Analytical and Numerical Treatment of the Mott--Hubbard Insulator in Infinite Dimensions

We calculate the density of states in the half-filled Hubbard model on a Bethe lattice with infinite connectivity. Based on our analytical results to second order in $t/U$, we propose a new `Fixed-Energy Exact Diagonalization' scheme for the numerical study of the Dynamical Mean-Field Theory. Corroborated by results from the Random Dispersion Approximation, we find that the gap opens at $U_{\rm c}=4.43 \pm 0.05$. Moreover, the density of states near the gap increases algebraically as a function of frequency with an exponent $\alpha=1/2$ in the insulating phase. We critically examine other analytical and numerical approaches and specify their merits and limitations when applied to the Mott--Hubbard insulator.Comment: 22 pages, 16 figures; minor changes (one reference added, included comparison with Falicov-Kimball model

Universal renormalization of saddle-point integrals for condensed Bose gases

When treating the ground-state contribution exactly, a variant of the saddle-point method emerges that works even for condensed Bose gases. Results thus obtained, such as canonical partition functions, differ by universal renormalization factors from those provided by the conventional but incorrect scheme. The amended method yields the statistical properties of ideal and very weakly interacting Bose gases with a fixed number of particles with particular simplicity. ͓S1063-651X͑99͒13012-5͔ PACS number͑s͒: 05.30. Ch, 05.30.Jp, 03.75.Fi The saddle-point method ͓1͔ is one of the true work horses of statistical mechanics. Greatly promoted by Schrö-dinger's lucid discussion ͓2͔, it serves as the essential tool for the comparison of different statistical ensembles. In the field of Bose-Einstein condensation, however, this work horse shows signs of illness, which already thwarts the attempt to compute canonical partition functions of ideal Bose gases ͓3͔. As detailed below, the usual answer to this textbook problem ͓4͔, probably given by generations of physicists, turns out to be incorrect in the condensate regime; moreover, the standard saddle-point approximation does not yield the correct fluctuation of the number of condensate particles ͓5,6͔. This shortcoming is particularly painful since there is now, in the wake of the impressive series of successful experiments on atomic Bose-Einstein condensates ͓7͔, enhanced interest in the statistical mechanics of mesoscopically small, isolated samples of Bosons ͓8-11͔ which cannot be described by the customary grand canonical ensemble. In this paper we discuss the reason for the failure of the standard approach, and modify the saddle-point approximation such that it works for all temperatures. The correct canonical N-particle partition functions will then allow us to demonstrate the large-N equality of grand canonical and canonical occupation numbers also in the condensate regime and to assess the sharpness of the onset of Bose-Einstein condensation in a gas with a fixed, finite number of particles. Comparing the results of the standard saddle-point scheme to those provided by the properly amended one, it is found that the error of the former is universal, that is, independent of the system's single-particle spectrum, so that correct results can be obtained even from the standard scheme by means of a simple, multiplicative renormalization. We start from the familiar expansion ͓4͔ of the grand canonical partition function ⌶(␤,z) of an ideal Bose gas with single-particle energies (ϭ0,1,2, . . . ) in terms of the canonical partition functions Z N (␤), ⌶͑␤,z ͒ϭ ͟ ϭ0

Evaluation of High Order Terms for the Hubbard Model in the Strong-coupling Limit

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 t^12/U^11. The results of the strong-coupling expansion deviate from numerical calculations as quantum Monte Carlo (or density-matrix renormalization-group) by less than 0.13% (0.32% respectively) for U>4.76.Comment: 8 pages, 5 figures, 2 flowcharts, 1 tabl

The saddle-point method for condensed Bose gases

The application of the conventional saddle-point approximation to condensed Bose gases is thwarted by the approach of the saddle-point to the ground-state singularity of the grand canonical partition function. We develop and test a variant of the saddle-point method which takes proper care of this complication, and provides accurate, flexible, and computationally efficient access to both canonical and microcanonical statistics. Remarkably, the error committed when naively employing the conventional approximation in the condensate regime turns out to be universal, that is, independent of the system's single-particle spectrum. The new scheme is able to cover all temperatures, including the critical temperature interval that marks the onset of Bose--Einstein condensation, and reveals in analytical detail how this onset leads to sharp features in gases with a fixed number of particles. In particular, within the canonical ensemble the crossover from the high-temperature asymptotics to the condensate regime occurs in an error-function-like manner; this error function reduces to a step function when the particle number becomes large. Our saddle-point formulas for occupation numbers and their fluctuations, verified by numerical calculations, clearly bring out the special role played by the ground state.Comment: 32 pages, 11 figures. Ann. Phys. (N.Y.), in pres