Improved summations of $n$-point correlation functions of projected entangled-pair states

Numerical treatment of two dimensional strongly-correlated systems is both extremely challenging and of fundamental importance. Infinite projected entangled-pair states (PEPS), a class of tensor networks, have demonstrated cutting-edge performance for ground state calculations, working directly in the thermodynamic limit. Furthermore, in recent years the application of PEPS has been extended to also low-lying excited states, using an ansatz that targets quasiparticle states above the ground state with high accuracy. A major technical challenge for those simulations is the accurate evaluation of summations of two- and three-point correlation functions with reasonable computational cost. In this work, we show how a reformulation of $n$-point functions in the context of PEPS leads to extra contributions to the results that prove to play an important role. Benchmarks for the frustrated $J_1-J_2$ Heisenberg model illustrate the improved precision, efficiency and stability of the simulations compared to previous approaches. Leveraging automatic differentiation to generate the most tedious and error-prone parts of the computation, the straightforward implementation presented here is a step towards broader adoption of the PEPS excitation ansatz in future applications.