# Subspace-Protected Topological Phases and Bulk-Boundary Correspondence

## TL;DR

This paper introduces subspace-protected topological phases in free-fermionic systems, defining new invariants and boundary phenomena, and establishes a bulk-boundary correspondence for these phases, expanding understanding beyond symmetry-based protection.

## Contribution

It proposes a novel subspace property for Hamiltonians that leads to new topological invariants and boundary phenomena, and demonstrates the bulk-boundary correspondence in these phases.

## Key findings

- Discovery of subspace-protected topological boundary modes
- Definition of new topological invariants based on subspace properties
- Establishment of bulk-boundary correspondence for subspace-protected phases

## Abstract

While tremendous research has revealed that symmetry enriches topological phases of matter, more general principles that protect topological phases have yet to be explored. In this Letter, we elucidate the roles of subspaces in free-fermionic topological phases. A subspace property for Hamiltonians enables us to define new topological invariants. It results in peculiar topological boundary phenomena, i.e., the emergence of an unpaired zero mode or zero-winding skin modes, characterizing subspace-protected topological phases. We establish and demonstrate the bulk-boundary correspondence in subspace-protected topological phases. We further discuss the interplay of the subspace property and internal symmetries. Toward application, we also propose possible platforms possessing the subspace property.