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Physical review. B, Solid state

Low- and high-temperature series expansions for quantum coupled oscillators systems or quantum fields on a lattice

F. T. Hioe

Published Nov 1, 1977• 7 citations
Publisher

Abstract

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, $m=2,3$,..., 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 ${A}_{n}$ and ${B}_{n}$ are certain combinatorial numbers which can be expressed in terms of the numbers of ways of placing certain diagrams on the $d$-dimensional $d=1,2,3$,... 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.

DOI
10.1103/physrevb.16.4112

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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 \ensuremathλ\ensuremathφ2m\ensuremath{\lambda}{\ensuremath{\varphi}}^{2m}\ensuremathλ\ensuremathφ2m 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=Zh(T)[1+A1(\ensuremathλTm\ensuremath−1)+A2(\ensuremathλTm\ensuremath−1)2+\ensuremath⋯]Z={Z}_{h}(T)[1+{A}_{1}(\ensuremath{\lambda}{T}^{m\ensuremath{-}1})+{A}_{2}{(\ensuremath{\lambda}{T}^{m\ensuremath{-}1})}^{2}+\ensuremath{\cdots}]Z=Zh​(T)[1+A1​(\ensuremathλTm\ensuremath−1)+A2​(\ensuremathλTm\ensuremath−1)2+\ensuremath⋯] and Z=Za(T)[1+B1(\ensuremathλTm\ensuremath−1)\ensuremath−1m+B2(\ensuremathλTm\ensuremath−1)\ensuremath−2m+\ensuremath⋯],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}] ,Z=Za​(T)[1+B1​(\ensuremathλTm\ensuremath−1)m\ensuremath−1​+B2​(\ensuremathλTm\ensuremath−1)m\ensuremath−2​+\ensuremath⋯], 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.
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.
The finite lattice method of series expansion is generalised to the qqq-state Potts model on the simple cubic lattice. It is found that the computational effort grows exponentially with the square of the number of series terms obtained, unlike two-dimensional lattices where the computational requirements grow exponentially with the number of terms. For the Ising (q=2q=2q=2) case we have extended low-temperature series for the partition functions, magnetisation and zero-field susceptibility to u26u^{26}u26 from u20u^{20}u20. The high-temperature series for the zero-field partition function is extended from v18v^{18}v18 to v22v^{22}v22. Subsequent analysis gives critical exponents in agreement with those from field theory.
We present simple tables of integers from which it is possible to reconstruct the high-temperature series coefficients through ${\mathrm{\ensuremath{\beta}}}^{14}$ for the susceptibility, for the second correlation moment, and for the second field derivative of the susceptibility of the O(N) classical Heisenberg model on a simple (hyper)cubic lattice in dimension d=2, 3, and 4 and for any N. To construct the tables we have used the recent extension of the high-temperature series by M. Luscher and P. Weisz and some analytic properties in N that we have derived from the Schwinger-Dyson equations of the O(N) model. We also present a numerical study of these series in the d=2 case. The main results are: (a) the extended series give further support to the Cardy-Hamber-Nienhuis exact formulas for the critical exponents when -2N2; (b) for N\ensuremath{\ge}3 there are no indications of any critical point at finite \ensuremath{\beta}; (c) the series are consistent with the low-temperature asymptotic forms predicted by the perturbative renormalization group.
The finite-lattice method of series expansion is generalized to the q-state Potts model on the simple cubic lattice. It is found that the computational effort grows exponentially with the square of the number of series terms obtained, unlike two-dimensional lattices where the computational requirements grow exponentially with the number of terms. For the Ising (q=2) case the authors have extended the low-temperature series for the partition functions, magnetization and zero-field susceptibility to u 26 from u 20 . The high-temperature series for the zero-field partition function is extended from nu 18 to nu 22 . Subsequent analysis gives critical exponents in agreement with those from field theory.
We suggest that renormalization-group recursion relations can profitably be combined with high-temperature series expansions in systems with critical points at zero temperature. Low-temperature recursion relations are integrated out to high temperatures, where conventional high-temperature series should accurately describe quantities such as the susceptibility. A "variational" principle is suggested which gives the optimal temperature at which to splice together the low- and high-temperature results. In this way, we obtain a sequence of approximants to the susceptibility of the classical two-dimensional Heisenberg model which apparently converges exponentially fast at all temperatures. Although further investigation reveals that these approximants are not as accurate as their exponential convergence suggests, they do appear to give a good quantitative description at all temperatures of the susceptibilities we have studied.
High-temperature expansions for the susceptibility and the second correlation moment of the classical N-vector model [also known as the O(N) symmetric Heisenberg classical spin model] on the square lattice are extended from order ${\mathrm{\ensuremath{\beta}}}^{14}$ to ${\mathrm{\ensuremath{\beta}}}^{21}$ for arbitrary N. For the second field derivative of the susceptibility the series expansion is extended from order ${\mathrm{\ensuremath{\beta}}}^{14}$ to ${\mathrm{\ensuremath{\beta}}}^{17}$. For -2\ensuremath{\le}N2, a numerical analysis of the series is performed in order to compare the critical exponents \ensuremath{\gamma}(N), \ensuremath{\nu}(N), and \ensuremath{\Delta}(N) to exact (though nonrigorous) formulas and to compute the ``dimensionless four point coupling constant'' g${\mathrm{^}}_{\mathit{r}}$(N). For N>2, we present a study of the analyticity properties of \ensuremath{\chi}, \ensuremath{\xi}, etc. in the complex \ensuremath{\beta} plane and describe a method to estimate the parameters which characterize their low-temperature behaviors. We compare our series estimates to the predictions of the perturbative renormalization group theory, to exact (but nonrigorous or conjectured) formulas and to the results of the 1/N expansion, always finding a good agreement. \textcopyright{} 1996 The American Physical Society.