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arXiv:cond-mat/0109535

Mott-Hubbard Insulator in Infinite Dimensions

Eva Kalinowski, Florian Gebhard

Published Sep 28, 2001• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.cond-mat/0109535

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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.
We calculate the Hubbard bands for the half-filled Hubbard model on a Bethe lattice with an infinite coordination number up to and including third order in the inverse Hubbard interaction. We employ the Kato-Takahashi perturbation theory to solve the self-consistency equation of the dynamical mean-field theory analytically for the single-impurity Anderson model in multichain geometry. The weight of the secondary Hubbard sub-bands is of fourth order so that the two-chain geometry is sufficient for our study. Our results for the Mott-Hubbard gap and the density of states of the lower Hubbard band agree very well with those from numerical dynamical density-matrix renormalization group calculations, apart from a resonance contribution at the upper band edge, which cannot be reproduced in low-order perturbation theory.
The ground-state energy of the Hubbard model on a Bethe lattice with infinite connectivity at half filling is calculated for the insulating phase. Using Kohn's transformation to derive an effective Hamiltonian for the strong-coupling limit, the resulting class of diagrams is determined. We develop an algorithm for an algebraic evaluation of the contributions of high-order terms and check it by applying it to the Falicov-Kimball model that is exactly solvable. For the Hubbard model, the ground-state energy is exactly calculated up to order t12/U11{t}^{12}/{U}^{11}t12/U11. The results of the strong-coupling expansion deviate from numerical calculations as quantum Monte Carlo (or density-matrix renormalization group) by less than 0.130.13%0.13 (0.320.32%0.32 respectively) for U>4.76U>4.76U>4.76.
We present a large-scale combinatorial-diagrammatic computation of high-order contributions to the strong-coupling Kato-Takahashi perturbation series for the Hubbard model in high dimensions. The ground-state energy of the Mott-insulating phase is determined exactly up to the 15th order in 1/U1/U1/U. The perturbation expansion is extrapolated to infinite order and the critical behavior is determined using the Domb-Sykes method. We compare the perturbative results with two dynamical mean-field theory (DMFT) calculations using a quantum Monte Carlo method and a density-matrix renormalization group method as impurity solvers. The comparison demonstrates the excellent agreement and accuracy of both extrapolated strong-coupling perturbation theory and quantum Monte Carlo based DMFT, even close to the critical coupling where the Mott insulator becomes unstable.
Starting from the exact solution of the Hubbard model in the atomic limit, and treating the hopping term as a perturbation by means of a diagram technique, we discuss the electronic structure of the model in some simple approximations. We show that a metal-insulator transition is obtained above a critical value of the on-site Coulomb interaction in a self-consistent evaluation of the one-particle Green's function to one-loop order. A relationship with results in infinite dimensions is briefly discussed.
Algebraic-Dynamical Perturbation Theory of Large-UUU Hubbard Models. Single Particle Spectrum of Antiferromagnetic Mott Insulating States
In this work, we present an analytical framework for studying antiferromagnetic (AFM) Mott insulating states in the Hubbard model. We first derive an analytical solution for the single-particle Green's functions in the atomic limit. Within a second-order perturbation approach, we compute the ground state energy and show that the ground state is antiferromagnetically ordered. Then we derive an analytical solution for single-particle Green's functions when effects of the hopping term are considered in the Néel state. With the analytical solution, we compute the spectral functions and explain various properties of the AFM Mott insulating state as observed both experimentally and numerically: i) magnetic blueshift of the Mott gap; ii) the low energy part in the parental compounds of cuprate high TcT_cTc​ superconductors, which corresponds to a single band Hubbard model description. This work comprehends the electronic properties of antiferromagnetic Mott states analytically and provides a foundation for future investigations of doped antiferromagnetic Mott insulators, aiming for the mechanism of cuprates high-TcT_cTc​ superconductivity.
The recently introduced concept of exact mean-field theories is applied to the extended simplified Hubbard model, which is a one-band extended Hubbard model (including nearest-neighbor interaction) where only the \ensuremath{\downarrow} electrons can hop. An exact mean-field Hamiltonian is derived for the model on a Bethe lattice in the limit of a large coordination number (Z\ensuremath{\rightarrow}\ensuremath{\infty}). The mean-field Hamiltonian is used to calculate the Green functions and the free energy at half-filling. This exact solution is analyzed analytically at small and large values of the on-site (U) and nearest-neighbor (V) interaction strengths. One finds that a phase transition occurs at sufficiently low temperatures for all U,V>0. The low-temperature phase is a charge-density wave for V>1/2U and a spin-density wave for V1/2U. Special attention is paid to (i) the critical temperature as a function of U and V, (ii) the order parameter as a function of temperature, and (iii) the density of states. The density of states shows that the low-temperature phase is an insulator for all U,V provided that V/U\ensuremath{\ne}1/2, while the high-temperature phase is a paramagnetic metal for all V if U2, and a paramagnetic insulator if U>2. Many detailed results are given.