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Journal of the Physical Society of Japan

Correlation Effects on the Spin Susceptibility in Two Dimensional Fermion Systems

Dai S. Hirashima, Hideaki Takahashi

Published Nov 1, 1998• 18 citations
Publisher

Abstract

Wavevector dependence and temperature dependence of magnetic susceptibility of clean interacting fermion systems in two dimensions are investigated by a second order perturbation theory. It is found that the static susceptibility varies linearly with wavevector q in the long wavelength limit at T =0. It is also found that the uniform susceptibility varies linearly with temperature at low temperatures.

DOI
10.1143/jpsj.67.3816

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Wavevector dependence and temperature dependence of magnetic susceptibility of clean interacting fermion systems in two dimensions are investigated by a second order perturbation theory. It is found that the static susceptibility varies linearly with wavevector q in the long wavelength limit at T =0. It is also found that the uniform susceptibility varies linearly with temperature at low temperatures.
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.
We calculate the basic parameters of the Fermi liquid: the scattering vertex, the Landau interaction function, the effective mass, and physical susceptibilities for a model of two-dimensional (2D) fermions with a short-ranged interaction at nonzero temperature. The leading temperature dependences of the spin components of the scattering vertex, the Landau function, and the spin susceptibility are found to be linear. T-linear terms in the effective mass and in the "charge-sector" quantities are found to cancel to second order in the interaction, but the cancellation is argued not to be generic. The connection with previous studies of the 2D Fermi-liquid parameters is discussed.
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.
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 compute the diagrammatic expansion of the particle-particle susceptibility via algorithmic Matsubara integration and compute the correlated pairing susceptibility in the thermodynamic limit of the two-dimensional Hubbard model. We study the static susceptibility and its dependence on the pair momentum q\mathbf{q}q for a range of temperature, interaction strength, and chemical potential. We show that ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave pairing is expected in the model in the $U/t\ensuremath{\rightarrow}{0}^{+}$ limit from direct perturbation theory. From this, we identify key second- and third-order diagrams that support pairing processes and note that the diagrams responsible are not a part of charge or spin susceptibility expansions. We find two key components for pairing at momenta (0,0) and $(\ensuremath{\pi},\ensuremath{\pi})$ that can be well fit as separate bosonic modes. We extract amplitudes and correlation length scales where we find a predominantly local $(\ensuremath{\pi},\ensuremath{\pi})$ pairing and nonlocal q=(0,0)\mathbf{q}=(0,0)q=(0,0) pairs and present the relative weights of these modes for variation in temperature, doping, and interaction strength.
We calculate, using perturbative calculations and Ward identities, the basic parameters of the Fermi liquid: the scattering vertex, the Landau interaction function, the effective mass, specific heat, and physical susceptibilities for a model of two-dimensional (2D) fermions with a short-ranged interaction at nonzero temperature. The leading temperature dependence of the spin components of the scattering vertex, the Landau function, and the spin susceptibility is found to be linear. We find that the standard T=0T=0T=0 relationships for a Galilean-invariant Fermi liquid are violated by finite-temperature terms. The coefficients in the temperature corrections to these relationships involve a subtle interplay between contributions from small and large $(\ensuremath{\sim}{2k}_{F})$ momentum processes. A connection with previous studies of the 2D Fermi-liquid parameters is discussed. We conclude that the linear leading temperature dependence of the parameters is a generic feature of the 2D Fermi liquid.