On gapped phases with a continuous symmetry and boundary operators

TL;DR

This paper explores how compact symmetry groups influence the classification of gapped quantum spin systems, demonstrating the equivalence of symmetry representations in the same phase and explicitly constructing boundary operators in one-dimensional models.

Contribution

It introduces a framework for understanding symmetry representations in gapped phases and constructs explicit boundary operators for certain 1D models, linking bulk and boundary symmetries.

Findings

Ground state spaces carry equivalent symmetry representations within the same phase.

Explicit construction of an 'excess spin' operator for 1D models with SU(2) symmetry.

Relation between boundary and bulk symmetry representations in frustration-free models.

Abstract

We discuss the role of compact symmetry groups, G, in the classification of gapped ground state phases of quantum spin systems. We consider two representations of G on infinite subsystems. First, in arbitrary dimensions, we show that the ground state spaces of models within the same G-symmetric phase carry equivalent representations of the group for each finite or infinite sublattice on which they can be defined and on which they remain gapped. This includes infinite systems with boundaries or with non-trivial topologies. Second, for two classes of one-dimensional models, by two different methods, for G=SU(2) in one, and G\subset SU(d), in the other we construct explicitly an `excess spin' operator that implements rotations of half of the infinite chain on the GNS Hilbert space of the ground state of the full chain. Since this operator is constructed as the limit of a sequence of observables, the representation itself is, in principle, experimentally observable. We claim that the corresponding unitary representation of G is closely related to the representation found at the boundary of half-infinite chains. We conclude with determining the precise relation between the two representations for the class of frustration-free models with matrix product ground states.