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Modern Physics Letters B

A SELF-ENERGY FUNCTIONAL FOR THE INFINITE-DIMENSIONAL HUBBARD MODEL

Y.M. Li, N. d’Ambrumenil

Published Dec 10, 1992• 7 citations
Publisher

Abstract

We present an approximate self-energy functional for the infinite-dimensional Hubbard model. This functional is a natural extension of the exact solution of the Falicov-Kimball model to the spin-symmetric case, and is exact in the uncorrelated and atomic limits. Using the functional we calculate the susceptibility and the specific heat for the Lorentzian density of states. We find that the susceptibility crosses over smoothly from that expected for an uncorrelated state with antiferromagnetic fluctuations to a Fermi liquid state at low temperature via a Kondo-type anomaly. The specific heat shows a peak at the corresponding Kondo temperature.

DOI
10.1142/s0217984992001526

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We present an approximate self-energy functional for the infinite-dimensional Hubbard model. This functional is a natural extension of the exact solution of the Falicov-Kimball model to the spin-symmetric case, and is exact in the uncorrelated and atomic limits. Using the functional we calculate the susceptibility and the specific heat for the Lorentzian density of states. We find that the susceptibility crosses over smoothly from that expected for an uncorrelated state with antiferromagnetic fluctuations to a Fermi liquid state at low temperature via a Kondo-type anomaly. The specific heat shows a peak at the corresponding Kondo temperature.
An approximate partition functional is derived for the infinite-dimensional Hubbard model. This functional naturally includes the exact solution of the Falicov-Kimball model as a special case, and is exact in the uncorrelated and atomic limits. It explicitly keeps spin symmetry. For the case of the Lorentzian density of states, we find that the Luttinger theorem is satisfied at zero temperature. The susceptibility crosses over smoothly from that expected for an uncorrelated state with antiferromagnetic fluctuations at high temperature to a correlated state at low temperature via a Kondo-type anomaly at a characteristic temperature T*. We attribute this anomaly to the appearance of the Hubbard pseudogap. The specific heat also shows a peak near T*. The resistivity goes to zero at zero temperature, in contrast to other approximations, rises sharply around T*, and has a rough linear temperature dependence above T*.
An approximate partition functional is derived for the infinite-dimensional Hubbard model. This functional naturally includes the exact solution of the Falicov-Kimball model as a special case, and is exact in the uncorrelated and atomic limits. It explicitly keeps spin symmetry. For the case of the Lorentzian density of states, we find that the Luttinger theorem is satisfied at zero temperature. The susceptibility crosses over smoothly from that expected for an uncorrelated state with antiferromagnetic fluctuations at high temperature to a correlated state at low temperature via a Kondo-type anomaly at a characteristic temperature T∗{\mathit{T}}^{\mathrm{*}}T∗. We attribute this anomaly to the appearance of the Hubbard pseudogap. The specific heat also shows a peak near T∗{\mathit{T}}^{\mathrm{*}}T∗. The resistivity goes to zero at zero temperature, in contrast to other approximations, rises sharply around T∗{\mathit{T}}^{\mathrm{*}}T∗, and has a rough linear temperature dependence above T∗{\mathit{T}}^{\mathrm{*}}T∗.
Employing the equation of motion method for the Green's function, we have obtained the self-energy for the Hubbard model in the weak correlation limit up to the 2nd order in U, the on-site interaction strength. The quasi-particle spectral density of states is determined numerically, using the local approximation which is known to be exact for the infinite dimensional Hubbard model. Further, assuming a linear dispersion for the bare electronic spectra for the 1D Hubbard model, we have evaluated analytically the imaginary part of the self-energy, Im Sigma, near the Fermi energy (E(F)) and the renormalization coefficient at E(F), Z(F). It is found that the 1D Hubbard model system exhibits a non-Fermi liquid behavior within the 2nd order perturbational treatment.
We present an asymptotically exact solution of the infinite-dimensional Hubbard model at a special interaction strength UT{\mathit{U}}_{\mathit{T}}UT​ corresponding to the strong-coupling Fermi-liquid fixed point. This solution is intimately related to the Toulouse limit of the single-impurity Kondo model and the symmetric Anderson model in its strong-coupling limit.
We present an exact mapping of the Hubbard model in infinite dimensions onto a single-impurity Anderson (or Wolff) model supplemented by a self-consistency condition. This provides a mean-field picture of strongly corrrelated systems, which becomes exact as d\ensuremath{\rightarrow}\ensuremath{\infty}. We point out a special integrable case of the mean-field equations, and study the general case using a perturbative renormalization group around the atomic limit. Three distinct Fermi-liquid regimes arise, corresponding to the Kondo, mixed-valence, and empty-orbitals regimes of the single-impurity problem. The Kondo resonance and the satellite peaks of the single-impurity model correspond to the quasiparticle and Hubbard-bands features of the Hubbard model, respectively.
Anomalies are found in the resistivity \ensuremath{\rho} and NMR rate 1/T1{\mathit{T}}_{1}T1​ of the infinite-dimensional Hubbard model using quantum Monte Carlo calculations and the noncrossing approximation. For temperatures greater than the ``Kondo scale'' T0{\mathit{T}}_{0}T0​, we obtain 1/T1{\mathit{T}}_{1}T1​\ensuremath{\sim}a+bT and \ensuremath{\rho}\ensuremath{\sim}c+dT (a, b, c, d constants). For temperatures T\ensuremath{\ll}T0{\mathit{T}}_{0}T0​ we infer from a saturation of the effective mass of the carriers that the ground state of the system is a Fermi liquid.