Landau theory for lattice higher gauge theory and Kramers-Wannier duality

TL;DR

This paper develops a Landau field theory framework for lattice higher gauge theories, analyzing their phases, dualities, and topological defects, and explores the implications of higher-form symmetries and their low-energy behaviors.

Contribution

It introduces a Landau theory for lattice higher gauge theories, including the continuum limit, phase analysis, and dualities, extending classical gauge concepts to higher-form symmetries.

Findings

Classical solutions show area and perimeter laws in different coupling regimes.

Constructed topological defects like vortices and domain walls for higher gauge theories.

Discussed dualities and low-energy effective theories related to higher-form symmetries.

Abstract

We derive a Landau field theory for a lattice higher gauge theory defined on $p$-dimensional open cells (i.e., sites, links, faces, cubes, etc.), and study its continuum-limit and phases. In this approach, the $p$-dimensional Wilson-surface operator of the higher gauge theory is promoted to a fundamental functional field that is charged under the $p$-form global symmetry. By explicitly solving the functional equation of motion, we show that the classical solution exhibits the area~(perimeter) law in the strong (weak) gauge coupling limit. In the deconfined phase, we also construct topological defects for both $\mathrm{U}(1)$ and $\mathbb{Z}_N^{}$ higher gauge theories, as analogs of vortex and domain-wall solutions in conventional field theories with $0$-form global symmetries. Besides, we examine low-energy effective theory by identifying the phase modulations of the functional field as low-energy modes, and discuss the Coleman-Mermin-Wagner theorem for higher-form global symmetries. Finally, we discuss infrared duality among Landau field theories, which originates from Kramers-Wannier duality in lattice higher gauge theories.