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Unified Bulk-Entanglement Correspondence in Non-Hermitian Systems

Xudong Zhang, Zhaoyu Sun, Bin Guo

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 $P_β$ 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 $P_β$ and the entanglement polarization $χ$ 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 $P_β \equiv χ(\tilde{H})\pmod 1$ 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 $χ(\tilde{H})$ 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.

Unified Bulk-Entanglement Correspondence in Non-Hermitian Systems

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 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 and the entanglement polarization of the biorthogonal ground state. Introducing a quasi-reciprocal Hamiltonian that removes the NHSE while preserving bulk topology, we rigorously prove the fundamental identity 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 becomes non-local due to the divergence of position variance, we reveal that 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.

Paper Structure

This paper contains 3 equations, 2 figures, 1 table.

Figures (2)

  • Figure 1: Unifying bulk-boundary correspondence via entanglement. (a) Evolution of the periodic boundary conditions entanglement spectrum (grey) with $t_1$. The standard PBC polarization $\chi$ (green dots) fails to capture the topology in the point-gapped phase, deviating from the Bloch invariant $P$ (red line). (b) Restoration of the bulk-boundary correspondence: By employing the quasi-reciprocal Hamiltonian $\tilde{H}$, the entanglement polarization $\chi(\tilde{H})$ (blue circles) is robustly quantized and matches the non-Bloch polarization $P_\beta$ (yellow triangles), confirming $P_\beta \equiv \chi(\tilde{H})\pmod 1$. (c) The GBZ (blue curve) at $t_1=0.8$. Its deviation from the unit circle (grey dashed line) induces long-range power-law hoppings in $\tilde{H}$. (d) Spatial localization of the entanglement edge modes ($\xi=0.5$). Parameters are fixed at $t_2=1, \gamma=0.4, t_3=0.2$, with system size $L=800$.
  • Figure 2: Topological phase transition and correspondence breakdown. (a) Verification of the correspondence: The coincidence of the non-Bloch polarization $P_\beta$ (yellow triangles) and the real-space entanglement polarization $\chi(\tilde{H})$ (blue circles) validates the identity $P_\beta \equiv \chi(\tilde{H})\pmod 1$ in the gapped regions. The grey dots represent the entanglement orbital spectrum (EOS) of $\tilde{H}$. The shaded region indicates the gapless Topological Semimetal (TSM) phase. (b) Collapse of the invariant: The sharp drop of the non-Bloch Wilson loop norm $|W_\beta|$ to zero marks the phase transition, driven by Exceptional Points (EPs) intersecting the Generalized Brillouin Zone (GBZ) contour. Parameters are fixed at $t_2 = 1$, $\gamma = 1$, $t_3 = 0.3$, with system size $L=800$.