A scaling hypothesis for projected entangled-pair states

TL;DR

This paper proposes a new scaling approach for PEPS simulations of critical systems, enabling reliable data extrapolation with small bond dimensions by using an effective correlation length to collapse data.

Contribution

It introduces a novel scaling hypothesis using effective correlation length for PEPS, allowing better extrapolation of data at small bond dimensions for critical systems.

Findings

Successful application to 3-D dimer model

Effective collapse of data points using correlation length

Enhanced extrapolation accuracy for critical models

Abstract

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 $\chi$ for inducing a collapse of data points, $f(D,\chi)=f(\xi(D,\chi))$, for arbitrary values of $D$ and the environment bond dimension $\chi$. As such we circumvent the need for extrapolations in $\chi$ and can use many distinct data points for a fixed value of $D$. Here, we need that the PEPS has been optimized using a fixed-$\chi$ gradient method, which can be achieved using a novel tensor-network algorithm for finding fixed points of 2-D transfer matrices, or by using the formalism of backwards differentiation. We test our hypothesis on the critical 3-D dimer model, the 3-D classical Ising model, and the 2-D quantum Heisenberg model.