Local indistinguishability and edge modes revealed by the sub-system fidelity

Abstract

We propose to use the sub-system fidelity, defined by comparing a pair of reduced density matrices, to identify and locate zero modes relating two degenerate ground states in one-dimensional interacting many-body systems, where the degeneracy arises from either symmetry protected topology (SPT), or from discrete symmetry breaking (DSB). A theorem is provided to construct locally indistinguishable (LI) states by linear recombination of the degenerate states of minimal bulk entanglement entropy. The theorem enables us to construct local operators that swap exactly the LI states and are therefore the zero energy modes in the many-body system. Interestingly, they can be located anywhere in the DSB case, but can only be accommodated near the edges in the SPT case. This can be used to identify or distinguish SPT states against the so-called cat states in the DSB case. We illustrate the results for the anisotropic Haldane chain and the interacting Kitaev model.