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arXiv:hep-lat/9612022

Computer Aided Series Expansions for Critical Phenomena

Hildegard Meyer‐Ortmanns, T. Reisz

Published Dec 20, 1996• 0 citations
arXivHTMLPDF

Abstract

Under quite general conditions critical phenomena can be described with high order linked cluster expansions. The coefficients of the series admit a graphical expansion that is generated with the aid of computers. Our generalization of linked cluster expansions from an infinite to a finite volume allows to perform a finite size scaling analysis. We also indicate a generalization to Dynamical Linked Cluster Expansions with possible applications to spin glasses and neural networks with coupled spin and interaction dynamics.

DOI
10.48550/arxiv.hep-lat/9612022

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Under quite general conditions critical phenomena can be described with high order linked cluster expansions. The coefficients of the series admit a graphical expansion that is generated with the aid of computers. Our generalization of linked cluster expansions from an infinite to a finite volume allows to perform a finite size scaling analysis. We also indicate a generalization to Dynamical Linked Cluster Expansions with possible applications to spin glasses and neural networks with coupled spin and interaction dynamics.
We consider 1/d1/d1/d expansions for classical spin systems based on the vertex renormalized linked cluster expansion (LCE). The free multiplicities of the LCE graphs on a hypercubic lattice in an arbitrary dimension d are calculated using generating functions. The technique is applied to the Ising model and to a two-component classical lattice gas corresponding to an extended Hubbard model at half filling in the zero-bandwidth limit. We use a resummation of the LCE to generate 1/d1/d1/d expansions for the equation of state and for the critical temperature. The method, which is rather general and applicable to a wide range of models, proves convenient for calculating asymptotic power series expansions in 1/d.1/d.1/d. The vertex renormalized expansion is shown to break down at low temperature in higher order approximations, barring attempts to construct simple approximations that are both self-consistent and exact to some finite order in 1/d.1/d.1/d.
We generalize the technique of linked cluster expansions on hypercubic lattices to actions that couple fields at lattice sites which are not nearest neighbors. We show that in this case the graphical expansion can be arranged in such a way that the classes of graphs to be considered are identical to those of the pure nearest neighbor interaction. The only change then concerns the computation of lattice imbedding numbers. All the complications that arise can be reduced to a generalization of the notion of free random walks, including hopping beyond nearest neighbor. Explicit expressions for combinatorical numbers of the latter are given. We show that under some general conditions the linked cluster expansion series have a nonvanishing radius of convergence.
A new method is proposed to study critical phenomena, which is a unification of Weiss' mean-field theory and Fisher's finite-size scaling theory. A systematic series of clusters is considered and generalized mean-fields are applied to the boundary spins of such clusters. From the size-dependence of the coefficients of mean-field singularities systematically obtained, the static critical exponents α, β and γ, and critical slowing down exponent Δ can be estimated. The critical point T C can be also evaluated from the size-dependence of mean-field critical points obtained in a systematic series of clusters. Some applications to the two- and three-dimensional Ising models are given explicitly to obtain T C =(2.24(0.036)) J / k B and γ=1.7(0.05) in the two-dimensional square lattice in our preliminary calculations.
The linked-cluster expansion technique for the high-temperature expansion of spin model is reviewed. A new algorithm for the computation of three-point and higher Green’s functions is presented. Series are computed for all components of two-point Green’s functions for a generalized 3D Ising model, to 25th order on the bcc lattice and to 23rd order on the sc lattice. Series for zero-momentum four-, six-, and eight-point functions are computed to 21st, 19th, and 17th order respectively on the bcc lattice. 1
There are many statistical systems in which degrees of freedom can be grouped according to their degree of correlation. We develop a cluster-expansion method for these systems based on selecting finite clusters of the most strongly interacting degrees of freedom. The starting point is Kubo's generalized cumulant formalism. We show how this expansion can be renormalized; that is, how the contributions of infinite classes of articulated finite clusters can be resummed. The elimination of articulated clusters from the renormalized expansion generates modified interactions in the finite clusters remaining in the expansion. When truncated at different orders, the final result is a hierarchy of mean-field approximations, which generally reproduce an increasing number of terms in exact series expansions for the system. High-order series-expansion coefficients can be obtained by perturbative renormalization of very large clusters, a convenient alternative to the complicated combinatorial analysis associated with standard expansion techniques for lattice models. We compare our work to previous finite-cluster techniques, such as the cluster-variation method of Kikuchi.
Perturbation series expansion methods are sophisticated numerical tools used to provide quantitative calculations in many areas of theoretical physics. This book gives a comprehensive guide to the use of series expansion methods for investigating phase transitions and critical phenomena, and lattice models of quantum magnetism, strongly correlated electron systems and elementary particles. Early chapters cover the classical treatment of critical phenomena through high-temperature expansions, and introduce graph theoretical and combinatorial algorithms. The book then discusses high-order linked-cluster perturbation expansions for quantum lattice models, finite temperature expansions, and lattice gauge models. Also included are numerous detailed examples and case studies, and an accompanying resources website, www.cambridge.org/9780521842426, contains programs for implementing these powerful numerical techniques. A valuable resource for graduate students and postdoctoral researchers working in condensed matter and particle physics, this book will also be useful as a reference for specialized graduate courses on series expansion methods.