A Six Vertex Model on a Fishnet

TL;DR

This paper directly computes the continuum properties of a 6-vertex model on a fishnet lattice, confirming its equivalence to a compactified scalar field and relating the circle radius to coupling ratios.

Contribution

It provides a direct calculation of the continuum limit of a 6-vertex model on a fishnet lattice, confirming its relation to a compactified scalar field and suggesting a method for non-Abelian generalizations.

Findings

Continuum properties match those of a scalar field on a circle.

Circle radius is related to coupling ratio v by R^{-2} = 2T_0 arccos(1-1/2v^2).

Method may extend to non-Abelian O(n) models.

Abstract

The flow of U(1) charge through dense fishnet diagrams, in a non-hermitian matrix scalar field theory g_1Tr(\Sigma^\dagger\Sigma)^2 + 2g_1vTr\Sigma^{\dagger 2}\Sigma^2, is described by a 6-vertex model on a ``diamond'' lattice [1]. We give a direct calculation of the continuum properties of the 6-vertex model on this novel lattice, explicitly confirming the conclusions of [1], that, for 1/2 < v< \infty, they are identical to those of a world-sheet scalar field compactified on a circle S_1. The radius of the circle is related to the ratio v of quartic couplings by R^{-2} = 2T_0 arccos(1-1/2v^2). This direct computational approach may be of value in generalizing the conclusions to the non-Abelian O(n) case.