Efficient optimization and conceptual barriers in variational finite Projected Entangled-Pair States

Projected entangled pair states (PEPS) on finite two-dimensional lattices are a natural ansatz for representing ground states of local many-body Hamiltonians, as they inherently satisfy the boundary law of entanglement entropy. In this paper, we propose the optimization of PEPS via an improved formulation of the time-dependent variational principle (TDVP), namely the minimum-step stochastic-reconfguration scheme recently introduced for neural quantum states. We further discuss possible numerical issues that might arise in such a sampling-based approach. In this context, investigate the entanglement properties of random PEPS and find an entanglement phase transition. We note that on one side of this transition, we can identify positive random tensors as product states. To demonstrate the power of the framework described in this paper, we apply the PEPS to study the notoriously challenging chiral spin liquids. Moreover, we exhibit our approach's capability to naturally handle long-range interactions by exploring the phase diagram of Rydberg atom arrays with long-range interactions. We further provide parallelized easy-to-use code, allowing the straightforward application of our method to general Hamiltonians composed of local interaction terms.