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Physical Review Letters

Mott-Hubbard transition in infinite dimensions

Marcelo J. Rozenberg, X. Y. Zhang, Gabriel Kotliar

Published Aug 24, 1992• 231 citations
Publisher

Abstract

We present a solution of the Hubbard model at half filling in the limit of infinite dimensions which corresponds to the Mott-Hubbard insulating phase. We describe the metal-insulator transition between this regime and the Fermi-liquid regime. Our analysis is based on an analytic consideration and exact Monte Carlo simulations at low but finite temperatures.

DOI
10.1103/physrevlett.69.1236

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We present a solution of the Hubbard model at half filling in the limit of infinite dimensions which corresponds to the Mott-Hubbard insulating phase. We describe the metal-insulator transition between this regime and the Fermi-liquid regime. Our analysis is based on an analytic consideration and exact Monte Carlo simulations at low but finite temperatures.
We study the Hubbard model in dimension d= infinity , within an approximation that simultaneously reproduces the atomic limit and the weak-coupling limit up to second order in the interaction strength U. This method, which is a slightly modified version of an approximation suggested by Edwards and Hertz, is applied to the calculation of the self-energy and of the spectral function in the paramagnetic regime for different U, band fillings n, and temperatures T. We find Fermi-liquid behaviour for small U and (for half-filling n=0.5) a metal-insulator (Mott) transition at a critical value U cr , but also a metallic non-Fermi-liquid phase for intermediate values U<U cr . We compare our results with results obtained within the second-order perturbation theory in U and (at finite temperature) with the (essentially exact) quantum Monte Carlo (QMC) results. Qualitatively, the approximation reproduces the correct tendencies obtained by QMC, but in detail there are still discrepancies, in particular for intermediate values of U.
The nature of the Mott-Hubbard metal-insulator transition in the infinite-dimensional Hubbard model is investigated by quantum Monte Carlo simulations down to temperature T=W/140T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}W/140T=W/140 ( W=bandwidthW\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\mathrm{bandwidth}W=bandwidth). Calculating with significantly higher precision than in previous work, we show that the hysteresis below ${T}_{\mathrm{IPT}}\ensuremath{\simeq}0.022W$, reported in earlier studies, disappears. Hence the transition is found to be continuous rather than discontinuous down to at least T=0.325TIPT{T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.325T}_{\mathrm{IPT}}T=0.325TIPT​. We also study the changes in the density of states across the transition, which illustrate that the Fermi liquid breaks down before the gap opens.
We present a powerful method for calculating the thermodynamic properties of infinite-dimensional Hubbard-type models using an exact diagonalization of an Anderson model with a finite number of sites. The resolution obtained for Green's functions is far superior to that of quantum Monte Carlo calculations. We apply the method to the half-filled Hubbard model for a discussion of the metal-insulator transition, and to the two-band Hubbard model where we find direct evidence for the existence of a superconducting instability at low temperatures.