Scaling Hypothesis for Projected Entangled-Pair States

We introduce a new paradigm for scaling simulations with projected entangled-pair states (PEPS) for critical strongly correlated systems, allowing for reliable extrapolations of PEPS data with relatively small bond dimensions D. The key ingredient consists of using the effective correlation length ξ for inducing a collapse of data points, f(D,χ)=f(ξ(D,χ)), for arbitrary values of D and the environment bond dimension χ. As such we circumvent the need for extrapolations in χ and can use many distinct data points for a fixed value of D. Here, we need that the PEPSs have been optimized using a fixed-χ gradient method, which can be achieved using a novel tensor-network algorithm for finding fixed points of 2D transfer matrices, or by using the formalism of backwards differentiation. We test our hypothesis on the critical 3D dimer model, the 3D classical Ising model, and the 2D quantum Heisenberg model.