Group Convolutional Neural Network for the Low-Energy Spectrum in the Quantum Dimer Model

We obtain the $\rm{p4m}$-symmetric Group Convolutional Neural Network (GCNN) representations of the lowest energy eigenstate of the quantum dimer model on $L{\times} L$ square-lattice in each of the ${(L^2+18L+72)}/{8}$ irreducible representations (irreps) of the lattice space group and use these to investigate the competition between columnar, plaquette and mixed phases. The networks are optimized within each irrep by minimizing the energy, which is estimated from samples obtained via a directed loop sampler. In extensive benchmarks, we show excellent agreement in energy estimates, order parameters and correlation functions with exact diagonalization or quantum Monte Carlo in systems of sizes $8\leq L\leq 32$. Analysis of the scaling of the gaps in different representation sectors with systems of sizes up to $L=32$ suggest a $4$-fold degenerate ground state for $V\leq 0.4$ narrowing the regime of possible mixed/plaquette phases to $0.4 < V< 1$. Our results show that GCNN is a powerful tool to investigate ground state phase diagrams. We also present ideas for significant further improvements via projection Monte Carlo methods assisted by the GCNN ansatzes.