Group convolutional neural network for the low-energy spectrum in the quantum dimer model

Abstract

We obtain the p⁢4⁢m-symmetric group convolutional neural network (GCNN) representations of the lowest energy eigenstate of the quantum dimer model on an 𝐿×𝐿 square-lattice in each of the (𝐿2+18⁢𝐿+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 an efficient 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 for systems of sizes 8≤𝐿≤32. Analysis of the scaling of the gaps in different representation sectors for systems of sizes up to 𝐿=32 suggest a fourfold-degenerate ground state for 𝑉≤0.4 narrowing the regime of possible mixed/plaquette phases to 0.4<𝑉<1. Our results show that GCNN is a powerful tool for investigating ground-state phase diagrams. The approach paves the way for even more accurate results by producing highly accurate variational baseline wave functions for quantum Monte Carlo approaches.