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arXiv:1802.08333 · Journal of Statistical Mechanics Theory and Experiment

Off-diagonal series expansion for quantum partition functions

Itay Hen

Published May 1, 2018• 13 citations
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Abstract

Abstract We derive an integral-free thermodynamic perturbation series expansion for quantum partition functions which enables an analytical term-by-term calculation of the series. The expansion is carried out around the partition function of the classical component of the Hamiltonian with the expansion parameter being the strength of the off-diagonal, or quantum, portion. To demonstrate the usefulness of the technique we analytically compute to third order the partition functions of the 1D Ising model with longitudinal and transverse fields, and the quantum 1D Heisenberg model.

DOI
10.1088/1742-5468/aabbe4

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In this paper, a new formulation which can be used to systematically extend and study the low- and high-temperature series for the thermodynamic functions of systems of coupled quantum oscillators corresponding to the quantum $\ensuremath{\lambda}{\ensuremath{\varphi}}^{2m}$ fields, m=2,3m=2,3m=2,3,..., on a lattice is proposed. In this formulation, the low- and high-temperature series for the partition function of the systems are given in the forms $Z={Z}_{h}(T)[1+{A}_{1}(\ensuremath{\lambda}{T}^{m\ensuremath{-}1})+{A}_{2}{(\ensuremath{\lambda}{T}^{m\ensuremath{-}1})}^{2}+\ensuremath{\cdots}]$ and $Z={Z}_{a}(T)[1+{B}_{1}{(\ensuremath{\lambda}{T}^{m\ensuremath{-}1})}^{\frac{\ensuremath{-}1}{m}}+{B}_{2}{(\ensuremath{\lambda}{T}^{m\ensuremath{-}1})}^{\frac{\ensuremath{-}2}{m}}+\ensuremath{\cdots}] ,$ respectively, where An{A}_{n}An​ and Bn{B}_{n}Bn​ are certain combinatorial numbers which can be expressed in terms of the numbers of ways of placing certain diagrams on the ddd-dimensional d=1,2,3d=1,2,3d=1,2,3,... discrete lattice. Note that the forms of these series are new and they are different from those of the more extensively studied classical models. A systematic study of these series will, as in the corresponding cases of the Ising and Heisenberg models, reveal the critical behaviors of this class of quantum-field models which is not only interesting in its own right, but is also important in our overall understanding of the critical behaviors of many other related models from the point of view of the renormalization-group approach.
A method is proposed for obtaining a systematic expansion of thermodynamic functions of spin systems with large spin S in powers of 1/ S . It uses the cumulant technique and a coherent-state representation of the partition function . The expansion of in terms of cumulants yields an effective classical Hamiltonian with temperature-dependent quantum corrections. For the Heisenberg quantum Hamiltonian, they have a non-Heisenberg form. The effective Hamiltonian can be solved by methods familiar for classical systems.