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Europhysics Letters (EPL)

Onsager Reaction Field Theory of the Heisenberg Model

David E. Logan, Yolande H. Szczech, Michael A. Tusch

Published May 10, 1995• 28 citations
Publisher

Abstract

We discuss a simple Onsager reaction field treatment of the Heisenberg model. For d = 2 in particular, this predicts correctly the absence of long-range order for any T > 0, produces an exponentially divergent correlation length as T → 0, and gives good quantitative agreement over a wide temperature range with both Monte Carlo and (where appropriate) high-temperature series expansions.

DOI
10.1209/0295-5075/30/5/010

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We discuss a simple Onsager reaction field treatment of the Heisenberg model. For d = 2 in particular, this predicts correctly the absence of long-range order for any T > 0, produces an exponentially divergent correlation length as T → 0, and gives good quantitative agreement over a wide temperature range with both Monte Carlo and (where appropriate) high-temperature series expansions.
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Onsager reaction field theory is used to investigate the one-dimensional ferromagnetic long-range interacting spin chain with the antiferromagnetic nearest-neighbor interaction (NNI) [Formula: see text]. The ferromagnetic long-range interactions considered in this paper decay as [Formula: see text] with the distance [Formula: see text] between lattice sites. It is found that both the zero temperature and finite-temperature phase diagrams of the system are strongly affected by the interplay between ferromagnetic long-range and antiferromagnetic NNIs. The critical temperature and the uniform susceptibility are obtained as a function of [Formula: see text] and [Formula: see text]. At finite temperatures and in the region [Formula: see text] in which [Formula: see text] is dependent of [Formula: see text], the ferromagnetic-paramagnetic phase transition survives for [Formula: see text] and no phase transition exists for [Formula: see text]. At [Formula: see text], the ferromagnetic-antiferromagnetic phase transition happens at zero temperature for [Formula: see text]. The ground state of the system keeps ferromagnetic when [Formula: see text]. But for [Formula: see text], the system becomes antiferromagnetic at all temperatures.
Monte Carlo renormalization-group methods were applied to the classical three-component Heisenberg model on a two-dimensional lattice. Expectation values of local correlations of spins and various sized block spins were computed using traditional Monte Carlo methods. By matching quantities at different length scales generated by different Monte Carlo Hamiltonians we directly determined the renormalization of the nearest-neighbor coupling. Using these data and the results of high-temperature expansions and low-temperature renormalization-group calculations we have determined that this model does not have a phase transition. We have also obtained the amplitude for the low-temperature divergence of the susceptibility and the correlation length.
Using an extended version of Handscomb's Monte Carlo method we study the spin-(1/2 quantum antiferromagnetic Heisenberg model on square lattices of various sizes. The calculated correlation length associated with antiferromagnetic order grows very rapidly as the temperature is lowered suggesting an essential singularity. Our results are consistent with those recently observed by neutron scattering done on Lo2{\mathrm{Lo}}_{2}Lo2​CuO4{\mathrm{CuO}}_{4}CuO4​.
Thermodynamic properties of the spin-1/2 Heisenberg ferromagnet are calculated by using Handscomb's Monte Carlo method. Several methods of analyses are used to determine critical properties of the model. For a square lattice we find that the susceptibility diverges exponentially at zero temperature. That is, \ensuremath{\chi}\ensuremath{\sim}exp[b(J/kB{\mathit{k}}_{\mathit{B}}kB​T)] with the constant b=4.5\ifmmode\pm\else\textpm\fi{}0.5, which is lower than the prediction (b=2\ensuremath{\pi}) of a modified spin-wave theory. For the simple-cubic lattice, we find that the critical temperature kB{\mathit{k}}_{\mathit{B}}kB​Tc{\mathit{T}}_{\mathit{c}}Tc​/J=1.68\ifmmode\pm\else\textpm\fi{}0.01, and the ratio of the exponents \ensuremath{\gamma}/\ensuremath{\nu}=2.0\ifmmode\pm\else\textpm\fi{}0.05, which are in good agreement with the estimates of the high-temperature series-expansion method.