We present an approach to the magnetic properties of the half-filled Hubbard model, based on an approximate mapping of its low-energy transverse spin excitations on to those of an effective underlying Heisenberg model, but with effective spin interactions which are self-consistently determined and not confined solely to nearest-neighbor couplings. The mapping is exact in strong-coupling and is found to be accurate over a very wide range of interaction strengths, down to weak coupling. At zero temperature, it permits ready evaluation at finite U of the one-loop effects of zero-point spin fluctuations on, e.g., the sublattice magnetization. At finite temperatures, thermodynamic properties of the system in the thermal paramagnet are studied via a physically transparent Onsager reaction field approach, which amounts to a self-consistent treatment of paramagnetic spin correlations. This is central not only in recovering the correct dimensionality dependence of antiferromagnetic long-ranged order, but also for the d=3 case of primary interest here yields a N\'eel temperature in close agreement with known strong- and weak-coupling limits. Spin correlation functions and magnetic susceptibilities also show very good agreement with quantum Monte Carlo calculations over an appreciable temperature range in which the low-lying transverse spin excitations are thermally dominant. \textcopyright{} 1996 The American Physical Society.
The exchange-correlation potential formalism previously introduced and applied to the one-dimensional Hubbard model has been extended to spin systems and applied to the case of the one-dimensional antiferromagnetic spin-
21 Heisenberg model. Within the spin exchange-correlation potential formulation, a sum rule for spin systems is derived. The exchange-correlation potential for the Heisenberg model is extrapolated from exact diagonalization results of small antiferromagnetic Heisenberg clusters. This procedure is also employed to revisit and computationally improve the previous investigation of the exchange-correlation potential of the half-filled Hubbard model, which was based on the exchange-correlation potential of the dimer. Numerical comparisons with exact benchmark calculations for both the Heisenberg and the Hubbard models indicate that, starting from the exchange-correlation potential of a finite cluster, the extrapolation procedure yields a one-particle spectral function with favorable accuracy at a relatively low computational cost. In addition, a comparison between the ground-state energies for the one-dimensional Hubbard and Heisenberg models displays how the well-known similarity in behavior of the two models at large interactions manifests itself within the exchange-correlation potential formalism.
Chains of quantum spins with open ends and isotropic Heisenberg exchange are studied. By diagonalizing the Hamiltonian for chains of finite length N and obtaining all the energy eigenvalues, the magnetic susceptibility X, the specific heat C v , and the partition function Z can be calculated exactly for these chains. The high-temperature series expansions of these are then evaluated. For X and C v it is found that the terms in the series consist of three parts. One is the normal high-T series already known in great detail for the N → ∞ ring (chain with periodic boundary conditions). The other two consist of a 'surface' term and a correction term of order (/T) N . The surface term is found as a series up to and including (1/T) 8 for spin S = 1/2 and 1. Simple Pade approximant formulae are given to extend the range of validity below T = 1.
The ground-state energy E and momentum distribution n k of the electrons are expanded from the atomic limit for the half-filled Hubbard model. The coefficients of expansion are represented by the spin correlation functions of the spin 1 / 2 Heisenberg model. Using the spin-wave theory, approximate values of coefficients are calculated for the square lattice and the simple cubic lattice. In one dimension, the theory shows a good agreement with the exact solution. An effective spin Hamiltonian for the half-filled Hubbard model with arbitrary hopping integrals is obtained up to the fifth order. It is shown that the fourth term contains four spin interactions.
We develop finite-temperature strong-coupling expansions for the
SU(N) Hubbard model in powers of $\ensuremath{\beta}t, w=exp(\ensuremath{-}\ensuremath{\beta}U)$ and $\frac{1}{\ensuremath{\beta}U}$ for arbitrary filling. The expansions are done in the grand canonical ensemble and are most useful at a density of one particle per site, where for
U larger than or of order of the bandwidth, the expansions converge over a wide temperature range ${t}^{2}/U\phantom{\rule{4pt}{0ex}}\ensuremath{\lesssim}\phantom{\rule{4pt}{0ex}}T\phantom{\rule{4pt}{0ex}}\ensuremath{\lesssim}\phantom{\rule{4pt}{0ex}}10U$. By taking the limit $w\ensuremath{\rightarrow}0$, valid at temperatures much less than
U, the expansions turn into a high-temperature expansion for a dressed
SU(N) Heisenberg model that includes nearest-neighbor exchange, further neighbor exchanges, and ring exchanges known from the
T=0 perturbation theory of the SU(2) Hubbard model. Below a filling of one particle per site, the $w\ensuremath{\rightarrow}0$ limit corresponds to an effective $t\text{\ensuremath{-}}J$ model. The onset of strong correlations can be identified by a plateaulike behavior in the entropy as a function of temperature. At small deviations from one particle per site, the expansions can be arranged in powers of a small parameter $\ensuremath{\delta}=1\ensuremath{-}n$, the deviation from one particle per site, where the leading $\ensuremath{\beta}t$-dependent terms correspond to holes sloshing around in a disordered
SU(N) background. We use these expansions to calculate the thermodynamic properties of the model at moderate and high temperatures over a wide parameter range.
With the aim of contributing to a better understanding of the finite-temperature properties of low-dimensional strongly correlated systems we study a one-dimensional half-filled-band extended Hubbard model with a term of exchange interaction between neighboring sites. We analyze the temperature dependence of the specific heat, internal energy, entropy, number of doubly occupied sites, magnetic susceptibility, spin-spin correlation functions, and also investigate a detailed diagram of magnetic correlations. We use the method of small-cluster exact diagonalization calculations with the application of the grand canonical ensemble and of extrapolation techniques to the infinite chain. Our results show that the examined model exhibits a rich structure of magnetic properties due to the competition between the onsite Coulombian repulsion and the exchange interaction.