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Physical review. B, Condensed matter

Observation of Fermi-liquid behavior in the two-dimensional Hubbard model for self-consistent perturbation theory

S. Wermbter

Published Apr 15, 1997• 12 citations
Publisher

Abstract

I present a numerical treatment for the retarded self-energy in two dimensions (2D), which is formally exact for the frequency and the momentum sums occurring in perturbation theory with respect to the Coulomb interaction U. In this frequency- and momentum-renormalization-group technique, which allows the sequential elimination of regions of high frequencies and momenta, I focus on the 2D fluctuation-exchange approximation for the Hubbard model. I present results for the self-energy and the spectral density down to temperatures t=${10}^{\mathrm{\ensuremath{-}}4}$/t (the next neighbor hopping amplitude). The obtained result shows evidence for a Fermi-liquid phase for bandfillings n near half filling for U/t=2 to U/t=8 for T\ensuremath{\rightarrow}0.

DOI
10.1103/physrevb.55.r10149

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I present a numerical treatment for the retarded self-energy in two dimensions (2D), which is formally exact for the frequency and the momentum sums occurring in perturbation theory with respect to the Coulomb interaction U. In this frequency- and momentum-renormalization-group technique, which allows the sequential elimination of regions of high frequencies and momenta, I focus on the 2D fluctuation-exchange approximation for the Hubbard model. I present results for the self-energy and the spectral density down to temperatures t=10\ensuremath−4{10}^{\mathrm{\ensuremath{-}}4}10\ensuremath−4/t (the next neighbor hopping amplitude). The obtained result shows evidence for a Fermi-liquid phase for bandfillings n near half filling for U/t=2 to U/t=8 for T\ensuremath{\rightarrow}0.
A frequency- and momentum-renormalization-group acceleration together with an analytical approach is used to obtain the retarded Green's function in the self-consistent and conserving fluctuation-exchange (FLEX) approximation for the two-dimensional Hubbard model in the normal state and in the superconducting state. The analytical expressions for this approach are given. For band-fillings near half filling the self-energy in the normal state exhibits Fermi-liquid behaviour for, low temperatures and frequencies near the chemical potential, if the momentum is chosen near the Fermi-surface. However due to the presence of large many body effects the observed Fermi-liquid region near the chemical potential and near the Fermi-surface is very small. Results for the single particle density of states, the occupation number and the spectral function are presented. The superconducting state with symmetry is obtained for U = 2 to U = 6 and a (U, n)-phase diagram for the transition temperature Tc is presented. A maximum Tc/t of 0.0275 is obtained for U = 6 near half filling.
Abstract A frequency‐ and momentum‐renormalization‐group acceleration together with an analytical approach is used to obtain the retarded Green's function in the self‐consistent and conserving fluctuation‐exchange (FLEX) approximation for the two‐dimensional Hubbard model in the normal state and in the superconducting state. The analytical expressions for this approach are given. For band‐fillings near half filling the self‐energy in the normal state exhibits Fermi‐liquid behaviour for, low temperatures and frequencies near the chemical potential, if the momentum is chosen near the Fermi‐surface. However due to the presence of large many body effects the observed Fermi‐liquid region near the chemical potential and near the Fermi‐surface is very small. Results for the single particle density of states, the occupation number and the spectral function are presented. The superconducting state with symmetry is obtained for U = 2 to U = 6 and a ( U , n )‐phase diagram for the transition temperature T c is presented. A maximum T c / t of 0.0275 is obtained for U = 6 near half filling.
Momentum- and frequency-dependent self-consistent field theories provide a potentially powerful tool for analyzing condensed-matter models for interacting electrons. Such theories generally require the solution of multidimensional integral equations. Solutions are limited by demands on computational time and storage, which increase rapidly for calculation of low-temperature properties. A renormalization-group technique which allows the sequential elimination of regions of high frequency and momentum is presented in detail, and is then applied to the solution of the fluctuation exchange approximation for the two-dimensional Hubbard model. Effective meshes with as many as 104{10}^{4}104 frequencies and 5122{512}^{2}5122 k points can be analyzed on a vector supercomputer using this approach. As a sample application, it is demonstrated that the fluctuation exchange approximation exhibits non-Fermi liquid behavior near half filling for sufficiently large interaction strengths.
I present a numerical treatment for a certain class of Feynman diagrams in perturbation theory with respect to the Coulomb correlation U. This method is formally exact for the frequency integrals occurring in the expression for the retarded Green's function. I specify this method to the self-consistent second-order perturbation theory (SOPT) for the single-impurity Anderson model and to the self-consistent fluctuation exchange approximation (FLEX) for the Hubbard model in infinite dimensions d\ensuremath{\rightarrow}\ensuremath{\infty}. The presented treatment is not only restricted to the limit d\ensuremath{\rightarrow}\ensuremath{\infty}. I present the resulting equations for the retarded self-energy, numerical results for the self-energy, density of states and the interaction spectrum in the FLEX for different scalings of the real frequency axis. The numerical solution of the SOPT and the FLEX could be obtained with very little numerical effort. In this form the calculation is nearly independent of the temperature and over a wide range nearly independent of the value of U. The computational time and the memory requirement for the solution of the self-energy equation are reduced to a minimum in this numerical treatment. \textcopyright{} 1996 The American Physical Society.
The two-dimensional Hubbard model on the square lattice is studied in the presence of lattice distortions in the adiabatic approximation. The self energy is computed within perturbation theory up to second order, which provides a way for studying the quasiparticle dispersion. We compute numerically the second order contribution to the self-energy using a standard Fast Fourier Transform Algorithm for finite system sizes. The stability of the lattice distortions is investigated and a schematic phase diagram is drawn. The Fermi surface is analyzed for densities near to half filling, the presence of lattice distortion change some spectral properties of the model and gives an anisotropic interacting FS. The spectral function is calculated along several lines in momentum space and the renormalized quasiparticle dispersion is obtained. The behavior of the density of states is shown for different values of the intrasite repulsion U in the different phases.