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arXiv:cond-mat/0205180 · Physical Review Letters

Integral Equation Generates High-Temperature Expansion of the Heisenberg Chain

Masahiro Shiroishi, Minoru Takahashi

Published Aug 22, 2002• 44 citations
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Abstract

Recently a new integral equation describing the thermodynamics of the 1D Heisenberg model was discovered by Takahashi. Using the integral equation we have succeeded in obtaining the high-temperature expansion of the specific heat and the magnetic susceptibility up to O[(J/T)(100)]. This is much higher than those obtained so far by the standard methods such as the linked-cluster algorithm. Our results will be useful to examine various approximation methods to extrapolate the high-temperature expansion to the low temperature region.

DOI
10.1103/physrevlett.89.117201

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Recently a new integral equation describing the thermodynamics of the 1D Heisenberg model was discovered by Takahashi. Using the integral equation we have succeeded in obtaining the high-temperature expansion of the specific heat and the magnetic susceptibility up to O[(J/T)(100)]. This is much higher than those obtained so far by the standard methods such as the linked-cluster algorithm. Our results will be useful to examine various approximation methods to extrapolate the high-temperature expansion to the low temperature region.
By combining conventional finite-temperature many-body perturbation theory with cluster expansions, we develop a systematic method to carry out high order arbitrary temperature perturbative calculations on a computer. The method is well suited to studying the thermodynamic properties of quantum disordered and quantum critical phases at finite temperatures. As an application, we calculate the magnetic susceptibility, internal energy, and specific heat of the bilayer Heisenberg model. It is shown that for a wide range of coupling constants these expansions show excellent convergence at all temperatures. Comparing the direct series (without extrapolations) for the bulk susceptibility to quantum Monte Carlo simulations we find an almost perfect agreement between the two methods even at the quantum critical coupling separating the dimerized and antiferromagnetic phases. The convergence fails only at very low temperatures, which are also difficult to reach by quantum Monte Carlo simulations.
Recently Rojas et al. [J. Math. Phys. 43, 1390 (2002)] obtained a closed analytical expression for the coefficients of arbitrary order in the cumulant expansion of a one-dimensional periodic chain model with nearest-neighbor interaction and spatial translation invariance; that approach can be applied equally well to both nonintegrable and exact integrable models. Here, we obtain the exact analytic expressions for the six lowest-order terms of the high-temperature expansion of the Helmholtz free energy per site of the nonintegrable one-dimensional spin-111 XXZ\mathrm{XXZ}XXZ Heisenberg model. Our analytical results for the specific heat and the static magnetic susceptibility are compared, up to order ${\ensuremath{\beta}}^{6}$ for the ferromagnetic and antiferromagnetic phases, with the respective numerical results of a periodic chain with ten sites (N=10)(N=10)(N=10) and numerical results in the literature [Bl\"ote, Physica B 79, 427 (1975)]. A very good agreement is obtained for both phases in the high-temperature regime.
We present a new application of the traditional thermodynamic Bethe ansatz to the spin-1/2 antiferromagnetic uniform Heisenberg chain and derive exact nonlinear integral equations for just two functions describing the elementary excitations. By using this approach, the magnetic susceptibility chi and specific heat C versus temperature T are calculated to high accuracy for 5x10(-25)</=T/J</=5. The chi(T) data agree very well at low T with the asymptotically exact theoretical low- T prediction of Lukyanov [Nucl. Phys. B522, 533 (1998)]. The unknown coefficients of the second and third lowest-order logarithmic correction terms in Lukyanov's theory for C(T) are estimated from the C(T) data.
A full description of the previously proposed method of calculating low temperature specific heat and susceptibility of quantum spin chains are given. By the use of internal energy as the expansion variable the domain of convergence of the high temperature series is remarkably extended. The low temperature properties are obtained by continuing it in a natural way to polynomials in internal energy, whose form is suggested by the spin wave theory. The method is checked in the S =1/2 XY model and is then applied to the Heisenberg model of S =1/2, 1, 3/2 and 2. The results are satisfactory in ferromagnets and reasonable in the S =1/2 antiferromagnet. In antiferromagnets of S ≧1 the results suggest inadequacy of our polynomial, which is, presumably, due to the existence of an energy gap.
The one-dimensional quantum Heisenberg model with random \ifmmode\pm\else\textpm\fi{}J bonds is studied for S=1/2 and S=1. The specific heat and the zero-field susceptibility are calculated by using high-temperature series expansions and the quantum transfer matrix method. The susceptibility shows a Curie-like temperature dependence at low temperatures as well as at high temperatures. The numerical results for the specific heat suggest that there are anomalously many low-lying excitations. The qualitative nature of these excitations is discussed based on the exact diagonalization of finite-size systems.