Quantum Criticality with the Multi-scale Entanglement Renormalization Ansatz

TL;DR

This paper introduces the multi-scale entanglement renormalization ansatz (MERA) for analyzing quantum critical systems in one dimension, demonstrating its ability to extract conformal data and compare favorably with matrix product states.

Contribution

It provides a detailed introduction to MERA and showcases its effectiveness in characterizing quantum critical points and conformal field theories in one-dimensional systems.

Findings

MERA accurately characterizes critical quantum spin chains.

It extracts conformal data such as scaling dimensions.

MERA compares favorably with matrix product state methods.

Abstract

The goal of this manuscript is to provide an introduction to the multi-scale entanglement renormalization ansatz (MERA) and its application to the study of quantum critical systems. Only systems in one spatial dimension are considered. The MERA, in its scale-invariant form, is seen to offer direct numerical access to the scale-invariant operators of a critical theory. As a result, given a critical Hamiltonian on the lattice, the scale-invariant MERA can be used to characterize the underlying conformal field theory. The performance of the MERA is benchmarked for several critical quantum spin chains, namely Ising, Potts, XX and (modified) Heisenberg models, and an insightful comparison with results obtained using a matrix product state is made. The extraction of accurate conformal data, such as scaling dimensions and operator product expansion coefficients of both local and non-local primary fields, is also illustrated.