Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions

TL;DR

This paper develops a finite correlation length scaling framework for iPEPS wave functions, enabling better approximation and extrapolation of gapless quantum states with Lorentz invariance, which are challenging for tensor networks.

Contribution

It introduces a novel finite correlation length scaling method for iPEPS, addressing limitations in representing Lorentz-invariant gapless states and facilitating extrapolation to the thermodynamic limit.

Findings

Best wave functions have finite correlation lengths.

Finite correlation length scaling allows extrapolation to infinite correlation length.

Framework applicable to various gapless quantum systems.

Abstract

It is an open question how well tensor network states in the form of an infinite projected entangled pair states (iPEPS) tensor network can approximate gapless quantum states of matter. Here we address this issue for two different physical scenarios: i) a conformally invariant $(2+1)d$ quantum critical point in the incarnation of the transverse field Ising model on the square lattice and ii) spontaneously broken continuous symmetries with gapless Goldstone modes exemplified by the $S=1/2$ antiferromagnetic Heisenberg and XY models on the square lattice. We find that the energetically best wave functions display {\em finite} correlation lengths and we introduce a powerful finite correlation length scaling framework for the analysis of such finite-$D$ iPEPS states. The framework is important i) to understand the mild limitations of the finite-$D$ iPEPS manifold in representing Lorentz-invariant, gapless many body quantum states and ii) to put forward a practical scheme in which the finite correlation length $\xi(D)$ combined with field theory inspired formulae can be used to extrapolate the data to infinite correlation length, i.e. to the thermodynamic limit. The finite correlation length scaling framework opens the way for further exploration of quantum matter with an (expected) Lorentz-invariant, massless low-energy description, with many applications ranging from condensed matter to high-energy physics.