High-Temperature Cluster Expansion for Classical and Quantum Spin Lattice Systems With Multi-Body Interactions

We develop a novel cluster expansion for finite-spin lattice systems subject to multi-body quantum -and, in particular, classical- interactions. Our approach is based on the use of "decoupling parameters", advocated by Park (J. Stat. Phys. 27, 553-576 (1982)), which relates partition functions with successive additional interaction terms. Our treatment, however, leads to an explicit expansion in a beta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-dependent effective fugacity that permits an explicit evaluation of free energy and correlation functions at small beta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}. To determine its convergence region we adopt a relatively recent cluster summation scheme that replaces the traditional use of Kikwood-Salzburg-like integral equations by more precise sums in terms of particular tree-diagrams Bissacot et al. (J. Stat. Phys. 139, 598-617 (2010)). As an application we show that our lower bound of the radius of beta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-analyticity is larger than Park's for quantum systems two-body interactions.