Subspace-protected Topological Phases

When a Hamiltonian is block-diagonalizable by a unitary matrix, the topological phase is classified by the topological number for each block component of the Hamiltonian [1]. If block-diagonalization is impossible, the phase is usually characterized by the topological number of the entire system. For block-triangular Hamiltonians, however, the eigenvalues depend only on the block-diagonal components. This suggests that topological characterization may be possible using just these components.

This study introduces topological phases protected by invariant subspaces for block-triangular matrices. The topological numbers are obtained by restricting the Hamiltonian to the invariant subspace in the Hilbert space and extracting the diagonal components. Beyond invariant subspaces, we can extend the notion of subspace-protected topological phases to a broader class of subspaces.

In this presentation, we construct the invariant-subspace-protected topological number for a triangular Hamiltonian the diagonal components of which are the Hatano-Nelson model [2]. We show that the non-Hermitian skin effect [3,4] is captured by our newly introduced number rather than the conventional topological number. We also discuss the effects of perturbations that break the triangularity of the Hamiltonian.