# Entanglement renormalization and symmetry fractionalization

## TL;DR

This paper investigates how symmetry fractionalization manifests differently in matrix product states and MERA representations of 1D quantum ground states, revealing that symmetry fractionalization does not occur in gapped MERA but may in critical cases.

## Contribution

It demonstrates that symmetry does not fractionalize in MERA for gapped ground states, contrasting with MPS, and explores potential fractionalization in critical states, informing tensor network algorithms.

## Key findings

- Symmetry does not fractionalize in gapped MERA.
- Symmetry can fractionalize in critical MERA.
- Results support using symmetric tensors in MERA algorithms.

## Abstract

It is well known that the matrix product state (MPS) description of a gapped ground state with a global on-site symmetry can exhibit "symmetry fractionalization". Namely, even though the symmetry acts as a linear representation on the physical degrees of freedom, the MPS matrices---which act on some virtual degrees of freedom---can transform under a projective representation. This was instrumental in classifying gapped symmetry protected phases that manifest in one dimensional quantum many-body systems. Here we consider the multi-scale entanglement renormalization ansatz (MERA) description of 1D ground states that have global on-site symmetries. We show that, in contrast to the MPS, the symmetry does not fractionalize in the MERA description if the ground state is gapped, assuming that the MERA preserves the symmetry at all length scales. However, it is still possible that the symmetry can fractionalize in the MERA if the ground state is critical, which may be relevant for characterizing critical symmetry protected phases. Our results also motivate the presumed use of symmetric tensors to implement global on-site symmetries in MERA algorithms.