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Singular corrections to the Fermi liquid theory

Andrey V. Chubukov, Dmitrii L. Maslov

Published Mar 1, 2003• 1 citation
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Abstract

We show that the singularities in the dynamical bosonic response functions of a two-dimensional Fermi liquid give rise to universal nonanalytic corrections to the results of the Fermi-liquid theory. In particular, we find a ${T}^{2}$ term in the specific heat, linear-in-$T$ terms in the effective mass and in the uniform spin susceptibility ${\ensuremath{\chi}}_{s}(Q=0,T),$ and $|Q|$ term in ${\ensuremath{\chi}}_{s}(Q,T=0).$ The existence of these terms has been the subject of recent controversy, which is resolved in this paper. We present exact expressions for all nonanalytic terms to second order in a generic interaction $U(Q)$ and show that the nonanalytic terms originate exclusively from forward- and backward-scattering of particles with zero total momentum.

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We show that the singularities in the dynamical bosonic response functions of a two-dimensional Fermi liquid give rise to universal nonanalytic corrections to the results of the Fermi-liquid theory. In particular, we find a T2{T}^{2}T2 term in the specific heat, linear-in-TTT terms in the effective mass and in the uniform spin susceptibility \ensuremathχs(Q=0,T),{\ensuremath{\chi}}_{s}(Q=0,T),\ensuremathχs​(Q=0,T), and ∣Q∣|Q|∣Q∣ term in \ensuremathχs(Q,T=0).{\ensuremath{\chi}}_{s}(Q,T=0).\ensuremathχs​(Q,T=0). The existence of these terms has been the subject of recent controversy, which is resolved in this paper. We present exact expressions for all nonanalytic terms to second order in a generic interaction U(Q)U(Q)U(Q) and show that the nonanalytic terms originate exclusively from forward- and backward-scattering of particles with zero total momentum.
We show that the singularities in the dynamical bosonic response functions of a two-dimensional Fermi liquid give rise to universal nonanalytic corrections to the results of the Fermi-liquid theory. In particular, we find a T2{T}^{2}T2 term in the specific heat, linear-in-TTT terms in the effective mass and in the uniform spin susceptibility ${\ensuremath{\chi}}_{s}(Q=0,T),$ and ∣Q∣|Q|∣Q∣ term in ${\ensuremath{\chi}}_{s}(Q,T=0).$ The existence of these terms has been the subject of recent controversy, which is resolved in this paper. We present exact expressions for all nonanalytic terms to second order in a generic interaction U(Q)U(Q)U(Q) and show that the nonanalytic terms originate exclusively from forward- and backward-scattering of particles with zero total momentum.
We consider a system of interacting fermions in two dimensions beyond the second-order perturbation theory in the interaction. It is shown that the mass-shell singularities in the self-energy, arising already at the second order of the perturbation theory, manifest a nonperturbative effect: an interaction with the zero-sound mode. Resumming the perturbation theory for a weak, short-range interaction and accounting for a finite curvature of the fermion spectrum, we eliminate the singularities and obtain the results for the quasiparticle self-energy and the spectral function to all orders in the interaction with the zero-sound mode. A threshold for emission of zero-sound waves leads a nonmonotonic variation of the self-energy with energy (or momentum) near the mass shell. Consequently, the spectral function has a kinklike feature. We also study in detail a nonanalytic temperature dependence of the specific heat $C(T)\ensuremath{\propto}{T}^{2}$. It turns out that although the interaction with the collective mode results in an enhancement of the fermion self-energy, this interaction does not affect the nonanalytic term in C(T)C(T)C(T) due to a subtle cancellation between the contributions from the real and imaginary parts of the self-energy. For a short-range and weak interaction, this implies that the second-order perturbation theory suffices to determine the nonanalytic part of C(T)C(T)C(T). We also obtain a general form of the nonanalytic term in C(T)C(T)C(T), valid for the case of a generic Fermi liquid, i.e., beyond the perturbation theory.
We calculate, using perturbative calculations and Ward identities, the basic parameters of the Fermi liquid: the scattering vertex, the Landau interaction function, the effective mass, specific heat, and physical susceptibilities for a model of two-dimensional (2D) fermions with a short-ranged interaction at nonzero temperature. The leading temperature dependence of the spin components of the scattering vertex, the Landau function, and the spin susceptibility is found to be linear. We find that the standard T=0T=0T=0 relationships for a Galilean-invariant Fermi liquid are violated by finite-temperature terms. The coefficients in the temperature corrections to these relationships involve a subtle interplay between contributions from small and large $(\ensuremath{\sim}{2k}_{F})$ momentum processes. A connection with previous studies of the 2D Fermi-liquid parameters is discussed. We conclude that the linear leading temperature dependence of the parameters is a generic feature of the 2D Fermi liquid.
We calculate the basic parameters of the Fermi liquid: the scattering vertex, the Landau interaction function, the effective mass, and physical susceptibilities for a model of two-dimensional (2D) fermions with a short-ranged interaction at nonzero temperature. The leading temperature dependences of the spin components of the scattering vertex, the Landau function, and the spin susceptibility are found to be linear. T-linear terms in the effective mass and in the "charge-sector" quantities are found to cancel to second order in the interaction, but the cancellation is argued not to be generic. The connection with previous studies of the 2D Fermi-liquid parameters is discussed.
We analyze how the logarithmic renormalizations in the Cooper channel affect the nonanalytic temperature dependence of the specific heat coefficient $\ensuremath{\gamma}(T)\ensuremath{-}\ensuremath{\gamma}(0)=A(T)T$ in a two-dimensional Fermi liquid. We show that A(T)A(T)A(T) is expressed exactly in terms of the fully renormalized backscattering amplitude, which includes the renormalization in the Cooper channel. In contrast to the one-dimensional case, both charge and spin components of the backscattering amplitudes are subject to this renormalization. We show that the logarithmic renormalization of the charge amplitude vanishes for a flat Fermi surface when the system becomes effectively one dimensional.
We analyze potential nonanalytic terms in the Landau diamagnetic susceptibility, ${\ensuremath{\chi}}_{\text{dia}}$, at a finite temperature TTT and/or in-plane magnetic field HHH in a two-dimensional (2D) Fermi liquid. To do this, we express the diamagnetic susceptibility as ${\ensuremath{\chi}}_{\text{dia}}={(e/c)}^{2}{lim}_{Q\ensuremath{\rightarrow}0}{\mathrm{\ensuremath{\Pi}}}_{\ensuremath{\perp}}^{\text{JJ}}(Q)/{Q}^{2}$, where ${\mathrm{\ensuremath{\Pi}}}_{\ensuremath{\perp}}^{\text{JJ}}$ is the transverse component of the static current-current correlator, and evaluate ${\mathrm{\ensuremath{\Pi}}}_{\ensuremath{\perp}}^{\text{JJ}}(Q)$ for a system of fermions with Hubbard interaction to second order in Hubbard UUU by combining self-energy, Maki-Thompson, and Aslamazov-Larkin diagrams. We find that at T=H=0T=H=0T=H=0, the expansion of ${\mathrm{\ensuremath{\Pi}}}_{\ensuremath{\perp}}^{\text{JJ}}(Q)/{Q}^{2}$ in UUU is regular, but at a finite TTT and/or HHH, it contains U2T{U}^{2}TU2T and/or U2∣H∣{U}^{2}|H|U2∣H∣ terms. Similar terms have been previously found for the paramagnetic Pauli susceptibility. We obtain the full expression for the nonanalytic $\ensuremath{\delta}{\ensuremath{\chi}}_{\text{dia}}(H,T)$ when both TTT and HHH are finite, and we show that the H/TH/TH/T dependence is similar to that for the Pauli susceptibility.