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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.