Sub-system fidelity for ground states in one dimensional interacting systems

TL;DR

This paper introduces a method using sub-system fidelity to identify and characterize symmetry protected topological states in one-dimensional interacting systems, revealing their local indistinguishability properties and edge behaviors.

Contribution

It develops a scheme based on sub-system fidelity to distinguish SPT states from symmetry-breaking states and locates zero-energy excitations in these systems.

Findings

SPT states are locally indistinguishable in a finite bulk region.

Zero modes can be positioned anywhere for cat states but are edge-localized for SPT states.

The method can identify phase transitions between SPT and symmetry-breaking states.

Abstract

We propose to utilize the sub-system fidelity (SSF), defined by comparing a pair of reduced density matrices derived from the degenerate ground states, to identify and/or characterize symmetry protected topological (SPT) states in one-dimensional interacting many-body systems. The SSF tells whether two states are locally indistinguishable (LI) by measurements within a given sub-system. Starting from two polar states (states that could be distinguished on either edge), the other combinations of these states can be mapped onto a Bloch sphere. We prove that a pair of orthogonal states on the equator of the Bloch sphere are LI, independently of whether they are SPT states or cat states (symmetry-preserving states by linear combinations of states that break discrete symmetries). Armed with this theorem, we provide a scheme to construct zero-energy exitations that swap the LI states. We show that the zero mode can be located anywhere for cat states, but is localized near the edge for SPT states. We also show that the SPT states are LI in a finite fraction of the bulk (excluding the two edges), whereas the symmetry-breaking states are distinguishable. This can be used to pinpoint the transition from SPT states to the symmetry-breaking states.