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Physical review. B, Condensed matter

High-temperature series expansions for the Hubbard model

D. F. B. ten Haaf, J. M. J. van Leeuwen

Published Sep 1, 1992• 28 citations
Publisher

Abstract

A method is described to obtain high-temperature series expansions for the grand potential and the nearest-neighbor correlation of the z-component of the spin in the Hubbard model. The method is applied to the square and simple cubic lattices. The series have been calculated up to eighth order in \ensuremath{\beta}t. The convergence of the series is discussed and a comparison with previous results in the literature is made. The present calculations contain an indication for the occurrence of a Nagaoka-like phase.

DOI
10.1103/physrevb.46.6313

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A method is described to obtain high-temperature series expansions for the grand potential and the nearest-neighbor correlation of the z-component of the spin in the Hubbard model. The method is applied to the square and simple cubic lattices. The series have been calculated up to eighth order in \ensuremath{\beta}t. The convergence of the series is discussed and a comparison with previous results in the literature is made. The present calculations contain an indication for the occurrence of a Nagaoka-like phase.
New algorithm of the finite lattice method is presented to generate the high-temperature expansion series of the Ising model. It enables us to obtain much longer series in three dimensions when compared not only to the previous algorithm of the finite lattice method but also to the standard graphical method. It is applied to extend the high-temperature series of the simple cubic Ising model from beta^{26} to beta^{46} for the free energy and from beta^{25} to beta^{32} for the magnetic susceptibility.
We present the high-temperature series for a nearest-neighbor model with O(2) symmetry on a simple cubic lattice with the most general single-site potential. In particular, the magnetic susceptibility and the second-moment correlation length are computed to 22nd order. The series specialized to some particular improved Hamiltonians have been already analyzed in the paper M. Campostrini, M. Hasenbusch, A. Pelissetto, and E. Vicari, Phys. Rev. B 74, 144506 (2006) [cond-mat/0605083], to determine the critical exponents and other universal quantities of the three-dimensional XY universality class.
25th-order high-temperature series are computed for a general nearest-neighbor three-dimensional Ising model with arbitrary potential on the simple cubic lattice. In particular, we consider three improved potentials characterized by suppressed leading scaling corrections. Critical exponents are extracted from high-temperature series specialized to improved potentials, obtaining gamma=1.2373(2), nu=0.63012(16), alpha=0.1096(5), eta=0.036 39(15), beta=0.326 53(10), and delta=4.78 93(8). Moreover, biased analyses of the 25th-order series of the standard Ising model provide the estimate Delta=0.52(3) for the exponent associated with the leading scaling corrections. By the same technique, we study the small-magnetization expansion of the Helmholtz free energy. The results are then applied to the construction of parametric representations of the critical equation of state, using a systematic approach based on a global stationarity condition. Accurate estimates of several universal amplitude ratios are also presented.
We study the single-band Hubbard model via high-temperature expansions to order (\ensuremath{\beta}t)9{)}^{9})9 for the standard two- and three-dimensional lattices. Series are derived for the grand potential and for the ferromagnetic susceptibility, for general values of the Coulomb repulsion parameter U and electron density n. The series are difficult to analyze with any degree of precision, but estimates are obtained for the ferromagnetic critical temperature.
The high-temperature expansion coefficients of the ordinary and the higher susceptibilities of the spin-1/2 nearest-neighbor Ising model are calculated exactly up to the 20th order for the general d-dimensional (hyper)simple-cubical lattices. These series are analyzed to study the dependence of critical parameters on the lattice dimensionality. Using the general d expression of the ordinary susceptibility, we have more than doubled the length of the existing series expansion of the critical temperature in powers of 1/d.
Abstract We investigated the two-dimensional Hubbard model using many-body perturbation theory, focusing on the systematic computation of thermodynamic properties. The contributions of generalized Hugenholtz (Feynman) diagrams were evaluated symbolically and reformulated using divided differences, enabling efficient and precise calculations. We derived the coefficients of the grand potential expansion up to the eighth order in both the interaction potential (U-expansion) and the inverse temperature (β-expansion). Additionally, we presented results for magnetic susceptibility and heat capacity at half-filling, providing critical insights into the thermodynamic behavior of the model in this regime.