The spectrum of a transfer matrix for loops

TL;DR

This paper analyzes the spectral properties of a transfer matrix for a 3D lattice loop model, revealing exact eigenvalues related to 2D Ising model quantities and explicit eigenfunctions on loop-space.

Contribution

It provides an exact evaluation of the transfer matrix eigenvalues and eigenfunctions for a 3D loop model using 2D Ising model data, a novel analytical approach.

Findings

Eigenvalues expressed in terms of Ising model partition function and energy

Eigenfunctions are explicit functions on loop-space

Transfer matrix invariance under loop-space motions

Abstract

We study the spectral properties of the transfer matrix for a gonihedric random surface model on a three-dimensional lattice. The transfer matrix is indexed by generalized loops in a natural fashion and is invariant under a group of motions in loop-space. The eigenvalues of the transfer matrix can be evaluated exactly in terms of the partition function, the internal energy and the correlation functions of the two-dimensional Ising model and the corresponding eigenfunctions are explicit functions on loop-space.