Group Convolutional Neural Network Ground State of the Quantum Dimer Model

We estimate the ground state of the square lattice Quantum Dimer Model in a $\rm{p4m}$-symmetric Group Convolutional Neural Network (GCNN) representation and show that results in agreement with exact diagonalization (ED) and quantum Monte Carlo (QMC) can be obtained with a $\mathcal{L}=2$ layer network. In systems of linear size $L=8$ with Hilbert space dimension $3.1\times 10^8$, GCNN shows fidelity as high as $0.99999$ with ED. For $12\leq L\leq 32$, we find excellent agreement with QMC estimates of energy, order parameters and correlation functions. The network is optimized by minimizing the energy estimated from a Metropolis algorithm assisted by a directed loop sampler. We analyze the quantum geometric tensor at the minima for $\mathcal{L}=1,2$ and $3$ and show that the empirical quantum dimension saturates with increasing network complexity due to Metropolis sampling constraints.