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arXiv:cond-mat/0506083 · Physical Review B

Thermodynamics of a Fermi liquid in a magnetic field

Joseph J. Betouras, Dmitri V. Efremov, Andrey V. Chubukov

Published Sep 14, 2005• 42 citations
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Abstract

We extend previous calculations of the nonanalytic terms in the spin susceptibility ${\ensuremath{\chi}}_{s}(T)$ and the specific heat $C(T)$ to systems in a magnetic field. Without a field, ${\ensuremath{\chi}}_{s}(T)$ and $C(T)∕T$ are linear in $T$ in two dimensions (2D), while in 3D, ${\ensuremath{\chi}}_{s}(T)\ensuremath{\propto}{T}^{2}$ and $C(T)∕T\ensuremath{\propto}{T}^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}T$. We show that in a magnetic field, the linear in $T$ terms in 2D become scaling functions of ${\ensuremath{\mu}}_{B}H∕T$. We present explicit expressions for these functions and show that at high fields ${\ensuremath{\mu}}_{B}H⪢T$, ${\ensuremath{\chi}}_{s}(T,H)$ scales as $\ensuremath{\mid}H\ensuremath{\mid}$. We also show that in 3D, ${\ensuremath{\chi}}_{s}(T,H)$ becomes nonanalytic in a field and at high fields scales as ${H}^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\ensuremath{\mid}H\ensuremath{\mid}$.

DOI
10.1103/physrevb.72.115112

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We use scaling and renormalization-group techniques to analyze the leading nonanalyticities in a Fermi liquid. We show that a physically motivated scaling hypothesis reproduces the results known from perturbation theory for the density of states, the density-of-states fluctuations, the specific heat, the spin susceptibility, and the nematic magnetic susceptibility. We also discuss the absence of nonanalytic terms in the density susceptibility. We then use a recent effective field theory for clean electron systems to derive the scaling hypothesis by means of renormalization-group techniques. This shows that the exponents (although not the prefactors) of the nonanalyticities that were previously derived by means of perturbative techniques are indeed exact, and can be understood as the leading corrections to scaling at the stable Fermi-liquid fixed point.
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.
We consider the nonanalytic temperature dependences of the specific heat coefficient, C(T)/T, and spin susceptibility, chi(s)(T), of 2D interacting fermions beyond the weak-coupling limit. We demonstrate within the Luttinger-Ward formalism that the leading temperature dependences of C(T)/T and chi(s)(T) are linear in T, and are described by the Fermi liquid theory. We show that these temperature dependences are universally determined by the states near the Fermi level and, for a generic interaction, are expressed via the spin and charge components of the exact backscattering amplitude of quasiparticles. We compare our theory to recent experiments on monolayers of He3.
We revisit the issue of the leading nonanalytic corrections to the temperature dependence of the specific heat coefficient, $\ensuremath{\gamma}(T)=C(T)∕T$, for a system of interacting fermions in three dimensions. We show that the leading temperature dependence of the specific heat coefficient $\ensuremath{\gamma}(T)\ensuremath{-}\ensuremath{\gamma}(0)\ensuremath{\propto}{T}^{3}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}T$ comes from two physically distinct processes. The first process involves a thermal excitation of a single particle-hole pair, whose components interact via a nonanalytic dynamic vertex. The second process involves an excitation of three particle-hole pairs which interact via the analytic static fixed-point vertex. We show that the single-pair contribution is expressed via the backscattering amplitude of quasiparticles at the Fermi surface. The three-pair contribution does not have a simple expression in terms of scattering in particular directions. We clarify the relation between these results and previous literature on both 3D and 2D systems, and discuss the relation between the nonanalyticities in $\ensuremath{\gamma}$ and those in spin susceptibilities.
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 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.