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Journal of Mathematical Physics

Low temperature expansions for the Gibbs states of quantum Ising lattice systems

Lawrence E. Thomas, Yin Zhong

Published Oct 1, 1984• 8 citations
Publisher

Abstract

By conditioning on a subalgebra of observables, we develop low temperature expansions for certain quantum Ising models, which include transverse Ising models and anisotropic xy models. These expansions are majorized by the series for functions solving certain nonlinear partial differential equations. Convergence for the majorizing series follows from Cauchy–Kowalevski theory applied to the partial differential equations. The low temperature expansions imply long-range order.

DOI
10.1063/1.526030

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By conditioning on a subalgebra of observables, we develop low temperature expansions for certain quantum Ising models, which include transverse Ising models and anisotropic xy models. These expansions are majorized by the series for functions solving certain nonlinear partial differential equations. Convergence for the majorizing series follows from Cauchy–Kowalevski theory applied to the partial differential equations. The low temperature expansions imply long-range order.
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Low-temperature expansion of Ising model has long been a topic of significant interest in condensed matter and statistical physics. In this paper we present new results of the coefficients in the low-temperature series of the Ising partition function on the square lattice, in the cases of a zero field and of an imaginary field $i(\ensuremath{\pi}/2){k}_{B}T$. The coefficients in the low-temperature series of the free energy in the thermodynamic limit are represented using the explicit expression of the density function of the Fisher zeros. The asymptotic behavior of the sequence of the coefficients when the order goes to infinity is determined exactly, for both the series of the free energy and of the partition function. Our analytic and numerical results demonstrate that the convergence radius of the sequence is dependent on the accumulation points of the Fisher zeros, which have the smallest modulus. In the zero field case this accumulation point is the physical critical point, while in the imaginary field case it corresponds to a nonphysical singularity. We further discuss the relation between the series coefficients and the energy state degeneracies, using the combinatorial expression of the coefficients and the subgraph expansion.
We exhibit a new method for calculating high-and low-temperature series expansions for diagonal spin-spin correlation functions in the d=2d=2d=2 Ising model. The method makes use of a differential equation obeyed by these functions.
We explain how and why all thermodynamic properties of spin systems can be computed in one and two dimensions in the whole range of temperatures overcoming the divergence towards zero temperature of the standard high-temperature series expansions (HTEs). The method relies on an approximation of the entropy versus energy (microcanonical potential function) on the whole range of energies. The success is related to the intrinsic physical constraints on the entropy function and a careful treatment of the boundary behaviors. This method is benchmarked against two one-dimensional solvable models: the Ising model in longitudinal field and the XY model in a transverse field. With ten terms in the HTE, we find a spin susceptibility within a few percent of the exact results over the entire range of temperatures. The method is then applied to two two-dimensional models: the supposedly gapped Heisenberg model and the J(1)-J(2)-J(d) model on the kagome lattice.
The finite-lattice method of series expansion has been used to extend low-temperature series for the partition function, order parameter and susceptibility of the spin-1 Ising model on the square lattice. A new formalism is described which uses two distinct transfer-matrix approaches in order to significantly reduce computer memory requirements and which permits the derivation of the series to 79th order. Subsequent analysis of the series clearly confirms that the spin-1 model has the same dominant critical exponents as the spin- 1/2 Ising model. Accurate estimates for both the critical temperature and non-physical singularities are obtained. In addition, evidence for a non-analytic confluent correction with exponent Delta 1 =1.1+or-0.1 is found.