysh approach to the renormalization group analysis of the disordered electron liquid
KKeldysh approach to the renormalization group analysis of the disordered electronliquid
G. Schwiete ∗ and A. M. Finkel’stein
2, 3 Dahlem Center for Complex Quantum Systems and Institut f¨ur Theoretische Physik,Freie Universit¨at Berlin, 14195 Berlin, Germany Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843-4242, USA Department 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 ofthe 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 reproducethe 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. Withthe 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 toobtain results for the electric conductivity and the spin susceptibility and thereby to re-derive themain results of the one-loop renormalization group analysis of the disordered electron liquid in theKeldysh formalism.
PACS numbers: 71.10.Ay, 72.10.-d, 72.15.Eb, 73.23.-b
I. INTRODUCTION
In disordered conductors, perturbations of charge andspin relax diffusively at low frequencies and large dis-tances. In a system obeying time-reversal symmetry, thelow-energy modes in the Cooper channel also have a dif-fusive form. These modes, Diffusons and Cooperons, de-scribe the low-energy dynamics of disordered electrons.The electron-electron (e-e) interaction causes a scatteringof the diffusion modes. As a result, the diffusion constant,frequency, and interaction constants acquire corrections,which in two dimensions are logarithmically divergent atlow temperatures.
The procedure that handles thesemutually coupled corrections corresponds to a renormal-ization group (RG) analysis.
The derivation of thecoupled RG equations is conveniently based on a gener-alized nonlinear sigma model (NL σ M) that includes theeffects of electron-electron interactions. The structureof the theory remains intact during the course of renor-malization, albeit with effective temperature-dependentparameters. Among other things, the RG analysis re-veals the importance of spin (as well as valley ) fluc-tuations for establishing the strange metallic phase atlow temperatures, which does not exist in two dimen-sions in the absence of e-e interactions. Based onthis theory, both quantitative and qualitative statementsabout transport and thermodynamic quantities close tothe metal-insulator transition in two-dimensional elec-tron systems can be obtained for the case when it isdriven by disorder and interactions.
By its essence, the NL σ M is a minimal microscopictheory, which incorporates all symmetry constraints andconservation laws relevant for the low-energy dynamics ofelectrons in disordered conductors. Phenomenologically,such a theory may be considered as an analog of theFermi liquid theory for the diffusion modes. As such, the range of applicability of the NL σ M can be broader thanthe conditions of its derivation.The original formulations of the NL σ M for non-interacting as well as for interacting systems werebased on the replica method, in combination with theimaginary time technique. In this scheme, the partitionfunction is replicated n times before the averaging overdisorder-configurations is performed; at the end of thecalculation, the limit n → n . As the main object of study is theequilibrium partition function, the theory can serve as aplatform for studying thermodynamic quantities as wellas the response to weak perturbations through the calcu-lation of equilibrium correlation functions. The replicasigma model is very convenient for perturbative RG cal-culations, which are at the heart of the mentioned suc-cesses of this approach.Despite these successes, the theory in its original for-mulation has certain limitations. The study of equilib-rium correlation functions may be obscured by the re-quired analytical continuation from imaginary frequen-cies to real ones, which can be very involved. Most no-tably, however, true non-equilibrium phenomena are be-yond the scope of this theory as it is constructed withthe help of the equilibrium imaginary time technique.An alternative approach to interacting many-body sys-tems, which is free of these limitations, is the so-calledKeldysh technique. It is closely related to real-timetechniques developed for classical systems.
In theseapproaches, correlation functions are calculated directlyin real time, thereby rendering the analytical contin-uation unnecessary. The range of applicability of theKeldysh approach includes systems in thermodynamicequilibrium as well as non-equilibrium problems. In thiscontext, the intimate connection to quantum kinetics is a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov of particular advantage. An additional property is veryconvenient when treating quenched disordered systems:the normalization of the Keldysh partition function is in-dependent of the disorder potential. The disorder aver-aging can therefore be performed straightforwardly with-out introducing replicated fields as was already noted inRefs. 27 and 30.In this work, we analyze a Keldysh NL σ M for e-e in-teractions in disordered electron systems. The KeldyshNL σ M was first employed for non-interacting electronsin Ref. 31. [A combination of replicas and the Keldyshapproach was already used in Ref. 32.] For disor-dered fermions with short-range interactions a Keldyshsigma model was constructed in Ref. 33, and the RGequations were re-derived for this case. A sigma modelfor electrons with long-range Coulomb interaction wasintroduced in Ref. 34, and generalized to include the in-teraction in the Cooper channel in a subsequent work. Our study differs from previous related works in sev-eral aspects. In contrast to Ref. 34, we account for boththe Coulomb interaction and Fermi liquid-type interac-tions in the singlet and the triplet channels in order tofind the Keldysh analog of the original model of Ref. 7.The obtained model allows us to perform the full RGanalysis in the presence of a perturbation that violatesthe time-reversal symmetry, i.e., when the Cooperons canbe neglected. Unlike Ref. 33, we implement the proce-dure directly in the Larkin-Ovchinnikov representation,(for a review, see Ref. 25), which is very convenient forthe calculation of retarded correlation functions. We alsointroduce source fields coupled to the particle and spindensities. They allow us to derive the density-density andspin-density spin-density correlation functions. This re-quires an analysis of the static and dynamic parts of thecorrelation functions, including vertex corrections, andenables us, in particular, to obtain the low-temperaturebehavior of the electric conductivity and the spin suscep-tibility. In this way, we re-derive the main results of theRG theory of the disordered electron liquid with the helpof the Keldysh sigma model. Whenever possible, we tryto highlight those aspects of the analysis that are specificfor the Keldysh approach. We conclude, that despite thedifferences related to working with Keldysh matrices in-stead of replicas, the RG-procedure in both schemes israther similar.The relevance of this study goes beyond a mere con-firmation of previously obtained results. We consider itas a step towards tackling problems that are sensitive tothe kinetics of the electronic system at energy scales ofthe order of the temperature or below. Such problems aretransparently treated within the Keldysh formalism. Therenormalized Keldysh NL σ M allows to analyze the sub-temperature regime with effective parameters encodingthe physics originating from the RG interval, i.e., fromenergies exceeding temperature. An important problemof this kind is the calculation of thermal transport. This paper is organized as follows. In Sec. II we de-scribe the main steps of the derivation of the Keldysh NL σ M and cast it into a form that is convenient for theRG analysis. Due to the complex structure of the ap-pearing fields and matrices in spin, Keldysh, time (fre-quency) and coordinate (momentum) spaces, the nota-tion can at times be involved. We therefore include, fromthe very beginning, a compact summary of our notationsas a reference point in Sec. II A. Section III is concernedwith the general structure of correlation functions forparticle-number densities and spin densities in the diffu-sive regime. We perform their calculation in the Keldyshformalism emphasizing the important role of conserva-tion laws. In Sec. IV we present the RG analysis of themodel. After introducing the general formalism, we dis-cuss in detail the renormalization of the parameters (RG-charges) appearing in the model, and derive the set ofcoupled RG equations. In Sec. V we return to the analy-sis of the correlation functions and calculate correctionsto the static parts as well as vertex corrections that arisein connection with the source fields for particle-numberand spin densities. This allows us to obtain the temper-ature dependence of the spin susceptibility in Sec. V A,and the electric conductivity in Sec. V C. Finally, we con-clude in Sec. VI.
II. KELDYSH SIGMA MODEL FORINTERACTING ELECTRON SYSTEMS
In this section, we present a derivation of the KeldyshNL σ M for the interacting electron liquid. We include theCoulomb interaction and Fermi-liquid type interactionsin the singlet and triplet channels as well as source fieldscoupling to density and spin, see Sec. II B. The resultingsigma model, which contains the Fermi liquid renormal-izations, is presented in Sec. II C. In Sec. II D we rewritethe sigma-model in a form that is convenient for the RGprocedure that will be presented later in Sec. IV. For theconvenience of the reader, we first summarize our nota-tions in Sec. II A.
A. Notations
In the approach we use, the original Keldysh contour disappears from the explicit formulation of the theorywhich, instead, is reformulated in terms of matrices. The 2 × γ or ˆ σ . Forthe Hubbard-Stratonovich (H-S) fields generating theelectron-electron interactions the lower index is also re-lated to the Keldysh space. We write, e.g., θ k , where k = 1 , (cid:126)σ = ( σ , σ , σ , σ ) T orthe three-component vector σ = ( σ , σ , σ ) T . For theH-S fields, e.g. for θ l , where l = (0 , − − (cid:126)θ =( θ , θ , θ , θ ) T , or three components, θ = ( θ , θ , θ ) T .Usually, each of the components of these vectors itself isa two-component vector in the Keldysh space, e.g., θ lk . Intotal, the vectors (cid:126)θ and θ acquire eight or six components,respectively. Besides the H-S fields, the auxiliary poten-tials (fields) (cid:126)ϕ = ( ϕ, ϕ , ϕ , ϕ ) T , ϕ = ( ϕ , ϕ , ϕ ) T areintroduced to generate the correlation functions describ-ing the density (singlet) and spin-density (triplet) chan-nels.We will use the symbols tr and Tr for traces. The sym-bol tr includes a trace in Keldysh space, an integrationover frequencies, and a summation over spin degrees offreedom. The symbol Tr, in addition to all above, in-cludes an integration over the spatial coordinates of allthe functions appearing under the trace.Underscoring of matrices and fields denote multipli-cation by the matrices ˆ u from the left and right, e.g.,ˆ Q = ˆ u ◦ ˆ Q ◦ ˆ u ; here the convolution is in the time do-main. After the Fourier transform, the convolution con-verts into an algebraic product, ˆ Q ε,ε (cid:48) = ˆ u ε ˆ Q ε,ε (cid:48) ˆ u ε (cid:48) . Thedefinition of the matrix ˆ u ε is given in Eq. (33); these ma-trices carry the information on the fermionic equilibriumdistribution function F ε = tanh( ε/ T ).Finally, in order to lighten the notation we will in thefollowing often write (cid:82) t = (cid:82) ∞−∞ dt and (cid:82) x = (cid:82) r ,t . When-ever the frequency integration is made explicit, we usethe symbol (cid:82) ε = (cid:82) dε/ π . Furthermore, ˆ ε acts triviallyon a matrix in the frequency space as ˆ ε ˆ Q εε (cid:48) = ε ˆ Q εε (cid:48) .The term irreducible correlation function in this papermeans that only those diagrams should be considered,which cannot be separated into two disconnected partsby cutting a single Coulomb interaction line. In orderto find the irreducible correlation function in the singletchannel, the long-range Coulomb interaction V ( q ) has tobe separated from the rest of the interaction amplitudes.The argument q in any amplitude of the electron-electroninteraction indicates that this amplitude is reducible withrespect to the Coulomb interaction. B. Derivation of the model
Starting point for the derivation is the Keldysh par-tition function for the interacting electron liquid in thecoherent state representation Z = (cid:90) D [ ψ † , ψ ] exp( iS [ ψ † , ψ ]) , (1)where the action S is defined as S [ ψ † , ψ ] = (cid:90) C dt L [ ψ † , ψ ] (2) L [ ψ † , ψ ] = (cid:90) r ψ † x i∂ t ψ x − K [ ψ † , ψ ] . (3) Here, C symbolizes the Keldysh contour, which con-sists of the forward (+) and backward ( − ) paths; x =( r , t ) and ψ x = ( ψ ↑ ( x ) , ψ ↓ ( x )) T , ψ † x = ( ψ ∗↑ ( x ) , ψ ∗↓ ( x )) arevectors of Grassmann fields comprising the two spin com-ponents. K is the grand canonical hamiltonian K = H − µN, H = H + H int . (4)The non-interacting part of the Hamiltonian is H = (cid:90) r ψ † x h ψ x , (5)where h = −∇ / m ∗ + u dis . Here, u dis ( r ) is the disorderpotential and m ∗ is the (renormalized) mass. The inter-action Hamiltonian H int can be subdivided into singletand triplet parts, H int = H int,ρ + H int,σ , where H int,ρ = 12 (cid:90) r , r (cid:48) n ( r , t ) V ρ ( r − r (cid:48) ) n ( r (cid:48) , t ) (6) H int,σ = 2 (cid:90) r , r (cid:48) s ( r , t ) V σ ( r − r (cid:48) ) s ( r (cid:48) , t ) . (7)We introduced the particle-number density and spin den-sities n ( x ) = ψ † x σ ψ x , s ( x ) = 12 ψ † x σ ψ x . (8)The interactions in the singlet and triplet channels aredescribed in terms of the amplitudes V ρ ( q ) = V ( q ) + F ρ ν , V σ = F σ ν . (9)Here, F ρ and F σ are the Fermi liquid parameters knownfrom the phenomenological Fermi liquid theory and ν is the single-particle density of states per spin direc-tion. In V ρ ( q ) the bare long-range part of the Coulombinteraction is separated from the short-range part. Thelatter determines the Fermi liquid renormalization of thepolarization operator.Next, we introduce fields on the forward and backwardpaths of the Keldysh contour, ψ ± , and group them intothe vector (cid:126)ψ = (cid:18) ψ + ψ − (cid:19) . (10)The corresponding action reads S [ (cid:126)ψ † , (cid:126)ψ ] = (cid:90) ∞−∞ dt (cid:16) L [ ψ † + , ψ + ] − L [ ψ †− , ψ − ] (cid:17) . (11)The interaction part can be decoupled with the help ofa four-component H-S field for each of the ± paths, ϑ l ± ,organized into a matrixˆ ϑ l = (cid:18) ϑ l + ϑ l − (cid:19) , l = (0 , − . (12)As a result, the partition function can be written as Z = (cid:90) D [ (cid:126)ϑ ] D [ (cid:126)ψ † , (cid:126)ψ ]exp( iS [ (cid:126)ψ † , (cid:126)ψ, (cid:126)ϑ ]) , (13)where S [ (cid:126)ψ † , (cid:126)ψ, (cid:126)ϑ ] = (cid:90) x (cid:126)ψ † x (cid:16) i∂ t − h + µ + ˆ ϑ l σ l (cid:17) ˆ σ (cid:126)ψ x + 12 (cid:90) r , r (cid:48) ,t (cid:126)ϑ T ( r , t ) V − ( r − r (cid:48) )ˆ σ (cid:126)ϑ ( r (cid:48) , t ) . (14)In the last formula, the sum over the repeated index l from 0 to 3 is implied, while ˆ σ is the third Pauli matrixin the space of forward and backward fields. As we havealready noted in Sec. II A, (cid:126)ϑ has eight components: eachof the l -components has two components in the Keldyshspace. (The same will hold for (cid:126)θ and (cid:126)ϕ introduced below.)We also introduced a matrix V comprising the interactionpotentials for the singlet and triplet channels V = diag( V ρ , V σ , V σ , V σ ) . (15)It is convenient to change the basis and perform theKeldysh rotation defined by (cid:126) Ψ † = (cid:126)ψ † ˆ L − , (cid:126) Ψ = ˆ L ˆ σ (cid:126)ψ, (16)where the rotation matrix L is given byˆ L = 1 √ (cid:18) −
11 1 (cid:19) , ˆ L − = ˆ L T = ˆ σ ˆ L ˆ σ . (17)Under the rotation ˆ L , the field ˆ ϑ transforms into ˆ θ (theupper index l is not shown),ˆ θ ≡ ˆ L ˆ ϑ ˆ L − = (cid:18) θ cl θ q θ q θ cl (cid:19) . (18)As a result, we come to a description in terms of theclassical (cl) and quantum (q) components of the bosonicfields θ icl/q = ( ϑ i + ± ϑ i − ) / . (19)With the help of two matrices in Keldysh space,ˆ γ = ˆ σ , ˆ γ = ˆ σ , (20)one may write ˆ θ l = (cid:88) k =1 , θ lk ˆ γ k , (21)where k = 1 denotes the classical component, while k = 2 denotes the quantum component. As a result,the Keldysh action in the rotated basis reads S [ (cid:126) Ψ † , (cid:126) Ψ , (cid:126)θ ] = (cid:90) x (cid:126) Ψ † x (cid:16) i∂ t − h + µ + ˆ θ l σ l (cid:17) (cid:126) Ψ x + (cid:90) r , r (cid:48) ,t (cid:126)θ T ( r , t )ˆ γ V − ( r − r (cid:48) ) (cid:126)θ ( r (cid:48) , t ) . (22) Working with classical and quantum fields is useful forthe calculation of physical quantities like correlationfunctions. The first step in the derivation of the NL σ M is theaveraging of the partition function over disorder config-urations. For the sake of simplicity, we will work witha delta-correlated impurity potential. This choice corre-sponds to the statistical weight (cid:104) . . . (cid:105) dis = N (cid:90) D [ u dis ]( . . . ) e − πντ (cid:82) d r u dis ( r ) . (23)The normalization factor N is chosen so that (cid:104) (cid:105) dis = 1.Averaging of the disorder-dependent part of the partitionfunction gives (cid:68) e − i (cid:82) x (cid:126) Ψ † x u dis ( r ) (cid:126) Ψ x (cid:69) dis = e iS dis , (24)where S dis = i πντ (cid:90) r ,t,t (cid:48) ( (cid:126) Ψ † r ,t (cid:126) Ψ r ,t )( (cid:126) Ψ † r ,t (cid:48) (cid:126) Ψ r ,t (cid:48) ) . (25)Following further the standard route for the derivationof the NL σ M, the four fermion term S dis is decoupledwith a H-S field ˆ Q ase iS dis = (cid:90) D [ ˆ Q ]e − τ (cid:82) r ,t,t (cid:48) (cid:126) Ψ † r ,t ˆ Q t,t (cid:48) ( r ) (cid:126) Ψ r ,t (cid:48) × e − πν τ (cid:82) r ,t,t (cid:48) tr[ ˆ Q t,t (cid:48) ( r ) ˆ Q t (cid:48) ,t ( r )] . (26)The matrix ˆ Q is Hermitian (note that the transpositioninvolves the interchange of the time arguments).To summarize, the Keldysh partition function has beenpresented in the form Z = (cid:90) D [ Q ] D [ (cid:126) Ψ † , (cid:126) Ψ] D [ (cid:126)θ ] exp( iS [ (cid:126) Ψ † , (cid:126) Ψ , (cid:126)θ, ˆ Q ]) , (27)where S [ (cid:126) Ψ † , (cid:126) Ψ , (cid:126)θ, Q ] (28)= (cid:90) x,x (cid:48) (cid:126) Ψ † x (cid:20) ˆ G − ( x − x (cid:48) ) + δ r − r (cid:48) i τ ˆ Q t,t (cid:48) ( r ) (cid:21) (cid:126) Ψ x (cid:48) + (cid:90) x (cid:126) Ψ † x ˆ θ l ( x ) σ l (cid:126) Ψ x + iπν τ (cid:90) r ,t,t (cid:48) tr[ ˆ Q t,t (cid:48) ( r ) ˆ Q t (cid:48) ,t ( r )]+ (cid:90) r , r (cid:48) ,t (cid:126)θ T ( r , t )ˆ γ V − ( r − r (cid:48) ) (cid:126)θ ( r (cid:48) , t ) . After the averaging, the matrix Green’s functionˆ G ( x, x (cid:48) ) = − i (cid:104) (cid:126) Ψ x (cid:126) Ψ † x (cid:48) (cid:105) (averaging is with respect to Ψ, Q and θ ) acquires the typical triangular structureˆ G = (cid:18) G R G K G A (cid:19) , (29)where G R , G A , and G K are the retarded, advanced andKeldysh components, respectively. Needless to say, thefree Green’s function ˆ G has the same structure.At this point it is convenient to introduce the auxiliarypotentials ϕ lcl,q ( x ) into the theory. To this end we replace (cid:126) Ψ † ˆ θ l σ l (cid:126) Ψ → (cid:126) Ψ † (cid:16) ˆ θ l − ˆ ϕ l (cid:17) σ l (cid:126) Ψ . (30)Here, ϕ cl ( x ) can be interpreted as a classical external po-tential, while ϕ icl ( i = 1 , ,
3) describes a magnetic cou-pling to the spin degrees of freedom. The correspondingquantum components do not have an immediate physicalinterpretation. They merely play the role of source fieldsused to generate correlation functions, see Sec. III.The main purpose of the manipulations presented inthis section so far was to perform the disorder averageand to cast the Keldysh partition function into a formthat is convenient for further analysis. No approxima-tions have been introduced. The resulting functionalwith action (28) is still very complicated. On the otherhand, as is well known, perturbations of charge and spinrelax diffusively at low temperatures. One may thereforeseek to find a low-energy theory of the disordered systemby integrating out the fast electronic degrees of freedomand focus on diffusion modes only. As described below,this eventually yields the low-energy field theory that de-scribes the diffusion modes including effects of their re-scattering, the so-called NL σ M. For non-interacting elec-trons, the NL σ M was first introduced by Wegner. In a system with time-reversal symmetry, the modesin the particle-particle channel (i.e. the Cooper channel)also have a diffusive form. Therefore, the two mentionedtypes of diffusion modes, known as Diffusons and Cooper-ons, should both be included in the effective description.Initially, the generalization of the sigma model descrip-tion to the interacting electron liquid with the help of thereplica approach concentrated on the charge and spin de-grees of freedom. Subsequently, both the electron inter-action in the Cooper channel and the Cooperon modeswere also included into the RG analysis. Compared tothe model presented in (28), this generalization requiresa further doubling of the size of vectors Ψ and matri-ces Q as to include the so-called time-reversal sector. For the sake of clarity, Cooperons and the interaction inthe Cooper channel will be ignored in the present work.Physically, this corresponds to the effect of a weak per-pendicular magnetic field.The next important step in the derivation of the NL σ Mis to find a saddle point for the field ˆ Q . In the presenceof the e-e interaction, this is a highly nontrivial task.One possible route to deal with this problem is to usethe saddle point of the non interacting theory (i.e., inthe absence of θ ) as a first approximation, and then toanalyze deviations with respect to this reference point.This is the strategy chosen by Finkel’stein in its orig-inal work and we also will follow this route here. (Analternative course was chosen in Ref. 34. There, a partof the effects of the electron interaction was accounted forby a modification of the equation determining the saddlepoint.)Let us, therefore, write the saddle point equation for the matrix field ˆ Q in the absence of the e-e interactionˆ Q t,t (cid:48) ( r ) = iπν (cid:18) ˆ G − + i τ ˆ Q (cid:19) − r , r ,t,t (cid:48) . (31)In equilibrium, it can be solved by the ansatz ˆ Q t,t (cid:48) ( r ) =ˆΛ t − t (cid:48) , where ˆΛ ε = (cid:18) F ε − (cid:19) , (32)and F ε = tanh( ε/ T ) is the fermionic equilibrium distri-bution function. It is sometimes important to rememberthat the saddle point ˆΛ inherits the analytical structureof the Keldysh Green’s function. In particular, the uni-ties in the 11 and 22 components should be interpreted asretarded and advanced elements, i.e., slightly displaced inthe time domain in accordance with the analytical prop-erties of the Green’s function. It is instructive to presentˆΛ in the form ˆΛ = ˆ u ◦ ˆ σ ◦ ˆ u (here, ◦ symbolizes a convo-lution), where ˆ u ε = ˆ u − ε ≡ (cid:18) F ε − (cid:19) . (33)In order to discuss slow (in space and time) fluctuationsaround this saddle point, we parametrize the matrix ˆ Q as ˆ Q = ˆ U ◦ ˆ σ ◦ ˆ U , (34)where ˆ U = ˆ U − . We will also often use the matrix ˆ Q defined as ˆ Q = ˆ u ◦ ˆ Q ◦ ˆ u. (35)Recall in this connection that ˆΛ = ˆ σ . The so defined ˆ Q and ˆ Q fulfill the constraintˆ Q ◦ ˆ Q = ˆ Q ◦ ˆ Q = ˆ1 . (36)The frequency representation of the matrix ˆ Q is formedaccording to the prescriptionˆ Q εε (cid:48) ( r ) = (cid:90) t,t (cid:48) ˆ Q tt (cid:48) ( r ) e iεt − iε (cid:48) t (cid:48) . (37)The matrices ˆ Q and ˆ U transform as ˆ Q does, followingthe same prescription. Naturally, we will consider theFourier transformed quantities ˆ Q εε (cid:48) , ˆ U εε (cid:48) , etc., as ma-trices in frequency space and write the parametrizationpresented in Eq. (34) as ˆ Q = ˆ U ˆ σ ˆ U , ˆ U ˆ U = ˆ1, so thatˆ Q = ˆ1. When choosing this parametrization, we imme-diately restrict ourselves to the so-called ”massless” man-ifold. Fluctuations that violate the constraint (36) aremassive and their dynamics is beyond our interest. The parametrization of Eq. (35) is very convenient for theRG procedure. For frequencies exceeding the tempera-ture, matrices ˆ u are almost frequency-independent. Onemay therefore integrate out ˆ U until the moment when ˆ u introduces the information about temperature.After integrating the fermionic fields Ψ, Ψ † , one canperform a gradient expansion in the slow fields ˆ U andˆ U and also expand in the fields (cid:126)θ and sources (cid:126)ϕ (whichare slowly varying by definition). The relevant steps havebeen described many times in the literature and we refer,e.g., to Refs. 39 and 25 for details. The result is thenonlinear sigma model in the form S = πνi (cid:104) D ( ∇ ˆ Q ) + 4 i (cid:16) ˆ ε + (ˆ θ l − ˆ ϕ l ) σ l (cid:17) ˆ Q (cid:105) + (cid:90) r , r (cid:48) ,t (cid:126)θ T ( r , t ) ˆ γ V − ( r − r (cid:48) ) (cid:126)θ ( r (cid:48) , t )+2 ν (cid:90) x ( (cid:126)θ − (cid:126)ϕ ) T ( x ) ˆ γ ( (cid:126)θ − (cid:126)ϕ )( x ) , (38)where ˆ ε acts trivially on a matrix in the frequency spaceas ˆ ε ˆ Q εε (cid:48) = ε ˆ Q εε (cid:48) . Note that for non-interacting elec-trons (in contrast to the case of e-e interactions), owingto the trace operation Tr(ˆ ε ˆ Q ) = Tr(ˆ ε ˆ Q ), only the sourceterm prevents one from removing the distribution func-tion from the action. The last term in Eq. (38) arisesas a result of integrating out fast electronic degrees offreedom with energies exceeding 1 /τ . The interval of en-ergies below 1 /τ down to temperature T is dominated bydiffusion modes, and it will be studied later in Secs. IIIand IV on the basis of the NL σ M. C. NL σ M after Fermi liquid renormalizations
The last term in Eq. (38) allows us to obtain the Fermiliquid (FL) renormalizations in the NL σ M in a systematicway, including the renormalization of the source fields. Asimilar treatment of the Fermi liquid corrections in theKeldysh formalism can be found, e.g., in Refs. 41 and 42.Upon integration in θ , one finds the action of the Keldyshsigma model for interacting electrons in the form S = S (cid:48) + S int + S (cid:48) ϕ , (39)where S (cid:48) = πνi (cid:90) r tr (cid:104) D ( ∇ ˆ Q ) + 4 i ˆ ε ˆ Q (cid:105) , (40) S int = − π ν (cid:90) rr (cid:48) t tr[ˆ γ i ˆ Q tt ( r )]ˆ γ ij ˜Γ ρ ( r − r (cid:48) )tr[ˆ γ j ˆ Q tt ( r (cid:48) )] − π ν (cid:90) r t tr[ˆ γ i σ ˆ Q tt ( r )]ˆ γ ij Γ σ tr[ˆ γ j σ ˆ Q tt ( r )] , (41)and S (cid:48) ϕ = S (cid:48) ϕQ + S (cid:48) ϕϕ with S (cid:48) ϕQ = πν (cid:90) r tr (cid:104) ˆ ϕ lF L ( r ) σ l ˆ Q ( r ) (cid:105) , (42) S (cid:48) ϕϕ = 2 ν (cid:90) r t (cid:126)ϕ TF L ( r , t )ˆ γ (cid:126)ϕ ( r , t ) . (43) Notation S (cid:48) indicates that the corresponding terms in theaction are not yet written in the final form suitable forthe RG analysis, and will be treated further in Sec. II D.As a result of the integration in (cid:126)θ , the source fields (cid:126)ϕ = ( ϕ, ϕ T ) T acquire static vertex corrections describingthe FL renormalizations and screening. Namely, we getfor the singlet component ϕ F L ( q , t ) = ϕ ( q , t )1 + F ρ + 2 νV ( q ) , (44)and for the triplet components ϕ iF L = ϕ i F σ , i = 1 , , . (45)Furthermore, the interaction amplitudes in the singletand triplet channels, symbolized by ˜Γ ρ and Γ σ respec-tively, acquire the desired form˜Γ ρ ( q ) = 2 νV ( q ) + F ρ νV ( q ) + F ρ ) , Γ σ = F σ F σ . (46)For future purposes it will be convenient to decomposethe interaction in the singlet channel into two parts. One of them is the statically screened Coulomb interac-tion Γ ( q ), while the other one is the short-range interac-tion Γ ρ which acts within the polarization operator alongwith Γ σ , ˜Γ ρ ( q ) = 2Γ ( q ) + Γ ρ , (47)whereΓ ( q ) = ν (1 + F ρ ) V − ( q ) + ∂n∂µ , Γ ρ = F ρ F ρ . (48)We also obtained the FL renormalization for ∂n∂µ , thequantity that determines the value of the polarizationoperator in the static limit ∂n∂µ = 2 ν F ρ . (49)This concludes the derivation of the Keldysh sigmamodel which, in principle, can be used as a starting pointfor the RG analysis of the disordered electron liquid. Inthe next section, we will nevertheless cast the NL σ M inan equivalent form that will turn out to be more suitablefor the renormalization group analysis.
D. NL σ M: Preparation for the RG-procedure
As a preparation for the RG analysis, we will nowpresent the model in a slightly modified form. We writethe action as S = S + S int + S ϕ , (50)and comment on the individual terms next.The second (i.e., the frequency) term in the expres-sion for S (cid:48) , Eq. (40), acquires logarithmic corrections atlow temperatures in the presence of the electron interac-tions. In other words, not only D , but also the dynamicsof the diffusion modes is modified in the course of therenormalization of the NL σ M. Following Refs. 7,9,43 wewill introduce the parameter z into the model in order toaccount for these changes. As a result, S takes the form S = πνi (cid:90) r tr (cid:104) D ( ∇ ˆ Q ) + 4 iz ˆ ε ˆ Q (cid:105) . (51)For technical reasons, it is convenient to rewrite theinteraction term, Eq. (41), in a different form. Instead oforganizing the short-range part of the interaction ampli-tudes into the singlet and triplet channel amplitudes, Γ ρ and Γ σ , we will pass to a representation that separatessmall-angle and large-angle scattering, described by Γ and Γ , respectively. The RG-analysis takes a simplerform in this representation. To this end we rewrite theinteraction terms with the help of the identityΓ σ (cid:126) σ αβ (cid:126) σ γδ = 2Γ σ δ αδ δ βγ − Γ σ δ αβ δ γδ , (52)where α , β , γ and δ are spin indices. The interactionamplitudes Γ and Γ are defined asΓ = 12 (Γ ρ − Γ σ ) , Γ = − Γ σ . (53)The amplitude Γ describes large angle scattering, whileΓ describes small angle scattering. It is therefore con-venient to define a new amplitude Γ( q ), which comprisesboth Γ and the screened Coulomb interaction Γ ( q ):Γ( q ) = Γ ( q ) + Γ . (54)In terms of the new amplitudes one finds the relation˜Γ ρ ( q ) = 2Γ( q ) − Γ , cf. Eq. (47). Note that in the limitof small q , the effective amplitude in the ρ channel canbe expressed in terms of Γ and Γ as follows˜Γ ρ ( q →
0) = 11 + F ρ + 2Γ − Γ . (55)Returning to the action, the interaction term can be(identically) rewritten as S int = − π ν (cid:90) rr (cid:48) t tr[ˆ γ i ˆ Q αα ; tt ( r )] γ ij Γ( r − r (cid:48) )tr[ˆ γ j ˆ Q ββ ; tt ( r (cid:48) )]+ π ν (cid:90) r t tr[ˆ γ i ˆ Q αβ ; tt ( r )] γ ij Γ tr[ˆ γ j ˆ Q βα ; tt ( r )] . (56)In order to obtain a more tractable form for the inter-action part of the action, let us introduce a set of H-Sfields: real φ ( x ), φ ( x ) and Hermitian φ ,αβ ( x ), eachwith classical and quantum components, which we char- acterize by their correlations (cid:104) φ i ( x ) φ j ( x (cid:48) ) (cid:105) = i ν Γ ( r − r (cid:48) ) δ ( t − t (cid:48) ) γ ij , (57) (cid:104) φ i ( x ) φ j ( x (cid:48) ) (cid:105) = i ν Γ δ ( x − x (cid:48) ) γ ij , (58) (cid:104) φ i ,αβ ( x ) φ j ,γδ ( x (cid:48) ) (cid:105) = − i ν Γ δ αδ δ βγ δ ( x − x (cid:48) ) γ ij . (59)This definition allows us to cast S int in a compact form S int = i ( πν ) (cid:88) n =0 (cid:90) rr (cid:48) (cid:104) tr[ ˆ φ n ( r ) ˆ Q ( r )]tr[ ˆ φ n ( r (cid:48) ) ˆ Q ( r (cid:48) )] (cid:105) . (60)Here, the frequency representation of the fields φ n hasbeen introduced in the matrix form, ˆ φ n ; εε (cid:48) , according tothe convention:ˆ φ n ; εε (cid:48) ( r ) = (cid:90) t ˆ φ n ( r , t ) e i ( ε − ε (cid:48) ) t . (61)We will sometimes use the notation ˆ φ = ˆ u ◦ ˆ φ ◦ ˆ u in analogyto Eq. (35), so that tr[ ˆ φ n ( r ) ˆ Q ( r )] = tr[ ˆ φ n ( r ) ˆ Q ( r )].We had to split the interaction in the singlet channelinto φ and φ , because for the calculation of the irre-ducible density-density correlation function (i.e., the po-larization operator) one needs to consider the Coulomband the short-range parts of the interaction separately.(Recall that the term irreducible in this context meansthat only those contributions should be considered, whichcannot be separated into two disconnected parts by cut-ting a single Coulomb interaction line.) We encounterthis problem considering the source terms associated withthe singlet channel, see Eq. (44). Source fields were in-troduced because they allow generating correlation func-tions by functional differentiation of the Keldysh parti-tion function, for details see Sec. III below. The potentialrelated to the singlet channel, ϕ , can be used to obtainthe density-density correlation function which, in turn,is related to electric conductivity through the Einsteinrelation, see Sec. V C. It is important to note, however,that only the knowledge of the irreducible density-densitycorrelation function is required for that purpose [for a de-tailed discussion of this point we refer to Ref. 44]. Forthis reason we will not work with the source term S (cid:48) ϕ ,but with a slightly modified one, S ϕ , for which the de-pendence on V ( q ) is removed. Note that the triplet partis unaffected by this change.Finally, we write S ϕ = S ϕQ + S ϕϕ , where S ϕQ = πν (cid:90) r tr (cid:104) ( γ ρ(cid:47) ˆ ϕ ( r ) + γ σ(cid:47) ˆ ϕ ( r ) σ ) ˆ Q ( r ) (cid:105) (62) S ϕϕ = 2 ν (cid:90) r t (cid:126)ϕ T ( r , t )ˆ γ diag( γ ρ • , γ σ • , γ σ • , γ σ • ) (cid:126)ϕ ( r , t ) . (63)Here, the constants γ ρ/σ(cid:47) and γ ρ/σ • have been introduced. γ ρ/σ(cid:47) characterize the (triangular) vertices and γ ρ/σ • thestatic part of the correlation function. By comparisonwith Eqs. (42)-(45) and keeping in mind the previousremarks one finds that the initial values for the renor-malization procedure read γ ρ(cid:47) = γ ρ • = 11 + F ρ , γ σ(cid:47) = γ σ • = 11 + F σ . (64)As one can see, γ ρ/σ(cid:47) = γ ρ/σ • initially coincide. It is a pri-ori not obvious, however, whether this important relationremains true under renormalization, and this is why thedifferent constants have been introduced.To summarize, the nonlinear sigma model contains sev-eral parameters (”charges”) that may in principle acquirelogarithmic corrections at low temperatures, D , z , Γ andΓ , γ ρ/σ(cid:47) and γ ρ/σ • . Let us state the initial values, whichfollow directly from the derivation presented in Sec. II,namely D = v F τ / , z = 1; (65)Γ = 12 (cid:18) F ρ F ρ − F σ F σ (cid:19) , Γ = − F σ F σ , (66)and the values for γ ρ/σ • , γ ρ/σ(cid:47) are written in Eq. (64). III. CORRELATION FUNCTIONS
In this section we first recall how retarded correlationfunctions can be generated from the Keldysh partitionfunction by taking derivatives with respect to the so-called quantum and classical components of suitably cho-sen source fields. Next, we discuss the general structureof the correlation functions for particle-number densitiesand spin densities in the diffusive regime. The conserva-tion laws for the total number of particles and for spinimpose important constraints on the structure of thesecorrelation functions.
A. Generalities
We are interested in the retarded correlation functions,which are defined as a commutator of operators: χ Roo ( x − x ) = − iθ ( t − t ) (cid:104) [ˆ o ( x ) , ˆ o ( x )] (cid:105) T . (67)In order to be in line with common notation, we usehats to denote operators in this section up to Eq. (71).Afterwards, the hat symbol will again be reserved forKeldysh matrices only. In Eq. (67), ˆ o can be either theoperator of the density ˆ n or of a component of the spindensity operator ˆ s ,ˆ n ( x ) = (cid:88) αβ ˆ ψ † α ( x ) σ αβ ˆ ψ β ( x ) , (68)ˆ s ( x ) = 12 (cid:88) αβ ˆ ψ † α ( x ) σ αβ ˆ ψ β ( x ) . (69) In Eq. (67), θ ( t − t (cid:48) ) is the Heaviside function and thermalaveraging is with respect to the grand canonical ensem-ble, (cid:104) . . . (cid:105) T = tr [ˆ ρ . . . ] , ˆ ρ = e − ˆ K/T tr(e − ˆ K/T ) , (70)where ˆ K = ˆ H − µ ˆ N , ˆ H is the Hamiltonian, and ˆ N thenumber operator. The field operators ˆ ψ and ˆ ψ † are writ-ten in the Heisenberg representation with respect to ˆ K .Using the time ordered product T [ˆ o ( t )ˆ o ( t )] = θ ( t − t )ˆ o ( t )ˆ o ( t ) + θ ( t − t )ˆ o ( t )ˆ o ( t ) and anti-time orderedproduct ˜ T [ˆ o ( t )ˆ o ( t )] = θ ( t − t )ˆ o ( t )ˆ o ( t ) + θ ( t − t )ˆ o ( t )ˆ o ( t ), one may present the correlation functionas χ Roo ( x − x ) = − i (cid:68) T [ˆ o ( x )ˆ o ( x )] − ˜ T [ˆ o ( x )ˆ o ( x )]+ˆ o ( x )ˆ o ( x ) − ˆ o ( x )ˆ o ( x ) (cid:69) T . (71)In the Keldysh formalism, this expression can conve-niently be represented with the help of the functionalintegral, namely χ Roo ( x − x ) = − i (cid:10) o + ( x ) o + ( x ) − o − ( x ) o − ( x )+ o − ( x ) o + ( x ) − o − ( x ) o + ( x ) (cid:11) = − i (cid:104) [ o + + o − ]( x )[ o + − o − ]( x ) (cid:105) , (72)where o ± are now the corresponding (bosonic) fieldson forward and backward paths of the Keldysh contourand averaging is with respect to the action S (compareEq. (11)). Introducing the classical and quantum com-ponents of the densities o as o cl/q = ( o + ± o − ), one maywrite the correlation function in the form χ Roo ( x − x ) = − i (cid:104) o cl ( x ) o q ( x ) (cid:105) . (73)The source term that has been introduced into the ac-tion in Eq. (30) can be re-written as follows: S source = − (cid:126) Ψ † ˆ ϕ i σ i (cid:126) Ψ (74)= − ϕ n cl + ϕ n q + 2 ϕ s cl + 2 ϕ s q )Therefore, the correlation functions for the density n andthe spin components s i can conveniently be written as χ Rnn ( x − x ) = i δ Z δϕ ( x ) δϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ = ϕ =0 , (75) χ Rs i s j ( x − x ) = i δ Z δ ϕ i ( x ) δ ϕ j ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ = ϕ =0 . (76)This is rather intuitive, as (cid:104) n cl ( x ) (cid:105) = i δZδϕ ( x ) , (cid:104) s icl ( x ) (cid:105) = i δZδ ϕ i ( x ) (77)are the average particle-number and spin densities in thepresence of the external (classical) potentials and, hence,the correlation functions describe the corresponding re-sponses.A very important observation can be made directlyfrom the definition of the correlation function, Eq. (67).To this end, let us first define the Fourier transformof the retarded correlation functions as χ Roo ( x − x ) = (cid:82) q χ Roo ( q , t − t ) exp( i q ( r − r )). Since for any giventime the operators of the total density and spin (cid:82) r ˆ n ( x )and (cid:82) r ˆ s ( x ) commute with ˆ K , as the total number ofparticles and the total spin are conserved, the correla-tion function χ ( q , t − t ) vanishes in the limit q → χ Roo ( q , t − t ) = (cid:82) ω χ Roo ( q , ω ) exp( − iω ( t − t )), it shouldbe appreciated that the limits q → ω → ω → B. Correlation functions from the sigma model
We will now discuss the density-density and spin-spincorrelation functions in the framework of the NL σ M. Thediscussion will be restricted to the so-called ladder ap-proximation, i.e., to an approximation, for which no in-ternal momentum and frequency integrations over diffu-sion modes are carried out. In fact, those integrationsgive rise to logarithmic corrections (arising from the in-terval
T < ε < /τ ), which is the essence of the RG-scheme. The logarithmic corrections may be absorbedinto the various charges of the model, while the form ofthe model is unchanged. The results for the correlationfunctions obtained in the ladder approximation are there-fore applicable at different scales (or temperatures) oncethe appearing charges are replaced by their renormalizedvalues. As already mentioned before, the conservationlaws for the number of particles and the total spin im-pose certain constraints on the relation between differentRG charges which must be obeyed at each step of therenormalization procedure. This observation serves as an important check for the correctness of the obtainedRG equations.In short, we now find the correlation functions for den-sity and spin in the Gaussian approximation with respectto fluctuations, i.e., with respect to diffusion modes. Wethereby assume that all non-Gaussian integrations thatlead to RG-corrections have been already performed. Asa preparation, let us start with the parametrization of thematrix ˆ U . A convenient choice of the parametrization isˆ U = e − ˆ P / , ˆ U = e ˆ P / , (78)with the additional constraint { ˆ P , ˆ σ } = 0 in order to avoid overcounting of the relevant degrees of freedom.The chosen parametrization is not the only possible one.In fact, it gives rise to a non-trivial Jacobian, which, how-ever, does not become relevant for the one-loop calcula-tion discussed in this manuscript. Other parametriza-tions exist; for an instructive discussion within the con-text of the Keldysh NL σ M we refer to Ref. 45. Returningto the exponential parametrization, Eq. (78), note thatˆ Q = ˆ σ exp( ˆ P ). Further, the matrices ˆ P can be writtenas ˆ P εε (cid:48) ( r ) = (cid:18) d cl ; εε (cid:48) ( r ) d q ; εε (cid:48) ( r ) 0 (cid:19) , (79)where d cl/q are hermitian matrices both in the frequencydomain and in spin space, [ d αβcl/q ; εε (cid:48) ] ∗ = d βαcl/q ; ε (cid:48) ε . Expand-ing S + S int (see Eqs. (51) and (60)) up to second orderin the generators ˆ P , one obtains S = − iπν (cid:90) tr[ D ( ∇ ˆ P ) − iz ˆ ε ˆ σ ˆ P ] (80)+ i πν ) (cid:88) n =0 (cid:90) rr (cid:48) (cid:104) tr[ ˆ φ n ( r )ˆ σ ˆ P ( r )]tr[ ˆ φ n ( r (cid:48) )ˆ σ ˆ P ( r (cid:48) )] (cid:105) . Recall that for the frequency representation of the fields φ n , the matrix form ˆ φ n ; εε (cid:48) has been introduced.By inverting the corresponding quadratic form, i.e.,in the Gaussian approximation, this action gives rise tocertain correlations for the components of ˆ P . The resultis most easily obtained after separation into singlet andtriplet channels. Defining d lcl/q ; εε (cid:48) = 12 (cid:88) αβ σ lβα d αβcl/q ; εε (cid:48) , l = (0 , − , (81)one obtains for the correlation functions describing dif-fusion of the particle-hole pairs in the singlet (indicatedby 0) and triplet (indicated by i, j ∈ { , , } ) channels,respectively: (cid:10) d cl ; ε ε ( q ) d q ; ε ε ( − q ) (cid:11) = − πν D ( q , ω ) × (82) (cid:16) δ ε ,ε δ ε ,ε − δ ω,ε − ε iπ ∆ ε ε ˜Γ ρ ( q ) ˜ D ( q , ω ) (cid:17) , and (cid:10) d icl ; ε ε ( q ) d jq ; ε ε ( − q ) (cid:11) = − πν δ ij D ( q , ω ) × (83) (cid:16) δ ε ,ε δ ε ,ε − δ ω,ε − ε iπ ∆ ε ε Γ σ D ( q , ω ) (cid:17) , where ω = ε − ε , ∆ ε,ε (cid:48) = F ε −F ε (cid:48) and δ ε,ε (cid:48) = 2 πδ ( ε − ε (cid:48) ).Obviously, on the level of the Gaussian fluctuations, thesinglet and triplet channels do not interfere with eachother. Note that three types of diffusons have beenintroduced in the above correlation functions: D ( q , ω ) = 1 D q − izω (84)0˜ D ( q , ω ) = 1 D q − i ˜ z ω (85) D ( q , ω ) = 1 D q − iz ω , (86)where ˜ z ( q ) = z − q ) + Γ = z − ˜Γ ρ ( q ), and z = z + Γ = z − Γ σ . We will see soon that actually ˜ z ( q ) ≈ D does not depend on ω . Transforming back to the original representation interms of spin projections, one finds (cid:68) d αβcl ; ε ε ( q ) d γδq ; ε ε ( − q ) (cid:69) (87)= − πν [ δ αδ δ βγ δ ε ,ε δ ε ,ε D ( q , ω )+ δ αδ δ βγ δ ω,ε − ε iπ ∆ ε ,ε D ( q , ω )Γ D ( q , ω ) − δ αβ δ γδ δ ω,ε − ε iπ ∆ ε ,ε D ( q , ω )Γ( q ) ˜ D ( q , ω ) (cid:105) . In order to demonstrate the general structure of the cor-relation functions for conserved quantities, we will beinterested in the irreducible correlation function in thesinglet channel, ˆ¯ χ nn ≡ ˆ χ nn | irr . For that, the ladderwhich is irreducible with respect to the Coulomb inter-action is required. It can be found simply by exclud-ing Γ ( q ), so that the expression for the irreducible av-erage (cid:68) d cl ; ε ε ( q ) d q ; ε ε ( − q ) (cid:69) irr coincides with the onestated in Eq. (82) up to the replacement ˜Γ( q ) → Γ ρ and˜ D → D , where D ( q , ω ) = 1 D q − iz ω (88)with z = z − + Γ = z − Γ ρ . (89)With this preparation, the correlation functions inthe ladder approximation can be calculated. In view ofEqs. (75) and (76), we may integrate out the ˆ P modesand keep resulting terms only up to quadratic order in ϕ . Therefore, we calculate the dressed term S ϕϕ ; d = S ϕϕ + i (cid:10)(cid:10) S ϕQ (cid:11)(cid:11) irr , (90)where for the second term both appearing matrices ˆ Q arereplaced by ˆ σ ˆ P , and the averaging is with respect to theaction (80) for which the contraction rules obtained abovecan be used. In Eq. (90), (cid:104)(cid:104) . . . (cid:105)(cid:105) denotes the connectedaverage. One may anticipate that S ϕϕ ; d has the followingform, S ϕϕ ; d = − (cid:90) xx (cid:48) (cid:126)ϕ T ( x ) ˆ X ( x − x (cid:48) ) (cid:126)ϕ ( x (cid:48) ) , (91)where ˆ X = diag( ˆ¯ χ nn , χ s x s x , χ s y s y , χ s z s z ), and the2 × χ oo have a structure that is typical for cor-relation functions in the Keldysh formalism. Indeed,ˆ χ oo = (cid:18) χ Aoo χ Roo χ Koo (cid:19) , (92) γ σ γ σ + γ σ Γ σ γ σ + γ σ Γ σ Γ σ γ σ + . . . γ ρ γ ρ + γ ρ Γ ρ γ ρ + γ ρ Γ ρ Γ ρ γ ρ + . . . FIG. 1: Dynamical correlation functions ¯ χ dyn , R nn (top) and χ dyn , R s k s k (bottom). where χ Aoo ( ω ) = χ Roo ( − ω ), and χ Koo ( ω ) = B ω (cid:0) χ Roo ( ω ) − χ Aoo ( ω ) (cid:1) . (93)Furthermore, the two terms in Eq. (90) for S ϕϕ ; d giverise to the static (st) and dynamical (dyn) parts of thecorrelation functions, respectively. As can directly beread off from Eq (63), the contribution from S ϕϕ is¯ χ st ,Rnn = − νγ ρ • , χ st ,Rs i s i = − νγ σ • , (94)while for the dynamical part one finds i (cid:10)(cid:10) S ϕQ (cid:11)(cid:11) irr (95)= i ( πν ) (cid:42)(cid:42)(cid:20)(cid:90) r tr (cid:104) ( γ ρ(cid:47) ˆ ϕ ( r ) + γ σ(cid:47) ˆ ϕ ( r ) σ )ˆ σ ˆ P ( r ) (cid:105)(cid:21) (cid:43)(cid:43) irr = − (cid:90) xx (cid:48) (cid:126)ϕ T ( x ) ˆ X dyn ( x − x (cid:48) ) (cid:126)ϕ ( x (cid:48) ) , where ˆ X dyn = diag( ˆ χ dyn nn , χ dyn s x s x , χ dyn s y s y , χ dyn s z s z ). Thecomponents of ˆ χ dyn have again the structure indicatedin Eq. (92), and¯ χ dyn , R nn ( q , ω ) = − ν ( γ (cid:47)ρ ) iω D ( q , ω ) , (96) χ dyn , R s i s i ( q , ω ) = − ν ( γ (cid:47)σ ) iω D ( q , ω ) . (97)For a diagrammatic illustration see Fig. 1.In order to obtain this result, the following relation hasbeen used 1 − F ε + ω F ε − ω = B ω (cid:0) F ε + ω − F ε − ω (cid:1) , (98)where B ω = coth (cid:16) ω T (cid:17) (99)is the bosonic equilibrium correlation function. A secondimportant identity is π (cid:90) ε (cid:0) F ε + ω − F ε − ω (cid:1) = ω. (100)The total correlation function is then found by addingthe static and the dynamical parts, χ Roo ( q , ω ) = χ st ,Roo + χ dyn ,Roo ( q , ω ) , (101)1with the result¯ χ Rnn ( q , ω ) = − νγ ρ • D q − iω (cid:16) z − ( γ ρ(cid:47) ) γ ρ • (cid:17) D q − iz ω , (102) χ Rs i s i ( q , ω ) = − νγ σ • D q − iω (cid:16) z − ( γ σ(cid:47) ) γ σ • (cid:17) D q − iz ω . (103)As discussed in Sec. III A, conservation of charge andspin demands that χ Roo ( q = 0 , ω →
0) = 0 . (104)In order to fulfill these conditions, the following relationsmust hold in view of Eqs. (102) and (103), z = ( γ ρ(cid:47) ) γ ρ • , z = ( γ σ(cid:47) ) γ σ • , (105)where the first relation is related to chargeconservation and the second one to the conser-vation of spin. One may readily check that for thebare values of z , γ ρ/σ • and γ ρ/σ(cid:47) , these relations arefulfilled. Below, we will discuss the renormalization ofthe NL σ M for interacting electrons. In the RG scheme,the parameters z , Γ , Γ (which determine z and z ) aswell as γ σ • and γ σ(cid:47) acquire logarithmic corrections andthereby become scale-dependent. It will be an importantcheck of the theory that the two conditions displayed inEq. (105) still hold after renormalization. Indeed, wewill find that γ ρ(cid:47) = γ ρ • = 11 + F ρ (106)are not renormalized, and that the relation z = 1 / (1 + F ρ ) holds under the RG flow. Therefore, the first relationin Eq. (105) is fulfilled. As a byproduct, it follows fromthese relations that 2Γ ( q ) = 1 / (1+ F ρ ) for small enough q when V − ( q ) ∂µ/∂n (cid:28)
1. Therefore, ˜ z ( q ) = 0 in thislimit and, hence, ˜ D ( q , ω ) = 1 /D q .Further, we will find that γ σ(cid:47) = γ σ • = z , (107)and the relation for the conservation of spin also holds,so that ¯ χ Rnn ( q , ω ) = − ∂n∂µ D q D q − i ω F ρ (108) χ Rs i s j ( q , ω ) = − νz D q D q − iz ω δ ij . (109)The correlation functions ¯ χ Rnn ( q , ω ) and χ Rss ( q , ω ) have auniversal form, which is typical for diffusive correlationfunctions of the densities of a conserved quantity. In aseparate publication, we show that the same structure,compare Eqs. (108) and (109), also holds for the heatdensity - heat density correlation function reflecting en-ergy conservation. Finally, a comment is in order. The vanishing of thecorrelation function χ nn ( q , ω ) in the limit q → χ nn is, in fact, the polarization operator. Furthermore,we need to know only the irreducible function ¯ χ nn ( q , ω )in order to extract the conductivity using the Einsteinrelation. IV. RENORMALIZATION
The renormalization group approach for the problemat hand follows a general philosophy that is common tomany problems in condensed matter physics. For the RGprocedure, the fields in the action are separated into fastand slow modes. Subsequently, the fast modes are inte-grated out with logarithmic accuracy, leading to an effec-tive action for the slow modes with scale-dependent pa-rameters, i.e., RG charges. A remark about the RG pro-cedure in the Keldysh technique is in order: for any the-ory in which a quenched disorder average is performed,diagrams that can be cut into separate parts by cuttingonly impurity lines should not appear. In the originalmodel of Ref. 7 the so-called replica trick was used in or-der to make sure that such contributions vanish. Whenusing the Keldysh approach, the vanishing is effected ina somewhat different way. Generally speaking, the mostimportant observation about the vanishing of unphysi-cal terms in the Keldysh technique is that the frequencyintegral over a product of several retarded or advancedfunctions (but not a mixture of them) vanishes. This ar-gument will frequently be used later on. The argument,however, does not carry over to the case when a singleretarded or advanced function is connected to the restby impurity lines only. This special case is discussed inconnection with Fig. 7 in Sec. IV C. [An alternative tothe replica and Keldysh approaches exists, the so-calledsupersymmetry technique. It is a very powerful toolfor noninteracting systems. Its application to interactingsystems, however, is a formidable challenge, and progressin this direction is so far limited. ]In order to lighten notations, starting from Sec. IV Cwe will leave out the hats symbolizing matrices inKeldysh space. A. Generalities
For the NL σ M the separation into fast and slow modesshould be done in such a way that the nonlinear con-straint ˆ Q = 1 is preserved ˆ Q = ˆ U ˆ Q ˆ U , ˆ Q = ˆ U ˆ σ ˆ U , ˆ U ˆ U = ˆ U ˆ U = ˆ1 . (110)Here, ˆ Q contains the fast variables, ˆ U and ˆ U representthe slow degrees of freedom. It is also convenient to in-2troduce the slow field ˆ Q s asˆ Q s = ˆ U ˆ σ ˆ U . (111)When inserting ˆ Q in the form specified in Eq. (110) intothe action S , one obtains S = πνi (cid:104) D ( ∇ ˆ Q ) + D [ ˆ Q , ˆΦ] +2 D ˆΦ[ ˆ Q , ∇ ˆ Q ] + 4 iz ˆ ε ˆ U ˆ Q ˆ U (cid:105) , (112)where ˆΦ = ˆ U ∇ ˆ U = −∇ ˆ U ˆ U . Using this notation, theinteraction reads S int = i ( πν ) (cid:88) n =0 (cid:68) Tr (cid:104) ˆ φ n ˆ U ˆ Q ˆ U (cid:105) Tr (cid:104) ˆ φ n ˆ U ˆ Q ˆ U (cid:105)(cid:69) . (113)For the RG-procedure, a particular parametrization forthe fast degrees of freedom needs to be chosen. In accordwith the previous Section, we will work with the exponen-tial parametrization ˆ U = exp( − ˆ P / Q = ˆ σ exp( ˆ P ), { ˆ σ , ˆ P } = 0. It turns out to be sufficient to expand up tosecond order in ˆ P . We left out terms linear in ˆ P . Suchterms describe the decay (or fusion) of a fast mode intoslow modes. These processes do not not play any role inthe RG-analysis. Then the result of the expansion reads S = πνi (cid:104) D ( ∇ ˆ Q s ) + 4 iz ˆ ε ˆ Q s (cid:105) (114)+ πνi (cid:104) D (ˆ σ P ˆΦ) + D ˆ P ( ˆΦˆ σ ) + D ˆΦ[ ∇ ˆ P , ˆ P ] + iz ˆ ε ˆ U ˆ σ ˆ P ˆ U (cid:105) − πνi (cid:104) D ( ∇ ˆ P ) (cid:105) ,S int = i πν ) (cid:88) n =0 (cid:68) Tr (cid:104) ˆ φ n ˆ Q s (cid:105) Tr (cid:104) ˆ φ n ˆ Q s (cid:105)(cid:69) (115)+ i πν ) (cid:88) n =0 (cid:68) Tr (cid:104) ˆ φ n ˆ Q s (cid:105) tr (cid:104) ˆ φ n ˆ U ˆ σ ˆ P ˆ U (cid:105)(cid:69) + i πν ) (cid:88) n =0 (cid:68) Tr (cid:104) ˆ φ n ˆ U ˆ σ ˆ P ˆ U (cid:105) (cid:104) ˆ φ n ˆ U ˆ σ ˆ P ˆ U (cid:105)(cid:69) . So far, the separation into fast and slow degrees waspurely formal. Let us now qualify this distinction:1. Frequencies in the interval λτ − < | ε | < τ − , 0 <λ < λτ − < Dk /z <τ − are referred to as fast.2. If at least one of the frequencies ε or ε (cid:48) for the slowfield ˆ U εε (cid:48) is fast, it has to be set equal to the unitmatrix.3. In the fast variables ˆ P εε (cid:48) at least one of the frequen-cies ε , ε (cid:48) or the momentum should be fast. For the frequency term in the action, one should explic-itly distinguish fast and slow frequencies, i.e., ˆ ε f and ˆ ε s .ThenTr (cid:104) z ˆ ε ˆ U ˆ σ ˆ P ˆ U (cid:105) = Tr (cid:104) z ˆ ε s ˆ U ˆ σ ˆ P ˆ U (cid:105) + Tr (cid:104) z ˆ ε f ˆ σ ˆ P (cid:105) . (116)We will now present a list of all the terms that are rele-vant for the one-loop RG-analysis. The following termscontain only slow modes S D = iπνD (cid:104) ( ∇ ˆ Q s ) (cid:105) (117) S z = − πνz Tr (cid:104) ˆ ε s ˆ Q s (cid:105) (118) S Γ = i πν ) (cid:68) Tr (cid:104) ˆ φ n ˆ Q s (cid:105) Tr (cid:104) ˆ φ n ˆ Q s (cid:105)(cid:69) (119) S γ (cid:47) = πν Tr (cid:104)(cid:0) γ ρ(cid:47) ˆ ϕ + γ σ(cid:47) ˆ ϕσ (cid:1) ˆ Q s (cid:105) (120) S γ • = 2 ν (cid:90) x (cid:126)ϕ T ( x )ˆ γ diag( γ ρ • , γ σ • , γ σ • , γ σ • ) (cid:126)ϕ ( x ) . (121)Terms S γ (cid:47) and S γ • arise from the source term S ϕ . Infact, S γ • is identical to S ϕϕ ; the present notation is usedto emphasize the dependence on the parameters γ ρ/σ • .Next, we come to the terms containing fast modes.The terms originating from S read S f, = − iπν (cid:104) D ( ∇ ˆ P ) − iz ˆ ε f ˆ σ ˆ P (cid:105) (122) S = − πνi (cid:104) D ˆΦ[ ˆ P , ∇ ˆ P ] (cid:105) (123) S = πνi (cid:104) D ˆ P ( ˆΦˆ σ ) + D (ˆ σ ˆ P ˆΦ) (cid:105) (124) S ε = − πν (cid:104) z ˆ ε s ˆ U ˆ σ ˆ P ˆ U (cid:105) . (125)Here, S has two parts, which we label as S a and S b inthe order of appearance.The interaction part of the action S int gives rise to thefollowing terms S int, = i πν ) (cid:88) n =0 (cid:68) Tr (cid:104) ˆ φ n ˆ U ˆ σ ˆ P ˆ U (cid:105) Tr (cid:104) ˆ φ n ˆ U ˆ σ ˆ P ˆ U (cid:105)(cid:69) S int, = i πν ) (cid:88) n =0 (cid:68) Tr (cid:104) ˆ φ n Q s (cid:105) Tr (cid:104) ˆ φ n ˆ U ˆ σ ˆ P ˆ U (cid:105)(cid:69) . (126)Note that the labeling of these two terms refers to theirdifferent structure with respect to ˆ P , and is not relatedto the fields φ and φ .Finally, the source term S ϕQ , see (62), generates a term S ϕ, = πν (cid:104)(cid:0) γ ρ(cid:47) ˆ ϕ + γ σ(cid:47) ˆ ϕσ (cid:1) ˆ U σ ˆ P ˆ U (cid:105) , (127)where the labeling is chosen in analogy to S int, .3 S f, S S ,a S ,b S " FIG. 2: The elements of the RG-procedure originating fromthe noninteracting part of the action. Open ends imply P .Closed sleeves correspond to U or U . When separated by anangle, a gradient acts on one of them. A slow frequency ε s stands in the vertex marked by a dot. S int, S int, d S int, S int, d FIG. 3: The elements of the RG-procedure originating fromthe interaction part of the action. A shaded square impliesone of the interaction amplitudes. A ladder means that theinteraction was dressed by ladder diagrams. Such terms areindicated by the subscript ”d”. S ϕ, FIG. 4: Source term
The terms containing fast modes are convenientlyrepresented in a diagrammatic language as depicted inFigs. 2, 3 and 4.We want to integrate out fast modes ˆ P in the Gaus-sian approximation, and in this way generate a new effec-tive action. Besides the slow part of the action, compareEqs. (117) to (121), corrections arise from the term∆ S = − i ln (cid:20)(cid:90) D [ ˆ P ] e iS + iS + iS ε + iS int + iS ϕ, e iS f, (cid:21) . (128)In general, if there are N different parts in the actionin which slow and fast modes couple to each other, onefinds ∆ S = − i ln (cid:18)(cid:90) D [ ˆ P ] (cid:16) e i (cid:80) Ni =1 S i (cid:17) e iS f, (cid:19) (129)= N (cid:88) i =1 (cid:104) S i (cid:105) + i N (cid:88) ij =1 (cid:104)(cid:104) S i S j (cid:105)(cid:105) − N (cid:88) ijk =1 (cid:104)(cid:104) S i S j S k (cid:105)(cid:105) + . . . Here, the connected average means that contractions be-tween different terms must be taken as (cid:104)(cid:104) AB (cid:105)(cid:105) = (cid:104) AB (cid:105) − (cid:104) A (cid:105) (cid:104) B (cid:105) , (130)and so on.When integrating out fast modes, two cases should bedistinguished. If at least one of the frequencies of the ˆ P -matrix is slow, then the contractions should be performedusing S f, alone. One can formulate two contraction rulesfor this case. Rule (i) applies when the two contractedˆ P s stand under different traces (cid:68) tr (cid:104) ˆ A ˆ P ε ε ( r ) (cid:105) tr (cid:104) ˆ B ˆ P ε ε ( r ) (cid:105)(cid:69) (131)= − πν tr (cid:104) ˆ A ⊥ ˆΠ ε ε ( r − r ) ˆ B ⊥ (cid:105) δ ε ,ε δ ε ,ε , where we denote ˆ A ⊥ = ( ˆ A − ˆ σ ˆ A ˆ σ ), andˆΠ ε + ω ε − ω ( q ) = (cid:18) D ( q , ω ) 00 D ( q , ω ) (cid:19) (132)contains a retarded diffuson D and an advanced one, D ( ω ) = D ( − ω ). A second contraction rule (ii) applieswhen two contracted ˆ P s appear within one trace. It readsas follows (cid:104) tr [ AP ε ε ( r ) BP ε ε ( r )] (cid:105) (133)= − πν (cid:16) tr[ A ˆΠ ε ε ( r − r )]tr[ B ] − tr[ A ˆ σ ˆΠ ε ε ( r − r )]tr[ B ˆ σ ] (cid:17) δ ε ε δ ε ,ε . In the second case, when both frequencies of the ˆ P matrix are fast, the free Gaussian action of the fast modesbesides S f, also contains a part originating from S int, .In the case in question, it takes the form S int, → S f,int ,where S f,int = i πν ) (cid:88) n =0 (cid:68) Tr (cid:104) ˆ φ n ˆ σ ˆ P (cid:105) Tr (cid:104) ˆ φ n ˆ σ ˆ P (cid:105)(cid:69) . (134)4Correspondingly, one should take the contraction withthe full quadratic form S f = S f, + S f,int . (135)The relevant contraction rule for the components of ˆ P has already been stated in Eqs. (82), (83) and (87).As is clear from the discussion presented in connectionwith these formulas in Sec. III B, the extension of thequadratic form corresponds to ”dressed” diffusons, whichinclude not only impurity scattering but also a rescatter-ing in the singlet and triplet channels as described bythe ampitudes Γ ρ and Γ σ . An example when this exten-sion becomes important is the dressing of the interactionwhich will be discussed next. B. Dressed interaction
Suppose that a certain average contains the interactionpart of the action, S int . Besides S int , one may as well in-sert in its place the second cumulant i (cid:104)(cid:104) S int (cid:105)(cid:105) , where foreach of the interaction terms one ˆ Q εε (cid:48) will be replaced bythe fast ˆ σ ˆ P εε (cid:48) with both frequencies fast, so that adja-cent ˆ U , ˆ U should be substituted by 1. The contraction ofsuch fast ˆ P s has to be taken with respect to S f . This casemay occur because the interaction fixes only the differ-ence of frequencies ε − ε (cid:48) rather than the two frequenciesindividually. It means that S int should be replaced byits dressed (extended) counterpart S int ; d = S int + i (cid:10)(cid:10) S int (cid:11)(cid:11) , (136)where specifically i (cid:10)(cid:10) S int (cid:11)(cid:11) = − i πν ) (cid:68)(cid:68) (cid:104) Tr[ φ n ˆ Q ]Tr[ φ n ˆ σ ˆ P ] (cid:105) φ (cid:69)(cid:69) S f (137)and we indicated by the labels φ and S f which kind ofaverage should be used. For the calculation of this objecta separation into singlet and triplet channel is useful, inclose analogy to the calculation of the correlation func-tions demonstrated before, see Fig. 1. The calculationgives S int ; d (138)= − π ν (cid:90) rr (cid:48) ,ε i tr[ˆ γ i σ ˆ Q ε ε ( r )]tr[ˆ γ j σ ˆ Q ε ε ( r (cid:48) )] × ˆΓ ijρ ; d ( r − r (cid:48) , ε − ε ) δ ε − ε ,ε − ε − π ν (cid:90) rr (cid:48) ,ε i tr[ˆ γ i σ ˆ Q ε ε ( r )]tr[ˆ γ j σ ˆ Q ε ε ( r (cid:48) )] × ˆΓ ijσ ; d ( r − r (cid:48) , ε − ε ) δ ε − ε ,ε − ε . The dressed ( d ) interaction can be obtained by thesubstitutions ˆ γ ij Γ ρ ( q ) → ˆΓ ijρ ; d ( q , ω ) and ˆ γ ij Γ σ ( q ) → ˆΓ ijσ ; d ( q , ω ), where the interaction matrices ˆΓ ijρ/σ ; d have a Γ ρ ; d = ˜Γ ρ + ˜Γ ρ ˜Γ ρ + ˜Γ ρ ˜Γ ρ ˜Γ ρ + . . . Γ σ ; d = Γ σ + Γ σ Γ σ + Γ σ Γ σ Γ σ + . . . FIG. 5: Dressed interactions.
Keldysh space structure (compare with Eq. (92)). As aresult, one getsˆΓ µ ; d ( q , ω ) = (cid:18) Γ Kµ ; d ( q , ω ) Γ Rµ ; d ( q , ω )Γ Aµ ; d ( q , ω ) 0 (cid:19) , µ = { ρ, σ } , (139)where Γ Aµ ; d ( q , ω ) = Γ Rµ ; d ( q , − ω ) , (140)Γ Kµ ; d ( q , ω ) = B ω (Γ Rµ ; d ( q , ω ) − Γ Aµ ; d ( q , ω )) , (141)and Γ Rρ ; d ( q , ω ) = ˜Γ ρ ( q ) (cid:16) − iω ˜Γ ρ ( q ) ˜ D ( q , ω ) (cid:17) (142)Γ Rσ ; d ( q , ω ) = Γ σ (cid:16) − iω Γ σ D ( q , ω ) (cid:17) . (143)Obviously, the difference between the dressed and bareamplitudes is in the dynamic properties; in the staticlimit the amplitudes are equal. A diagrammatic illustra-tion of dressing is shown in Fig. 5.Clearly, Γ Rρ/σ ; d describe rescattering in the singlet andtriplet channels with intermediate sections composed ofa pair of retarded and advanced Green’s functions (some-times referred to as RA sections). Each RA section givesrise to a window function ∆ ε + ω/ ,ε − ω/ which, when in-tegrated in ε , produces a factor of ω (compare relation(100)). This is why the coefficients of the frequencies ofthe diffusion modes ˜ D and D are modified by the in-teraction amplitudes, see Eqs (84)-(86). An importantdifference to the calculation of the correlation function isthat in the present case the interaction may be reduciblewith respect to the Coulomb interaction, and ˜Γ ρ ( q ) and˜ D appear in the singlet channel.A somewhat simplified way to express the same resultis Γ Rρ ; d ( q , ω ) = ˜Γ ρ ( q ) ˜ D D , Γ Rσ ; d ( q , ω ) = Γ σ D D . (144)In order to obtain Γ d and Γ d , one may use the relationsΓ Rd = (cid:16) Γ Rρ ; d − Γ Rσ ; d (cid:17) and Γ R d = − Γ Rσ ; d to findΓ Rd ( q , ω ) = Γ( q ) ˜ D D D , Γ R d ( q , ω ) = Γ D D . (145)Needless to say, Γ Rd and Γ R d are components of interac-tion matrices ˆΓ d and ˆΓ d with a structure as indicatedin Eq. (139).5If a model for a disordered Fermi liquid with shortrange interactions is considered, one may use the replace-ment Γ( q ) → Γ , ˜ D → D in the final expressions. Forthe Coulomb case, it is useful to single out the screenedCoulomb interaction explicitly. To this end, one may usethe identityΓ Rd = Γ( q ) ˜ D D D = Γ ( q ) ˜ D D D + Γ D D D . (146)After defining ˜Γ R d = Γ ( q ) ˜ D D , (147)one may single out the Coulomb interactionΓ Rd = Γ R d + Γ R d , (148)where Γ R d = ˜Γ d D D , Γ R d = Γ D D D . (149)Note that the entire dependence on the Coulomb interac-tion is delegated to ˜Γ d . Furthermore, with the use of theidentities ∂ µ n = 2 ν/ (1 + F ρ ) as well as z = 1 / (1 + F ρ ),one can obtain ˜Γ d in the form˜Γ R d ( q , ω ) = ν (1 + F ρ ) (cid:20) V − ( q ) + ∂n∂µ D q D q − iz ω (cid:21) − . (150)We observe that ˜Γ R d is the dynamically screenedCoulomb interaction. Following this decomposition ofthe dressed interaction, we elevate relations (57), (58)and (59) to (cid:104) φ i ( x ) φ j ( x (cid:48) ) (cid:105) = i ν ˆΓ ij d ( x − x (cid:48) ) , (151) (cid:104) φ i ( x ) φ j ( x (cid:48) ) (cid:105) = i ν ˆΓ ij d ( x − x (cid:48) ) , (152) (cid:104) φ i ,αβ ( x ) φ j ,γδ ( x (cid:48) ) (cid:105) = − i ν ˆΓ ij d ( x − x (cid:48) ) δ αδ δ βγ , (153)whenever the dressed interaction is used. We remindthat Γ d and Γ d are defined in Eq. (149) and Γ d in Eq. (145). The appearing interaction matrices havethe typical Keldysh structure, compare Eq. (139). Whendressing is not needed (such as for external vertices de-fined below), the static limit may be taken and Γ Rn ; d → Γ n for n = 0 − C. Renormalization of the diffusion coefficient
In this section, we discuss the renormalization of thediffusive term S D in the one loop approximation. Thisterm contains two slow momenta (spatial gradients). It ( a ) S int, ( b ) i S S int, ( c ) i S S int, ( d ) − S S int, FIG. 6: The four different terms contributing to ∆ S D . Forthe terms ( a ) and ( b ) a gradient expansion is needed. means that we can use S at most twice or S once. Addi-tionally, gradients can be generated by Taylor expansionof the slow fields U , ¯ U . As a result, one should consider∆ S D = (cid:104) S int (cid:105) + i (cid:104)(cid:104) S S int (cid:105)(cid:105) + i (cid:104)(cid:104) S S int (cid:105)(cid:105) (154) − (cid:10)(cid:10) S S int (cid:11)(cid:11) . We will discuss these terms one by one and use the oppor-tunity to highlight some aspects that are specific for theRG procedure in the Keldysh formalism. For a diagram-matic illustration of the four terms, see Fig. 6. Recallthat for notational simplicity, we will from now on leaveout hats for matrices in Keldysh space. (cid:104) S int (cid:105) S int consists of two parts, S int, and S int, . First con-sider (cid:104) S int, (cid:105) . The corresponding expression contains thefollowing average (cid:90) ε (cid:104) P ε ε P ε ε (cid:105) ∝ (cid:18) D ( ω ) 00 D ( ω ) (cid:19) (155)The diagram for (cid:104) S int, (cid:105) is displayed in Fig. 7. It is imme-diately obvious that this diagram can be cut into separateparts by cutting only impurity lines. As is well known,such diagrams should not appear for any theory in whicha quenched disorder average is performed. The so-calledreplica method was invented to eliminate such contri-butions. Indeed, the internal Green’s function allows fora free summation over the replica index, and therefore thediagram vanishes in the zero-replica limit. In the Keldyshtechnique, the vanishing of unphysical terms mostly oc-curs because the frequency integral over a product of sev-eral retarded or advanced functions (but not a mixture ofthem) vanishes. This argument, however, does not carry6 FIG. 7: Diagram for (cid:104) S int, (cid:105) . This term vanishes as discussedin the text. over to the case of a single retarded or advanced func-tion as is relevant for the discussed term. In this caseone needs to argue that the contribution of the unphys-ical diagram to the calculation of any physical quantitywill always contain the frequency integral of the sum ofone retarded and one advanced function, and it is simpleto see that their sum vanishes. In the example at hand,the retarded and advanced diffuson appear as separateelements of the matrix M ε ε = (cid:82) ε (cid:104) P ε ε P ε ε (cid:105) . When-ever physical quantities are calculated, all modes haveto be integrated out, which implies that eventually thesum of retarded and advanced functions will appear. An-ticipating this fact, diagrams as encountered for (cid:104) S int, (cid:105) may safely be dropped; Fig. 7 illustrates this importantpoint.For the other term, (cid:104) S int, (cid:105) , see Fig. 6(a), one finds (cid:104) S int, (cid:105) (156)= iπν (cid:88) n =0 (cid:90) r r ,ε ε (cid:68) tr (cid:104)(cid:0) ¯ U ( r ) φ n ( r ) U ( r ) (cid:1) ⊥ ε ε (cid:0) ¯ U ( r ) φ n ( r ) U ( r ) (cid:1) ⊥ ε ε Π ε ε ( r − r ) (cid:105)(cid:69) . Here and in the following we denote M ⊥ = ( M − σ M σ ) /
2, and M (cid:107) = ( M + σ M σ ) /
2, so that M = M (cid:107) + M ⊥ . M (cid:107) is the diagonal part of M in Keldyshspace, and M ⊥ the off-diagonal part; [ M (cid:107) , σ ] = 0, { M ⊥ , σ } = 0. We will mostly work in such a way thatcontractions in P are performed first, while the choiceof fast and slow frequencies for the P -matrices is madea posteriori. This is a straightforward procedure sincethe frequency arguments of P always reappear explic-itly as arguments of the diffusion propagators Π. Forthe renormalization of the diffusion coefficient, in thediscussed contribution precisely one frequency argumentof the P -matrices is fast. This might be either ε or ε , see Fig. 8 for an illustration. Due to the identity X ⊥ ˆΠ ε ε = ˆΠ ε ε X ⊥ for any matrix X in Keldysh spaceboth possibilities are equivalent. For definiteness, wechoose here ε as fast and write ε = ε f . This leadsto the intermediate result (cid:104) S int, (cid:105) = 2 iπν (cid:88) n =0 (cid:90) r r ,ε ε f (cid:68) tr (cid:104) ( ¯ U ( r ) φ n ( r )) ⊥ ε ε f × ( φ n ( r ) U ( r )) ⊥ ε f ε Π ε ε f ( r − r ) (cid:105)(cid:69) . (157)We do not evaluate this term right now, but first proceedwith the other terms. ε ε ε f ε ε ε f FIG. 8: This figure illustrates the two choices of ε f for theaverage (cid:104) S int, (cid:105) , where ε f symbolizes the fast frequency. Infact, both choices are equivalent and in this way one comesfrom Eq. (156) to Eq. (157). i (cid:104)(cid:104) S S int (cid:105)(cid:105) The relevant contribution comes from S int, only. Onefinds i (cid:104)(cid:104) S S int, (cid:105)(cid:105) = − iπν (cid:88) n =0 (cid:90) r i ,ε i D ( ∇ r (cid:48)(cid:48) − ∇ r (cid:48) ) (158) (cid:68) tr (cid:104)(cid:0) ¯ U ( r ) φ n ( r ) U ( r ) (cid:1) ⊥ ε ε (cid:0) ¯ U ( r ) φ n ( r ) U ( r ) (cid:1) ⊥ ε ε × Φ (cid:107) ε ε ( r ) ˆΠ ε ε ( r − r (cid:48) ) ˆΠ ε ε ( r (cid:48)(cid:48) − r ) (cid:105)(cid:69)(cid:12)(cid:12)(cid:12) r (cid:48)(cid:48) = r (cid:48) = r , where ε and ε are necessarily slow, because of Φ ε ε .Recall that Φ = U ∇ U = −∇ U U , and it is clear that itcan only have two slow indices or vanish. Therefore ε needs to be fast and i (cid:104)(cid:104) S S int, (cid:105)(cid:105) = − iπν (cid:88) n =0 (cid:90) r i ,ε i D ( ∇ r (cid:48)(cid:48) − ∇ r (cid:48) ) (159) (cid:68) tr (cid:104)(cid:0) ¯ U ( r ) φ n ( r ) (cid:1) ⊥ ε ε f (cid:0) φ n ( r ) U ( r ) (cid:1) ⊥ ε f ε × Φ (cid:107) ε ε ( r )Π ε ε f ( r − r (cid:48) )Π ε ε f ( r (cid:48)(cid:48) − r ) (cid:105)(cid:69)(cid:12)(cid:12)(cid:12) r (cid:48)(cid:48) = r (cid:48) = r . Fig. 6(b) illustrates the structure of this term. One mayalready notice the structural similarity to Eq. (157); thesame observation also holds for the remaining contribu-tions to ∆ S D . This is why the further evaluation is post-poned until all four terms have been discussed. i (cid:104)(cid:104) S S int (cid:105)(cid:105) S contains two terms, S a and S b . In S b , all fre-quencies of the P matrices are forced to be slow dueto the presence of Φ and this does not lead to an RG-contribution to the diffusion coefficient. The relevant7contribution comes from a combination of S a and S int, : i (cid:104)(cid:104) S a S int, (cid:105)(cid:105) = 2 iπνD (cid:88) n =0 (cid:90) r i ,ε i (160) (cid:68) tr (cid:104)(cid:0) ¯ U ( r ) φ n ( r ) (cid:1) ⊥ ε ε f (cid:0) φ n ( r ) U ( r ) (cid:1) ⊥ ε f ε × (Φ( r )ΛΦ( r )Λ) (cid:107) ε ε Π ε ε f ( r − r )Π ε ε f ( r − r ) (cid:105)(cid:69) . For an illustration of this contribution see Fig. 6(c). Theexpression will be evaluated further together with the other contributions to ∆ S D . − (cid:10)(cid:10) S S int (cid:11)(cid:11) Similarly to the previously discussed terms, the dom-inant contribution comes from S int, . The contractionscan be performed in several ways, as indicated below − (cid:10)(cid:10) S S int, (cid:11)(cid:11) = i ( πν ) D (cid:88) n =0 (cid:16) P ↔ ∇ P ]Tr[Φ P ↔ ∇ P ] (cid:104) Tr[ φ n U σ P ¯ U ]Tr[ φ n U σ P ¯ U ] (cid:105) (161)+4Tr[Φ P ↔ ∇ P ]Tr[Φ P ↔ ∇ P ] (cid:104) Tr[ φ n U σ P ¯ U ]Tr[ φ n U σ P ¯ U ] (cid:105) +2Tr[Φ P ↔ ∇ P ]Tr[Φ P ↔ ∇ P ] (cid:104) Tr[ φ n U σ P ¯ U ]Tr[ φ n U σ P ¯ U ] (cid:105) (cid:17) . FIG. 9: Terms of the kind displayed in this figure arise whenevaluating the average − (cid:10)(cid:10) S S int, (cid:11)(cid:11) , compare the first andthird terms in Eq. (161). All frequencies involved are bound tobe small. This makes the contributions of this type irrelevant. For the first and last terms, all frequencies of P are fixedto be slow by the presence of two Φ-fields. This is whyterms of this kind are irrelevant for the RG; see Fig. 9 foran illustration. Out of the three terms, the relevant oneis the second which reduces to the contribution displayedin Fig. 6(d). It gives − (cid:10)(cid:10) S S int (cid:11)(cid:11) = (162)2 iπνD (cid:88) n =0 (cid:90) r i ,ε i ( ∇ r (cid:48)(cid:48) − ∇ r (cid:48) )( ∇ r (cid:48)(cid:48) − ∇ r (cid:48) ) × tr (cid:104) ( ¯ U ( r ) φ n ( r )) ⊥ ε ε f ( φ n ( r ) U ( r )) ⊥ ε f ε Π ε ε f ( r , r (cid:48)(cid:48) ) × Φ (cid:107) ε ε ( z )Π ε ε f ( r (cid:48) , r (cid:48)(cid:48) )Φ (cid:107) ε ε ( r )Π ε ε f ( r (cid:48) , r ) (cid:105) .
5. The correction ∆ D In the previous sections, expressions were obtained forthe four different contributions to the RG-corrections to S D . They can be found in Eqs. (157), (159), (160) and(162). As is obvious from these formulas, and also fromthe diagrammatic representation in Fig. 6, the followingblock is common to all four terms (cid:88) n =0 (cid:68)(cid:0) ¯ U ( r ) φ n ( r ) (cid:1) ⊥ ε ε f (cid:0) φ n ( r ) U ( r ) (cid:1) ⊥ ε f ε (cid:69) (163)= i ν (cid:90) ε ˆ V ijε ε f ( r − r ) × (cid:0) U ε ε ( r ) γ i u ε f (cid:1) ⊥ (cid:0) u ε f γ j U ε ε ( r ) (cid:1) ⊥ , where U = uU , ¯ U = ¯ U u and V = Γ d − d .The gradient expansion of U and ¯ U mentioned at thebeginning of the calculation is necessary for (cid:104)(cid:104) S int (cid:105)(cid:105) and i (cid:104)(cid:104) S S int (cid:105)(cid:105) only, since the expressions for i (cid:104)(cid:104) S S int (cid:105)(cid:105) and − (cid:104)(cid:104) S S int (cid:105)(cid:105) already contain two slow gradients (via Φ).Since ε f is fast and all other frequencies are slow, we canneglect, with the logarithmic accuracy, the slow frequen-cies ε i compared to ε f in the RG-integrals. Putting theseremarks into effect, one finds8∆ S D = (cid:104) S int (cid:105) + i (cid:104)(cid:104) S S int (cid:105)(cid:105) + i (cid:104)(cid:104) S S int (cid:105)(cid:105) − (cid:10)(cid:10) S S int (cid:11)(cid:11) (164)= − π (cid:90) r , p ,ε i tr (cid:104)(cid:110) − δ ε ε D ∇ r (cid:48) ∇ r (cid:48)(cid:48) − Φ (cid:107) ε ε ( r ) D ∇ r (cid:48)(cid:48) + Φ (cid:107) ε ε ( r ) D ∇ r (cid:48) + D (Φ ε ε ( r ) σ Φ ε ε ( r ) σ ) (cid:107) (cid:111)(cid:0) ¯ U ε ε ( r (cid:48) ) γ i u ε f (cid:1) ⊥ ˆ V ij − ε f ( p )Π ε f ( p ) (cid:0) u ε f γ j U ε ε ( r (cid:48)(cid:48) ) (cid:1) ⊥ (cid:105)(cid:12)(cid:12)(cid:12) r (cid:48)(cid:48) = r (cid:48) = r − d π (cid:90) r , p ,ε i tr (cid:104)(cid:110) δ ε ε D ∇ r (cid:48) ∇ r (cid:48)(cid:48) + Φ (cid:107) ε ε ( r ) D ∇ r (cid:48)(cid:48) − Φ (cid:107) ε ε ( r ) D ∇ r (cid:48) − D Φ (cid:107) ε ε ( r )Φ (cid:107) ε ε ( r ) (cid:111)(cid:0) ¯ U ε ε ( r (cid:48) ) γ i u ε f (cid:1) ⊥ ˆ V ij − ε f ( p ) D p Π ε f ( p ) (cid:0) u ε f γ j U ε ε ( r (cid:48)(cid:48) ) (cid:1) ⊥ (cid:105)(cid:12)(cid:12)(cid:12) r (cid:48)(cid:48) = r (cid:48) = r , where d is the dimension. An additional term, which doesnot contain any gradients, was left out here. Fortunately,such terms need to cancel once all corrections are consid-ered, as they would make the diffuson massive. (We havechecked this cancellation by a perturbative calculation.)In order to further evaluate this expression, we study thequantity˜ R mab ( p ) = (cid:90) ε f [ γ i u ε f ] a V ij − ε f ( p )Π mε f ( p )[ u ε f γ j ] b , (165)where a, b ∈ {(cid:107) , ⊥} , m = 2 , R mab ( p ) is a matrix inKeldysh space. For example, ˜ R m (cid:107)(cid:107) ( p ) is a diagonal matrixwith entries˜ R m (cid:107)(cid:107) ( p ) = (cid:90) ε f (cid:16) B ε f V Rε f + ( F ε f − B ε f ) V Aε f (cid:17) D mε f , (166)˜ R m (cid:107)(cid:107) ( p ) = (cid:90) ε f (cid:16) −B ε f V Aε f + ( B ε f − F ε f ) V Rε f (cid:17) D mε f . (167)For the RG calculation in 2 d , these integrals need to befound with logarithmic accuracy only. To this end notethat for the purpose of the RG analysis, we may set F ε f ≈ B ε f ≈ sign( ε f ) . (168)Due to the frequent occurrence of the sign-factor, let usintroduce the notation σ f = sign( ε f ) . (169)As a consequence˜ R m (cid:107)(cid:107) ( p ) ≈ (cid:90) ε f σ f D mε f V Rε f . (170)In a similar way one finds ˜ R m ⊥⊥ ( p ) = ˜ R m (cid:107)(cid:107) ( p ).Next, consider the off-diagonal matrix ˜ R m (cid:107)⊥ with en-tries˜ R m (cid:107)⊥ ( p ) = (cid:90) ε f (cid:16) F ε f B ε f V Rε f + V Aε f (1 − F ε f B ε f ) (cid:17) D mε f , ˜ R m (cid:107)⊥ ( p ) = (cid:90) ε f D nε f V Aε f . (171) Employing again the approximations of Eq. (168), we seethat both components reduce to integrals over a productof only retarded or only advanced functions. A similarstructure, obviously, holds for ˜ R m ⊥(cid:107) ( p ). In perturbativecalculations such terms vanish after integration in fre-quency a discussed earlier. In the RG procedure it is alittle bit more complicated. After integration in momen-tum, such terms are odd functions in frequency. Thus,although the integration over the fast frequency is per-formed within limited intervals, the sum over the positiveand negative frequency-intervals vanishes. It is useful inthis connection to compare the expressions for the diag-onal and off-diagonal matrices ˜ R . The diagonal ones, seeEq. (170), contain an additional factor σ f which makesthe ε f -integrals finite.Therefore, we need to keep only the (cid:107)(cid:107) and ⊥⊥ com-ponents. Coming back to ∆ S D as given in Eq. (164), oneobtains (cid:90) ε ,ε f (cid:0) ¯ U ε ε γ i u ε f (cid:1) ⊥ V ij − ε f Π nε f (cid:0) u ε γ j U ε f ε (cid:1) ⊥ = (cid:88) a,b = ⊥(cid:107) (cid:104) ¯ U a ˜ R ma (cid:48) b (cid:48) U b (cid:105) ε ε = (cid:16) ¯ U (cid:107) U (cid:107) + ¯ U ⊥ U ⊥ (cid:17) ε ε (cid:90) ε f σ f D mε f V Rε f , (172)where in the second line we denoted ⊥ (cid:48) = (cid:107) , (cid:107) (cid:48) = ⊥ for a (cid:48) and b (cid:48) , and used the obvious fact that the off-diagonal part of the product C = AB is given by C ⊥ = (cid:80) a = ⊥ , (cid:107) A a B a (cid:48) . As only the parallel componentof the total matrix considered in Eq. (172) enters thetrace in Eq. (164), we may effectively replace¯ U (cid:107) ( r (cid:48) ) U (cid:107) ( r (cid:48)(cid:48) ) + ¯ U ⊥ ( r (cid:48) ) U ⊥ ( r (cid:48)(cid:48) ) → ¯ U ( r (cid:48) ) U ( r (cid:48)(cid:48) ) . (173)It was used that the matrices u cancel. Let us furtherintroduce the notation R = (cid:90) p ˜ R (cid:107)(cid:107) ( p ) = (cid:90) p ,ε f σ f D ε f ( p ) V Rε f ( p ) , (174) R = (cid:90) p D p ˜ R (cid:107)(cid:107) ( p ) = (cid:90) p ,ε f σ f D p D ε f ( p ) V Rε f ( p ) . S D = − π R Tr (cid:104) − D ∇ U ∇ U − D ∇ U Φ (cid:107) U + DU Φ (cid:107) ∇ U + D [(Φ (cid:107) ) − (Φ ⊥ ) ] (cid:105) − πd R Tr (cid:104) D ∇ U ∇ U + D ∇ U Φ (cid:107) U − DU Φ (cid:107) ∇ U − D Φ (cid:107) Φ (cid:107) (cid:105) = 4 πd R Tr (cid:2) D (Φ ⊥ ) (cid:3) . (175)We see that the two-diffuson contributions cancel out (asit may be expected from general arguments ), and theremaining term comes from the three-diffuson term only.Using Tr[(Φ ⊥ ) ] = − Tr[( ∇ Q s ) ], one finds∆ S D = − πd (cid:90) tr[ D ( ∇ Q s ) ] × (176) (cid:90) p ,ε f σ f D p D ε f ( p ) (cid:2) Γ Rd ( p , ε f ) − R ,d ( p , ε f ) (cid:3) , This leads to the following result for the correction to thediffusion coefficient∆ D = 4 iDdν (cid:90) p ,ε f σ f D p D ε f ( p ) × (cid:2) Γ Rd ( p , ε f ) − R ,d ( p , ε f ) (cid:3) . (177)The factor d in the denominator results from an aver-aging over the direction of momentum. The logarithmicintegral will be evaluated in Sec. IV F below.Finally, the situation with the abandoned terms, whereall frequencies were forced to be slow, is worth comment-ing. See Fig. 9 as an example. Such terms have a hybridstructure, as they resemble at the same time the S D -termand the interaction term of the action: they contain gra-dients and mix frequencies. The remaining momentumintegrals are not logarithmic, and are determined by thelower cutoff λτ − of the RG-interval. Compared to theelectron-electron interaction terms, the discussed termscontain a small parameter ρDk / ( λτ − ), which is notcompensated by a large logarithm. Here, the small pa-rameter ρ is the only small parameter introduced for theRG analysis: ρ = 1(2 π ) νD . (178)It has the meaning of the sheet resistance measured indimensional units; note an extra factor π as compared tothe quantum resistance. D. Renormalization of z There are two corrections to S z ,∆ S z = i (cid:104)(cid:104) S ε S int (cid:105)(cid:105) + (cid:104) S int (cid:105) . (179) FIG. 10: Diagrammatic representation for i (cid:104)(cid:104) S ε S int, (cid:105)(cid:105) . Thisterm contributes to ∆ z . Below we present some details of the calculation. As itturns out, the dominant contributions arise from thoseterms for which S int is replaced by S int, . i (cid:104)(cid:104) S ε S int, (cid:105)(cid:105) After evaluating the relevant contractions in the P -matrices, one obtains the expression i (cid:104)(cid:104) S ε S int, (cid:105)(cid:105) = 2 πν (cid:88) n =0 (cid:90) r i ,ε i (180) (cid:10) tr (cid:2) ( U ( r ) φ n ( r ) U ( r ) σ ) ⊥ ε ε Π ε ε ( r − r ) × Π ε ε ( r − r )( U ( r ) φ n ( r ) U ( r )) ⊥ ε ε × ( U ( r ) zε s U ( r )) (cid:107) ε ε ] (cid:69) . The frequencies ε and ε are bound to be slow due topresence of ε s , while ε is fast. This observation directlyleads to the result i (cid:104)(cid:104) S ε S int, (cid:105)(cid:105) = − πi R Tr [ zε s Q s ] . (181)The corresponding diagram is displayed in Fig. 10. (cid:104) S int, (cid:105) This term is somewhat special, as it contains a contri-bution from the boundaries of the frequency integrationinterval. Starting point is formula (157), see also Fig. 11,where (unlike previously) r may directly be set equal to r , but an expansion in slow frequencies is performed. Inorder to see how it works, it is convenient to first performthe average in φ (cid:104) S int, (cid:105) ≈ − π Tr (cid:104) ( U ε ε γ i u ε f ) ⊥ ˆΠ ε f − ε ( p ) V ijε − ε f ( p ) × ( u ε f γ j U ε ε ) ⊥ (cid:3) . (182)An expansion in slow frequencies could be either in ε orin ε . When expanding in ε , the matrices U , ¯ U cancelfollowing the previous arguments. Therefore, one shouldconsider an expansion in ε and study˜ R ab ( p ; ε ) = (cid:90) ε f [ γ i u ε f ] a (cid:2) V ε − ε f ( p ) − V − ε f ( p ) (cid:3) ij × Π ε f ( p )[ u ε f γ j ] b , a, b ∈ {⊥ , (cid:107)} . (183)0 FIG. 11: The average (cid:104) S int, (cid:105) as relevant for the calculationof ∆ z . Expansion in the slow frequency is needed to be per-formed. Only the ⊥⊥ and (cid:107)(cid:107) components give a logarithmic con-tribution. Further, it should be noted that an expansionof the distribution function in ε is not necessary sincesuch terms would be exponentially suppressed in the RGregime. Defining R = 1 z (cid:90) ε f ,p σ f D ε f ( p ) ∂ ε f V Rε f ( p ) , (184)one obtains ˜ R (cid:107)(cid:107) ,ε ≈ − ˜ R ⊥⊥ ,ε ≈ − zε R σ , (185)and further on (cid:104) S int, (cid:105) ≈ − π R Tr [ zεQ s ] . (186)The integral R may be rearranged with the use of apartial integration in ε f : R = 12 πz (cid:90) p σ f D ( p ) V Rε f ( p ) (cid:12)(cid:12)(cid:12) bound − i R , (187)where the index bound indicates that expression shouldbe evaluated at the boundaries of the frequency integra-tion interval.
3. The correction ∆ z When combining the two contributions, Eqs. (181) and(186), a partial cancellation occurs and only the bound-ary terms remain. For the total correction to z one readsoff ∆ z = 12 πν (cid:90) p σ f D ε f ( p ) V Rε f ( p ) (cid:12)(cid:12)(cid:12) bound . (188)It is important to note that once the integrand is eval-uated at the two boundaries, i.e., the upper and lowerlimits of the frequency integral, the momentum integralis convergent and yields a logarithmic correction. E. Renormalization of the interaction amplitudes
The interaction term S Γ contains three interaction am-plitudes, Γ ( q ), Γ and Γ . The amplitude Γ differs by the spin structure from the other two and, therefore,corrections to either of these two classes are easily iden-tified. The amplitudes Γ ( q ) and Γ have the same spinstructure, but they differ in another aspect. Recall thatΓ ( q ) is the statically screened long-range Coulomb inter-action, while Γ is short-range as it is directly related tothe Fermi liquid amplitudes. A correction to Γ ( q ) couldarise only from diagrams, for which the Coulomb inter-action is not part of the logarithmic integration. Suchtype of diagrams can be generated with the help of S int, and closely resemble vertex corrections for a scalar ver-tex. Importantly, such corrections, although they arisefrom individual diagrams, eventually cancel, once all con-tributions are summed up. Indeed, it turns out that thecancellation occurs between certain pairs of diagrams.The calculation will, therefore, be organized in such away that these pair diagrams are treated together. Asalready indicated, the cancellation of the corrections toΓ ( q ) also reflects itself in the fact that the scalar tri-angular vertex γ ρ(cid:47) remains unrenormalized. This will bedemonstrated explicitly below in Secs. V A and V B. Incontrast, the correction to the amplitude Γ , which isshort-range in character, is finite.Generally, the RG-equations at the one-loop levelsum the series of logarithmic corrections of the kind( ρ ln 1 /T τ ) n , where ρ , the small parameter of the RG ex-pansion, has been introduced in Eq. (178). Correctionsto the interaction amplitude may contain a product ofseveral interaction amplitudes, with some of them beingdressed. Even on the level of the one-loop approximation,it is a priori not clear whether the number of diagramsthat needs to be considered in order to derive such a sys-tem of equations is finite. As has first been demonstratedby Finkel’stein in Ref. 7, it is fortunately the case and theproduct of at most four (dressed and undressed) interac-tion amplitudes is involved in the calculation. The mainguiding rule here is that the order of the RG-equationis determined by the number of momentum integrations:each integration generates the small parameter ρ . Therecannot be too many dressed amplitudes, because other-wise it is impossible to arrange them without an addi-tional momentum integration.In order to structure the calculation, we will presentthe correction to S Γ as the sum of 6 individual contri-butions. Apart from the first one, all of them consistof pairs of diagrams. These pairs arise as a result of adifferent choice of the fast frequency for the logarithmicintegration. The above mentioned cancellation of cor-rections to Γ ( q ) takes place between the two partnerdiagrams forming a pair [whenever such correction ap-pears]. For the corrections to Γ and Γ the cancellationis not complete, and these corrections remain finite. Wewrite ∆ S Γ = (cid:88) i =0 (∆ S Γ ) i , (189)1where (∆ S Γ ) = (cid:104) S int, (cid:105) (190)(∆ S Γ ) = i (cid:10)(cid:10) S int, (cid:11)(cid:11) (191)(∆ S Γ ) = i (cid:104)(cid:104) S int, S int, (cid:105)(cid:105) (192)(∆ S Γ ) = − (cid:10)(cid:10) S int, S int, (cid:11)(cid:11) (193)(∆ S Γ ) = − (cid:10)(cid:10) S int, S int, (cid:11)(cid:11) (194)(∆ S Γ ) = − i (cid:10)(cid:10) S int, S int, (cid:11)(cid:11) . (195)We will present details of the calculation of the first twocontributions, the other ones can be considered in a sim-ilar way, but we will only state the results and displaythe corresponding diagrams. As already mentioned, thecalculation of vertex corrections presented in Secs. V Aand V B have a close similarity to some of the diagramsthat are important here. The interested reader may findadditional information there, in particular about the can-celations for pair diagrams. (cid:104) S int, (cid:105) This term has been considered before and we may useformula (156) for (cid:104) S int, (cid:105) as our starting point. In thepresent context, we consider the case that the two fre-quency arguments ε and ε are slow, while the mo-mentum entering Π is fast, see Fig. 12. Therefore wecan approximate it by just Π( p , p that are of interest, the frequency de-pendence may be neglected. In this approximation.Π( p , ≈ D ( p , ≡ D ( p ) becomes proportional to theunit matrix in Keldysh space and additionally the sum-mation in ε and ε may be performed. As no expansionin slow momenta is required, we may put r → r for thearguments of the slow modes: (cid:104) S int, (cid:105) = iπν (cid:88) n =0 (cid:90) rr (cid:48) (cid:68) tr[ (cid:0) ¯ U ( r ) φ n ( r ) U ( r ) (cid:1) ⊥ (196) × (cid:0) ¯ U ( r ) φ n ( r (cid:48) ) U ( r (cid:48) ) (cid:1) ⊥ ] (cid:69) D ( r − r (cid:48) )= iπν (cid:88) n =0 (cid:90) r , r (cid:48) tr[ φ n ( r ) φ n ( r (cid:48) ) − Q s ( r ) φ n ( r ) Q s ( r ) φ n ( r (cid:48) )] D ( r − r (cid:48) ) . The first term in the last equation is just a constant andcan be dropped. After performing the average in φ one FIG. 12: (cid:104) S int, (cid:105) as relevant for the renormalization of theinteraction amplitudes. In this case, all frequencies involvedare slow, the logarithmic correction arises from an integrationover fast momenta. obtains (cid:104) S int, (cid:105) = π (cid:90) ε i Tr (cid:104) Q s,αβ ; ε ε γ i Q s,βα ; ε ,ε γ j (cid:105) (197) × (cid:90) r ˆΓ ij ( p ) D ( p ) δ ε − ε ,ε − ε − π (cid:90) tr (cid:104) Q s,αα ; ε ε γ i Q s,ββ ; ε ,ε γ j (cid:105) × (cid:90) r ˆΓ ij D ( p ) δ ε − ε ,ε − ε . As the frequency arguments of Γ, Γ are slow (whilethe momenta are fast), no dressing of the interaction linewas included, and the static amplitudes can be used. Aswas already noted before, in such a case ˆΓ and ˆΓ are off-diagonal matrices in Keldysh space and take the simpleform ˆΓ = (cid:18) (cid:19) , ˆΓ = (cid:18) Γ (cid:19) . (198)We can use the relation (recall that γ is the unit matrix)Tr[ Q Q γ ] + Tr[ Q γ Q ]= Tr[ Q ]Tr[ γ Q ] + Tr[ γ Q ]Tr[ Q ] , (199)where all appearing Q -matrices have fixed frequency ar-guments and spin indices. The result is (cid:104) S int, (cid:105) = π (cid:90) r ,ε i tr (cid:104) γ i Q s,αβ ; ε ε ( r ) (cid:105) γ ij × (200)tr (cid:104) γ j Q s,βα ; ε ,ε ( r ) (cid:105) δ ε − ε ,ε − ε (cid:90) p Γ( p ) D ( p ) − π (cid:90) r ,ε i tr (cid:104) γ i Q s,αα ; ε ε ( r ) (cid:105) γ ij ×× tr (cid:104) γ j Q s,ββ ; ε ε ( r ) (cid:105) δ ε − ε ,ε − ε (cid:90) p Γ D ( p ) . Comparing to the original interaction term, Eq. (56), onefinds that the structure of the Γ and Γ terms are repro-duced, leading to the resulting corrections from (∆ S Γ ) :(∆Γ ) = 1 πν Γ (cid:90) p D ( p ) , (201)(∆Γ ) = 1 πν (cid:90) p Γ( p ) D ( p ) . (202)2
2. Pairs of diagrams
As we have already mentioned, pairs of diagrams ariseas a result of a different choice of the fast frequency ε f forthe logarithmic integration. These pairs of diagrams aredisplayed as two columns in Fig. 13. As an illustration,we discuss in detail one pair of diagrams, labeled as 1(a)and 1(b). This pair gives rise to the correction (∆ S Γ ) ,and originates from i (cid:10)(cid:10) S int, (cid:11)(cid:11) = (203) − i ( πν ) (cid:68)(cid:68)(cid:10) Tr[ φ U σ P U ]Tr[ φ U σ P U ] (cid:11) φ ×(cid:104) Tr[ φ (cid:48) U σ P U ]Tr[ φ (cid:48) U σ P U ] (cid:105) φ (cid:48) (cid:69)(cid:69) . Note that φ (cid:48) has the same correlation as φ (As it willbecome clear later, only the φ -contractions have to beconsidered in all diagrams presented in Fig. 13. Other-wise, the contributions are canceled out within each ofthe pairs.)We perform the contractions, and introduce a symme-try factor two: i (cid:10)(cid:10) S int, (cid:11)(cid:11) = − i ( πν ) (cid:90) r i ,ε i (204) (cid:104) tr[( U φ U ) ⊥ ε ε ( r )Π ε ε ( r − r )( U φ (cid:48) U ) ⊥ ε ε ( r )]tr[( U φ (cid:48) U ) ⊥ ε ε ( r )Π ε ε ( r − r )( U φ U ) ⊥ ε ε ( r )] (cid:105) φ φ (cid:48) . The different ways in which the occurring frequencies canbe chosen as being fast are as follows:(a) ( ε , ε ) fast or equivalently ( ε , ε ) fast → (∆Γ ) (b) ( ε , ε ) fast or equivalently ( ε , ε ) fast → (∆Γ ) .These two possibilities lead to the diagrams displayed inFigs. 13 as 1(a) and 1(b), respectively. For case (a), acorrection to Γ arises; for case (b) a correction to Γ .(a) Let ( ε , ε ) be fast: We account for the equivalentchoices by a factor of two, neglect slow frequencies inthe diffusion propagators, and take the slow U modes atcoinciding points. In this way one obtains∆ S = (205) − i ( πν ) (cid:90) (cid:68) tr[( φ (cid:48) ( r ) Q ( r ) φ ( r ) σ ) ε ε Π ε ( r − r ) − ( φ (cid:48) ( r ) φ ( r )) ε ε Π ε ( r − r )] × tr[( φ (cid:48) ( r ) Q ( r ) φ ( r ) σ ) ε ε Π ε ( r − r ) − ( φ (cid:48) ( r ) φ ( r )) ε ε Π ε ( r − r )] (cid:69) . The term of interest is the one containing two Q ’s andfor this term one obtains∆ S = π i (cid:90) r , p ,ε i Γ ij d ( p , ε f )Γ kl d ( p , − ε f ) δ ε − ε ,ε − ε × tr (cid:104) ( γ k Λ ε f Π ε f ( p ) γ i ) Q αα ; ε ε ( r ) (cid:105) × tr (cid:104) ( γ l Λ − ε f Π − ε f ( p ) γ j ) Q y,ββ ; ε ε ( r ) (cid:105) , (206) a ) 1( b )2( a ) 2( b )3( a ) 3( b )4( a ) 4( b )5( a ) 5( b ) FIG. 13: The pairs of diagrams related to (∆ S Γ ) i , i = 1 − a ) give rise to the corrections (∆Γ ) i anddiagrams labeled as ( b ) to the corrections (∆Γ ) i . Only thosecontributions remain, for which all interaction amplitudes areof the Γ -type. All other contributions, which contain theamplitudes Γ or Γ at least once, cancel between the twodiagrams forming a pair. An important consequence is thatthe amplitude Γ remains unrenormalized. where we remind that Λ ε = u ε σ u ε and we defined Π ε = u ε Π ε u ε . After a somewhat tedious but straightforwardcalculation one may show that the following expressionemerges∆ S = − iπ (cid:90) p ,ε f σ f [ D Rε f ( p )Γ R d ( p , ε f )] (207) × (cid:90) r ,ε i tr[ γ Q αα ; ε ε ( r )]tr[ γ Q ββ ; ε ε ( r )] δ ε − ε ,ε − ε . We see that the typical structure of the Γ -type interac-tion term is reproduced.(b) Now, let ( ε , ε ) be fast: In a similar way we findthat we should evaluate the following expression∆ S = (208) − i ( πν ) (cid:90) (cid:104) tr[( φ (cid:48) ( r ) Q ( r ) φ ( r ) σ ) ε ε Π ε ( r − r )] × tr[( φ ( r ) Q ( r ) φ (cid:48) ( r ) σ ) ε ε Π ε ( r − r )] (cid:105) φ φ (cid:48) . After performing the averaging with respect to φ and φ (cid:48) S = π i (cid:90) r , p ,ε i Γ il ,d ( p , ε f )Γ jk ,d ( p , − ε f ) (209) × tr[( γ j Λ ε f Π ε f γ i ) Q ε ε ; αβ ] × tr[( γ l Λ ε f Π ε f γ k ) Q ε ε ; βα ] δ ε − ε ,ε − ε . The origin of the additional factor 2 compared to formula(206) is the spin degree of freedom. Further evaluationgives∆ S = iπ (cid:90) p ,ε f σ f [ D Rε f ( p )Γ R d ( p , ε f )] (210) (cid:90) r ,ε i tr[ γ Q αβ ; ε ε ( r )]tr[ γ Q βα ; ε ε ( r )] δ ε − ε ,ε − ε . Here, the structure of the Γ -type interaction term isreproduced.The result for the corrections to Γ and Γ from thefirst pair of diagrams can easily be found by comparingthe obtained results to S Γ ,(∆Γ ) = iν (cid:90) p ,ε f σ f [ D Rε f ( p )Γ R d ( p , ε f )] (211)(∆Γ ) = 2 iν (cid:90) p ,ε f σ f [ D Rε f ( p )Γ R d ( p , ε f )] . (212)Now let us clarify the cancellation within each pairwhen a contraction is not of the φ -type. If one or bothof the H-S fields φ and φ (cid:48) are replaced by φ ( φ (cid:48) ) or φ ( φ (cid:48) ), then the overall spin structure of both typesof terms corresponding to diagrams (a) and (b) coincideas well as their spin factors. However, the relative signapparent from formulas (207) and (210) remains, andthus those terms cancel. Therefore, the only contributionthat remains is the one shown above with two amplitudesΓ .The remaining pairs of diagrams can be treated in asimilar way. The diagrams are displayed in Fig. 13. Eachpair is formed by the two diagrams labeled as (a) and (b).The results are(∆Γ ) = − iν Γ (cid:90) ε f σ f Γ ,d ( p , ε f ) D ε f ( p ) (213)(∆Γ ) = 2 ν Γ (cid:90) | ε f | Γ ,d ( p , ε f ) D ε f ( p ) (214)(∆Γ ) = − ν Γ (cid:90) | ε f | Γ ,d ( p , ε f ) D ε f ( p ) (215)(∆Γ ) = − iν Γ (cid:90) | ε f | ε f Γ ,d ( p , ε f ) D ε f ( p ) . (216)The previously explained relation holds for all five pairsof diagrams:(∆Γ ) i = 2(∆Γ ) i , i = 1 − . (217)Note that the amplitudes which appear in connectionwith the external legs of the diagrams are not dressed.As encountered already for the first pair of diagrams, acancellation takes place if at least one of the amplitudesΓ is replaced by Γ or Γ . F. Logarithmic integrals
Here, we present a list of logarithmic integrals thatappear as a result of the RG transformations. As shownabove, the NL σ M preserves its original form during thecourse of this procedure. This implies that the obtainedcorrections can be rewritten in the form of RG equationsfor the flowing (i.e., scale-dependent) parameters of themodel. As we have already mentioned, the only smallparameter needed for the RG analysis is ρ , which has themeaning of the sheet resistance determined at a givenscale and measured in dimensional units. ρ = π ) νD We concentrate on the long-range Coulomb interac-tion. In this limit, the effective interaction in the singletchannel is controlled by the inverse of the polarizationoperator. Even despite the screening, the resulting cor-rection to D differs substantially from the case of theshort-range interaction due the frequency dependence ofthe polarization operator, as given by Eq. (108), whichcannot be ignored. One has to start with Eq. (177), andto write inside the integral i (cid:90) p ,ε f σ f D p D ε f ( p ) (cid:2) Γ Rd ( p , ε f ) − R ,d ( p , ε f ) (cid:3) (218)all dressed amplitudes in the explicit form: D ε f ( p ) (cid:2) Γ Rd ( p , ε f ) − R ,d ( p , ε f ) (cid:3) =Γ ˜ D D D + Γ DD D − D D . (219)For brevity, we omitted the arguments p , ε f in the secondline. At a given ε f , the integral over p is convergentfor each of the three terms, both in the limits of largeand small momenta. One can, therefore, safely performthe integration over p ; the result is real and inverselyproportional to ε f . The last fact is clear, if one takesa look at the dimension of the integrands. Next, owingto σ f , the remaining integral over ε f is twice the integralover the positive frequencies only. To present the integralin a form suitable for the RG-treatment, it remains tointegrate within the energy shell λ Λ τ < ε f < Λ τ , where λ < τ is the current scale in the RG-procedure.The upper cutoff of the scaling process is Λ τ ∼ /τ ;the lower one is discussed below. We will present allcorrections as proportional to (cid:90) Λ τ λ Λ τ dε f ε f = ln λ − . (220)Performing the integrations described above, one getsthe result∆ ρρ = (cid:104)
11 + F ρ f ( z, z ) + Γ f ( z, z , z ) − f ( z, z, z ) (cid:105) ln λ − , (221)4where f ( a, b ) = 1 a − b ln ab , (222) f ( a, b, c ) = 2 bb − c f ( a, b ) − cb − c f ( a, c ) , (223)together with the definition f ( a, a ) = 1 /a . The termsin Eq. (221) arise from the Γ , Γ and Γ -contributionsas given in the second line of the expression (219). Ob-viously, for a short-range interaction, the Γ -term shouldbe excluded. z The most natural way to get the RG-equation for z isto rewrite Eq. (188) as follows∆ z = 12 πν (cid:90) p σ f D ε f ( p ) V Rε f ( p ) (cid:12)(cid:12)(cid:12) Λ τ λ Λ τ . (224)The integral in p becomes convergent once the upper andlower limits are considered together. Then, the straight-forward integration yields∆ z = ρ (cid:34) − F ρ − Γ + 2Γ (cid:35) ln λ − . (225)In the case of a short-range interaction, the first termshould be abandoned.Note that another way to perform the RG-procedureis to introduce a momentum cutoff besides the one infrequency. Γ and Γ After uncovering the dressed amplitudes, and perform-ing the necessary integrations, one gets:(∆Γ ) = Γ ρ ln λ − (226)and(∆Γ ) = − Γ z ρ ln λ − (227)(∆Γ ) = 2Γ f ( z , z ) ρ ln λ − (∆Γ ) = − Γ z f ( z, z, z ) ρ ln λ − (∆Γ ) = Γ z f ( z , z , z ) ρ ln λ − (∆Γ ) = Γ ( z − z ) (cid:18) z + 1 z − f ( z, z ) (cid:19) ρ ln λ − . Remarkably, the sum of the five terms i = 1 − (cid:88) (∆Γ ) i = Γ z ρ ln λ − . (228) For Γ we have(∆Γ ) = (cid:20) F ρ ) + Γ (cid:21) ρ ln λ − , (229)and (∆Γ ) i = 2(∆Γ ) i for i = 1 − ) and (∆Γ ) can be summarizedas follows:(∆Γ ) = (cid:20) Γ + Γ z (cid:21) ρ ln λ − , (230)(∆Γ ) = (cid:20) F ρ ) + Γ + 2 Γ z (cid:21) ρ ln λ − . (231) z = z − + Γ It follows from the above results that z , which de-termines the dynamics in the ρ -channel (e.g., in the po-larization operator), remains unchanged during the RGtransformations. Indeed, by comparing Eq. (225) withEqs. (230) and (231), one immediately observes that∆ z = 0 . (232)The initial values stated in Eq. (66) in Sec. II, there-fore allow to determine the value of this unrenormalizedcombination: z = z − + Γ = 11 + F ρ . (233)This Ward identity is important for finding the cor-rect form of ¯ χ Rnn ( q , ω ), and also for establishing theuniversal form of the RG equations in the case of thescreened long range Coulomb interaction. Indeed, in viewof Eq. (55), where the interaction amplitude in the ρ -channel for small momenta has been defined, ˜Γ ρ ( q →
0) = F ρ + 2Γ − Γ , one can read the obtained relation(233) as ˜Γ ρ ( q →
0) = z. (234)Thus, the renormalized interaction amplitude and the pa-rameter describing the renormalization of the frequencyterm in the case of the screened long-range interactioncoincide, and do not depend on the nonuniversal Fermiliquid amplitudes. This is the reason why the RG equa-tions in this case acquire a universal form.
5. Final form of the RG equations
We will write now the RG equations for the case of thescreened Coulomb interaction. To make the equationsuniversal, we exclude the combination F ρ ) + Γ us-ing identity (233) discussed above. As a result, on canrewrite Eq. (221) in the form∆ ρρ = (cid:20) − (cid:18) z + Γ Γ ln z + Γ z − (cid:19)(cid:21) ln λ − , (235)5where the two terms in the square brackets represent con-tributions of the ρ (singlet) and σ (triplet) channels, re-spectively. Note that the factor 3 is typical for the tripletchannel, and that these two contributions have oppositesigns. With the help of Eq. (233), the equation describingthe renormalization of Γ acquires the following form(∆Γ ) = (cid:20) z z (cid:21) ρ ln λ − . (236)Finally, the equation for ∆ z simplifies, and takes a formin which the contribution of the two channels becomesimmediately recognizable∆ z = 12 [ − z + 3Γ ] ρ ln λ − . (237)The corresponding RG equations can be obtained by tak-ing derivatives with respect to ln λ − , with all the coef-ficients understood as flowing parameters. These three( instead of four ) equations constitute a complete set ofRG-equations describing the disordered electron liquid inthe presence of the long-range Coulomb interaction. Thelong range character of the Coulomb interaction, i.e., theinfinite amplitude in the limit q →
0, leads to a universalform of the RG-equations. Moreover, one may introducea new variable w = Γ /z which allows to decouple theequations for ρ and the interaction in the σ -channel (rep-resented now by w ) from the equation for z :1 ρ d log ρd ln λ − = 4 − w w ln(1 + w ) (238)1 ρ dw d ln λ − = (1 + w ) , (239)and 1 ρ dzd ln λ − = z (3 w − . (240)Although these equations were derived in the one-loop(first order in ρ ) approximation, the observed decou-pling of the equation for z from the equations describ-ing the other two RG charges (as well as the possibilityof presenting the equations in terms of the ratio Γ /z )reflects the general structure of the NL σ M. Thisfact is important for the analysis of the Metal-Insulatortransition. The fixed point existing in the phase plane ρ − w determines the equation for z which, in turn, con-trols the critical behavior (as a function of temperature)at the metal-insulator transition.
6. Lowest cutoff
Finally, let us comment on the lowest cutoff for theRG-procedure. In the replica NL σ M the lower cutoff ap-pears from the discreteness of the Matsubara frequencies,which are used to describe electron interactions at finite temperatures. In the Keldysh technique it happens dif-ferently. The matrix ˆ Q = ˆ u ◦ ˆ U ◦ ˆ σ ◦ ˆ U ◦ ˆ u , which is themain object of study in theory of interacting electrons,contains a superposition of two kinds of rotations. Ma-trices U, U describe fluctuations that correspond to dif-fusons, while matrices u establish the connection of thediffusion modes with temperature. The latter matriceslimit rotations of U at energies smaller than T , and this isthe way how the low-energy cutoff enters the RG-scheme.Technically, the cutoff enters due to the smoothening ofthe function σ f at ε f ∼ T . The whole RG-procedure canbe reformulated as a process of gradual sharpening of σ f ,starting from 1 /τ and up to T . V. CORRELATION FUNCTIONS ANDCONDUCTIVITY
We now combine the analysis presented in Secs. III andIV; the RG-equations derived above will be connectedwith the observable quantities, such as the correlationfunctions and electric conductivity.As it will be shown below, there is an important differ-ence between the static part of the density-density cor-relation function ¯ χ st,Rnn , and the static part of the spin-density spin-density correlation function χ st,Rs i s i . Namely,¯ χ st,Rnn = − νγ σ • = − ν/ (1 + F ρ ) remains unrenormal-ized, whereas χ st,Rs i s i becomes scale-dependent. The rea-son for the particular behavior of χ st,Rnn lies in the wellknown Ward identity: χ st,Rnn = − ∂n/∂µ . It has beenargued that the cancellation of corrections to ∂n/∂µ isrelated to the fact that it is the much smaller quantity1 /τ and not µ that determines the ultraviolet cut-off forthe logarithmic singularities originating from the diffu-sive regime. As a consequence, the dependence of thedensity n on the chemical potential µ cannot be modifiedby the discussed logarithmic corrections and, therefore, ∂n/∂µ remains unchanged. (We shall demonstrate belowthat, technically, it is due to the cancellation of the loga-rithmic corrections.) No protection of this type exists forthe spin susceptibility that is determined by the staticpart of the spin-density spin-density correlation functionand, indeed, the spin-susceptibility is renormalized. Fi-nally, we use the density-density correlation function toobtain the Einstein relation for interacting electrons, andto relate the electric conductivity to the scaling parame-ter ρ . A. Corrections to γ ρ/σ • and the spin susceptibility The static parts of the correlation functions are de-termined by γ ρ/σ • ; compare the discussion in Sec. III, inparticular Eq. (94). We now show how these quantitiesare modified by the RG-corrections. One needs to find∆ S ϕϕ = − (cid:10)(cid:10) S ϕ, S int, (cid:11)(cid:11) − i (cid:10)(cid:10) S ϕ, S int, (cid:11)(cid:11) . (241)6 a ) 4( b )5( a ) 5( b ) FIG. 14: These diagrams give rise to the corrections to γ ρ/σ • .They are organized into two pairs, in close analogy to thecorresponding diagrams in Fig. 13 with the same labels. The corresponding diagrams are closely related to thosepresented in Fig. 13, in particular to contributions 4 and5 for the renormalization of the interaction amplitudes.In a similar way, when calculating the corrections to γ ρ/σ • ,one also deals with pairs of diagrams, see Fig. 14.We present some details for the first term in Eq. (241).As mentioned, the correction consists of two parts, − (cid:10)(cid:10) S ϕ, S int, (cid:11)(cid:11) = A + B, (242)corresponding to the diagrams labeled as 4(a) and 4(b)in Fig. 14, respectively. A and B take the form A = − i πν ) (cid:88) n =0 (243) (cid:18) (cid:104)(cid:104) Tr[ ϑσ P P ]Tr[ φ n σ P ]tr[ φ n σ P ]tr[ ϑσ P P ] (cid:105)(cid:105) φ + (cid:104)(cid:104) Tr[ ϑσ P P ]Tr[ φ n σ P ]tr[ φ n σ P ]tr[ ϑσ P P ] (cid:105)(cid:105) φ (cid:19) = − (cid:90) x (cid:126)ϑ Tαβ ( x ) γ (cid:126)ϑ βα ( x ) (cid:90) p ,ε f | ε f | Γ Rd ( p , ε f ) D ε f ( p )+2 (cid:90) x (cid:126)ϑ Tαα ( x ) γ (cid:126)ϑ ββ ( x ) (cid:90) p ,ε f | ε f | Γ R ,d ( p , ε f ) D ε f ( p ) , and B = − i πν ) (cid:88) n =0 (244) (cid:104)(cid:104) Tr[ ϑσ P P ]Tr[ φ n σ P ]tr[ φ n σ P ]tr[ ϑσ P P ] (cid:105)(cid:105) φ = 2 (cid:90) x (cid:126)ϑ Tαβ ( x ) γ (cid:126)ϑ βα ( x ) × (cid:90) p ,ε f | ε f | (cid:2) Γ Rd ( p , ε f ) − R ,d ( p , ε f ) (cid:3) D ε f ( p ) . In these expressions, we abbreviated ϑ = γ ρ(cid:47) ϕ + γ σ(cid:47) ϕσ . Summing contributions A and B , one gets − (cid:10)(cid:10) S ϕ, S int, (cid:11)(cid:11) (245)= − γ σ(cid:47) ) (cid:90) x (cid:126) ϕ T ( x ) γ (cid:126) ϕ ( x ) (cid:90) p ,ε f | ε f | Γ R ,d ( p , ε f ) D ε f . Two remarks are in order here. First, we see that theamplitude Γ disappears from the final result due to a can-cellation between A and B . Second, a logarithmic cor-rection exists only for the triplet component, the singletpart remains untouched. These two observations carryover to the calculation of the other contribution to ∆ S ϕϕ ,which is also organized into a pair of diagrams; see di-agrams 5(a) and 5(b) in Fig. 14. The total result canconveniently be written in the form∆ S ϕϕ = 4 ν ( γ σ(cid:47) ) Γ (cid:88) i =4 , (∆Γ ) i (cid:90) x (cid:126) ϕ T ( x ) γ (cid:126) ϕ ( x ) . (246)Comparing with S ϕϕ , and using the relations for (∆Γ) i stated in Eq. (226), one finds∆ γ ρ • = 0 , ∆ γ σ • = 2Γ zz ( γ σ(cid:47) ) g ln λ − . (247)The correction to γ σ • depends on the vertices γ σ(cid:47) . InSec. V B, we will show that the RG-equations generalizethe Fermi-liquid relations for γ σ(cid:47) and γ σ • as follows γ σ(cid:47) = γ σ • = z + Γ . (248)As a result, we observe that the renormalization of theelectron-electron interaction in the triplet channel leadsto the scale-dependent spin susceptibility χ σ = ( z + Γ ) χ σfree , (249)where χ σfree = 1 / g L µ B ) ν is the unrenormalized spinsusceptibility of the free electron gas. B. Vertex corrections
As we have seen in Eq. (247), the knowledge of the tri-angular vertices γ σ(cid:47) is crucial for finding the static vertex γ σ • . In addition, γ ρ/σ(cid:47) also determines the dynamical cor-relation functions, see Eqs. (96) and (97). We will discussthe renormalization of the vertices in this section.First of all, it is important to stress that γ ρ/σ(cid:47) has beenchosen as the common charge for two vertices: the oneassociated with the quantum source and the one associ-ated with the classical one. It is crucial for the overallstructure of the theory that both of them are renormal-ized in the same way. As will be seen below, it is indeedthe case.In order to find the vertex corrections, one needs tofind corrections to the term S ϕQ defined in Eq. (62):∆ S ϕQ = i (cid:104)(cid:104) S ϕ, S int, (cid:105)(cid:105) − (cid:10)(cid:10) S ϕ, S int, (cid:11)(cid:11) (250) − (cid:104)(cid:104) S ϕ, S int, S int, (cid:105)(cid:105) − i (cid:10)(cid:10) S ϕ, S int, S int, (cid:11)(cid:11) . S ϕ, and S int, ,the calculation is very similar to the one performed forthe renormalization of the interaction amplitudes, com-pare the corrections (∆ S Γ ) − in Eqs. (192)-(195).Again, the diagrams come in pairs, see Fig. 15, whichis structured in analogy to Fig. 13. Here, we merely statethe result, which can be expressed in terms of the correc-tions to the interaction amplitudes stated in Eq. (226):∆ S ϕQ (251)= 2 πν Tr[ γ σ(cid:47) ˆ ϕσ Q ] (cid:34) (cid:88) i =2 (∆Γ ) i Γ + (cid:88) i =4 (∆Γ ) i Γ (cid:35) . It turns out that the final result is very simple∆ γ ρ(cid:47) = 0 , ∆ γ σ(cid:47) = 2Γ z γ σ(cid:47) g ln λ − . (252)It is instructive to compare ∆ γ σ(cid:47) with the correction to z , ∆ z = ∆ z + ∆Γ = 2Γ z z g ln λ − . (253)Since initially z = γ σ(cid:47) = 1 / (1 + F σ ), as it follows fromEqs. (64) and (66), we may conclude that the relation γ σ(cid:47) = z (254)holds also for the renormalized quantities. With this in-formation at hand, one may return to the calculation of∆ γ σ • , and finds∆ γ σ • = 2Γ z z g ln λ − = ∆ z . (255)Since initially γ σ • = 1+Γ , one obtains that γ σ(cid:47) = γ σ • = z as it was already stated in Eq. (107). Besides, the abovecalculations confirm that ∆ γ ρ(cid:47) = ∆ γ ρ • = 0.Importantly, these results imply that the relations(105) are indeed fulfilled. These relations make sure thatthe conservation laws hold at any stage of the renormal-ization procedure. Let us note that for the triplet chan-nel not only the ratio ( γ σ (cid:52) ) /γ σ • equals z but, besides,each of the quantities γ σ (cid:52) and γ σ • separately. For the sin-glet channel, the statement ∆ γ ρ(cid:47) = ∆ γ ρ • = 0 should besupplemented with the observation that ∆ z = 0. Thisis sufficient for the relation z = ( γ ρ(cid:47) ) /γ ρ • to hold un-changed. C. Electric conductivity
Combination of the continuity equation and the Kuboformula allows to extract the electric conductivity fromthe retarded density-density correlation function as fol-lows: σ = − e lim ω → lim q → (cid:20) ω q Im ¯ χ Rnn ( q , ω ) (cid:21) . (256) a ) 2( b )3( a ) 3( b )4( a ) 4( b )5( a ) 5( b ) FIG. 15: The four pairs of diagrams relevant for the vertexcorrections.
Formula (108) can be conveniently written as¯ χ Rnn ( q , ω ) = − ∂n∂µ D F L q D F L q − iω , (257)where D F L = D (1 + F ρ ) . (258)As a result, Eq. (256) leads to the Einstein relation σ = e ∂n∂µ D F L = 2 νe D. (259)One can see that the Fermi liquid correction 1+ F ρ can-cels between D F L and ∂n/∂µ = 2 ν/ (1 + F ρ ), so that therenormalized diffusion coefficient D in the NL σ M yieldsdirectly the electric conductivity with minimal dimen-sional coefficients. VI. CONCLUSION
We, thus, re-derived using the Keldysh technique themain results of the RG theory of the disordered electronliquid.
Besides the set of the RG equations, the dis-cussed items include: (i) the derivation of the Einsteinrelation which allows to connect the electric conductiv-ity to the scale-dependent diffusion coefficient D in theNL σ M, (ii) the expression for the renormalized spin sus-ceptibility, (iii) a number of relations between the vertices8and the interaction parameters, which in essence are theWard identities. For understanding the overall structureof the Keldysh NL σ M, it was crucial to observe that thetwo vertices, the one associated with the quantum sourceand the one associated with the classical one, are bothrenormalized in the same way.The validity of the theory has been confirmed exper-imentally by measuring resistance along with in-planemagnetoresistance in Si-MOSFETs at various tempera-tures and densities.
We concentrated here mainly on the peculiarities in-duced by the matrix structure of the NL σ M in theKeldysh technique. We conclude, that apart from differ-ences related to working with Keldysh matrices insteadof replicas, the RG-procedure in both schemes are rathersimilar. In subsequent papers we apply the developed technique for the calculation of the heat density-heat den-sity correlation function, which allows us to analyze heattransport at low temperatures.
Acknowledgments
G. S. would like to thank K. Takahashi and K. B. Efe-tov for work on related subjects. The authors gratefullyacknowledge the support by the Alexander von HumboldtFoundation. G. S. also acknowledges financial supportby the Albert Einstein Minerva Center for TheoreticalPhysics at the Weizmann Institute of Science. A. F.is supported by the National Science Foundation grantNSF-DMR-1006752. ∗ Electronic address: [email protected] B. L. Altshuler and A. G. Aronov,
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