Scaling of entanglement support for Matrix Product States

TL;DR

This paper investigates how matrix product states approximate critical spin chains, revealing a universal scaling of entanglement entropy with matrix size and introducing a finite correlation length concept that aids in extracting critical properties.

Contribution

It provides a detailed analysis of the scaling behavior of MPS at criticality, establishing a finite-$ extit{chi}$ scaling framework and deriving critical exponents from this scaling.

Findings

Entanglement entropy scales as S ~ (1/6) log chi for the quantum Ising model.

Finite correlation length xi_chi scales as chi^kappa with kappa=2 for Ising.

Finite-$ extit{chi}$ scaling enables extraction of critical exponents.

Abstract

The power of matrix product states to describe infinite-size translational-invariant critical spin chains is investigated. At criticality, the accuracy with which they describe ground state properties of a system is limited by the size $\chi$ of the matrices that form the approximation. This limitation is quantified in terms of the scaling of the half-chain entanglement entropy. In the case of the quantum Ising model, we find $S \sim {1/6}\log \chi$ with high precision. This result can be understood as the emergence of an effective finite correlation length $\xi_\chi$ ruling of all the scaling properties in the system. We produce five extra pieces of evidence for this finite-$\chi$ scaling, namely, the scaling of the correlation length, the scaling of magnetization, the shift of the critical point, and the scaling of the entanglement entropy for a finite block of spins. All our computations are consistent with a scaling relation of the form $\xi_\chi\sim \chi^{\kappa}$, with $\kappa=2$ for the Ising model. In the case of the Heisenberg model, we find similar results with the value $\kappa\sim 1.37$. We also show how finite-$\chi$ scaling allow to extract critical exponents. These results are obtained using the infinite time evolved block decimation algorithm which works in the thermodynamical limit and are verified to agree with density matrix renormalization group results.