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It is known that the normal three-dimensional (3D) Ising model on a cubiclattice is dual to the Wegner's 3D $Z_2$ lattice gauge theory. Here we find anunusual $Z_2$ lattice gauge theory which is dual to the 3D Ising model with notonly nearest-neighbor (nn) coupling, but also next-nearest-neighbor (nnn)coupling. Our gauge theory has on each edge four $Z_2$ variables that haveproduct $+1$, each located on a vector perpendicular to the edge. The nncoupling in the Ising model maps to the plaquette term in the gauge theorywhere the four variables multiplied have their vectors pointing inward, whilethe nnn coupling maps to the coupling between the $Z_2$ variables on nearbyvectors on each edge in the gauge theory. A Wilson loop observable in the gaugetheory depends on a framing of a loop, and maps to a surface of flipped-sign nnand nnn couplings in the Ising model. Further numerical simulations could bemade to explore the universality at the phase transition.

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