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Physical review. B, Condensed matter

Logarithmic temperature contributions to long-wavelength spin susceptibility from particle-particle scattering

B. Sriram Shastry, Subodh R. Shenoy

Published Sep 1, 1979• 1 citation
Publisher

Abstract

Based on a second-order perturbation calculation with a $\ensuremath{\delta}$-function interaction, Geldart and Rasolt have recently shown that the long-wavelength static spin susceptibility is nonanalytic in $T$ at low temperatures. We find the nonanalyticity is suppressed by the summation of particle-particle scattering terms to all orders. Commonly held analyticity assumptions regarding the long-wavelength and low-temperature behavior of the susceptibility are thus not invalidated. We suggest that for the nonuniform susceptibility, a Kohn-Luttinger-Ward-like equivalence between the zero- and low-temperature perturbation series exists between classes of terms, rather than order by order.

DOI
10.1103/physrevb.20.2183

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Based on a second-order perturbation calculation with a \ensuremathδ\ensuremath{\delta}\ensuremathδ-function interaction, Geldart and Rasolt have recently shown that the long-wavelength static spin susceptibility is nonanalytic in TTT at low temperatures. We find the nonanalyticity is suppressed by the summation of particle-particle scattering terms to all orders. Commonly held analyticity assumptions regarding the long-wavelength and low-temperature behavior of the susceptibility are thus not invalidated. We suggest that for the nonuniform susceptibility, a Kohn-Luttinger-Ward-like equivalence between the zero- and low-temperature perturbation series exists between classes of terms, rather than order by order.
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.
The analytical structure of the wave-number-dependent magnetic susceptibility $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}})$ is studied at small wave number $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ and low temperature. It is found to exhibit a nonanalytic behavior in contrast to commonly assumed forms for $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}})$ of interacting systems. The importance and implications of these results are discussed for the cases of short-range point interactions and long-range Coulomb interactions.
We consider a two-dimensional electron system with Coulomb interaction between particles at a finite temperature TTT. We show that the dynamic Kohn anomaly in the response function at 2kF2{k}_{F}2kF​ leads to a nonanalytic, linear-in-TTT correction to the spin susceptibility, $\ensuremath{\delta}\ensuremath{\chi}(T)=aT$, the same as in systems with short-range interaction. We show that the singularity of the Coulomb interaction at q=0q=0q=0 does not invalidate the expansion of the prefactor aaa in powers of rs{r}_{s}rs​, but makes the expansion nonanalytic.
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