Transformer wave function for two dimensional frustrated magnets: Emergence of a spin-liquid phase in the Shastry-Sutherland model

Understanding quantum magnetism in two-dimensional systems represents a lively branch in modern condensed-matter physics. In the presence of competing superexchange couplings, magnetic order is frustrated and can be suppressed down to zero temperature. Still, capturing the correct nature of the exact ground state is a highly complicated task, since energy gaps in the spectrum may be very small and states with different physical properties may have competing energies. Here, we introduce a variational ansatz for two-dimensional frustrated magnets by leveraging the power of representation learning. The key idea is to use a particular deep neural network with real-valued parameters, a so-called transformer, to map physical spin configurations into a high-dimensional feature space. Within this abstract space, the determination of the ground-state properties is simplified and requires only a shallow output layer with complex-valued parameters. We illustrate the efficacy of this variational ansatz by studying the ground-state phase diagram of the Shastry-Sutherland model, which captures the low-temperature behavior of SrCu2(BO3)2 with its intriguing properties. With highly accurate numerical simulations, we provide strong evidence for the stabilization of a spin liquid between the plaquette and antiferromagnetic phases. In addition, a direct calculation of the triplet excitation at the P point provides compelling evidence for a gapless spin liquid. Our findings underscore the potential of neural-network quantum states as a valuable tool for probing uncharted phases of matter, and open up new possibilities for establishing the properties of many-body systems.