A fermionic path integral for exact enumeration of polygons on the simple cubic lattice

Abstract

Enumerating polygons on regular lattices is a classic problem in rigorous statistical mechanics. The goal of enumerating polygons on the square lattice via fermionic path integration was achieved using a free-fermion quadratic action in the late 1970s. Given that polygon edges only link 2 vertices, it is considered plausible, if not natural, that an action of degree 2 in the Grassmann variables might suffice to enumerate lattice polygons in any dimension. Nevertheless, on nonplanar lattices the problem has remained open for more than four decades. Here we derive the Grassmann action for exact enumeration of polygons on the simple cubic lattice. Moreover, we prove that this action is not quadratic but quartic -- corresponding to a model of interacting fermions.

arXiv commentSubmitted to Physical Review B

References in corpus (11)

• An RVB phase in the triangular lattice quantum dimer model
• Topological Order and Conformal Quantum Critical Points
• Low-energy sector of the S=1/2 Kagome antiferromagnet
• Continuous Phase Transition without Gap Closing in Non-Hermitian Quantum Many-Body Systems
• Series studies of the Potts model. I: The simple cubic Ising model
• Quasi-local holographic dualities in non-perturbative 3d quantum gravity I - Convergence of multiple approaches and examples of Ponzano-Regge statistical duals
• Spin-liquid behaviour and the interplay between Pokrovsky-Talapov and Ising criticality in the distorted, triangular-lattice, dipolar Ising antiferromagnet
• Classical vertex model dualities in a family of 2D frustrated quantum antiferromagnets
• Loops, Surfaces and Grassmann Representation in Two- and Three-Dimensional Ising Models
• Grassmannization of classical models
• The double hypergeometric series for the partition function of the 2D anisotropic Ising model