Gauge Equivariant Neural Networks for Quantum Lattice Gauge Theories

Gauge symmetries play a key role in physics appearing in areas such as quantum field theories of the fundamental particles and emergent degrees of freedom in quantum materials. Motivated by the desire to efficiently simulate many-body quantum systems with exact local gauge invariance, gauge equivariant neural-network quantum states are introduced, which exactly satisfy the local Hilbert space constraints necessary for the description of quantum lattice gauge theory with ${\mathbb{Z}}_d$ gauge group and non-Abelian Kitaev $D(G)$ models on different geometries. Focusing on the special case of ${\mathbb{Z}}_2$ gauge group on a periodically identified square lattice, the equivariant architecture is analytically shown to contain the loop-gas solution as a special case. Gauge equivariant neural-network quantum states are used in combination with variational quantum Monte Carlo to obtain compact descriptions of the ground state wave function for the ${\mathbb{Z}}_2$ theory away from the exactly solvable limit, and to demonstrate the confining or deconfining phase transition of the Wilson loop order parameter.

Figure 1

(a) Gauge equivariant neural network architecture. (b) Gauge equivariant block. (c) Gauge invariant block. For (b) and (c), the convolution neural network (CNN) component uses $\mathrm{channel}=2$, $\mathrm{stride}=1$, activation $\mathrm{function}=\text{leaky}$ relu and periodic boundary padding. The kernel size in CNN is different for different systems sizes.

Figure 2

The variance (top) and the energy difference between the exact ground state (bottom) of the gauge equivariant network, gauge invariant network (no equivariant layers) and RBM on a $3\times 3$ lattice of Eq. (1). The details of the above networks are provided in Sec. VI of the Supplemental Material [37].

Figure 3

The variance (top) and the energy difference between the exact ground state (bottom) of the gauge equivariant network, gauge invariant network (no equivariant layers) and RBM on a $3\times 3$ lattice of $H=-{\sum }_{f\in F}{\prod }_{e\in f}{Z}_e-{\sum }_{e\in E}{X}_e-J_y{\sum }_{f\in F}{\prod }_{e\in f}{Y}_e$. The details of the above networks are the same as Fig. 2.

Figure 4

The ground state expectation value of rectangle Wilson loops of size $l_1\times l_2$ ($l_1$, $l_2\le 4$) as a function of the enclosed area $A_C$ in the confining phase for $h>h_{\mathrm{c}}$ (top) and of the enclosed perimeter $P_C$ in the deconfined phase for $h<h_{\mathrm{c}}$ (bottom) on a $12\times 12$ lattice. The linear fit to the log-linear plot is consistent with area law scaling $⟨{\widehat{W}}_C⟩\sim e^{-\alpha A_C}$ with best fit parameter $\alpha =0.185$ and a perimeter law scaling $⟨{\widehat{W}}_C⟩\sim e^{-{\alpha }^{\prime }{P}_C}$ with best fit parameter ${\alpha }^{\prime }=0.00718$. Because of the smallness of ${\alpha }^{\prime }$, we note that a linear fit of $⟨{\widehat{W}}_C⟩$ versus $P_C$ is also consistent with the data.

Figure 5

Based on the optimized gauge equivariant states on $L\times L$ lattices as a function of $h$, it shows (top) derivative of the energy per site calculated by the Hellmann-Feynman theorem. (Bottom) Magnitude of $k_1$ (solid) and $k_2$ (dotted) for fitting $\log ⟨{\widehat{W}}_C⟩=k_1{A}_C+k_2{P}_C+b$ with area $A_c$ and perimeter $P_c$. The changes at around $h=0.30$ in the top figure and the increase in $A_c$ in the bottom figure suggest the confined or deconfined phase transition.