Unifying All Classical Spin Models in a Lattice Gauge Theory

The partition function of all classical spin models, including all discrete standard statistical models and all Abelian discrete lattice gauge theories (LGTs), is expressed as a special instance of the partition function of the 4D ${\mathbb{Z}}_2$ LGT. This unifies all classical spin models with apparently very different features in a single complete model. This result is applied to establish a new method to compute the mean-field theory of Abelian discrete LGTs with $d\ge 4$, and to show that computing the partition function of the 4D ${\mathbb{Z}}_2$ LGT is computationally hard (#P hard). The 4D ${\mathbb{Z}}_2$ LGT is also proved to be approximately complete for Abelian continuous models. The proof uses techniques from quantum information.

Figure 1

(a) The state $|{\psi }_{\mathrm{LGT}}⟩$ (here, on a 2D square lattice) places one quantum particle (blue dots) at every face, which characterizes the interaction of classical spins (black dots) on that face. (b) The left face is merged with the right one by setting $J_{1234}=\infty$ (marked with a shaded face).

Figure 2

Spins fixed by the gauge are marked in red. A single-body, a two-body and a three-body Ising-type interaction with coupling strengths $J_1$, $J_{12}$, and $J_{123}$ are shown in (a), (b), and (c), respectively.

Figure 3

Replication of spins in four dimensions. Yellow faces represent the fourth dimension, and they have the same meaning as blue faces, that is, $s_2$ propagates into $s_3$ by the same method as it propagates into $s_4$.