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arXiv:cond-mat/9611099 · Physical review. B, Condensed matter

Nonanalytic behavior of the spin susceptibility in clean Fermi systems

D. Belitz, T. R. Kirkpatrick, Thomas Vojta

Published Apr 15, 1997• 192 citations
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Abstract

The wave vector and temperature-dependent static spin susceptibility, ${\mathrm{\ensuremath{\chi}}}_{\mathrm{s}}$(Q,T), of clean interacting Fermi systems is considered in dimensions 1\ensuremath{\leqslant}d\ensuremath{\leqslant}3. We show that at zero temperature ${\mathrm{\ensuremath{\chi}}}_{\mathrm{s}}$ is a nonanalytic function of |Q|, with the leading nonanalyticity being |Q${\mathrm{|}}^{\mathrm{d}\mathrm{\ensuremath{-}}1}$ for 13, and ${\mathbf{Q}}^{2}$ln |Q| for d=3. For the homogeneous spin susceptibility we find a nonanalytic temperature dependence ${\mathrm{T}}^{\mathrm{d}\mathrm{\ensuremath{-}}1}$ for 13. We give qualitative mode-mode coupling arguments to that effect, and corroborate these arguments by a perturbative calculation to second order in the electron-electron interaction amplitude. The implications of this, in particular for itinerant ferromagnetism, are discussed. We also point out the relation between our findings and established perturbative results for one-dimensional systems, as well as for the temperature dependence of ${\mathrm{\ensuremath{\chi}}}_{\mathrm{s}}$(Q=0) in d=3.

DOI
10.1103/physrevb.55.9452

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Spin susceptibility of a clean Fermi gas with repulsion in any dimension is considered using a supersymmetric low energy theory of interacting spin excitations and renormalization scheme recently proposed by Aleiner and Efetov. We generalize this method to include the coupling to the magnetic field. As a result, we obtain for the correction $\ensuremath{\delta}\ensuremath{\chi}$ to the Pauli susceptibility a nonanalytic temperature dependence of the form ${T}^{d\ensuremath{-}1}{\ensuremath{\gamma}}_{b}^{2}(T)$ in dimensions d=2d=2d=2, 3, where ${\ensuremath{\gamma}}_{b}(T)$ is an effective ddd-dependent logarithmically renormalized backscattering amplitude. In one dimension, $\ensuremath{\delta}\ensuremath{\chi}$ is proportional to ${\ensuremath{\gamma}}_{b}(T)$, and we reproduce a well known result obtained long ago by a direct calculation.
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.
A long-range interaction via virtual particle-hole pairs between Fermi-liquid quasiparticles leads to a nonanalytic behavior of the spin susceptibility $\ensuremath{\chi}$ as a function of the temperature (T)(T)(T), magnetic field (B)(\mathbf{B})(B), and wave number. In this paper, we study the effect of the Rashba spin-orbit interaction (SOI) on the nonanalytic behavior of $\ensuremath{\chi}$ for a two-dimensional electron liquid. Although the SOI breaks the SU(2) symmetry, it does not eliminate nonanalyticity but rather makes it anisotropic: while the linear scaling of ${\ensuremath{\chi}}_{zz}$ with TTT and ∣B∣|\mathbf{B}|∣B∣ saturates at the energy scale set by the SOI, that of ${\ensuremath{\chi}}_{xx}$ $(={\ensuremath{\chi}}_{yy})$ continues through this energy scale until renormalization of the electron-electron interaction in the Cooper channel becomes important. We show that the renormalization group flow in the Cooper channel has a nontrivial fixed point and study the consequences of this fixed point for the nonanalytic behavior of $\ensuremath{\chi}$. An immediate consequence of SOI-induced anisotropy in the nonanalytic behavior of $\ensuremath{\chi}$ is a possible instability of a second-order ferromagnetic quantum phase transition with respect to a first-order transition to an XYXYXY ferromagnetic state.
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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.