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arXiv:cond-mat/9403030 · Physical review. B, Condensed matter

Equation-of-motion approach to the Hubbard model in infinite dimensions

Claudius Gros

Published Sep 15, 1994• 29 citations
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Abstract

We consider the Hubbard model on the infinite-dimensional Bethe lattice and construct a systematic series of self-consistent approximations to the one-particle Green's function, ${\mathit{G}}^{(\mathit{n})}$(\ensuremath{\omega}), n=2,3,.... The first n-1 equations of motion are exactly fullfilled by ${\mathit{G}}^{(\mathit{n})}$(\ensuremath{\omega}) and the nth equation of motion is decoupled following a simple set of decoupling rules. ${\mathit{G}}^{(2)}$(\ensuremath{\omega}) corresponds to the Hubbard-III approximation. We present analytic and numerical results for the Mott-Hubbard transition at half filling for n=2,3,4.

DOI
10.1103/physrevb.50.7295

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We consider the Hubbard model on the infinite-dimensional Bethe lattice and construct a systematic series of self-consistent approximations to the one-particle Green's function, G(n){\mathit{G}}^{(\mathit{n})}G(n)(\ensuremath{\omega}), n=2,3,.... The first n-1 equations of motion are exactly fullfilled by G(n){\mathit{G}}^{(\mathit{n})}G(n)(\ensuremath{\omega}) and the nth equation of motion is decoupled following a simple set of decoupling rules. G(2){\mathit{G}}^{(2)}G(2)(\ensuremath{\omega}) corresponds to the Hubbard-III approximation. We present analytic and numerical results for the Mott-Hubbard transition at half filling for n=2,3,4.
We calculate the one-particle density of states for the Mott-Hubbard insulating phase of the Hubbard model on a Bethe lattice in the limit of infinite coordination number. We employ the Kato-Takahashi perturbation theory around the strong-coupling limit to derive the Green function. We show that the Green function for the lower Hubbard band can be expressed in terms of polynomials in the bare hole-hopping operator. We check our technique against the exact solution of the Falicov-Kimball model and give explicit results up to and including second order in the inverse Hubbard interaction. Our results provide a stringent test for analytical and numerical investigations of the Mott-Hubbard insulator and the Mott-Hubbard transition within the dynamical mean-field theory. We find that the Hubbard-III approximation is not satisfactory beyond lowest order, but the local-moment approach provides a very good description of the Mott-Hubbard insulator at strong coupling.
The Mott-Hubbard transition in the half-filled Hubbard model is studied analytically on a Bethe lattice for the paramagnetic state and the classical N\'eel state. The single-particle density of states is obtained by calculating the one-particle Green's function represented by the Lanczos infinite continued fraction. The paramagnetic solution shows that the Mott-Hubbard transition is signaled by both collapsing Hubbard bands at Uc{\mathit{U}}_{\mathit{c}}Uc​=2t∗{\mathit{t}}_{\mathrm{*}}t∗​ and a sharp Lorentzian peak appearing at midgap at UUc{\mathit{U}}_{\mathit{c}}Uc​. For the N\'eel state, however, the system becomes an insulator for U>0. We also obtain the specific heat and entropy for both the metallic and insulating regimes. Our analytic work reproduces the key results obtained by numerical method in infinite dimensions.
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.
The exact solution of a quantum Bethe lattice model in the thermodynamic limit amounts to solve a functional self-consistent equation. In this paper we obtain this equation for the Bose-Hubbard model on the Bethe lattice, under two equivalent forms. The first one, based on a coherent-state path integral, leads in the large connectivity limit to the mean-field treatment of Fisher et al. [Phys. Rev. B 40, 546 (1989)] at the leading order, and to the bosonic dynamical mean field theory as a first correction, as recently derived by Byczuk and Vollhardt [Phys. Rev. B 77, 235106 (2008)]. We obtain an alternative form of the equation using the occupation number representation, which can be easily solved with an arbitrary numerical precision, for any finite connectivity. We thus compute the transition line between the superfluid and Mott insulator phases of the model, along with thermodynamic observables and the space and imaginary-time dependence of correlation functions. The finite connectivity of the Bethe lattice induces a richer physical content with respect to its infinitely connected counterpart: a notion of distance between sites of the lattice is preserved, and the bosons are still weakly mobile in the Mott insulator phase. The Bethe lattice construction can be viewed as an approximation to the finite-dimensional version of the model. We show indeed a quantitatively reasonable agreement between our predictions and the results of Quantum Monte Carlo simulations in two and three dimensions.
Using second-order perturbation theory in U for the self-energy, we solve the self-consistent equations for the local Green function of the infinite-dimensional Hubbard model. We consider the cases of Gaussian and semicircular free densities of states (DOS), and calculate the spectral densities rho ( omega ) for various U. In the semicircular case we investigate the existence of a Mott transition at half-filling. Even for large U we obtain a metallic solution. Treating the free DOS as a variable parameter, the method also gives an approximate solution of the finite-dimensional Hubbard model. We consider the case of a (regularized) logarithmic DOS appropriate for the two-dimensional case.
In view of a recent controversy we investigated the Mott-Hubbard transition in D = ∞ with a novel cluster approach. i) We show that any truncated Bethe lattice of order n can be mapped exactly to a finite, Hubbard-like cluster. ii) We evaluate the self-energy numerically for n = 0, 1, 2 and compare with a series of self-consistent equation-of-motion solutions. iii) We find the gap to open continuously at the critical U c ∼ 2.5 t * ( t ≡ t * /√(4 d )). iv) A low-energy theory for the Mott-Hubbard transition is developed and relations between critical exponents are presented.