Loops, Surfaces and Grassmann Representation in Two- and Three-Dimensional Ising Models

TL;DR

This paper develops an algebraic hopping expansion of the Ising model's partition function, providing a simplified, exact representation of loop and surface expansions in 2D and 3D, with insights into convergence and geometry factors.

Contribution

It introduces an algebraic hopping expansion that simplifies the geometric loop and surface representations of the Ising model in two and three dimensions, including convergence analysis.

Findings

Exact algebraic representation of loop and surface expansions.

Radius of convergence linked to critical temperature.

Simplified derivation of geometry factors and 2n-point functions.

Abstract

Starting from the known representation of the partition function of the 2- and 3-D Ising models as an integral over Grassmann variables, we perform a hopping expansion of the corresponding Pfaffian. We show that this expansion is an exact, algebraic representation of the loop- and surface expansions (with intrinsic geometry) of the 2- and 3-D Ising models. Such an algebraic calculus is much simpler to deal with than working with the geometrical objects. For the 2-D case we show that the algebra of hopping generators allows a simple algebraic treatment of the geometry factors and counting problems, and as a result we obtain the corrected loop expansion of the free energy. We compute the radius of convergence of this expansion and show that it is determined by the critical temperature. In 3-D the hopping expansion leads to the surface representation of the Ising model in terms of surfaces with intrinsic geometry. Based on a representation of the 3-D model as a product of 2-D models coupled to an auxiliary field, we give a simple derivation of the geometry factor which prevents overcounting of surfaces and provide a classification of possible sets of surfaces to be summed over. For 2- and 3-D we derive a compact formula for 2n-point functions in loop (surface) representation.