# Dual $Z_2$ Lattice Gauge Theory of the 3D Ising Model with both Nearest- and Next-Nearest-Neighbor Couplings

## TL;DR

This paper introduces a novel $Z_2$ lattice gauge theory dual to a 3D Ising model with both nearest- and next-nearest-neighbor couplings, expanding the understanding of dualities in lattice models.

## Contribution

It develops an unusual $Z_2$ gauge theory dual to a 3D Ising model with extended couplings, including a new mapping of interactions and observables.

## Key findings

- Defines a gauge theory with four $Z_2$ variables per edge
- Maps nnn couplings to interactions between nearby vectors
- Suggests further numerical studies for phase transition analysis

## Abstract

It is known that the normal three-dimensional (3D) Ising model on a cubic lattice is dual to the Wegner's 3D $Z_2$ lattice gauge theory. Here we find an unusual $Z_2$ lattice gauge theory which is dual to the 3D Ising model with not only nearest-neighbor (nn) coupling, but also next-nearest-neighbor (nnn) coupling. Our gauge theory has on each edge four $Z_2$ variables that have product $+1$, each located on a vector perpendicular to the edge. The nn coupling in the Ising model maps to the plaquette term in the gauge theory where the four variables multiplied have their vectors pointing inward, while the nnn coupling maps to the coupling between the $Z_2$ variables on nearby vectors on each edge in the gauge theory. A Wilson loop observable in the gauge theory depends on a framing of a loop, and maps to a surface of flipped-sign nn and nnn couplings in the Ising model. Further numerical simulations could be made to explore the universality at the phase transition.