Unified Bulk-Entanglement Correspondence in Non-Hermitian Systems

TL;DR

This paper establishes a universal correspondence between non-Bloch polarization and entanglement polarization in non-Hermitian systems, providing a robust real-space bulk diagnostic that overcomes the limitations of traditional topological invariants.

Contribution

It introduces a quasi-reciprocal Hamiltonian linking non-Bloch and entanglement polarizations, unifying geometric and entanglement approaches in non-Hermitian topological phases.

Findings

Proves $P_{\beta} \equiv \chi(\tilde{H}) \pmod 1$ in the thermodynamic limit.

Shows entanglement polarization remains quantized even when Resta polarization fails.

Restores bulk-boundary correspondence across various non-Hermitian phases.

Abstract

The non-Hermitian skin effect (NHSE) fundamentally invalidates the conventional bulk-boundary correspondence (BBC), leading topological diagnostics into a crisis. While the non-Bloch polarization $P_{\beta}$ defined on the generalized Brillouin zone restores momentum-space topology, a direct, robust real-space bulk probe has remained elusive. We resolve this by establishing a universal correspondence between $P_{\beta}$ and the entanglement polarization $\chi$ of the biorthogonal ground state. Introducing a quasi-reciprocal Hamiltonian $\tilde{H}$ that removes the NHSE while preserving bulk topology, we rigorously prove the fundamental identity $P_{\beta} \equiv \chi(\tilde{H})\pmod 1$ in the thermodynamic limit under the quasi-locality assumption. Crucially, we demonstrate that this equivalence transcends the locality constraints that limit traditional topological invariants. While the conventional Resta polarization fails when $\tilde{H}$ becomes non-local due to the divergence of position variance, we reveal that $\chi(\tilde{H})$ remains robustly quantized, protected by the Fredholm index of Toeplitz operators. Our work thus identifies entanglement as the unique real-space diagnostic capable of capturing non-Bloch topology beyond the breakdown of locality, successfully restoring the BBC across diverse non-Hermitian systems such as line-gap, point-gap, and gapless phases, thereby unifying the geometric and entanglement paradigms in non-Hermitian physics.