PHYSICAL REVIE%' B VOLUME 34, NUMBER 8 15 OCTOBER 1986 Effective Landau theory for disordered interacting electron systems: SpeciTic-heat behavior C. Castellani Dipartimento di Fisiea, Universita del/'Aquila, I-67100 L'Aquila, Italy and Dipartimento di Fisiea, Universita "La Sapienza, " I-00185 Roma, Italy C. Di Castro Dipartimento di Fisica, Universita "La Sapienza, " I-00185' Roma, Italy (Received 10 February 1986; revised manuscript received 8 August 1986) The frequency renormalization parameter of the generalized nonlinear o model introduced to describe the interacting disordered electron system is identified in terms of the specific heat. This allows us to complete the effective Landau Fermi-liquid picture for this system and to give the asymptotic behavior of the electronic specific heat in the various universality classes of the metal- insulator transition. The electron-electron interaction in a disordered medi- um has been found' to introduce relevant corrections to the Landau theory of the normal Fermi liquid. Many of these corrections are logarithmically divergent in two di- mensions as the temperature T decreases to zero. ' Since that discovery, z theorists have been looking for a renormalization-group approach which could sum the correction terms in the physical quantities to obtain power-law behaviors near the metal-insulator transition, at least in the a expansion (a d-2, d being the dimen- sionality). Theorists have had to face the problem of deriving the renormalization group equations for the couplings describ- ing the electron-electron interaction in the different chan- nels. This problem was first solved by Finkelstein in the simplified case where particle-particle (hole-hole) chan- nels are suppressed, so that only the singlet (I, ) and trip- let (I, ) interaction amplitudes must be considered. Finkelstein' mapped the interacting disordered electron system into an effective nonlinear cr model. As in the stan- dard weak-localization regime, ' the expansion parameter of the model is the dimensionless inverse conductivity r A'/(2')zvaD, where D is the diffusion coefficient, va is the bare density of states, and A is the ultraviolet cutoff of order (Doro) ', ra being the bare scattering time. To- gether with I „ I „and r, the effective nonlinear o model is specified by the frequency or temperature renormalization parameter Z which is needed to take care, in a consistent way, of the corrections introduced by the electron-electron interaction in the diffusive mode (the ladder in the particle-hole channels). s s We shall show in this report that, at least to lowest order in a and to one-loop expansion, Z is related to the electron- ic specific-heat renormalization induced by the interaction in the presence of disorder Z s CP' j T (1) yo' ' where yu (2' vu/3) is the coefficient of the linear term of the Landau specific heat in the absence of disorder (cf' ycT). This completes the identification of the renor- C$I ZC fp Z 'X Z2 L 8n tin Zzo ' 8p 8p (3) where the renormalization parameters Z and Zz are ex- pressed in terms of the renormalized couplings of Finkelstein's nonlinear o model4 and can be analyzed by the renormalization-group approach. To identify y/yu with Z we shall first evaluate the lead- ing corrections to cy in terms of the renormalized parame- ters of the effective nonlinear o model to lowest order in r, thus generalizing the perturbative results of Ref. 9. For this purpose we use the standard procedure of multiplying the interaction amplitudes appearing in the Lagrangian of the effective nonlinear o model" by a parameter g. By in- tegrating the derivative with respect to rl of the logarithm of the grand partition function in the interval (0, 1), we ob- malization parameters of the effective nonlinear o model in terms of physical quantities. It has, in fact, been showns s that, while Z renormalizes the frequency in the diffusive mode, the two combinations Zi Z — 2vo(I; — I 0) (I u being the static screened Coulomb amplitude) and Z2 Z+vui, renormalize the frequency associated with the density and the spin-fluctuation modes, respec- tively. They can be expressed in terms of the thermo- dynamic density of states Bn/8p and the spin susceptibility z: Zi ~ Zz 1 8n E (2) vo Ii where Xp pjjvo/2 is the Pauli susceptibility. If one substi- tutes Zi and Zz in Eq. (2) with their bare values Zi 1 — 2vu(I; — I II), Z2 1+ vol, one recovers the standard Landau expressions (8n/8p)~ and X for 8n/8p and X in the absence of disorder. According to the analysis carried out in Refs. 4-8, Zi remains unrenormalized and equal to its initial value Zi. Z2 instead is strongly renor- malized, leading to a pronounced spin susceptibility enhancement in the nonmagnetic impurity case. The problem of disordered interacting electrons may therefore be modeled in terms of a "renormalized" Landau theory 34 5935 1986 The American Physical Society