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Physical Review Letters

Hubbard model in infinite dimensions: A quantum Monte Carlo study

Mark Jarrell

Published Jul 6, 1992• 626 citations
Publisher

Abstract

An essentially exact solution of the infinite-dimensional Hubbard model is made possible by a new self-consistent Monte Carlo procedure. Near half filling antiferromagnetism and a pseudogap in the single-particle density of states are found for sufficiently large values of the intrasite Coulomb interaction. At half filling the antiferromagnetic transition temperature obtains its largest value when the intrasite Coulomb interaction U\ensuremath{\approxeq}3.

DOI
10.1103/physrevlett.69.168

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An essentially exact solution of the infinite-dimensional Hubbard model is made possible by a new self-consistent Monte Carlo procedure. Near half filling antiferromagnetism and a pseudogap in the single-particle density of states are found for sufficiently large values of the intrasite Coulomb interaction. At half filling the antiferromagnetic transition temperature obtains its largest value when the intrasite Coulomb interaction U\ensuremath{\approxeq}3.
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A detailed quantitative study of the physical properties of the infinite-dimensional Hubbard model at half filling is presented. The method makes use of an exact mapping onto a single-impurity model supplemented by a self-consistency condition. This coupled problem is solved numerically. Results for thermodynamic quantities (specific heat, entropy, . . .), one-particle spectral properties, and magnetic properties (response to a uniform magnetic field) are presented and discussed. The nature of the Mott-Hubbard metal-insulator transition found in this model is investigated. A numerical solution of the mean-field equations inside the antiferromagnetic phase is also reported.
We present a numerical solution of the infinite-dimensional Hubbard model at finite temperature in the paramagnetic phase. The problem reduces to a single-impurity Anderson model supplemented by a self-consistency condition. Using Monte Carlo methods and complete enumeration we determine the imaginary-time Green's function, the density of doubly occupied sites, and the compressibility close to half filling. All three quantities present direct evidence for a Mott insulating phase above a critical value of U.
The Quantum Monte Carlo simulations of the ionic Hubbard model on a two-dimensional square lattice at half filling were performed. The method based on the direct-space, proposed by Suzuki and al., Hirsch and al., was used. Cycles of increasing and decreasing values of the Coulomb interaction $ U $ were performed for fixed temperature ($ kT=0.01 $). Results indicate that, at low temperature, the two insulator phases are separated by a metallic phase for weak to intermediate values of the staggered potential $ Δ$. For large Coulomb repulsion the system is in a Mott insulator with an antiferromagnetism order. On increasing and decreasing the Coulomb interaction $ U $ the metal-Mott insulator transition shows an hysteresis phenomenon while the metal-band insulator transition is continue. For large $ Δ$ it seems that the metallic region shrinks to a single metallic point. However, the band insulator to the Mott insulator transition is not direct for the studied model. A phase diagram is drawn for the temperature $ kT=0.01 $. For $ Δ=0.5 $ cycles of increasing and decreasing temperature were programmed for different values of the Coulomb interaction $ U $ . A behaviour change appears for $ U\simeq 1.75 $. This suggests that a crossover line divides the metallic region of the phase diagram.
We present a powerful method for calculating the thermodynamic properties of the Hubbard model in infinite dimensions, using an exact diagonalization of an Anderson model with a finite number of sites. At finite temperatures, the explicit diagonalization of the Anderson Hamiltonian allows the calculation of Green's functions with a resolution far superior to that of Quantum Monte Carlo calculations. At zero temperature, the Lanczòs method is used and yields the essentially exact zero-temperature solution of the model, except in a region of very small frequencies. Numerical results for the half-filled case in the paramagnetic phase (quasi-particle weight, self-energy, and also real-frequency spectral densities) are presented.