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 \(\beta \)-dependent effective fugacity that permits an explicit evaluation of free energy and correlation functions at small \(\beta \). 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 \(\beta \)-analyticity is larger than Park’s for quantum systems two-body interactions.