Interaction corrections at intermediate temperatures: Longitudinal conductivity and kinetic equation G´bor Zala, B.N. Narozhny, and I.L. Aleiner a arXiv:cond-mat/0105406v2 14 Sep 2001 Department of Physics and Astronomy, SUNY at Stony Brook, Stony Brook, NY, 11794 (June 11, 2018) It is well known that electron-electron interaction in two dimensional disordered systems leads to logarithmically divergent Altshuler-Aronov corrections to conductivity at low temperatures (T τ ≪ 1; τ is the elastic mean-free time). This paper is devoted to the fate of such corrections at intermediate temperatures T τ > 1. We show that in this (ballistic) regime the temperature dependence of conduc∼ tivity is still governed by the same physical processes as the Altshuler-Aronov corrections - electron scattering by Friedel oscillations. However, in this regime the correction is linear in temperature; the value and even the sign of the slope depends on the strength of electron-electron interaction. (This sign change may be relevant for the “metal-insulator” transition observed recently.) We show that the slope is directly related to the renormalization of the spin susceptibility and grows as the system approaches the ferromagnetic Stoner instability. Also, we obtain the temperature dependence of the conductivity in the cross-over region between the diffusive and ballistic regimes. Finally, we derive the quantum kinetic equation, which describes electron transport for arbitrary value of T τ . PACS numbers: 72.10.-d, 71.30.+h, 71.10.Ay tion (1.2) predicts always metallic sign of the interaction correction. The absence of a rigorous calculation at intermediate temperatures, T τ /¯ ≃ 1, may have contributed to the h notion that those two limits are governed by different physical processes. In this paper we prove that notion erroneous: the results (1.1) and (1.2) are due to the same physical process, namely elastic scattering of electrons by the self-consistent potential created by all the other electrons. Therefore, these two different expressions are in fact the two limits of a single interaction correction. We calculate the correction within assumptions of the Fermi liquid theory (other limitations of our approach we discuss below) and present the cross-over function between the diffusive and ballistic limits. Moreover, we show that the existing theory for the ballistic limit (1.2) is incomplete. First, the results of Ref. 4–7 account only for Hartree-like interaction terms missing the exchange or Fock terms. Second, this theory essentially employs a perturbative expansion in terms of the interaction strength, which breaks down for stronger coupling. Both issues lead to the change in the theoretical prediction even on a qualitative level. The consequence of the first point is that the correction to conductivity (1.2) is always negative unlike the correction in the diffusive limit that changes sign depending on σ the value of F0 . This sign change is due to competition between the universal (and positive) Fock correction and the coupling-specific (and negative) Hartree contribution. If the Fock potential (or, to be more precise, singlet channel) is properly taken into account, then the sign of the correction in the ballistic limit is also not universal (being positive for weak interaction in contrast to Refs. 4–7). As follows from the second point, for the stronger interaction the Hartree correction should be modified to include higher order processes. For this case we show I. INTRODUCTION Temperature dependent corrections to conductivity due to electron-electron interactions has been a subject of theoretical1–7 and experimental8–10 studies for more than two decades. Recently the interest in the matter was renewed with appearance of new data11,12 showing a sign change in the temperature dependence of conductivity in two dimensions (2D). Theoretical discussions12 that followed emphasized the question of whether that data indicated a non-Fermi liquid behavior. However, the experiments were performed in a regime where the temperature T was of the same order of magnitude as the inverse scattering time τ −1 (obtained from the Drude conductivity), while pre-existing calculations were focused on the two limiting cases: the diffusive regime1–3 T τ ≪ 1, and the ballistic regime6,7 T τ ≫ 1. In the diffusive limit one finds1–3 for the logarithmically divergent correction to the diagonal conductivity δσ: δσ = − e2 ln 2π 2 ¯ h h ¯ Tτ 1+3 1− σ ln(1 + F0 ) σ F0 , (1.1) σ where F0 is the interaction constant in the triplet channel which depends on the interaction strength. It is clear, that the sign of this logarithmically divergent correction may be positive (metallic) or negative (insulating), deσ pending on the value of F0 13 . 6,7 The result for the ballistic region frequently cited in literature reads δσ = − e2 π¯ h Tτ h ¯ f (rs ), (1.2) where f (rs ) is a positive function of the gas parameter of the system, rs . In a sharp contrast to Eq. (1.1), equa1 (see Section III) that in fact it should be replaced by the triplet channel correction, which is characterized by the σ Fermi liquid constant F0 . This constant measures the σ strength of the spin-exchange interaction. If F0 < 0, the spin-exchange interaction tends to align electron spins and (if it is strong enough) leads to the ferromagnetic Stoner instability13 . Even though this constant is unknown, it can be found experimentally by means of independent measurement of the spin susceptibility of the system. As a function of temperature the interaction correction to conductivity is almost always monotonous, except for a narrow region of parameters (where it is so small that it can hardly be observed). The remainder of the paper is organized as follows: the following section is devoted to qualitative discussion of the physics involved. In the same section we summarize our results. Then we present two alternative approaches to the microscopic calculation. In Section III we use the traditional perturbation theory1 to derive the results presented in Section II, while in Section IV the same results are obtained using the quantum kinetic equation that we derive. The advantage of the kinetic equation approach is that it can be readily used to discuss the temperature behavior of quantities other than conductivity. These results are advertised in Conclusions and will be published elsewhere14 . Consider a single impurity localized at some point, taken as the origin. The impurity potential U (r) induces a modulation of electron density close to the impurity. The oscillating part of the modulation is known as the Friedel oscillation, which in 2D can be written as δρ(r) = − νλ sin(2kF r). 2πr2 (2.1) Here r denotes the distance to the impurity and its potential is treated in the Born approximation λ = U (r)dr. In 2D the free electron DoS is given by ν = m/π¯ 2 and h m is the electron mass, kF is the Fermi momentum. Taking into account electron-electron interaction V0 (r1 − r2 ) one finds additional scattering potential due to the Friedel oscillation Eq. (2.1). This potential can be presented as a sum of the direct (Hartree) and exchange (Fock) terms17 δV (r1 , r2 ) = VH (r1 )δ(r1 − r2 ) − VF (r1 , r2 ); (2.2a) VH (r1 ) = (2.2b) VF (r1 , r2 ) = II. QUALITATIVE DISCUSSION AND RESULTS dr3 V0 (r1 − r3 )δρ(r3 ); 1 V0 (r1 − r2 )δn(r1 , r2 ), 2 (2.2c) where by ρ(r) we denote diagonal elements of the one electron density matrix n, In this section we describe the scattering processes contributing to the temperature dependence of conductivity. We show that unlike the standard Fermi liquid T 2 corrections, the leading correction to conductivity is accumulated at large distances, of the order vF /min(T, T /τ ). In the ballistic limit such correction is linear in temperature and we derive this result here using a text-book quantum mechanical approach. The diffusive limit is discussed in detail in Ref. 1. The resulting correction δσ ∼ ln T seems to be rather different from the linear one, but we show that both corrections arise due to the same physics - coherent scattering by Friedel oscillations. Throughout the paper we keep the units such that h = 1, ¯ except for the final answers. Ψ∗ (r1 )Ψk (r2 ). k n(r1 , r2 ) = (2.3) k The factor 1/2 indicates that only electrons with the same spin participate in exchange interaction. As a function of distance from the impurity the Hartree-Fock energy δV oscillates similarly to Eq. (2.1). B 11 00 11 00 11 00 C A. Scattering by Friedel oscillations A We start with the simplest case of a weak short-range interaction V0 (r1 − r2 ) and show how one can obtain the correction to conductivity in the ballistic limit, i.e. due to a single scatterer. This discussion is similar to that of Ref. 15, where the correction to the one-particle density of states (DoS) was discussed, and also of Ref. 16, which describes the correction to the conductivity in the diffusive limit. FIG. 1. Scattering by the Friedel oscillation. Interference between the two paths A and B contributes mostly to backscattering. The Friedel oscillation is created due to backscattering on the impurity, path C. 2 eikr νλ δΨ(r) = −i √ V0 (q) √ 2π kr The leading correction to conductivity is a result of interference between two semi-classical paths depicted on Fig. 1. If an electron follows path “A”, it scatters off the Friedel oscillation created by the impurity and path “B” corresponds to scattering by the impurity itself. Interference is most important for scattering angles close to π (or for backscattering), since the extra phase factor accumulated by the electron on path “A” (ei2kR with R being the length of the extra path interval relative to “B” and 2k being the difference between initial and final momenta for that extra path interval) is canceled by the phase of the Friedel oscillation e−i2kF R so that the amplitudes corresponding to the two paths are coherent. As a result, the probability of backscattering is greater than the classical expectation (taken into account in the Drude conductivity). Therefore, taking into account interference effects leads to a correction to conductivity. We note that the interference persists to large distances, limited only by temperature R ≈ 1/|k − kF | ≤ vF /T . Thus there is a possibility for the correction to have a non-trivial temperature dependence. The sign of the correction depends on the sign of the coupling constant that describes electron-electron interaction. To put the above argument onto more rigorous footing and to find the temperature dependence of the correction consider now a scattering problem in the potential Eq. (2.2). Following the textbook approach18 , we write a particle’s wave function as a sum of the incoming plane wave and the out-coming spherical wave (in 2D it is given by a Bessel function, which we replace by its asymptotic form) Ψ = eik·r + if (θ) 2π ikr e . kr where q = k − kr/r, dr1 VH (r1 )eik·r1 |q| = 2k sin(θ/2), with θ being the angle of scattering. Comparing to Eq. (2.4) we find the scattering amplitude as a function of θ (it also depends on the electron’s energy ǫ = k 2 /2m) f (θ) = − νλ V0 (q) 2π dr sin(2kF r)eiq·r . r2 (2.7) The integral can be evaluated exactly19 and the result is given by f (θ) = − νλ V0 (q) 2π π 2, arcsin 2kF q |q| < 2kF ; , |q| > 2kF . (2.8) Let us examine this expression more closely. Since |q| ≤ 2k, the scattering amplitude Eq. (2.8) for small k weakly depends on the angle through the Fourier component of the interaction V0 (q), see background value of f (θ) on Fig. 2. However, we are dealing with electronic excitations close to the Fermi surface, so in fact k is close to kF , |k − kF |/kF ≪ 1. If k > kF , then the scattering amplitude Eq. (2.8) has a non-trivial angular dependence around θ = π shown on Fig. 2. According to Eq. (2.8) such dependence is only possible in the region |q| > 2kF . This translates into the condition |θ − π| < [2(k − kF )/kF ]1/2 , which determines the singular dependence of the width of the feature in the scattering amplitude on the energy of the scattered electron. Finally, using the fact that arcsin(1 − x) = √ π/2 − 2x, we find that the dependence of the height of the feature in the scattering amplitude is also singular: δf (θ) ≃ [(k − kF )/kF ]1/2 . (2.4) Here f (θ) is the scattering amplitude, which we will discuss in the Born approximation. For the impurity potential itself the amplitude f (θ) weakly depends on the angle. At zero temperature it determines the Drude conductivity σD , while the leading temperature correction is proportional to T 2 , as is usual for Fermi systems. We now show that this is not the case for the potential Eq. (2.2). In fact, taking into account Eq. (2.2) leads to enhanced backscattering and thus to the conductivity correction that is linear in temperature. First, we discuss the Hartree potential Eq. (2.2b). Far from the scatterer the wave function of a particle can be found in the first order of the perturbation theory as Ψ = eik·r + δΨ(r), where the correction is given by18 δΨ(r) = i dr1 iq·r1 , (2.6) 2 sin(2kF r1 )e r1 f( θ) π θ (ε−ε /ε )1/2 F F 2π eik|r−r1 | . (2.5) k|r − r1 | 1/2 (ε−ε /ε ) Here |r − r1 | ≈ r − r · r1 /r, since we are looking at large distances. Substituting the form of the potential Eq. (2.2b) and introducing the Fourier transfer of the electron-electron interaction V0 we can rewrite Eq. (2.5) as F F FIG. 2. Scattering amplitude. The singularity for backscattering is due to interference of paths “A” and “B” on Fig. 1. 3 T δσ = −2νV0 (2kF ) . σD ǫF The transport scattering rate τ −1 is determined by the scattering cross-section and can be found with the help of the amplitude Eq. (2.8), as well as the constant amplitude f0 of the scattering by the impurity itself dθ (1 − cos θ)|f0 + f (θ)|2 . 2π τ −1 (ǫ) = The conductivity Eq. (2.12) is the same correction as the one calculated in Ref. 6, see also Eq. (1.2), up to a numerical factor. It is also clear that Eq. (2.12) is not the full story. We have forgotten about the Fock part of the potential Eq. (2.2)! Substituting Eq. (2.4) into Eq. (2.3), we find the perturbation of the density matrix [which appears in the Fock potential Eq. (2.2c)] δn(r1 , r2 ) ≈ δρ[(r1 + r2 )/2]. Then the argument can be repeated. The only difference is that the leading temperature correction comes from the Fourier component at q = 0, rather than q = 2kF . What is most important, the Fock potential enters with the opposite sign. Therefore the expression for the conductivity Eq. (2.12) has to be corrected (2.9) The leading energy dependence of τ −1 comes from the interference term, which is proportional to f (θ). Then integration around θ = π is dominated by the feature of f (θ) resulting in a term of order (ǫ − ǫF )/ǫF . It is this term that gives rise to the linear temperature dependence we are after. Since we are interested in this leading correction only, in all other terms we can set k ≈ kF and write the scattering rate as −1 τ −1 (ǫ) = τ0 + νλ ǫ − ǫF η(ǫ − ǫF )f0 . V0 (2kF ) 2 ǫF (2.10) −1 Here η(x) is the Heaviside step function and τ0 is the zero-temperature rate that determines the Drude conductivity (indeed, the θ = π feature in f (θ) only exists for k > kF and at T = 0 there are no electrons with k > kF ). To obtain the scattering time we have to integrate the energy-dependent rate Eq. (2.10) with the derivative of the Fermi distribution function nF (ǫ) τ= dǫτ (ǫ) σ = σD 1 − ν 2V0 (2kF ) − V0 (0) T . ǫF (2.13) The sign of the correction is thus not universal and depends on the details of electron-electron scattering. If the weak interaction is reasonably long-ranged, then V0 (0) ≫ V0 (2kF ), so that the correction in Eq. (2.13) has the sign opposite to that in Ref. 6. ∂ nF (ǫ). ∂ǫ Then the second term in Eq. (2.10) leads to a linear correction to the Drude conductivity, small as T /ǫF . However this is not the only contribution to the temperature dependence. At finite temperatures we also have to modify the Friedel oscillation Eq. (2.1) as follows: δρ(r) = − (2.12) A 111 000 111 000 111 000 111 000 111 000 111 000 νλT 2 2 2πvF sinh2 B rT vF sin(2kF r). Consequently, the scattering amplitude Eq. (2.7) becomes temperature dependent f (θ) = − νλ V0 (q) 2π =− p= 2T vF C dr2 −2r2 vT iq·r2 F sin(2k r )e F 2 2 e r2 νλ V0 (q) arcsin 2π 2 + (q + 2kF )2 + 4kF p 2T vF ; FIG. 3. Scattering process with two impurities and the Friedel oscillation. Scattering to all angles is affected by interference. The relevant Friedel oscillation is created by the self-intersecting path C. (2.11) 2 + (q − 2kF )2 . So far we have considered the effect of a single impurity. The extension of the above arguments to the case of many impurities is straightforward. In particular, one can consider a scattering process, which involves two impurities and the Friedel oscillation shown on Fig. 3. It is clear that this process contributes to the scattering amplitude at any angle, and not just for backscattering as the single impurity process on Fig. 1 (which is typical for Neglecting the small temperature dependent term in the denominator in Eq. (2.11) brings us back to Eq. (2.8). Keeping this term leads to the same feature in f (θ) as the one on Fig. 2, only now its width and magnitude are √ proportional to T . The resulting correction to the conductivity is therefore similar to the one discussed above. Up to a numerical coefficient, 4 the diffusive motion of electrons). Such processes were discussed in detail (although using a slightly different language) in Ref. 16. Scattering by Friedel oscillations created by multiple impurities results in a conductivity correction (1.1) that is logarithmic in temperature and is typical for 2D diffusive systems1 . Comparing the scattering processes on Figs. 1 and 3, one can clearly see that conductivity corrections, which arise from these processes are governed by the same physics: coherent scattering by the Friedel oscillation, which means that the ballistic and diffusive regions should be analyzed on the same footing. In the next subsection we present the results of such anlysis, postponing the actual calculations until Secs. III and IV. w(x ≫ 1) ≈ 8 1 ln(2x) − (ln x − 1)(6 ln 2 − 1) . 3πx 4 In the opposite limit T τ ≪ 1 it approaches a constant value (C ≈ 0.577 . . . is the Euler’s constant and ζ ′ (x) is a derivative of the Riemann zeta function), w(x ≪ 1) ≈ 1 + 2πx 9 ln x − ln 2 − C + 3 + 6ζ ′ (2) , 4 so that the linear correction does not completely vanish in the diffusive limit, but competes with the logarithmic term and in semiconductor structures with low Fermi energy it might be important except for the lowest temperatures. The full function w(x) is plotted on Fig 4. B. Results Let us first consider the case of a weak, short range interaction potential. Then the interaction can be treated in the lowest order of perturbation theory, so that the resulting correction is proportional to the interaction constant: 1.0 0.8 w(x) 0.6 Tτ 3 e2 γ2 EF γ1 1 − w(T τ ) − . δσw = ln π¯ h h ¯ 8 4π T (2.14) 0.4 0.2 Similarly to Eq. (2.13), it can be written as a sum of Hartree and Fock contributions (similar expression for γ1 in one-dimensional systems was obtained in Ref. 20): 0.0 0 FS 4 6 8 10 12 14 x γ1 = ν V0 (0) − 2V0 (2kF ) , γ2 = ν V0 (0) − 2 V0 (k) 2 FIG. 4. Dimensionless function w(x), which is defined so that w(0) = 1. , (2.15) If the Coulomb interaction is considered, then the lowest order in interaction is not sufficient since for the long range interaction νV0 (q ≈ 0) ≫ 1. Although the interaction itself is still independent of the electron spin, summation of the perturbation theory depends on the spin state of the two electrons involved. In the first order correction Eq. (2.14) all spin channels gave identical contributions. The total number of channels is 4 and they can be classified by the total spin of the two electrons: one state with the total spin zero (“singlet”) and three states with the total spin 1 (“triplet”; the three states differ by the value of the z-component of the total spin). For long range interaction the perturbation theory for the Hartree correction singlet and triplet channels is different. It is known1,2 (see also Section III), that the singlet channel contribution should be combined with the Fock correction as a renormalization of the coupling constant. However, the final result is universal due to dynamical screening: the singlet channel modification of the coupling does not affect the result. What remains of the Hartree term is the triplet channel contribution, which now depends on the corresponding Fermi liquid constant σ F0 . Thus, the total correction to the conductivity can be where . . . F S stands for the average over the Fermi surface. Here we kept the notation for the electron-electron interaction adopted in the previous Section. Then the Hartree correction is proportional to the Fourier component of V0 (q) at q = 2kF , while for the Fock correction q = 0. The two corrections have different sign as we discussed above. The extra factor of 2 in the Hartree correction is due to electron spin degeneracy. Note, that Eq.(2.14) is defined only up to a temperature-independent constant which is determined by the ultraviolet contribution. We have chosen the argument of the logarithm to be EF /T instead of the usual 1/T τ to emphasize that contrary to the naive expectations the logarithmic term persists up to temperatures much larger than 1/τ , see also Ref. 15. The different expressions for the Hartree terms in γ1 and γ2 are related to the fact that the single impurity scattering, see Fig. 1, and multiple impurity case, see Fig. 3, allow for different possible scattering angles. The dimensionless function w(T τ ) describes the crossover between ballistic and diffusive regimes. In the ballistic limit T τ ≫ 1 it vanishes as 5 written as a sum of the “charge” (which combines Fock and singlet part of Hartree) and triplet contributions σ = σD + δσT + δσC , in the first line of Eq. (2.16c) and √ 2 2 + 2 − rs rs 1 σ ln √ F0 → − 2 2 2π 2 − rs 2 − 2 − rs (2.16a) where the charge channel correction is given by δσC = e2 T τ 3 e2 EF 1 − f (T τ ) − 2 ln , π¯ ¯ h h 8 2π ¯ h T σ F0 → − (2.16b) σ 3 e2 T τ 3F0 σ 1 − t(T τ ; F0 ) σ ) π¯ ¯ (1 + F0 h h 8 −3 1 − 1 σ σ ln(1 + F0 ) F0 EF e2 ln . 2¯ 2π h T rs arctan 2 rs − 2 1 2 2 r − 1, rs > 2 2 s in the second line. For rs > 1 our replacament is well ∼ justified even within weak coupling scheme. Similar to Eq. (2.14) the dimensionless functions f (x) σ and t(x; F0 ) describe the cross-over between ballistic and diffusive limits. They are plotted on Figs. 5 and 6 and full expressions are given by Eqs. (3.35). The universal function f (x) has the following limits and the triplet channel correction is δσT = 1 π 2 , rs < 2; (2.16c) f (x ≫ 1) ≈ − here the factor of three in the triplet channel correction Eq. (2.16c) is due to the fact that all three components of the triplet state contribute equally. We reiterate that the corrections Eqs.(2.16) are defined only up to a temperature independent (however not necessarily Fermi-liquid constant independent) term, see also discussion after Eq. (2.14). 7 1 2(ln x − 1) ln 2 − ln(2x) ; (2.17a) 3πx 2 f (x ≪ 1) ≈ 1 − γ1 x + γ1 = − ζ ′ (2) π + π 6 π x ln x; 6 C+ 1 ln 2 3 (2.17b) ≈ 0.7216; 1.0 F0=100 F0=0.1 F0=-0.3 F0=-0.8 2.5 0.8 2.0 0.6 F0=1 F0=-0.1 F0=-0.6 f(x) t(x) 1.5 0.4 1.0 0.5 0.2 0.0 0.0 0 2 4 6 8 10 12 14 0 5 10 X x FIG. 5. Dimensionless function f (x), defined so that f (0) = 1 σ FIG. 6. Dimensionless function t(x, F0 ) defined so that σ t(0, F0 ) = 1. We should warn the reader here, that we describe the interaction in the triplet channel by one coupling conσ stant F0 . For the weak coupling limit, it corresponds to the approximation V0 (2kF ) ≃ V0 (k) F S . This approximation overestimates the√ triplet channel contribution to the ballistic case for rs = 2e2 /(κ¯ vF ) ≪ 1. However, in h this limit contribution itself is much smaller than the singlet one. For better estimates in this regime one should use σ The function t(x; F0 ) depends on the coupling constant and therefore its asymptotic form also depends on σ σ F0 . For very small x ≪ 1 + F0 the asymptotic form is σ F0 → − σ t(x ≪ 1 + F0 ) ≈ 1 − γ2 x + γ2 = − rs 1 √ 2 rs + 2 + 6 ζ ′ (2) 3π 3+ 1 σ 1 + F0 1 π C 3+ σ 18 1 + F0 π 1 x ln x 3 + σ 18 1 + F0 ; πγ3 σ 9(1 + F0 ) (2.18) − + ln 2 1 + 3 σ 1 + F0 ; 5F σ − 3 γ3 = 1 − 0 σ − 1 + F0 5 σ − 2F0 2 and the magneto-resistance in the parallel field. The theory for interaction corrections in the magnetic field will be addressed in the forthcoming paper14 . The correction in Fig. 7 is almost always monotonous, σ except for a narrow region −0.45 < F0 < −0.25. A typical curve in this region is shown in Fig. 8. Note, however, that the overall magnitude of the correction in the range of T τ in Fig. 7 is so small that it can hardly be observed. σ ln(1 + F0 ) . σ F0 Notice that at T τ → 0, Eqs. (2.16) reproduce the known result (1.1). Let us point out that for numerical reasons contributions of scaling functions w, f, t change the result only by few percents and they can be neglected for all the practical purposes. Notice that while the charge channel correction Eq. (2.16b) is universal, the triplet channel correction σ Eq. (2.16c) is proportional to F0 , which might be negative. That leads to the conclusion, that the overall sign of the total correction Eqs. (2.16) depends on value of σ F0 : it can be either positive or negative, see Fig. 7. 2 h e2 0.0 −0.2 σ F0 =−0.3 −0.4 2 h e2 10.0 σ −0.1 −0.2 F0 =0.0 0.0 −0.6 σ −0.3 F0 =−0.4 −0.8 −0.4 −10.0 −0.5 −1.0 0.0 −20.0 0.3 0.4 0.5 FIG. 8. The non-monotonous correction to conductivity. Note the difference in the overall scale relative to the previous figure. −30.0 −0.7 −50.0 0.0 0.2 T =h −0.6 −40.0 0.1 σ F0 =−0.8 1.0 2.0 3.0 4.0 When the interaction becomes so strong that the sysσ tem approaches the Stoner instability, F0 ceases to be a constant and becomes momentum-dependent. Thus the result Eq. (2.16) is no longer valid. Although the simple condition δσT < σD suggests that this happens at σ T ≈ (1 + F0 )EF , the more detailed analysis (see Section III E) shows that it happens much earlier. In fact, σ the approximation of the constant F0 is valid in the parameter region defined by the inequality 5.0 T =h FIG. 7. Total interaction correction to conductivity. The divergence at T τ /¯ → 0 is due to the usual logarithmic h σ correction1 . Curve F0 = 0 corresponds to the universal behavior of completely spin polarized electron gas. The correction is defined up to a temperature independent part, see Eq. (3.32) and discussion after Eq. (2.14). T σ < (1 + F0 )2 , EF Combining together all of the above results we plot the total correction to the conductivity on Fig. 7 for differσ ent values of F0 . The divergence at low temperature is due to the usual logarithmic correction1 . Although the σ exact value of F0 can not be calculated theoretically (in particular, its relation to the conventional measure of the interaction strength, rs , is unknown for rs > 1), in principle it can be found from a measurement of the Pauli spin susceptibility ν χ= σ, 1 + F0 (2.20) see Section III E for the origin of this inequality. We were not able to make a reliable calculation of δσ(T ) at higher temperatures. III. PERTURBATION THEORY In this section we show how the announced results Eq. (2.16) can be obtained with the help of the traditional perturbation theory. We try to explain the most important points of the calculation in detail. The comprehensive review of the diagrammatic technique for disordered systems can be found in Ref. 1. We start by a brief discussion of the case of a weak, short-range interaction potential. Although this case is artificial and is (2.19) where the density of states ν should be obtained from a measurement of the specific heat (at τ −1 ≪ T ≪ EF ). σ The constant F0 is the only parameter in our theory which describes all the data, including the Hall coefficient 7 unrelated to any experiment, it is governed by the same physics as the general problem, and it is simple enough to allow a transparent presentation. To generalize to stronger coupling, we need to recall the basic ideas of the Landau Fermi liquid theory and to identify the soft modes in the system. Then we present the calculation leading to Eq. (2.16). Finally, to establish the relation of our results to existing literature, we briefly discuss scattering on a single impurity (this discussion is completely analogous to the one in Section II but uses the language of diagrams). Hartree term corresponds to the diagrams “a”, while the Fock contribution corresponds to diagrams “b”. Evaluation of the correction consists of two main steps: (i) analytic continuation to real time, and (ii) disorder averaging. While these two steps can be performed in any order without affecting the result, it is more convenient (for technical reasons) to start with step (i). Although analytic continuation in Eq. (3.1) is now a textbook task, we include a brief discussion of the standard procedure in the Appendix to make the paper selfcontained. After the continuation any physical quantity is expressed in terms of exact (i.e. not averaged over disorder) retarded and advanced Green’s functions of the electronic system, which are defined as A. Hartree-Fock considerations. R(A) The static conductivity of a system of electrons is given by the Kubo formula σαβ = G12 j  1/T  1 − lim Re  ω→0 Ωn 0  dτ Tτ ˆα (τ )ˆβ (0) eiΩn τ  j j Ψ∗ (r1 )Ψj (r2 ) j , ǫ − ǫj ± ı0 (3.2) where j labels the exact eigenstates of the system and ǫj are the exact eigenvalues, counted from the Fermi energy: (3.1)  (ǫ) = −∇2 + U (r) Ψj (r) = (ǫj + ǫF )Ψj (r). 2m , Here U (r) is the disorder potential. The resulting expression for the correction to the symmetric part of the conductivity (the Hall conductivity will be discussed in a separate publication14 ) can be written as16 iΩn →ω where ˆα (τ ) is the operator of the electric current at j imaginary time τ and the analytic continuation of the function defined at Matzubara frequencies Ωn = 2πT n to function analytic at Imω > 0 is performed. ∞ δσαβ = dΩ ∂ 8π 2 ∂Ω Ω coth Ω 2T d2 r3 d2 r4 (3.3) −∞ αβ αβ ×Im V (r3 − r4 ) BF (Ω; r3 , r4 ) − 2BH (Ω; r3 , r4 ) jα jα jβ jα jβ 1 jβ +{α ↔ β} 3 2 a jα jα jα jβ where the extra factor of 2 in the Hartree term is due to the summation over electron spin. Here we denoted products of four Green’s functions as BF (H) . For the Fock term we have jβ jβ 1 2 , 3 B αβ (Ω; r3 , r4 ) = F b FIG. 9. Interaction correction to conductivity in the lowest order of perturbation theory. Here solid lines correspond to Matsubara Green’s functions −G(iǫn ; r1 , r2 ) and the wavy line represents the interaction potential, −V (r1 − r2 ). d2 r1 d2 r5 V ˆα 15 ˆβ 53 × J1 GR (ǫ)J5 GA (ǫ)GR (ǫ − Ω)GA (ǫ) 34 41 (3.4a) ˆα 15 ˆβ 53 +J1 GA (ǫ)J5 GR (ǫ)GR (ǫ − Ω)GR (ǫ) 34 41 ˆα ˆβ +2J1 GR (ǫ)GR (ǫ − Ω)J5 GR (ǫ − Ω)GA (ǫ) 54 41 13 35 (3.4c) ˆα ˆβ −J1 GA (ǫ)J5 GA (ǫ)GR (ǫ − Ω)GA (ǫ) 15 53 34 41 Assuming that electrons interact by means of a weak, short-range interaction (range shorter than vF min(τ, 1/T ), V (r) it is sufficient to consider the lowest order of the perturbation theory. The perturbation theory can be conveniently expressed in terms of Feinman diagrams. The lowest order diagrams for the interaction correction to the conductivity are shown on Fig. 9. The (3.4b) (3.4d) ˆα ˆβ −J1 GA (ǫ)GR (ǫ − Ω)J5 GR (ǫ − Ω)GA (ǫ) , (3.4e) 13 35 54 41 8 diagrams for averaged quantities can be constructed using the four “building blocks” (we use the momentum representation since translational invariance is restored after averaging): (1) the average electronic Green’s function (denoted as a solid line; there should be no confusion with the previous use of the solid line for exact Green’s functions before averaging), which in momentum space can be written as where V is the area of the system. Equations (3.4c) and (3.4e) come from the diagram “b3” on Fig. 9 and the rest of Eq. (3.4) correspond to diagrams “b1” and “b2”. For the Hartree term the expression is similar, B αβ (Ω; r3 , r4 ) = H d2 r1 d2 r5 V ˆα 15 ˆβ 53 × J1 GR (ǫ)J5 GA (ǫ)GR (ǫ − Ω)GA (ǫ) 44 31 (3.5a) ˆα ˆβ +J1 GA (ǫ)J5 GR (ǫ)GR (ǫ − Ω)GR (ǫ) 15 53 44 31 (3.5b) ˆα ˆβ +2J1 GR (ǫ)GR (ǫ − Ω)J5 GR (ǫ − Ω)GA (ǫ) 13 45 54 31 (3.5c) ˆα 15 ˆβ 53 −J1 GA (ǫ)J5 GA (ǫ)GR (ǫ − Ω)GA (ǫ) 44 31 GR(A) (k, ǫ) = (3.5d) U (r1 )U (r2 ) = A k ∇f1 f2 − f1 ∇f2 eA(r) f1 (r)f2 (r). m ; (3.8) 1 δ(r1 − r2 ). 2πντ R A k k-q A R p k-q R p-q k-q k FIG. 10. Dressed interaction vertex. (3.6) (3) the dressed interaction vertex Γ (q and Ω are momentum and frequency of the interaction propagator), which represents a geometric series in disorder potential shown on Fig. 10; In the above expressions terms corresponding to diagrams “b3” and “a3” on Fig. 9 allow for at least one of the spatial integrations to be performed with the help of the identity ˆβ 54 dr5 GR (ǫ)J5 GR (ǫ) = −ie(r3 − r4 )β GR (ǫ). 34 35 i 2τ In the diagrams this correlator is represented by the dotted line; Again, Eqs. (3.5c) and (3.5e) correspond to the diagram “a3” in Fig. 9. The current operator is defined as − ǫ − ξk ± (2) the disorder potential, which is assumed to be Gaussian with the correlator ˆβ 54 ˆα 13 −J1 GA (ǫ)GR (ǫ − Ω)J5 GR (ǫ − Ω)GA (ǫ) . (3.5e) 31 45 ie ˆ f1 (r)Jf2 (r)= 2m 1 Γ(q; Ω) = 1 + (3.7) Now it is clear that Hartree terms Eqs. (3.5c) and (3.5e) vanish identically, since there the identity (3.7) should be applied with coordinates r3 and r4 being equal to each other. In the Fock terms Eqs. (3.4e) and (3.4c) one needs to further multiply the result of Eq. (3.7) by the interaction potential V (r3 − r4 ). In the case of the short range interaction potential this also gives vanishing contribution. Thus we conclude, that the diagram “a3” on Fig. 9 does not contribute for any form of the interaction, while the diagram “b3” vanishes for the short-range interaction. The same identity can also be applied to terms Eqs. (3.4d) and (3.5d), which also vanish by the same reason. Thus the task of averaging over disorder is now simplified because we only need to average two Fock terms Eqs. (3.4a) and (3.4b) and two Hartree terms Eqs. (3.5a) and (3.5b). These expressions contain only Green’s functions of non-interacting electrons and can be averaged using the standard diagrammatic technique of the theory of disordered systems (see Ref. 1 for review). The 1/τ 1 S−τ (3.9a) where we denote S= iΩ + 1 τ 2 2 + vF q 2 , (3.9b) (4) the averaged product of a retarded and an advanced Green’s functions (sometimes referred to as the diffuson), where we have summed up a geometric series shown on Fig. 11 R Γ 2πντ A FIG. 11. Diffuson - geometric series of impurity lines 9 Using these building blocks we can average the products of Green’s functions as shown on Fig. 12. It is convenient to write the averaged BF (H) in the momentum representation. The product BF , which appears in the Fock term, can be viewed as a function of coordinates of the two interaction vertices and can be transformed to the momentum space as where nk corresponds to the direction of the momentum k on Fig. 10. The final expression for BH is similar to Eq. (3.11) −2πδ n1 n2 τ 1 BH (Ω; n1 , n2 , q) = − σD S d2 q iq(r3 −r4 ) e BF (q, Ω) . (3.10) (2π)2 + Using the explicit expressions Eqs. (3.8) and (3.9) we can write the analytic form of the averaged BF + BF (Ω; r1 , r2 ) = 1 (Γ2 − 1)τ Γ(Γ + 1) BF (q, Ω) = + 2 σD S vF q 2 1 τ iΩ + S A R j jα A R β j R R jα A A β A j R R R jα R R and, therefore, according to Eqs. (3.10) and (3.11) the relation BF (Ω; q) = jβ FIG. 12. Averaged product of four Green’s functions. The wavy lines indicate δ(pin − pout + q) for the Fock contribution BF (q) and 2πδ(pin − pout + q)δ[npin ] for the Hartree contribution BH (q, n1 , n2 ). The Hartree contribution is considered analogously. One can write dθ1 dθ2 d2 q ikF (n1 −n2 )(r1 −r2 ) e (2π) (2π) (2π)2 d2 r3 d2 r4 V (r3 − r4 )eiq(r3 −r4 ) ≈ V (0); V (3.11) d2 r3 d2 r4 V (r3 − r4 )eikF (n1 −n2 )(r3 −r4 ) V where ni = (cos θi , sin θi ) indictates the direction of the momentum. Then, disorder averaging of BH (Ω; n1 , n2 , q) is performed with the help of the same diagrams (see Fig. 12) but the expression for the vertices changed as indicated in the figure caption. Accordingly, the expression for the dressed vertex (3.9a), see also Fig. 10, is changed to S 1 ΓH (n, nk ; q; Ω) = 2πδ(nnk ) + , SH Sτ − 1 dθ1 dθ2 BH (Ω; n1 , n2 , q) 2π 2π must hold [this can be easily verified using explicit expresions (3.11) and (3.13)]. We are now prepared to calculate the temperature dependence of the conductivity from Eq. (3.3). We substitute Eqs. (3.11) and (3.13) into Eq. (3.3). As we will see, the main contribution to the temperature dependence is provided by wave-vectors qT ≃ max(T, (T /τ )1/2 )/vF . On the other hand the potential V (r) has a range much shorter than 1/qT . This enables us to use the following approximations A ×BH (Ω; n1 , n2 , q), Γ , SH (n1 )SH (n2 ) BF (Ω; r, r) = BH (Ω; r, r), b BH (Ω; r1 , r2 ) = dθ4 ΓH (n1 , n3 )ΓH (n2 , n4 ) (n3 n4 ) 2π SH (n3 )SH (n4 ) β R jβ A (3.13) and we suppressed the arguments q, Ω in the right-handside of the equation. We note in passing, that by construction of Eqs. (3.4a), (3.4b), (3.5a) and (3.5b) that a jα dθ3 2π + (n1 n2 ) . R R dθ3 ΓH (n1 , n3 )ΓH (n2 , n3 ) 2π 2 −1 αβ In the absence of magnetic field, BF (H) = δ αβ BF (H) , which is why we did not include the Greek indices in Eq. (3.11). jα A τ S ≈V 2kF sin n1 n2 2 , where V (k) in the right-hand-side of the above equations denotes the Fourier transform of the interaction potential. Altogether, we now write the conductivity correction as (3.12) SH (n, q; Ω) = iΩ − ivF qn + 1/τ , 10 ∞ δσ = −∞ dΩ ∂ 8π 2 ∂Ω Ω coth d2 q (2π)2 Ω 2T infrared behavior of the system where Fi can be considered as fixed. Therefore, our first step is to identify the terms in the interaction Hamiltonian, which may produce the biggest contributions at temperatures much smaller than the Fermi energy. This procedure contains nothing new in comparison with the standard identification of singlet, triplet and Cooper channels, see Ref. 22, and we present here the main steps to make the paper self-contained. The original interaction Hamiltonian has the form (3.14) ×Im V0 (0)BF (q, Ω) dθ1 2π −2 dθ2 V0 2π 2kF sin n1 n2 2 BH (Ω; n1 , n2 , q) . Evaluating this integral (where we only keep the temperature dependent part, see Section III F for details) one arrives to the same result Eq. (2.14), but with the coefficient in the form Eq. (2.15), in agreement with the discussion of Section II. Let us now turn to the case of the Coulomb potential, where the scheme of the calculation (as described so far) breaks down. In the Fock term we have V (0), which diverges for the Coulomb interaction (V (q) ∼ 1/q). To obtain meaningful results one needs to take into account the effect of dynamical screening. The Hartree term seems to work better since using just the static screening makes the result finite. However, this is wrong also, since in this case diagrams with extra interaction lines do not contain any smallness (see e.g. Fig. 13; there the correction is ∼ V (2kF )V (0)). Thus one can not justify the perturbation theory in the interaction potential. The way out of this problem is the standard theory of Landau Fermi liquid, which we briefly discuss in the following subsection. ˆ Hint = q,pi V (q) † † ψσ1 (p1 )ψσ2 (p2 )ψσ2 (p2 + q)ψσ1 (p1 − q), 2 and we imply summation over repeated spin indices. Soft modes of the system correspond to the situation when two of the fermionic operators have momenta close to each other. The difference of the momenta q∗ defines the scale 1/q∗ ≫ λF , which is the smallest length scale allowed in the theory. Therefore, we explicitly separate the Hamiltonian into a part that contains all the soft modes ˆ (first three terms) and a correction δ H, which does not contain such pairs of fermionic operators: ˆ ˆ ˆ ˆ ˆ Hint = Hρ + Hσ + Hpp + δ H. (3.15) The explicit expressions for the entries of the Hamiltonian (3.15) are the following. The interaction in the singlet channel (charge dynamics) is described by 1 ˆ Hρ = 2 |q| 1 holds. On the other hand, all the ∼ temperature and disorder dependence is determined by F σ n1 n2 1 ˆ Hσ = 2 pi j=x,y,z ν × (3.18) |q| 1, but still far from the ∼ Wigner crystal instability23 , rs < 37, exact calculation of ∼ the parameters F from the first principles (as well as their explicit expressions in terms of rs ) is not possible. Nevertheless, to study the behavior of the system at distances much larger than λF , one can still disregard the term δH in Eq. (3.15). Then parameters F are no longer bound by Eqs. (3.17), (3.19), and (3.21) [or by Eq. (3.22) for the Coulomb interaction] but rather should be treated as starting parameters for the low energy theory. The form of Eqs. (3.16), (3.18) and (3.20) is guarded by symmetries of the system: Eq. (3.16) is guarded by translational symmetry and charge conservation; Eq. (3.18) is guarded by translational symmetry and symmetry with respect to spin rotations; and Eq. (3.20) is guarded by all above symmetries and the electron-hole symmetry, which holds approximately at low energies. All the consideration above essentially repeats the basics of the Landau Fermi-liquid theory21 . We reiterate, that this theory does not imply that the interaction is weak; the only assumption here is that no symmetry is broken at small distances. ψσ4 (q − p2 ) describes singlet, F e , and triplet, F o , pairing fluctuations. The parameters in this Hamiltonian are F e,o (θ) = √ (3.22) . We also used the identity ˆj ˆj 2δσ1 σ2 δσ3 σ4 = δσ1 σ3 δσ2 σ4 + σσ1 σ3 σσ4 σ2 C. Disorder averaging = σσ1 σ3 σσ2 σ4 ˆy ˆy + σσ1 σ3 ˜j † σ j σ4 σ2 ˜ . To study the interaction correction to conductivity due to charge and triplet channel interactions introduced in the previous subsection, we follow the same route as in the case of the short-range interaction. In particular, the charge channel correction is a direct generalization of the Fock term. We start however with the discussion of disorder averaging. The correction to conductivity Eq. (3.3) represents the first order perturbation theory in the original potential V (q), valid when the potential is weak. For stronger coupling we make use of the effective Hamiltonian Eq. (3.15). Although the diagrams for conductivity look similar to So far, the representation (3.15) of original interaction is exact. The only advantage of this representation is that it explicitly separates the term δH which does not contain coupling to the low energy excitations of the fermionic system. Therefore, the contribution of δH to physical quantities is regular and not infrared divergent [like (T /vF q∗)2 ]. Therefore, for the electron system with weak short range interaction, δH can be disregarded at all. Moreover, even if the interaction is not weak or long range, δH can be treated in all the orders of perturbation 12 the Fock term “b” on Fig. 9, their content is now quite different. First, the wavy line now represents the propagator for one of the soft modes in Eq. (3.15). Therefore the expression for the conductivity Eq. (3.3) should be rewritten as ∞ δσαβ = − −∞ ×Im dΩ ∂ 8π 2 ∂Ω 1 τ 2 Γ3 vF q 2 . τ S6 −1 + (3.25) (3.24) A R A DA (Ω, r3 , r4 ) + TrDT (Ω, r3 , r4 ) A jα ∞ −∞ dΩ ∂ 4π 2 ∂Ω Ω coth jβ R a , jα A R R jβ A b R j d R α A j A β j jβ R A c 1 τ R e FIG. 14. Conductivity diagrams, group I. Diagrams (a), (b), and (d) were evaluated for the short range interaction in Sec. III A. In the diffusive regime1 only diagrams (a), (d), (e), were considered at ω, qvF ≪ 1/τ . R jα A A jα R R R jβ a R R A A b A jβ R A jβ A jα R jα A jβ R d R jα A R α jβ j A R jβ A R e FIG. 15. Conductivity diagrams, group II. Diagrams (a) and (b) were evaluated for the short range interaction in Sec. III A. In the diffusive regime1 only diagrams (a), (d), (e), were considered at ω, qvF ≪ 1/τ . 1 2τ (iΩ + τ )Γ (Γ2 − 1)τ 2 ˜ + BF (Ω, q) = − S3 S iΩ + S R α c Γ2 τ Γ(Γ + 1) + 2 2 vF q 2 A R jβ A d2 q (2π)2 Ω 2T R jα A ˜ DA (Ω, q) + TrDT (Ω, q) BF (Ω, q) ; 1 2 τ A R jβ R A where DA and DT are advanced propagators for charge and triplet channels [DT is a 3 × 3 matrix as follows from Eq. (3.18), see also Sec. III E] and BF is the product of electronic Green’s functions given by Eq. (3.4), the same as in the Fock term. Deriving Eq. (3.24) we assumed that the spin rotational symmetry is preserved, i.e. no Zeeman splitting or the spin-orbit interaction is present. We also neglected the dependence of the interaction propagators on the direction of the electron momenta. Lifting of those two assumptions is straightforward but it will not be done in the present paper. To the leading order in 1/kF l we can average the propagators independently of BF (see e.g. Ref. 1). Here we proceed with averaging BF and the discussion of the propagators follows. We have already averaged the product BF of four Green’s functions for the case of the short-range potential. There the three terms Eqs. (3.4e), (3.4c), and (3.4d) vanished due to the particular form of the potential. Now we have to take these terms into account and consider the full set of diagrams shown on Figs. 14 and 15. These diagrams can be evaluated in exactly the same way as those in Section III A (where we considered a subset of these diagrams). As a result, the averaged BF has a form similar to Eq. (3.11) and can again be expressed in terms of the dressed vertex Γ [see Eq. (3.9)]. We are still interested in the longitudinal conductivity and thus disregard the Hall contribution. Thus, after averaging the correction Eq. (3.24) takes the form v 2 q 2 − 2 iΩ + + F S5 iΩ + S where quantities Γ and S are defined in Eq. (3.9). Ω Ω coth 2T αβ × BF (Ω, r3 , r4 ) + {α ↔ β} ×Im 2Γ2 S3 jα A d2 r3 d2 r4 V 2 δσ = −e2 vF πν − It is important to emphasize that 2 −1 ˜ BF (Ω, q = 0) = 0, 13 (3.26) [to see this one should use explicit expressions (3.9) in Eq. (3.25)]. This property is not accidental – it is guarded by the gauge invariance of the system: no interaction with zero momentum transfer can affect the value of the closed loop. To proceed further with the actual calculation of the correction (3.25) we need to specify the interaction propagator. It will be done in the following two subsections. dsc being the screening radius, νV (1/dsc ) = 1) and we will not take it into account. Second, at frequencies smaller than the plasmon frequency we can neglect the unity in the denominator in Eq. (3.28), which corresponds to the unitary limit, i.e. D. Charge channel Thus the original coupling V (q) as well as the renormalization of the coupling by the Fermi liquid parameter Eq. (3.16) does not affect the resulting propagator. In other words the propagator becomes universal. It is important to emphasize that Eq. (3.29) gives the upper bound for the strength of the repulsive interaction. This is guaranteed by stability of the electron system with respect to the Wigner crystallization, i.e. by ρ the condition νV (q) + F0 > 0 at q < q∗. Therefore, we always have DA = − In this section we discuss the charge channel correction, described by the Hamiltonian (3.16). Because the effective interaction is characterized by the momentum transfer much smaller than the Fermi wave vector, the Random Phase Approximation (RPA), see Fig. 16, is applicable. To simplify further considerations, we approximate the Fermi liquid parameter F ρ by its zero angular harmonic ρ F ρ (θ) ≈ F0 , ρ νV (q) + F0 , ρ ν + (νV (q) + F0 )ΠA so that Eq. (3.29) is indeed the upper bound for the propagator Eq. (3.28a). Note, that the above condition is ρ satisfied regardless of the sign of F0 . In particular, it is ρ possible to have F0 < −1 so that the so-called compressρ ibility of the system ν/(1 + F0 ) is negative. This fact, however, has nothing to do with stability of the Fermi liquid and does not affect transport phenomena24 . Using the propagator Eq. (3.29) in the expression for the correction Eq. (3.25) we obtain after momentum integration (3.28a) where the polarization operator is given by ΠA (Ω, q) = ν 1 − iΩ 1 , S−τ (3.28b) ∞ using the notation (3.9b). The polarization operator (3.28b) differs from the more standard one (used for instance in Ref. 16) since the diffusion approximation has not been made yet. Indeed, expanding the polarization operator in small Ω and q we can recover the usual diffusive form. In terms of the scattering time it corresponds to the limit T τ ≪ 1. We do not do that here since we want to calculate the conductivity for all values of T τ . The form of the propagator (3.28) and expression for the conductivity correction (3.25) suggests that there could be two contributions. First, the propagator Eq. (3.28) has a pole which corresponds to the 2D plasmon. However, the plasmon dispersion relation is (vF qpl )2 νV (qpl ) = 2Ω Ω + i τ (3.29) ρ νV + F0 1 < , ρ ν + (νV + F0 )Π Π (3.27) this approximation does not affect the final result because of the long range nature of the Coulomb potential V (q → 0) → ∞. Consequently, we write the charge channel propagator in the form DA (Ω, q) = − 1 1 S−τ 1 =− . 1 ΠA ν S − τ − iΩ δσC = −e2 τ dΩ ∂ 2π ∂Ω Ω coth Ω 2T 0 1 Ωτ 2 arctan Ωτ + + H(Ωτ ) ln 2 π πΩτ 2π + 1 1 1 + H(Ωτ ) arctan π Ωτ + Ωτ 4π 1 1 + H(Ωτ ) ln 1 + 2 2 2 Ω τ , (3.30) where the dimensionless function H(x) is defined as H(x) = , i i.e. (vF qpl )2 ≪ |Ω Ω + τ | at all distances larger than the screening radius. According to the gauge invariance condition (3.26) this contribution is strongly suppressed (by a factor of the order of max[T, (T /τ )1/2 ]dsc /vF , with 1 . 4 + x2 In the frequency integral Eq. (3.30) we single out the first two terms as being dominant in the ballistic and diffusive limits respectively with the rest being the crossover function. The diffusive limit is given by 14 ∞ δσC (T τ ≪ 1) = −e2 τ dΩ ∂ 2π ∂Ω Ω coth Ω 2T E. Triplet channel 1 = πΩτ 0 e2 = − 2 ln 2π EF T . In this section we discuss the correction in the triplet channel. Similar to the case of the charge channel, we need to derive the interaction propagator in the triplet channel and then use Eq. (3.25). As follows from the Hamiltonian Eq. (3.18), the triplet channel propagator is now a 3 × 3 matrix. Apart from this minor complication, the propagator can be found using the same RPA approximation as the one used in Section III D, see Fig. 16. (3.31) In the opposite limit we can replace arctan Ωτ by π/2. Then the integral is divergent in the ultra-violet, but that large constant can be incorporated in the definition of τ . This is done as follows: ∞ dΩ ∂ ∂Ω Ω coth Ω 2T → −2T + EF coth 0 EF , 2T (3.32) D where EF is put for the upper limit of the integral. This is consistent with the approximations in momentum integration, where one typically relies on fast convergence in order to set the integration limit (otherwise determined by the Fermi energy) to infinity and to set all momenta in the numerator to the Fermi momentum in magnitude. Since we are interested in temperatures T ≪ EF , the second term is essentially a temperature independent (although infinite) constant. The temperature dependent correction to the conductivity is determined by the first term. As a result ∞ 2 δσC (T τ ≫ 1) = −e τ = e2 dΩ ∂ 2π ∂Ω -F D EF T 8 f (x) = 3 Ω 2T 0 + Tτ . π σ σi j σj Similarly to the charge channel, we take the Fermi liqˆ uid coupling F σ to be independent of electron momenta σ F σ (θ) ≈ F0 . (3.36) Unlike the case of the charge channel, this approximation slightly affects final results (see discussion after Eqs. (2.16) for the drawbacks of this approximation as well as for its remedies). Then the matrix equation for the triplet propagator has the form Tτ 3 + e2 1 − f (T τ ) , (3.34) π 8 [DT ]ij = −δij xz H(2xz) ln 2 π σ F0 Fσ − 0 ν ν ˆ Πik [DT ]kj (3.37) k=x,y,z where i, j = x, y, z. In the absence of the magnetic field and spin-orbit scattering each electronic Green’s function is a diagonal matrix in the spin space, and therefore xz 1 1 + H(2xz) ln 1 + 2π 2 (2xz)2 2 arctan 2xz − 1 . π a FIG. 16. Interaction propagator in the (a) singlet and (b) triplet channel 1 1 + 1 + H(2xz) arctan π 2xz + σi = 0 ∂ dz (z coth z) − 1 ∂z σ ij (3.33) Ω coth where the dimensionless function f (x) is defined as a dimensionless integral: ∞ ij -F ρ b Integrating the full expression Eq. (3.30) we find the correction valid at all values of T τ , e2 δσC = − 2 ln 2π -V ΠA (q, Ω) = δik ΠA (q, Ω), ik (3.35) (3.38) where ΠA (q, Ω) is the polarization operator given by Eq. (3.28b). Altogether, using Eq. (3.38) in the equation (3.37), we find the triplet channel propagator as The factor 3/8 is introduced for convenience, so that f (0) = 1. The integral can be evaluated analytical in the two limiting cases and the result is given by Eq. (2.17). In the intermediate regime the integral can be evaluated numerically and the result is plotted on Fig. 5. A DT (Ω, q) 15 ij = −δij σ F0 . σ ν + F0 ΠA (Ω, q) (3.39) σ σ σ ˜ h3 (x) = H(x; 1 + 2F0 ) − 1 − 2F0 + (2 + 3F0 )H(x) Before we continue, let us discuss the validity of the approximation Eq. (3.36). Consider the situation close σ to the Stoner instability F0 → −1. In this case the pole of the propagator Eq. (3.39) describes a magnetic excitation in the system. In the ballistic case (q > 1/l) it is a slow, over-damped spin wave + σ ˜ h4 (x) = H(x; F0 ) σ −iω ≈ (1 + F0 )vF |q|. ∞ δσT = −3e τ 1− + + Ωτ 2π dΩ ∂ 4π ∂Ω Ω coth Ωτ 4π (3.41f) 1 σ σ σ − 2F0 (1 + F0 )2 − (F0 )2 x2 2 ˜ H(x; y) = Ω 2T σ 2 F0 1 + arctan Ωτ σ πΩτ (1 + F0 ) π δσT = −3 1 − (3.41a) + , σ t(x; F0 ) (1 + 1 , + (xy)2 y)2 1 σ σ ln(1 + F0 ) F0 e2 ln 2π 2 EF T σ 3 3F0 2Tτ σ 1 − t(T τ ; F0 ) . σ) e π (1 + F0 8 8 = 3 ∞ dz 0 where we introduce notations: σ ˜ h1 (x) = H(x; 1 + 2F0 ) × (3.42) xz π ∂ (z coth z) − 1 ∂z H(2xz) + h1 (2xz) ln 2 + h4 (2xz) + (3.41b) 1 σ σ ˜ h2 (x) = h1 (x) + H(x; F0 ) − (1 + F0 ) 2 , 1 1 1 + H(2xz) + 4x2 z 2 h3 (2xz) arctan π 2xz + σ σ × 5 + 6F0 − 4(2 + 3F0 )H(x) , 1 σ ˜ σ − (1 + F0 )H(x; F0 ) 2 . σ where the dimensionless function t(x; F0 ) is defined as 1 1 + H(Ωτ ) + h2 (Ωτ ) ln 1 + 2 2 2 Ω τ σ +F0 x2 1 σ + 2F0 2 which is related to the function H(x) introduced in Sec˜ tion III D simply by H(x) = H(x; 1). The expression in brackets turns into its counterpart σ in the charge channel in the unitary limit (F0 → ∞). Its first term describes the diffusive limit described in Ref. 1 (the formal difference in the coefficient stems from the difference in the definition of the coupling constant). The frequency integral is evaluated in the same way as in Eq. (3.31). Similar to our discussion of the charge channel correction [see e.g. Eq. (3.33)], we identify the second term in Eq. (3.41a) with the ballistic limit (which we discuss in more detail in the next Section). The intermediate temperature regime is described by the expression [which appeared previously in Section II B, Eq. (2.16c)]: 1 1 1 + H(Ωτ ) + (Ωτ )2 h3 (Ωτ ) arctan π Ωτ + (3.41e) ˜ Here we introduce a dimensionless function H(x; y) (3.40) H(Ωτ ) + h1 (Ωτ ) ln 2 + h4 (Ωτ ) σ 1 + F0 σ ln(1 + F0 ), σ F0 × −∞ 1 σ σ ln(1 + F0 ) F0 σ σ 5F0 − 3 1 − (F0 )2 σ ln(1 + F0 ) + σ 2 F0 +h5 (x) At higher temperatures T > T ∗ our theory is not applicable. Having discussed the validity of the approach, we proceed with the straightforward calculation: one has to substitute the propagator Eq. (3.39) into the expression for the correction Eq. (3.25) and evaluate the integral. The result of the momentum integration is given by 2 (3.41d) σ σ σ ˜ ˜ h5 (x) = (2F0 − 1)H(x; 1 + 2F0 ) + H 2 (x; F0 ) The main contribution to the temperature dependent conductivity correction comes from this pole at ω ∼ T . The corresponding typical momenta are k ∗ ∼ T /[(1 + σ F0 )vF ]. Although we are using the momentum indepenσ dent F0 , it is known25 that fluctuations in the triplet channel produce a non-analytic correction to the spin susceptibility, so up to a numerical coefficient F σ ≈ σ F0 (1 − |q|/kF ). Such momentum dependence can only σ be neglected if k ∗ ≤ kF (1 + F0 ). This translates into a limitation for the temperature range where the results listed in Section II B are valid26 : σ T ≪ T ∗ ≈ (1 + F0 )2 EF . σ F0 ˜ σ σ σ ˜ H(x; F0 ) 1 + F0 x2 H(x; F0 ) , 2 1 xz 1 + H(2xz) + h2 (2xz) ln 1 + 2π 2 (2xz)2 + (3.41c) 16 2 arctan(2xz) − 1 π . (3.43) interaction potential. In this case the diagram “a3” of Fig. 9 is equal to zero even before the averaging (as a total derivative) and we only need to average the diagrams “a1” and “a2”. The rigorous procedure would involve dressing the interaction vertices according to Fig. 10 and adding diffusons Fig. 11 as it was done in Section III A (see Fig. 12), evaluating the resulting expression, and finally taking the limit T τ → ∞. However, the same result can be obtained by making the expansion by noticing that impurity line brings smallness 1/T τ . Therefore, high temperature limit may be studied by considering diagrams on Fig. 17 directly. Such approach is completely equivalent to that of Ref. 6. The result [which can also be obtained from the general expression Eq. (2.16c)] is similar to the one obtained in Ref. 6 (the difference is the extra factor of ln 2 found in Ref. 6 due to an error in this reference, which consists in putting the energy of the scattered electron on the Fermi shell rather than integrating over it): Except for the limiting cases [see Eq. (2.18)] the integral in Eq. (3.43) has to be evaluated numerically. We plot σ the result for several values of F0 in Fig. 6. F. Single impurity limit In the previous Sections we obtained the expression for the correction to conductivity averaged over disorder. To complete the calculation we needed to separately average the interaction propagator and use the result to evaluate the integral in Eq. (3.25). In doing this we assumed that the dimensionless conductance of the system is large or in terms of the scattering time τ EF ≫ 1. We have not, however, assumed anything about the relative value of the scattering rate and temperature. In other words, the correction Eq. (3.25) is valid in both the diffusive T τ ≪ 1 and ballistic T τ ≫ 1 limits. It also describes the crossover behavior at intermediate temperatures. The temperature behavior of the interaction correction in the limiting cases can of course be obtained from the general result Eq. (2.16). As we pointed out in Section II B, in the diffusive limit our results coincide with the standard theory, Ref. 1. On the other hand the correction in the ballistic limit is subject to conflicting claims in literature6,27 . Unfortunately, neither result is completely correct. Therefore we discuss the ballistic limit in some detail, starting with diagrams before averaging (i.e. diagrams on Figs. 9). This way we are able to point out exactly which diagram produces the dominant result and which diagrams were missed in existing theories. We begin by discussing the Hartree term. This contribution was considered in Ref. 6 in the framework of the temperature dependent dielectric function. The physical idea was that electrons tend to screen the charged impurities and thus modify the scattering rate. In what follows we show which diagrams describe this process and how to calculate the resulting correction, which appears to be the same (up to a numerical factor miscalculated in Ref. 6; see below for detailed explanation). The important difference between the two approaches is that the impurity screening picture described only the direct (Hartree) interaction, while missing on the exchange part. The latter was later considered in Ref. 27. We think that this consideration is erroneous, and we discuss the Fock term in Section III F 2. T EF δσH = −4σD [−νD(2kF )] (3.44) (for weak coupling D(2kF ) ≡ −V (2kF ) The factor of 4 in Eq. (3.44) can be interpreted as a result of a summation over four spin configurations. Although correct for weak coupling, this factor should be modified when stronger interaction is considered, see discussion above. R jα A A R R R R R jα A A A jα A jβ jβ R R a R R β A j A b FIG. 17. Single impurity diagrams for Hartree channel. 2. Fock contribution In the similar manner one can discuss the single impurity contribution to the Fock term. Again, for weak interaction we could simply expand the result of disorder averaging for the Fock term Eq. (3.11) to the leading order in 1/T τ . For Coulomb interaction we would expand Eq. (3.25), since in Eq. (3.11) the special form of the delta-function potential was utilized to eliminate the diagram “b3” on Fig. 9. Diagrammatically, such expansions equivalent to direct evaluation of diagrams without impurity lines (but with averaged electron Green’s functions) shown in Fig. 3 and diagrams with only one impurity line shown on Figs. 19, 20, and 21. 1. Single impurity limit for Hartree term The goal of this discussion is to show which diagrams correspond to the ballistic limit of the Hartree term (as discussed in Section II) and how it relates to other interaction corrections we discuss in this paper. The Hartree term corresponds to averaging the two diagrams on Fig. 9, where the wavy line represents a weak 17 The evaluation of the single impurity diagrams for the Fock term is straight-forward and is completely analogous to the Hartree term discussed in the previous subsection. The result can be written as jα A R A jα A jβ R R e a jα A A j jα A R β j A d A R jβ β A A 2 e τT σF = . π A R a R jβ jβ R jα A A jα A R R e d (3.45) jα R R jβ A R c FIG. 19. Single impurity diagrams for Fock channel with the impurity line dressing one interaction vertex. This result contradicts (even in sign) that of Ref. 27. Here we briefly discuss the reason for this contradiction. We notice that one has to be careful to keep track of gauge invariance while evaluating diagrams for the Fock term. Gauge invariance manifests itself in the fact that any interaction at zero momentum gives no contribution to physical quantities, which are expressed diagrammatically as closed loops, see Eq. (3.26). This is indeed the case for Eq. (3.25), where we summed up all the diagrams. On the other hand, any individual diagram is not gauge invariant. In particular, each subset of diagrams in Figs. 18-21 is not gauge invariant. Therefore to obtain the result Eq. (3.45) from these diagrams one has to disregard terms which contain higher than second powers of the scattering rate 1/τ . jα A A R jα R R R jβ jα A R jβ R A R jα R jβ jβ A a R A R jβ A A b jα A R jβ R d R jα A R R A α jβ j jβ A R e c R e FIG. 20. Single impurity diagrams for Fock channel with the impurity line connecting a retarded and an advanced Green’s functions across the diagram. jα A A R A jβ A b jα A R R jβ A a jα A R d a R R jα jβ A R c FIG. 21. The single impurity diagram for Fock channel with the impurity line connecting two advanced Green’s functions. R A jα jβ A R e IV. KINETIC EQUATION APPROACH FIG. 18. Fock channel diagrams without impurity lines. Our purpose in this Section is to put the treatment of the interaction effects in disordered systems into the framework of the kinetic equation. Even though at this point this will not produce any further physical results, this proves to be more convenient for practical calculations of more sophisticated quantities, such as the Hall coefficient, the thermal conductivity, energy relaxation, etc. The kinetic equation approach is also applicable for the description of non-linear effects. The main technical advantage of the kinetic equation is that it operates with gauge invariant quantities from the very beginning, unlike the perturbation theory, where each diagram taken As we already mentioned, the contribution from the plasmon pole is small due to the condition (3.26). However, in Ref. 27 it was claimed otherwise. Namely, diagrams in Fig. 18 were claimed to be important for the plasmon correction and to give a large result, while diagrams in Figs. 19-21 were alleged to be not important for the plasmon correction. This claim explicitly violates gauge invariance and leads to incorrect conclusions. In particular, the plasmon contribution to the conductivity was overestimated by a factor of order of (vF /ds T ) ≃ (EF /T ). 18 separately is not gauge invariant (and may produce nonphysical divergences). We will present the final form of the kinetic equation in subsection IV A, and show how to operate with this equation for the conductivity calculation in subsection IV B. The derivation of this equation based on the Keldysh technique for non-equilibrium systems28 is presented in subsection IV C. point-like scattering; generalization to the finite range is straightforward) as well as by the self-consistent field generated by all the other electrons: Stel {f } = − 8 + I0 (t; ǫ, n, r) f (t; ǫ, r, n) τ I0 (t; ǫ, n, r) = − ∞ (4.1a) n + −∞ and the average current is dǫ nf (t; ǫ, r, n) n (4.1b) and so on. Here ν is the density of states (entering into linear specific heat of the clean system) at the Fermi surface and vF is the Fermi velocity, n = (cos θ, sin θ) is the unit vector in the direction of the electron momentum and angular averaging is introduced as n = dθ ... 2π The Boltzmann-like equation for the distribution function has the form ∂t + vF n∇ + evF (nE) ∂ ∂ + ωc n × ∂ǫ ∂n f = St {f } , (4.2) αβ K1 (ω) = Im where E denotes the external electric field and ωc is a vector with the magnitude equal to the cyclotron frequency corresponding to an external magnetic field perpendicular to the plane and the direction along the field. Equations (4.1) and (4.2) neglect energy dependence of the velocity of electrons, which makes it inapplicable for quantities associated with electron-hole asymmetry, such as the thermopower. On the other hand, any component of the thermal and electrical conductivities is still within our description. All of the interaction effects are taken into account in the collision integral St {f } = Stel {f } + Stin {f } . f (t; ǫ − ω, r, n) n n (4.5a) dω αβ K (ω) f (t; ǫ − ω, r, n) n . (4.5b) 2π 1 The collision integral Eq. (4.4) preserves the number of particles on a given energy shell: integrating Stel {f (t; ǫ, r, n)} over directions of n gives zero for any value of ǫ [see also Eq.(4.58)]. The term I1 expresses enhanced momentum relaxation due to static disorder. The physics of this term was discussed in detail in Section II. The term I0 describes electron scattering by non-equilibrium non-local Fock like potential created by all other electrons. This process is responsible for generation of the finite drift velocity of electrons. One can easily see that I0 vanishes in the equilibrium situation f (ǫ, n, r) = f [ǫ + eϕ(r)], ∇α ϕ = −Eα . The kernels K0 , K1 , and L0 entering into Eqs. (4.5) can be expressed in terms of interaction propagators and the propagators describing semi-classical dynamics of noninteracting electrons. Explicitly: −∞ ... dω αβ nα K0 (ω) nβ f (t; ǫ − ω, r, n) 2π ∂ nα Lαβ (ω) 0 ∇β + eEβ 2 ∂ǫ αβ I1 (t; ǫ, r) = − ∞ J(t, r) = eνvF (4.4) n The effect of the self-consistent field is described by the last two terms, where we introduce notations: As usual in the kinetic equation approach, averages of observable quantities are expressed as certain integrals of the distribution function f (t; ǫ, r, n). For instance the averaged density is dǫ f (t; ǫ, r, n) n 8 αβ + nα I1 (t; ǫ, r) nβ f (t; ǫ, r, n) n . τ A. Final form of the kinetic equation ρ(t, r) = ν f (t; ǫ, r, n) − f (t; ǫ, r, n) τ × nα D Dnβ − αβ K0 (ω) = Im × The elastic part of the collision integral describes scattering of electrons by static impurities (we assume × 19 D δαβ 2 D D +i d2 q R D (ω, q) (2π)2 nα Dnβ D − Lαβ (ω) = −Re 0 (4.3) d2 q R D (ω, q) (2π)2 (4.6a) ∂ D ∂ω (4.6b) i ∂ Dnβ − Dnα Dnβ vF ∂qα d2 q R D (ω, q) (2π)2 (4.6c) ∂ ∂ ∂ nα D − Dnα D − Dnα D ∂qβ ∂qβ ∂qβ Equations (4.2) - (4.8) constitute the complete system of transport equations with the leading interaction corrections taken into account. They may be used to study both linear and non-linear response. We reiterate that they do not include effects of electron-hole asymmetry and in this form can not produce finite thermopower. The Hall effect, the thermal conductivity, and energy relaxation, however, are included and will be studied in a subsequent publication14 . In the following Subsection we apply the kinetic equation approach to study the interaction correction to the conductivity at intermediate and low temperatures and reproduce the results obtained in Section III by means of diagrammatic technique. The reason for doing so is to show how the kinetic equation works and to demonstrate explicitly that both approaches are equivalent. Closing our description of the structure of the kinetic equation, we discuss the range of its applicability. Any kinetic equation implies that the distribution function changes slowly on the spatial scale of the Fermi wave length λF and on the time scale 1/ǫF . In our case, the conditions are more restrictive. First, in the interaction correction to the elastic collision integral we take into account only the effect of the interaction on the zeroth and first angular harmonics of the distribution function. This implies that the equation gives the correct description for the interaction effects on the conductivity and diffusion, whereas it is not correct for description of the quantities involving higher angular harmonics. Second, we made a gradient expansion in Eq. (4.5a) and only took into account terms linear in the electric field. This implies that the distribution function changes slowly on the spatial scale LT = min(¯ vF /T, vF (¯ τ /T )1/2 ), and on the time h h scale of the order of ¯ /T . The electric field expansion h is justified by the condition eELT ≪ T . One can check that both these conditions are satisfied, if the energy relaxation time is much longer than the time for the elastic collisions. We also did not include quantum effects of the magnetic field. This is justified at ωc ≪ max(T /¯ , τ −1 ). h Finally, the interaction part of Eq. (4.4) is calculated in the first loop approximation. It means, that it has to be considered as the first order correction to 1/τ . If one is interested in the next order interaction correction to the elastic part, one should take into account the second loop correction, which is not considered in the present paper. On the contrary, the inelastic part (4.8) can be considered in all orders to find the zero angular momentum part of the distribution function; the only assumption here is the validity of the Fermi liquid description at energies smaller than ǫF . Here, DR (ω) = DA (−ω) denotes the retarded interaction propagator [see i.e. Eq. (3.29)] and we introduce the short hand notation for the angular averaging dθdθ′ a(n)D(n, n′ ; ω, q)b(n′ ), (2π)2 aDb ≡ dθdθ′ dθ′′ a(n)D(n, n′ )b(n′ )D(n′ , n′′ )c(n′′ ) (2π)3 aDbDc ≡ for arbitrary functions a, b. The function D(n, n′ ; ω, q) describes the classical motion of a particle on the energy shell ǫF in a magnetic field: − iω + ivF nq + ωc n × + ∂ ∂n D(n, n′ ; ω, q) 1 D(n, n′ ; ω, q) − D(n, n′ ; ω, q) τ (4.7) = 2πδ(nn′ ). n As we have already mentioned, the elastic part of the collision integral is nulled by a distribution function of the form f [ǫ + eϕ(r)] for an arbitrary f . It is the inelastic term that is responsible for establishing the local thermal equilibrium and it has the standard form Stin {f } = dω dǫ1 A(ω)f (ǫ1 ) [1 − f (ǫ1 − ω)] (4.8a) × − f (ǫ) [1 − f (ǫ + ω)] + [1 − f (ǫ)] f (ǫ − ω) f (ǫ) = f (t; ǫ, r, n) n . The kernel A(ω) describes matrix elements for inelastic processes in both ballistic and diffusive limits. The explicit expression for this kernel is A(ω) = d2 q 2 2 [Re D ] DR (ω, q) , (2π)2 2ν π (4.8b) where D is given by the solution to Eq. (4.7) averaged over angles. The above equations are written for the interaction in the singlet channel only. In a situation where both triplet and singlet channels are present, but the distribution function does not have a spin structure (no Zeeman splitting or non-equilibrium spin occupation present), one has to replace R DR → DR + TrDT (4.9) in Eqs. (4.6) and R R |DR |2 → |DR |2 + Tr DT DT B. Conductivity calculation † (4.10) In order to calculate the conductivity at zero magnetic field ωc = 0 we look for the solution of Eqs. (4.2) - (4.4) in a form in Eq. (4.8b). 20 f (n, ǫ) = fF (ǫ) + nΓ(ǫ), ∞ (4.11) 2 ǫ/T where fF (ǫ) = 1/(e +1) is the Fermi distribution function (all the energies are counted from the Fermi level), and Γ is the quantity to be found and it is proportional to the electric field. We substitute Eq. (4.11) into Eqs. (4.2), (4.4), (4.5) and (4.8). The inelastic part of the collision integral [see Eq. (4.8)] obviously vanishes, as effects of the heating are proportional to at least the second power of the electric field. As a result, we obtain an equation for Γ: evF Eα Γα (ǫ) ∂fF (ǫ) =− ∂ǫ τ − 4 τ −∞ −∞ +4evF τ C , Cτ − 1 where D0 denotes the solution of Eq. (4.7) without the angular averaged term (and in the absence of the magnetic field) ∂ dω αβ L (ω)eEβ fF (ǫ − ω) 2π 0 ∂ǫ D0 (n, ω, q) = C= δσ = σD −∞ (4.13) K1 (ω) = −Im × dω αβ ∂ L0 (ω)eEβ fF (ǫ − ω) 2π ∂ǫ 1 2 q2 vF . 2 2 (−iω + 1/τ ) + vF q 2 , × qdq R D (ω, q) 4π ω dω ∂ ω coth π ∂ω 2T (4.16a) 2 + C − (−iω + 1/τ ) C(C − 1/τ )2 qdq R D (ω, q) 4π , (4.16b) C − (−iω + 1/τ ) (C − (−iω + 1/τ ))2 1 + 2 C(C − 1/τ )2 C(C − 1/τ ) vF q 2 L0 (ω) = −Im vF τ (4.14a) qdq R D (ω, q) 4π C − (−iω + 1/τ ) C − 1/τ K0 (ω) = Im × × K0 (ω) − K1 (ω) − −iω + ivF nq + 1/τ which is similar to the notation S used in Section III [in fact, C = S ∗ , see Eq. (3.9b)]. Substituting Eq. (4.15) into Eqs. (4.6a) – (4.6c) and performing the angular integration we arrive to Substituting Eqs. (4.13) into Eq. (4.11) and the result into Eq. (4.1b), we integrate over ǫ and find the conductivity ∞ 1 Here we used the short-hand notation ∂fF (ǫ) dω αβ K1 (ω)fF (ǫ − ω) 2π ∂ǫ σ = σD + δσ, (4.15) +D0 (n, ω, q)D0 (n′ , ω, q) ∂fF (ǫ − ω) αβ Eβ +K0 (ω)fF (ǫ) ∂ǫ −4fF (ǫ) −∞ D(n, n′ ; ω, q)= 2πδ(nn′ )D0 (n, ω, q) (4.12) ∂fF (ǫ) ∂ǫ dωL0 (ω) = 0. In order to derive explicit expressions for the kernels Ki and L0 we have to solve Eq. (4.7) for the function D in the absence of the magnetic field. The result can be written as We solve Eq. (4.12) by iterations. As usual for kinetic equations, the solution is expressed in terms of the unperturbed distribution function fF (ǫ) and the kernels, which in this case are given by Eq. (4.6): Γα (ǫ)= −evF τ Eα , ∞ dωKi (ω) = dω αβ K1 (ω)fF (ǫ − ω)Γβ (ǫ) 2π 4fF (ǫ) τ ω ∂ ∂fF (ǫ − ω) ω coth = −1 + ∂ǫ ∂ω 2T ∞ αβ +K0 (ω)fF (ǫ)Γβ (ǫ − ω) − dǫfF (ǫ) , (4.16c) 2 3 vF q 2 1/τ 2 v2 q2 + F3 3 (C − 1/τ )2 2τ C C (C − 1/τ )3 . Together with the conductivity correction Eq. (4.14b) the above expressions Eq. (4.16) are identical to Eq. (3.25) obtained in Section III by means of the standard perturbation theory. Thus the kinetic equation approach is completely equivalent to such diagrammatic calculation. Integration over the wave vector q requires the knowledge of the interaction propagator. Substituting Eq. (3.29) for the singlet channel and Eq. (3.39) for the triplet channel and performing the straightforward integration we arrive to the results in Section II B. L0 (ω) , (4.14b) vF τ 2 where the Drude conductivity is σD = e2 νvF τ /2. Here we used the fact that in the absence of the magnetic field αβ all the kernels are diagonal, Ki = δαβ Ki , Lαβ = δαβ L0 . 0 We also used the identities 21 In the above expressions all the fields are defined on the Keldysh time contour shown in Fig. 22. In particular, the fermionic fields ψ † and ψ (as well as the bosonic field φ) can be treated as doublets C. Derivation of the kinetic equation In this section we derive the kinetic equation discussed in Section IV A. For simplicity we show the derivation for the case of the singlet channel interaction Eq. (3.16). The case of the triplet channel can be treated in the same manner with minor differences (introduction of extra spin indices) described in the end of this Section. To keep the discussion at the same level as in Section III, we treat the Fermi liquid parameter F ρ in Eq. (3.16) as a constant, similar to our treatment of the triplet channel in Section III E. ψ+ ψ− ψ= , (4.21) where we adopt the notation that fields with a − (+) subscript (also referred to by Greek letters in this Section) reside on the lower (upper) part of the contour on Fig. (22). The time dependent fermionic operators ψ are taken in the interaction represntation ˆ −i∂t ψ(t) = H1 (t); ψ(t) , 1. Keldysh formalism ˆ where H1 is the one-electron Hamiltonian which includes the static disorder potential as well as external fields. Consequently, the Green’s function in Eq. (4.17) is a 2 × 2 matrix. Time ordering along the contour is denoted in Eq. (4.19) by TC . Matrices γ α in Eq. (4.20) are ˆ defined as Here we summarize the results originally obtained by Keldysh28 that enable us to calculate correlation functions for any non-equilibrium distribution. Let us first consider a Green’s function of electrons before disorder averaging. The electron-electron interaction is described by the Hamiltonian Eq. (3.16). In the pathintegral formulation it can be decoupled from fermion operators using an auxiliary bosonic field φ. Then the Green’s function can be written as [Dφ] G(x1 , x2 |φ)e−iSB [φ] , G(x1 , x2 ) = G(x1 , x2 |φ) = (4.17) −∞ xi ≡ (ti , ri ). The bosonic action Eq. (4.18) can be treated in the saddle point approximation: e−iSB [φ] = e−iF [φ] , dtd2 r ψ † φα γ α ψ , ˆ (4.23b) where Π is the electronic polarization operator, defined as (4.20) Παβ (x1 , x2 ) = −∞ where σz = diag(−1, 1) is the Pauli matrix in the Keldysh ˆ space. G(x1 , x2 ) 8 8 + δ2 F δφα (x1 )δφβ (x2 ) . (4.24) φ=0 The quadratic expansion in Eq. (4.23b) is justified, provided that the fields φ are slowly changing on the scale much larger than λF . Let us now average the Green’s function Eq. (4.17) over disorder: C - (4.23a) 1 F [φ] = F [φ = 0] + φT Πφ + O(φ3 ), 2 (4.19) ∞ SF [φ, ψ] = 0 0 0 1 Here, as well as in Eq. (4.19) the angular brackets . . . denote quantum-mechanical averaging. In this section we will use the short hand notation where −V0 is the (bare; following Eq. (3.16) V0 = V (q) + ρ F0 /ν) electron-electron interaction propagator and Z is the partition function, Z[φ] = TC e−iSF [φ,ψ] γ− = ˆ 1 † TC ψα (x1 )ψβ (x2 )e−iSF [φ,ψ] . (4.22) Z[φ] 1 T −1 φ V0 σ3 φ + i log Z[φ], (4.18) 2 dtd2 r SB [φ] = ; The Green’s function G(x1 , x2 |φ) in Eq. (4.17) is given by with the action defined as ∞ −1 0 0 0 γ+ = ˆ dis = [Dφ] G(x1 , x2 |φ) dis e −i SB [φ] dis , (4.25) Fig. 22. The Keldysh contour 22 where . . . dis hereafter denotes averaging over disorder. Here we average the electronic Green’s function Eq. (4.22) separately from the bosonic action Eq. (4.23). This approximation means that we neglect correlations between mesoscopic fluctuations of the polarizability in Eq. (4.23b) and the fermionic operators in Eq. (4.22) (which describe the motion of conduction electrons). This is the same approximation we used in Section III. It is justified by the well known fact that mesoscopic fluctuations are smaller than average quantities by a factor of the order 1/(EF τ )2 . It is convenient31 to rotate the Keldysh basis as follows G→ 1 σx ˆ 2 1 1 −1 1 G 1 1 −1 1 . The coupling Eq. (4.20) between the fermionic and bosonic fields in the rotated basis has the form: ψ † φα γ α ψ → ψ † ˆ G(x1 , x2 |φ) = G (x1 , x2 |φ) GZ (x1 , x2 |φ) ˆ ˆ D(1, 2) = D0 (1, 2) + (4.26) ˆ D= ψ. (4.31) DR 0 DK DA ˆ ˆ ˆ d3d4D0 (1, 3)Π(3, 4)D(4, 2) ˆ Π= , ΠR 0 ΠK ΠA (4.32) and we introduced the short hand notation (i) ≡ (ti , ri ). The bare interaction propagators are K G (x1 , x2 |φ) GA (x1 , x2 |φ) φ2 φ1 The propagators Eq. (4.30) are solutions of the Dyson equations In the new basis the Green’s function Eq. (4.22) has the form R φ1 φ2 . (4.27) ρ R A D0 = D0 = − [V (r1 − r2 ) + F0 δ(r1 − r2 )] δ(t1 − t2 ), After the averaging over the bosonic field and over the disorder according to Eq. (4.25) the entries in Eq. (4.27) acquires the following meaning/ where after integrating over the bosonic field φ the diagonal elements GR(A) become the retarded (advanced) Green’s functions of the electron system K D0 = 0. (4.33) Any classical external field takes identical values on the two branches of the contour and, hence, in the rotated basis has only a diagonal component. The matrix Green’s function (4.27) satisfies the equation GR (t1 , t2 ) = −iη(t1 − t2 ) ψ(t1 )ψ † (t2 ) + ψ † (t2 )ψ(t1 ) , GA (t1 , t2 ) = iη(t2 − t1 ) ψ(t1 )ψ † (t2 ) + ψ † (t2 )ψ(t1 ) , where η(t) is the Heaviside step function. The lower diagonal element vanishes due to the causality, i∂t1 + EF − −i∇r1 + Aext (x1 ) 2 ˆ − φ(x1 ) 2m (4.34) GZ (t1 , t2 ) = 0, even before the disorder averaging. Finally, the upper off-diagonal element (the so-called Keldysh Green’s function) is related to the one particle density matrix GK (t1 , t2 ) = −i ψ(t1 )ψ † (t2 ) − ψ † (t2 )ψ(t1 ) , where U (r) is the potential due to the static disorder, Aext (x1 ) and ϕext (x1 ) are the vector and scalar potential due to the external electric and magnetic fields. (4.28) the quantum mechanical averaging is performed with an arbitrary distribution function to be found from the solution of the kinetic equation. The bosonic field in the rotated basis has the two components 1 φ1(2) = (φ+ ± φ− ) 2 which are described by the propagators φ1 (t1 , r1 )φ1 (t2 , r2 ) = i K D (t1 , t2 ; r1 , r2 ), 2 eE = ∂t Aext − ∇ϕext , (4.30a) i A D (t1 , t2 ; r1 , r2 ), 2 i∂t1 + EF − (4.30b) (4.30c) φ2 (t1 , r1 )φ1 (t2 , r2 ) = φ2 (t1 , r1 )φ2 (t2 , r2 ) = 0. 1 eB = − ∇ × Aext c (4.35) The equation (4.34) is the basis for the further consideration. One can perform the disorder average in Eq. (4.34) in the leading in 1/(EF τ ) approximation, which amounts to summation over all the non-intersecting impurity lines one obtains (4.29) i φ1 (t1 , r1 )φ2 (t2 , r2 ) = DR (t1 , t2 ; r1 , r2 ), 2 ˆ G(x1 , x2 |φ) = Iδ(x1 − x2 ) −U (r1 ) − ϕext (x1 ) −ϕext (x1 ) −i∇r1 + Aext (x1 ) 2m G(x1 , x2 |φ) = ˆ Iδ(x1 − x2 ) + (4.30d) 23 2 ˆ − φ(x1 ) (4.36) dx3 Σ(x1 , x3 |φ)G(x3 , x2 |φ); Σ(x1 , x2 |φ) = δ(r1 − r2 ) G(x1 , x2 |φ). 2πντ and we will suppress the coordinate and the time arguments unless otherwise is stated. A product of such matrices should be understood as a matrix product in Keldysh space and a convolution in time: The equation (4.36) allows for semi-classical treatment introduced in Refs. 29,30, and described in great details in Ref. 31. Since we have already averaged the equation of motion over disorder, the semi-classical approximation now amounts to averaging the Green’s function G(x1 , x2 |φ) over the distance from the Fermi surface. This is done in two steps: g(n, r)ˆ(n1 , r) ˆ g G(t1 , t2 ; p; R) = d re r = r1 − r2 ; iP ·r R= G(x1 , x2 |φ), ≡ (4.42) ∞ dt3 g(t1 , t3 ; n, r) ˆ (4.37) ik g (t3 , t2 ; n1 , r) ˆ kj , k −∞ 2 ij and solutions of the homogeneous equation (4.39) are subject to the constraints ∞ 1 (r1 + r2 ); 2 ˆK g(n, r)ˆ(n, r) = I , ˆ g dt Tr g (t, t; n, r) = 0, (4.43) ˆ −∞ 1 P =p− Aext t1 , R + Aext t2 , R 2 g(t1 , t2 ; n, r) = ˆ i π ; where [I K ]ij = δij δ(t1 − t2 ). (4.38) ∞ dξ G t1 , t2 ; n pF + −∞ The scalar and vector potentials in Eq. (4.40) have the following structure in the Keldysh space ξ ;r , vF ˆ A(t, r) = ϕ(t, r) = ˆ Since we follow the avenue of Ref. 31, we will skip further intermediate steps, and use the semiclassical equation written in the next subsection. g g ˆ ˆ n − g ng ˆ ˆ , 2τ where angular averaging is defined as before n = dθ ..., 2π n = (cos θ, sin θ), and the covariant derivatives in Eq. (4.39) are defined as ˜ˆ ˆ ˆ g gˆ ˆ ∂t g = ∂t1 g + ∂t2 g + iϕ(t1 , r)ˆ − iˆϕ(t2 , r), ˆ ˜ ˆ ∇ˆ = ∇ˆ + iA(t1 , r)ˆ − iˆA(t2 , r). g g g g (4.40a) O(t, r) = −ν (4.40b) gR gZ gK gA t1 →t , , (4.45) dθ O(pF n, r) 2π × lim Here g is a matrix in Keldysh space, ˆ g (t1 , t2 ; n, r) = ˆ (4.44) acting on the electron system, and φ1,2 (t, r) are the auxiliary fluctuating fields decoupling the interaction in the singlet channel. Because the singlet channel describes processes with small momentum transfers (smaller than q∗, see Section III B), the fields φ1,2 (t, r) vary slowly on the scale of the 1/q∗. The condition (4.30d) enforces causality of the physical response functions. It is worth noticing that the decoupling of interaction can be performed also using a fluctuating vector potential; our choice is strictly a matter of taste. In this formalism any observable quantity described by one electron operator O(ˆ, r ) is given by [see Eqs. (4.28) p ˆ and (4.38)] (4.39) ... ϕext (t, r) + φ1 (t, r) φ2 (t, r) φ2 (t, r) ϕext (t, r) + φ1 (t, r) , eE = ∂t Aext − ∇ϕext The dynamics of the electron matrix Green’s function is then described by the Eilenberger equation29 : g= ˆ 0 Aext (t, r) where ϕext and Aext are the external (classical) potentials due to the electric field E, 2. Eilenberger equation ∂ ˜ ˜ ∂t + vF n∇ + ωc n × ∂n Aext (t, r) 0 π K g (t1 , t; n, r) 2 (4.46) φ + ϕext (t, r) , where . . . φ stands for averaging over both auxiliary fields φ1,2 fluctuating according to Eqs. (4.30). The (4.41) 24 and we redefine the covariant derivatives Eq. (4.40) to include only the external scalar and vector potentials: last term in brackets is a consequence of the ultraviolet anomaly, and its form is enforced by the requirement of the gauge invariance. Finally, the electronic polarization operators are determined [see Eqs. (4.24) and (4.38)] as variational derivatives of the solutions to the Eilenberger equation (4.39): R A Π (1, 2) = Π (2, 1) dθ 2π =ν ˜ ∂t g = ∂t1 g + ∂t2 g + i [ϕext (t1 , r) − ϕext (t2 , r)] g, ˜ ∇g = ∇g + i Aext (t1 , r) − Aext (t2 , r) g. (4.47) π δg K (t1 , t1 ; n, r1 ) δ12 + 2δφ1 (t2 , r2 ) φ Now we are prepared to derive the collision integral. We notice that due to causality gZ φ = 0 in all orders of the perturbation theory. We separate slow and fast degrees of freedom as follows ; ΠK (1, 2) = νπ gK = gK K Z dθ δg (t1 , t1 ; n, r1 ) + δg (t1 , t1 ; n, r1 ) 2π 2δφ2 (t2 , r2 ) × φ . g = 1− A g =− 1− gZ gK , ∂ ˜ ˜ ∂t + vF n∇ + ωc n × ∂n (4.48) ∂ ˜ ˜ ∂t + vF n∇ + ωc n × ∂n gK φ −i φ1 (t1 , r) − φ1 (t2 , r) g Z − iφ2 (t1 , r)g R + ig A φ2 (t2 , r) − 1 4τ gK φ δg n − gZ ng R ∂ ˜ ˜ ∂t + vF n∇ + ωc n × ∂n + gA gZ n (4.49b) −i φ1 (t1 , r) − φ1 (t2 , r) g K − iφ2 (t1 , r)g A + ig R φ2 (t2 , r) + 1 K A g g 2τ n − gK ng A + gR gK n 1 δg Z − δg Z τ δg K + δg Z g K (4.53a) 1 δg K − δg K τ Z gK φ n − gK n φ δg n φ − gK φ δg Z n Z gK φ n gK φ n φ (4.53b) , Solutions to the equations (4.53) should be substituted into the equation (4.49b) for the smooth part. Than the equation for the smooth part should be averaged over the fluctuating fields φ1,2 with the help of Eq. (4.30). As a result − gA n gZ , gK = δg Z − = 2iφ2 (t1 , r)δ(t1 − t2 ) − i [φ1 (t1 , r) − φ1 (t2 , r)] g K 1 4τ 1 Z R g g 2τ (4.52) = −2iφ2 (t1 , r)δ(t1 − t2 ), + + (4.51) [expansion up to the second order in δg Z is unnecessary because terms of such kind vanish due to Eq. (4.30d)]. We now substitute Eqs. (4.51) and (4.52) into Eqs. (4.49) and obtain equations governing the behavior of the fluctuating parts (4.49a) gZ = gZ = δgZ , 1 g A = −δ(t1 − t2 ) + δg Z g K , 2 where the square root should be understood in operator sense: as a sum of its Taylor series, with all arising products hereafter being time convolutions, similar to Eq. (4.42). The two remaining independent components of the Eilenberger equation have the explicit form ∂ ˜ ˜ ∂t + vF n∇ + ωc n × ∂n + δgK ; 1 g R = δ(t1 − t2 ) − g K δg Z ; 2 Our goal now is to obtain an equation for the Keldysh function averaged over the fluctuating fields, g K (t1 , t1 ; n, r1 ) φ . It is this quantity that determines physical observables, see Eq. (4.46). We will do this using the non-crossing approximation for bosonic propagators (i.e. the first loop approximation for the collision integral), see Fig. 16. This approximation is justified provided that the resulting dynamics for the electrons (characterized by time τǫ ) is slow in comparison with motion of relevant bosonic mode, T τǫ ≫ 1. To do so, we notice that only two components of the matrix g are independent, and the other two are fixed by ˆ the constraint (4.43). For our purposes, we choose to fix the diagonal components gK gZ ; φ where δg is the contribution fluctuating with the auxiliary fields and we calculate it to first order in φ. In the same approximation Eq. (4.48) becomes 3. Derivation of the kinetic equation R (4.50) ∂ ˜ ˜ ∂t + vF n∇ + ωc n × ∂n − gR ngK , = Stin 25 gK φ + Stel gK gK φ φ . (4.54) Here we separate the collision integrals into two contributions. The physical meaning of such separation will be discussed shortly. The first, inelastic part has the structure Stin gK Stel φ (t1 , t2 ; n, r) gK φ = −i [φ1 (t1 , r) − φ1 (t2 , r)] δg K (t1 , t2 ; n, r) φ . The second, elastic contribution has the form (4.55) (t1 , t2 ; n; r) = 1 τ g K (t1 , t2 ; n; r) φ n − g K (t1 , t2 ; n; r) + dt3 dθ1 2π + dt3 dθ1 F R (t1 , t3 ; n, n1 ; r) g K (t3 , t2 ; n1 , r) 2π (4.56) φ g K (t1 , t3 ; n1 , r) φ F A (t3 , t2 ; n1 , n; r) − g K (t1 , t3 ; n, r) φ F A (t3 , t2 ; n, n1 ; r) φ − F R (t1 , t3 ; n1 , n; r) g K (t3 , t2 ; n, r) φ , where the first term is just the ordinary impurity scattering and the remaining terms characterize interaction effects. The kernels in Eq. (4.56) are defined as F R (t1 , t2 ; n, n1 ; r) = 1 4τ dt3 δg K (t1 , t3 ; n, r) δg Z (t3 , t2 ; n1 , r) − δg Z (t3 , t2 ; n, r) φ F A (t1 , t2 ; n, n1 ; r) = 1 4τ dt3 δg Z (t1 , t3 ; n1 , r) − δg Z (t1 , t3 ; n, r) δg K (t3 , t2 ; n, r) φ The equations (4.54), (4.55), (4.56), (4.53), (4.30), and (4.32) constitute a closed system of kinetic equations. Although sufficient for description of interaction effects in disorder systems, these equations are inconvenient for analytical calculations because the expressions for the collision integral are nonlocal in space and time. To simplify further calculations we will use the assumption that g K (t1 , t2 ; n, r) φ is a smooth function so that a gradient expansion will be possible. Before embarking on such calculation we pause to discuss the physical distinction between the elastic (4.56) and inelastic (4.55) collision terms. One immediately notices from Eq. (4.56) that dθ Stel (t1 , t2 ; n; r) = 0, Let us now perform the actual calculation of the collision integrals. We solve Eq. (4.53a) and obtain δg Z (t1 , t2 ; n, r) = 2iδ(t1 − t2 ) × gK φ (t1 , t1 ; n, r) = 0, dr1 dt3 φ2 (r1 , t3 ) D(t; n, n′ ; r1 , r2 ) = (4.60) dθ′ D(t3 − t1 , n′ , n; r1 , r) 2π dωd2 q iq(r1 −r2 )−iωt e D(n, n′ ; ω, q), (2π)3 where the diffuson propagator D is defined in Eq. (4.7). To simplify the analytic solution of Eq. (4.53b), we assume without loss of generality that g K φ varies slowly on the spatial scale LT = vF min(1/T, τ /T ), and also a slow function of t1 + t2 on the time scale ≃ 1/T . These assumptions are consistent with the first loop approximation we already invoked. In what follows we will keep only the zeroth and first angular harmonics (which is consistent with assumption about the spatial smoothness) in the direction dependence of the Keldysh function: (4.58) for any t1 and t2 . This indicates that this part of the collision integral preserves the number of particles on a given energy shell [see below for explicit connections between time representation and energy representation Eq. (4.65)]. The inelastic term (4.55) does not vanish after angular averaging. Therefore this part does promote electrons between energy shells. However, we notice that Stin (4.57) g (t1 , t2 ; n, r) φ ≈ g (t1 , t2 ; n, r) n +2n n′ g (t1 , t2 ; n′ , r) (4.59) n′ (4.61) This approximation does not affect results for any relevant quantities. From now on we will suppress the explicit sign of averaging over the fluctuating fields because we will not be dealing with non-averaged quantities anymore. We now substitute Eq. (4.61) into the right-hand side of Eq. (4.53b) and obtain for any direction n. Taking coinciding time arguments is equivalent to integrating over the whole energy spectrum [see Eq. (4.65)], so that not only the total number of particles is conserved, but the total number of particles moving along a given direction n is conserved (i.e. inelastic forward scattering). 26 K δg K (t1 , t2 ; n, r) = δg1 (t1 , t2 ; n, r) K +δg2 (t1 , t2 ; n, r) inelastic parts of the collision integral. To obtain nonvanishing contribution to the latter we have to do each one of the following: (i) take into account the first angular harmonic; (ii) perform the first order gradient expansion; (iii) expand up to the first order in external fields, Aext . The result is (4.62) The first term in Eq. (4.62) is proportional to the field φ1 and gives contributions to both the elastic and the K δg1 (t1 , t2 ; n, r) = −i × dn′ D(t, n, n′ ; r, r1 ) 2π dt [φ1 (r1 , t1 − t) − φ1 (r1 , t2 − t)] g K (t1 − t, t2 − t; n1 , r) (4.63a) n1 +2n′ n1 g K (t1 − t, t2 − t; n1 , r) (4.63b) n1 ˜ + (r1 − r) ∇ g K (t1 − t, t2 − t; n1 , r) n1 (4.63c) where the covariant derivative is defined in Eq. (4.50) and we neglected higher order derivatives of the external fields. ˜ Expansion in the time coordinate t1 + t2 (using the covariant derivative ∂t ) is not necessary because it produces a negligible correction to the inelastic collision integral and does not affect the elastic one. The second term in the RHS of Eq. (4.62) is proportional to the field φ2 , and according to Eqs. (4.57) and (4.30d) it does not contribute to the elastic collision integral. Therefore, it is sufficient to keep only the zeroth angular component and neglect gradient terms at all. This yields K δg2 (t1 , t2 ; n, r) = + dθ′ dθ′′ 2π 2π i g (t1 − t, t3 ; n1 , r) τ n1 dr1 dtD(t, n, n′ ; r, r1 ) 2iφ2 (r1 , t1 − t)δ(t1 − t2 ) g (t3 , t2 − t; n1 , r) n1 [ D(t4 − t3 , n′′ , n1 ; r2 , r1 ) As we already mentioned, g(t1 , t2 ) has a much faster dependence on the difference t1 − t2 then on the sum t1 + t2 . Therefore it is more convenient to use a temporal transformation of the Green’s function dǫ t1 + t2 g K (t1 , t2 ; n, r) = g , ǫ; n, r eiǫ(t2 −t1 ) , 2π 2 n1 (4.64) − D(t4 − t3 , n′′ , n′ ; r2 , r1 )] .φ2 (r2 , t4 ) g(t, ǫ) = −g(t, −ǫ); (4.67) DK (t, ω; r1 , r2 ) = DK (t, −ω; r2 , r1 ) ; DR (t, ω; r1 , r2 ) = DA (t, −ω; r2 , r1 ) . (4.65) which defines the precise notion of energy ǫ in this context. We introduce the same transformation for the propagators of auxiliary fields (4.30) D(t1 , t2 ) = dω D 2π t1 + t2 , ω eiω(t2 −t1 ) 2 Now, we are ready to obtain the explicit form of the collision integral. We start with the inelastic contribution and perform the following three steps: (1) substitute Eq. (4.63a) and (4.64) into Eq. (4.8a); (2) average over the fields φ1,2 with the help of Eq. (4.30); (3) perform the temporal transformation (4.65) of the result. As a result we obtain with the help of Eqs. (4.66) and (4.67) the following form of the collision integral: (4.66) The transformed functions have the symmetry property (hereafter we omit the K superscript for brevity since we are only dealing with the Keldysh function) Stin g K (t, ǫ; r) = − i 2 d2 r1 dω K D (t, ω; r, r1 ) [ D(ω; r, r1 ) + D(−ω; r1 , r) ] 2π × [ g (t, ǫ; n, r) + i 2τ dr1 dr2 n − g (t, ǫ − ω; n, r) n ] dω DR (t, ω; r1 , r2 ) − DA (t, ω; r2 , r1 ) [ D(ω; r, r1 ) D(−ω; r, r2 ) − D(ω; r, r1 )D(−ω; r, r2 ) ] 2π × g (t, ǫ + ω; n, r) n g (t, ǫ; n, r) 27 n (4.68) where the angular averaging of the diffusons is defined after Eqs. (4.6). Now, we have to express the bosonic propagator in terms of the fermionic polarization operators. The polarization operators are given by Eqs. (4.47), where we now substitute Eqs. (4.60), (4.63a), and (4.64). After the temporal transformation (4.65) we find ΠR (ω; t, r1 , r2 ) = ΠA (−ω; t, r2 , r1 ) = ν δ(r1 − r2 ) + i D(ω; r1 , r2 ) 4 dǫ [ g (t, ǫ; n, r) n − g (t, ǫ − ω; n, r) n ] = ν [δ(r1 − r2 ) + iω D(ω; r1 , r2 ) ] ΠK (t, ω; r1 , r2 ) = iν 4τ × (4.69b) dr D(ω; r, r1 ) D(−ω; r, r2 ) − D(ω; r, r1 )D(−ω; r, r2 ) dǫ [ g (t, ǫ + ω; n, r) g (t, ǫ; n, r) n (4.69a) n (4.70) − 4] The last step in the calculation of the interaction propagators is to solve Eq. (4.32) with the polarization operators Eq. (4.69a). This gives DR,A in the form given by Eq. (3.28a) and for the Keldysh component we obtain DK = DR ΠK DA . (4.71a) Also we can relate the difference of the retarded and advanced propagators which enters the collision integral Eq. (4.68) to the polarization operators: DR − DA = DR ΠR − ΠA DA . (4.71b) To obtain the final form of the inelastic part of the collision integral Eqs. (4.8) we need to substitute Eq. (4.71) into Eq. (4.68), while using Eq. (4.69a) for ΠR − ΠA . In addition, we note that 2 τ D(ω; q)D(−ω; −q) − D(ω; q) D(−ω; −q) = D(ω; q) + D(−ω; −q) . Finally, we introduce the gauge invariant distribution function f as f (ǫ, t; n, r) = 1 1 − g(ǫ + ϕext (r), t; n, r) 2 4 (4.72) and obtain Eqs. (4.8). The calculation of the elastic part of the collision integral is completely analogous. We substitute Eqs. (4.63a)– (4.63c) and Eq. (4.60) into Eqs. (4.57) and average over fluctuating fields with the help of Eq. (4.30). After the temporal transformation (4.65) we find F R (ǫ, t; n1 , n2 , r) = F A (ǫ, t; n1 , n2 , r)∗ = × i 4τ dω 2π dr1 dr2 DR (ω, r1 , r2 ) dn3 dn4 [D(ω; n3 , n2 , r2 , r) − D(ω; n3 , n1 , r2 , r)] D(ω; n1 , n4 , r, r1 ) 2π 2π × g(t, ǫ − ω; n′ , r) +2n4 n′ g(t, ǫ − ω; n1 , r) + (r1 − r) ∇ + Deriving Eq. (4.73) we use the fact that (4.73a) n′ ∂ Aext ∂ ∂t ∂ǫ dωDR (ω)D(ω) = 0. 28 (4.73b) n′ g(t, ǫ − ω; n′ , r) n′ (4.73c) We substitute Eqs. (4.73) and (4.61) into Eq. (4.58). We expand the result into angular harmonics. The zeroth angular harmonic vanishes because of the conservation law (4.58), and the first harmonic gives (we write only interaction correction to the collision integral) Stel (t; ǫ; r) = 2 τ dω 2π αβ nα K1 (ω) nβ g(t; ǫ, r, n) n g(t; ǫ − ω, r, n) αβ + nα K0 (ω) nβ g(t; ǫ − ω, r, n) + nα Lαβ (ω) 0 g(t; ǫ, r, n) 2 n n ∇β + g(t; ǫ, r, n) n (4.74a) n (4.74b) ∂[Aext ]β ∂ ∂t ∂ǫ g(t; ǫ − ω, r, n) n . (4.74c) Here the kernels K and L are given by Eq. (4.6). Each labeled separately term in Eqs. (4.74) corresponds to ones in Eqs. (4.73) and in Eqs. (4.63) so the origin of terms can be easily traced. fields in the Gorkov-Nambu space. We will not discuss this question further in the present paper. Finally, we use the gauge invariant distribution function (4.72) instead of g and we arrive to Eqs. (4.5). Closing this section, we remark that the above treatment can be easily generalized to include other channels as well as the higher angular harmonics of the Fermi liquid constant. For the latter task one has to introduce angle dependent auxiliary fields φ1,2 (r, n, t) and use ρ F0 → F ρ (n1 n2 ). The triplet channel requires introduction of the couˆ pling of the form h1,2 (r, t)σ, where σ j , j = x, y, z are ˆ the Pauli matrices in the spin space, and h1,2 (r, t) are the auxiliary fields. Accordingly, each bosonic propagator from Eq. (4.30) becomes a 3 × 3 matrix. The equation (4.32) retains the same form with the matrix multiplication in Keldysh and spin spaces implied. The equation (4.33) becomes R A σ [D0 ]ij = [D0 ]ij = −F0 δij δ(r1 − r2 )δ(t1 − t2 ), V. CONCLUSIONS This paper is an attempt to consistently describe the effect of electron-electron interaction on longitudinal conductivity of disordered 2D electron gas at T ≪ EF . Our results are valid for an arbitrary relation between T and h ¯ /τ and are summarized in Section II B. At low temperatures T τ ≪ ¯ we reproduce the logarithmically dih vergent Altshuler-Aronov correction. At higher temperatures T τ > ¯ , i.e. in the ballistic region, we found the h linear temperature dependence in accord with Refs. 6,27. However, even the sign of the slope of this dependence depends on the strength of electron-electron interaction in contradiction to the results of Refs. 6,27 (see Sections II and III F for discussions of this discrepancy). We deliberately did not compare the theory with experimental data, postponing this comparison until the publication of theoretical results for Hall conductivity and magneto-resistance in the parallel field. For comparison with data obtained for Si-MOSFET samples the valley degeneracy should be taken into account (the degenaracy may increase the numerical factor in Eq. (2.16c) by as much as a factor of 5 in the case of low intervalley scattering). We also relegate the corresponding discussion to a separate publication. Finally, we derived a kinetic equation to describe the effect of electron-electron interaction at arbitrary T τ . The advantage of this approach is that it turns out to be more convenient for practical calculations of transport properties in magnetic field as well as thermal transport properties. (4.75) and Eq. (4.47) is modified to ΠR (1, 2) = ΠA (2, 1) ij ji =ν dθ 2π δ12 δji + (4.76) π Trσi δg K (t1 , t1 ; n, r1 ) 4δhj (t2 , r2 ) 1 φ ; ΠK (1, 2) = πν dθ Trσi δg K (t1 , t1 ; n, r1 ) + Trσi δg Z (t1 , t1 ; n, r1 ) 2π 4δhj (t2 , r2 ) 2 φ where trace is performed in spin space. Further derivation consists of a repetition of the steps described in this section, and in the absence of the spin structure of the distribution function, fij = δij f , results in Eqs. (4.9) and (4.10). The spin-orbit interaction or Zeeman splitting by external magnetic field slightly changes the results, but we will postpone the corresponding analysis until the future publication14 . Finally, the Cooper channel interaction (3.20) can be treated in the same manner by introducing auxiliary ACKNOWLEDGMENTS We are grateful to B.L. Altshuler, M.E. Gershenson, L.I. Glazman, A.I. Larkin, and M.Yu. Reizer for nu29 We perform the analytic continuation in each term (A1a) and (A1b) separately. In Eq. (A1a) we use the standard procedure merous discussions. I.A. acknowledges support of the Packard Foundation. APPENDIX: (. . .) = T ǫn 1 4πi dǫ tanh ǫ (. . .), 2T (A2) C where integration contour is shown on Fig. 23. We deform this contour to form C1 , use the facts that tanh(ǫ + iΩn )/2T = tanh ǫ/2T , and G(ǫ ± i0) = GR(A) (ǫ) and we obtain iΩn Jα G(iǫn + iΩn )Jβ G(iǫn ) M1 (iΩn ) = T ǫn C = C1 dǫ ǫ Jα G(ǫ + iΩn )Jβ GR (ǫ) − GA (ǫ) tanh 4πi 2T +Jα GR (ǫ) − GA (ǫ) Jβ G(ǫ − iΩn ) FIG. 23. Integration contours for analytic continuation of Eqs. (A1). In the form (A3) frequency Ω is present only in functions which may have singularities only on the real axis, so that the required analytic continuation can be easily performed: In this Appendix we show in some detail the procedure of analytical continuation that leads to the expression for the interaction correction Eq. (3.3) to the conductivity in terms of exact Green’s function of noninteracting disordered system. The structure of the current correlator is M1 (ω) = M1 (iΩn → ω + i0) ǫ dǫ Jα GR (ǫ + ω)Jβ GR (ǫ) − GA (ǫ) tanh = 4πi 2T +Jα GR (ǫ) − GA (ǫ) Jβ GA (ǫ − ω) 1/T dτ Tτ ˆα (τ )ˆβ (0) e j j iΩn τ = −T (A4) Thus one obtains for the quantity entering into conductivity (3.1) 0 −T (A3) Jα G(iǫn + iΩn )Jβ G(iǫn ) (A1a) M1 (ω) ω dǫ ǫ = Re tanh Jα GA (ǫ)Jβ ∂ǫ GA (ǫ) 2π 2T dǫ d ǫ Jα GR (ǫ)Jβ GA (ǫ) + tanh 4π dǫ 2T ǫn N1 = − lim Im ω→0 Jα G(iǫn + iΩn )Γβ (iǫn + iΩn , iǫn )G(iǫn ), (A1b) ǫn where ǫn = πT (2n + 1) is the fermionic Matsubara frequency, G(iǫn ) is the exact Green’s function of the interacting system (diagrams 1 and 2 on Fig. 9 are the first order correction to the Green’s function), Γβ (iǫn +iΩn , iǫn ) is the vertex function (not to be confused with disorder averaged interaction vertex Γ from Sec. III). Diagrams 3 on Fig. 9 are the first order correction to the vertex function. The current operator is defined in Eq. (3.6). Note, that we omit the spatial coordinates and the integration whenever it should cause no confusion. (A5) Equation (A5) can be further simplified for the calculation of the symmetric part of the conductivity sym N1 = Re dǫ 8π d ǫ tanh dǫ 2T (A6) × − Jα GA (ǫ)Jβ GA (ǫ) + Jα GR (ǫ)Jβ GA (ǫ) + (α ↔ β) Term (A1b) is considered analogously. We find similarly to Eq. (A3) Jα G(iǫn + iΩn )Γβ (iǫn + iΩn , iǫn )G(iǫn ) M2 (iΩn ) = T ǫn = dǫ ǫ Jα G(ǫ + iΩn ) Γβ (ǫ + iΩn , ǫ + i0)GR (ǫ) − Γβ (ǫ + iΩn , ǫ − i0)GA (ǫ) tanh 4πi 2T +Jα GR (ǫ)Γβ (ǫ + i0, ǫ − iΩn ) − GA (ǫ)Γβ (ǫ − i0, ǫ − iΩn ) G(ǫ − iΩn ) (1.7) Using analytic properties of the Green’s function and the vertex function, we perform the analytic continuation and 30 obtain M2 (ω) = M2 (iΩn → ω + i0) ǫ dǫ Jα GR (ǫ + ω) Γβ (ǫ + ω + iδ1 , ǫ + iδ2 )GR (ǫ) − Γβ (ǫ + ω + i0, ǫ − i0)GA (ǫ) tanh = lim+ lim+ 4πi 2T δ1 →0 δ2 →0 +Jα GR (ǫ)Γβ (ǫ + i0, ǫ − ω − i0) − GA (ǫ)Γβ (ǫ − iδ2 , ǫ − ω − iδ1 ) GA (ǫ − ω) . (1.8) In the appropriate frequency limit, we find N2 = − lim Im ω→0 dǫ 4π + M2 (ω) ω d ǫ tanh dǫ 2T = Re dǫ ǫ ∂ tanh Jα GA (ǫ) 2π 2T ∂ǫ1 R dǫ 4π Jα G (ǫ)Γβ (ǫ + i0, ǫ − i0)G (ǫ) [Γ(iǫn , iǫm )β ]12 = d ǫ tanh dǫ 2T (1.10) δ1 →0+ D12 (iΩm )G12 (iǫn − iΩm ) lim Γβ (ǫ − iδ2 , ǫ1 − iδ1 )= − δ2 →0+ +i +i Γβ (ǫ + i0, ǫ − i0)= − +i +i (1.12) D12 (Ωm ) [G(iǫn − iΩm )Jβ G(iǫm − iΩm )]12 . After analytic continuation similar to that in the derivation of Eq. (A4) we find dΩ Ω A ImD12 (Ω) GA (ǫ − Ω) (1.13) coth 12 2π 2T ǫ−Ω A dΩ tanh D12 (Ω) GA (ǫ − Ω) − GR (ǫ − Ω) 12 12 4π 2T ΣA (ǫ) = − 12 For brevity we consider only the “Fock” contribution of Fig. 9 b: lim T Ωm +Jα GR (ǫ)Jβ GA (ǫ)ΣA (ǫ)GA (ǫ) + (α ↔ β) . Ωm (1.9) where D is the bosonic propagator, and we restored the notation for spatial coordinates. In the same order × − Jα GA (ǫ)ΣA (ǫ)GA (ǫ)Jβ GA (ǫ) Σ(iǫn )12 = T ǫ1 =ǫ lim Γβ (ǫ − iδ2 , ǫ1 − iδ1 )GA (ǫ1 ) A Further calculation requires specification of the form of the self-energy and the vertex function. We have to find both in the first order in interaction propagator, and expand Eq. (A6) up to the first order: sym δN1 = Re lim δ1 →0+ δ2 →0+ +i (1.11) and for the vertex function we have two cases: Ω dΩ A ImD12 (Ω) GA (ǫ − Ω)Jβ GA (ǫ1 − Ω) coth 2π 2T 12 dΩ ǫ−Ω A tanh D12 (Ω) GA (ǫ − Ω) − GR (ǫ − Ω) Jβ GA (ǫ1 − Ω) 4π 2T 12 ǫ1 − Ω A dΩ tanh D12 (Ω) GR (ǫ − Ω)Jβ GA (ǫ1 − Ω) − GR (ǫ1 − Ω) 4π 2T 12 dΩ Ω A ImD12 (Ω) GR (ǫ − Ω)Jβ GA (ǫ − Ω) coth 2π 2T 12 dΩ ǫ−Ω R A R tanh D12 (Ω) G (ǫ − Ω) − G (ǫ − Ω) Jβ GA (ǫ − Ω) 4π 2T dΩ ǫ−Ω A tanh D12 (Ω) GR (ǫ − Ω)Jβ GA (ǫ − Ω) − GR (ǫ − Ω) 4π 2T (1.14a) 12 12 (1.14b) We now substitute Eq. (1.13) into Eq. (1.10). We use the fact that the combination containing only retarded or only advanced Green’s functions vanish upon the disorder averaging. Moreover, the average of the combinations like G(ǫ)G(ǫ − Ω1 ) . . . G(ǫ − ΩN ) does not depend on the energy ǫ, which enable us to perform the integration over ǫ using dǫ tanh ǫ d ǫ−Ω d tanh = −2 2T dǫ 2T dΩ We find using DA (Ω) = [DA (−Ω)]∗ 31 Ω coth Ω 2T . dΩ d 8π 2 dΩ sym δN1 = Im Ω coth Ω 2T A D12 (Ω) (1.15a) × Jα GA (ǫ)GR (ǫ − Ω)GA (ǫ)Jβ GA (ǫ) − Jα GA (ǫ)GR (ǫ − Ω)GA (ǫ)Jβ GR (ǫ) − Jα GR (ǫ)GR (ǫ − Ω)GR (ǫ)Jβ GA (ǫ) 12 12 12 Ω dΩ 4π 2 2T sinh2 − Ω 2T A ImD12 (Ω) Re Jα GA (ǫ)GA (ǫ − Ω)GA (ǫ)Jβ GR (ǫ) + (α ↔ β). 12 (1.15b) The same manipulations are performed with substituition of Eqs. (1.14) into Eq. (1.9). One finds for the symmetrized part sym δN2 = Im dΩ d 8π 2 dΩ × GA (ǫ)Jα GA (ǫ) − Ω dΩ 8π 2 2T sinh2 Ω 2T A D12 (Ω) GR (ǫ − Ω)Jβ GR (ǫ − Ω) 12 Ω 2T Ω coth A ImD12 (Ω) (1.16a) − 2 GR (ǫ)Jα GA (ǫ) 21 GR (ǫ)Jα GA (ǫ) 12 GR (ǫ − Ω)Jβ GR (ǫ − Ω) GA (ǫ − Ω)Jβ GR (ǫ − Ω) 12 21 + (α ↔ β). 21 (1.16b) Total correction to the conductivity is just N1 + N2 . In the Hartree-Fock approximation DA = −V (q) and we obtain Eq. (3.3). In the case for the stronger interaction terms (1.15a) and (1.16a) are added to produce Eq. (3.24) and terms (1.15b), (1.16b) give rise to the inelastic or so-called dephasing term: dΩ Ω 2 8π 2T sinh2 deph δσαβ = − + GR (ǫ)Jα GA (ǫ) 12 Ω 2T A ImD12 (Ω) Re GA (ǫ − Ω)Jβ GR (ǫ − Ω) In our leading approximation in 1/EF τ this term vanishes, see Fig. 24. The role of this term in the temperature dependence of weak localization correction is discussed in detail in Ref. 16. jα R R R jα R jβ 21 + (α ↔ β). (1.17) (1985). A. Gold and V.T. Dolgopolov, Phys. Rev. B 33, 1076 (1986). 7 S. Das Sarma and E.H. Hwang, Phys. Rev. Lett. 83, 164 (1999). 8 B.L. Altshuler, A.G. Aronov, M.E. Gershenson, and Yu.V. Sharvin, Sov. Sci. A. Phys. 9, 223 (1987). 9 G. Bergman, Physics Reports, 107, 1 (1984). 10 K.M. Cham and R.G. Wheeler, Phys. Rev. Lett. 44, 1472 (1980). 11 For a review see B.L. Altshuler, D.L. Maslov, and V.M. Pudalov, Physica E, 9 (2), 209 (2001). 12 For a recent review and references see E. Abrahams, S. V. Kravchenko, and M. P. Sarachik, Rev. Mod. Phys. 73, 251 (2001). 13 σ Logarithmic renormalization of F0 , see Ref. 2,3 will not be relevant for the further discussion. 14 G. Zala, B.N. Narozhny, and I.L. Aleiner (in preparation). 15 A.M. Rudin, I.L. Aleiner, and L.I. Glazman, Phys. Rev. B, 55, 9322 (1997). 16 I.L. Aleiner, B.L. Altshuler, and M.E. Gershenson, Waves Random Media 9, 201 (1999). 17 C. Kittel, “Quantum Theory of Solids” (Wiley, New York, 1963). 18 L.D. Landau, I.M. Lifshits, “Quantum Mechanics” (Pergamon, New York, 1977). 19 I.S. Gradstein I.M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, San Diego, 1994). 20 K.A. Matveev, D. Yue, and L. I. Glazman, Phys. Rev. Lett. 6 R jβ A 2Jα GA (ǫ)GA (ǫ − Ω)GA (ǫ)Jβ GR (ǫ) 12 0 A FIG. 24. Cancellation of inelastic term (1.17) in the leading ladder approximation. 1 B.L. Altshuler and A.G. Aronov in Electron-Electron Interactions in Disordered Systems, eds. A.L. Efros, M. Pollak (North-Holland, Amsterdam, 1985). 2 A.M. Finkelstein, Zh. Eksp. Teor. Fiz. 84, 168 (1983) [Sov. Phys. JETP 57, 97 (1983)]; Z. Phys. B 56, 189 (1984). 3 C. Castellani, C. Di Castro, P. A. Lee, and M. Ma, Phys. Rev. B 30, 527 (1984); ibid 30, 1596 (1984); C. Castellani, C. Di Castro, P.A. Lee, ibid 57, R9381 (1998). 4 F. Stern, Phys. Rev. Lett. 44, 1469 (1980). 5 F. Stern and S. Das Sarma, Solid State Electron., 28, 158 32 lani, Phys. Rev. Lett. 84, 4779 (2000). D. Belitz, T.R. Kirkpatrick, and T. Vojta, Phys. Rev. B 55, 9452 (1997); D. Belitz, T.R. Kirkpatrick, A.J. Millis, and T. Vojta, Phys. Rev. B 58, 14155 (1998). 26 We are grateful to A.I. Larkin for calling our attention to σ the momentum dependence of F0 . 27 M.Yu. Reizer, Phys. Rev. B 57, 12338 (1998). 28 L.V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1945 (1964) [Sov. Phys. JETP 20, 1018 (1964)]. 29 G. Eilenberger, Z. Phys. Bd. 214, 195 (1968). 30 A.I. Larkin and Y.N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 55, 2262 (1968) [Sov. Phys. JETP 28, 1200 (1969)]. 31 A.I. Larkin and Y.N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 73, 299 (1977) [Sov. Phys. JETP 46, 155 (1977)]. 71, 3351 (1993); D. Yue, L.I. Glazman, and K.A. Matveev, Phys. Rev. B 49, 1966 (1994); 21 L.D. Landau, Zh. Eksp. Teor. Fiz. 30, 1058 (1956); ibid 32, 59 (1957). 22 A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski, “Methods of quantum field theory in statistical physics” (Prentice-Hall, Englewood Cliffs, N.J., 1963). 23 B. Tanatar and D. M. Ceperley, Phys. Rev. B 39, 5005 (1989) 24 This statement contradicts conclusions of Q. Si, C.M. Varma, Phys. Rev. Lett. 81, 4951 (1998). We believe that this paper mishandles the Fermi liquid renormalizations, ρ which amounts to omission of F0 in the numerator of Eq. (3.29), see also comment by P. Schwab and C. Castel- 25 33