Lattice Gauge Theories and Spin Models

TL;DR

This paper explores dualities between lattice gauge theories and spin models, deriving transformations and solutions for various gauge groups, and introduces a new magnetic disorder operator with a variational ground state analysis.

Contribution

It generalizes the Wegner $Z_2$ gauge-spin duality to SU(N) theories and provides exact solutions for Gauss law constraints in terms of spin or dual operators.

Findings

Derived dualities for $Z_2$, $U(1)$, and $SU(N)$ gauge theories.

Constructed a gauge-invariant magnetic disorder operator for SU(N).

Developed a variational ground state for SU(2) spin model.

Abstract

The Wegner $Z_2$ gauge theory-$Z_2$ Ising spin model duality in $(2+1)$ dimensions is revisited and derived through a series of canonical transformations. The Kramers-Wannier duality is similarly obtained. The Wegner $Z_2$ gauge-spin duality is directly generalized to SU(N) lattice gauge theory in $(2+1)$ dimensions to obtain the SU(N) spin model in terms of the SU(N) magnetic fields and their conjugate SU(N) electric scalar potentials. The exact and complete solutions of the $Z_2, U(1), SU(N)$ Gauss law constraints in terms of the corresponding spin or dual potential operators are given. The gauge-spin duality naturally leads to a new gauge invariant magnetic disorder operator for SU(N) lattice gauge theory which produces a magnetic vortex on the plaquette. A variational ground state of the SU(2) spin model with nearest neighbor interactions is constructed to analyze SU(2) gauge theory.