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Physical review. B, Solid state

Analytical structure of the wave-number-dependent susceptibility of many-fermion systems at low temperature and long wavelength

D. J. W. Geldart, Mark Rasolt

Published Feb 1, 1977• 13 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.15.1523

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Analytical structure of the wave-number-dependent susceptibility of many-fermion systems at low temperature and long wavelength. II

D. J. W. Geldart, Mark Rasolt

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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.

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

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Journal of the Physical Society of Japan•Nov 1, 1998•18 citations
The analytical structure of the wave-number-dependent magnetic susceptibility \ensuremathχ(k\ensuremath→)\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}})\ensuremathχ(k\ensuremath→​) is studied at small wave number k\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}k\ensuremath→​ and low temperature. It is found to exhibit a nonanalytic behavior in contrast to commonly assumed forms for \ensuremathχ(k\ensuremath→)\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}})\ensuremathχ(k\ensuremath→​) 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.
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.
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 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 analytical forms of free-electron susceptibilities ${\ensuremath{\chi}}_{d}(q)$ and their range functions ${\ensuremath{\chi}}_{d}(r)$ are derived at nonzero temperature starting from the Green's function representation by properly evaluating the contributions from the poles of free-electron temperature Green's function for each dimension d=1,d=1,d=1, 2, and 3. The present formalism produces not only ${\ensuremath{\chi}}_{d}(q)$ and ${\ensuremath{\chi}}_{d}(r)$ which show more accurate temperature-dependent behavior than our previous results for d=1d=1d=1 and 3, but also temperature-dependent two-dimensional ${\ensuremath{\chi}}_{2}(q)$ and ${\ensuremath{\chi}}_{2}(r)$ for a wide range of temperature. Our analytical results show that irrespective of dimension, the singular behavior of ${\ensuremath{\chi}}_{d}(q)$ at $q=\ifmmode\pm\else\textpm\fi{}{2k}_{F}$ becomes suppressed at nonzero temperature as the singular points transit to complex wave vectors ${2k}_{0}^{\ifmmode\pm\else\textpm\fi{}},$ and this transition causes ${\ensuremath{\chi}}_{d}(r)$ to be exponentially damped with common damping factor ${e}^{\ensuremath{-}2{\ensuremath{\eta}}_{0}\mathrm{sin}{\ensuremath{\varphi}}_{r}{k}_{F}r}\ensuremath{\sim}{e}^{\ensuremath{-}\ensuremath{\pi}{T}^{\ensuremath{'}}{k}_{F}r}$ for low enough temperature, where the exponent of the damping factor corresponds to an imaginary part of wave vectors ${2k}_{0}^{\ifmmode\pm\else\textpm\fi{}}.$ We also show that the causality relation of the response function is essential in understanding the correct behavior of ${\ensuremath{\chi}}_{d}(q)$ and ${\ensuremath{\chi}}_{d}(r)$ for all dimensions.
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.