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arXiv:cond-mat/0606389 · Physical Review B

Temperature dependence of the spin susceptibility of a clean Fermi gas with repulsion

Georg Schwiete, K. B. Efetov

Published Oct 17, 2006• 24 citations
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Abstract

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 $d=2$, 3, where ${\ensuremath{\gamma}}_{b}(T)$ is an effective $d$-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.

DOI
10.1103/physrevb.74.165108

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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 \ensuremathδ\ensuremathχ\ensuremath{\delta}\ensuremath{\chi}\ensuremathδ\ensuremathχ to the Pauli susceptibility a nonanalytic temperature dependence of the form Td\ensuremath−1\ensuremathγb2(T){T}^{d\ensuremath{-}1}{\ensuremath{\gamma}}_{b}^{2}(T)Td\ensuremath−1\ensuremathγb2​(T) in dimensions d=2d=2d=2, 3, where \ensuremathγb(T){\ensuremath{\gamma}}_{b}(T)\ensuremathγb​(T) is an effective ddd-dependent logarithmically renormalized backscattering amplitude. In one dimension, \ensuremathδ\ensuremathχ\ensuremath{\delta}\ensuremath{\chi}\ensuremathδ\ensuremathχ is proportional to \ensuremathγb(T){\ensuremath{\gamma}}_{b}(T)\ensuremathγb​(T), and we reproduce a well known result obtained long ago by a direct calculation.
We discuss a recent bosonization method developed to study clean Fermi gases with a repulsion in any dimensions. The method enables one to consider both density and spin excitations. It is demonstrated that due to a non-abelian structure of the effective theory, the spin excitations interact with each other, which leads to new logarithmic in temperature corrections to physical quantities. Using a renormalization group scheme constructed for the effective low energy field theory these logarithms are summed up in all orders. Temperature dependent corrections to the specific heat and spin susceptibility are obtained for all dimensions d=1,2,3.
We suggest a method of calculations for a clean Fermi gas with a repulsion in any dimension. This method is based on writing equations for quasiclassical Green functions and reducing them to equations for collective spin and charge excitations. The spin excitations interact with each other, and this leads to nontrivial physics. Writing the solution of the equations and the partition function in terms of a functional integral over supervectors and averaging over fluctuating fields we come to an effective field theory describing the spin excitations. In some respects, the theory is similar to bosonization but also includes the ``ghost'' excitations, which prevents overcounting of the degrees of freedom. Expansion in the interaction reveals logarithmic in temperature corrections. This enables us to suggest a renormalization group scheme and derive renormalization group equations. Solving these equations and using their solutions for calculating thermodynamic quantities we obtain explicit expression for the specific heat containing only an effective amplitude of the backward scattering. This amplitude has a complicated dependence on the logarithm of temperature, which leads to a nontrivial temperature dependence of the specific heat.
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 Q2{\mathbf{Q}}^{2}Q2ln |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.
A free Fermi gas has, famously, a superconducting susceptibility that diverges logarithmically at zero temperature. In this paper we ask whether this is still true for a Fermi liquid and find that the answer is that it does not. From the perspective of the renormalization group for interacting fermions, the question arises because a repulsive interaction in the Cooper channel is a marginally irrelevant operator at the Fermi liquid fixed point and thus is also expected to infect various physical quantities with logarithms. Somewhat surprisingly, at least from the renormalization group viewpoint, the result for the superconducting susceptibility is that two logarithms are not better than one. In the course of this investigation we derive a Callan-Symanzik equation for the repulsive Fermi liquid using the momentum-shell renormalization group, and use it to compute the long-wavelength behavior of the superconducting correlation function in the emergent low-energy theory. We expect this technique to be of broader interest.
We consider the nonanalytic terms in the spin susceptibility arising as a result of rescattering of pairs of quasiparticles. We emphasize the importance of rescattering in the Cooper channel for the analysis of the temperature dependences in the two-dimensional electron systems in the ballistic regime. In the calculation of the linear in TTT term we use angular harmonics in the Cooper channel, because for each harmonic the interaction amplitude is renormalized independently. We observe, that as a consequence of strong renormalizations in the Cooper ladder, the temperature derivative of the spin susceptibility may change its sign at low temperatures.
Using a recently suggested method of bosonization in an arbitrary dimension, we study the anomalous contribution of the low-energy spin and charge excitations to thermodynamic quantities of a two-dimensional (2D) Fermi liquid. The method is slightly modified for the present purpose such that the effective supersymmetric action no longer contains the high-energy degrees of freedom but still accounts for effects of the finite curvature of the Fermi surface. Calculating the anomalous contribution $\ensuremath{\delta}c(T)$ to the specific heat, we show that the leading logarithmic in temperature corrections to $\ensuremath{\delta}c(T)/{T}^{2}$ can be obtained in a scheme combining a summation of ladder diagrams and renormalization group equations. The final result is represented as the sum of two separate terms that can be interpreted as coming from singlet and triplet superconducting excitations. The latter may diverge in certain regions of the coupling constants, which should correspond to the formation of triplet Cooper pairs.