Geometry of reduced density matrices for symmetry-protected topological phases

TL;DR

This paper investigates the geometric structure of reduced density matrices in symmetry-protected topological phases, revealing ruled surface structures that signal topological order in the thermodynamic limit.

Contribution

It demonstrates that ruled surfaces in the convex set of reduced density matrices can signal SPT order, requiring symmetry-breaking terms and finite-size analysis for detection.

Findings

Ruled surfaces appear on the boundary of reduced density matrices in SPT phases.

Symmetry-breaking terms are necessary to observe ruled surfaces.

Finite systems can reveal ruled surface structures despite thermodynamic limit requirements.

Abstract

In this paper, we study the geometry of reduced density matrices for states with symmetry-protected topological (SPT) order. We observe ruled surface structures on the boundary of the convex set of low dimension projections of the reduced density matrices. In order to signal the SPT order using ruled surfaces, it is important that we add a symmetry-breaking term to the boundary of the system---no ruled surface emerges in systems without boundary or when we add a symmetry-breaking term representing a thermodynamic quantity. Although the ruled surfaces only appear in the thermodynamic limit where the ground-state degeneracy is exact, we analyze the precision of our numerical algorithm and show that a finite system calculation suffices to reveal the ruled surface structures.