Customizable neural-network states for topological phases

Obtaining an accurate ground state wave function is a key question in the quantum many-body problem. Variational methods have proven to be excellent computationally scalable tools for the efficient approximation of ground states. Recently, generic tools such as neural networks have been established as versatile ground state ansatzes. Here, we introduce an interpretable physically motivated variational neural network ansatz based on a tunable extension of the Restricted Boltzmann Machine architecture. We illustrate its success on the example of Kitaev's toric code in the presence of magnetic fields and the transverse field Ising model. We are able to identify critical perturbation strengths leading to a transition out of the topological phase of the toric code model whose first- or second order nature depends on the direction of the magnetic fields. The flexibility of our variational ansatz allows us in addition to study more subtle properties of the phase diagram, such as first-order transition lines outside of the topological phase. We achieve this by formulating a novel algorithm for identification of excited states in the framework of variational Monte Carlo.