The breakdown of conventional bulk-boundary correspondence (BBC) in non-Hermitian system can be resolved by the generalized Brillouin zone (GBZ) theory. However, extending the GBZ theory to interacting many-body systems remains an open problem. Here, we consider an interacting non-Hermitian model characterized by a circular GBZ. We show that, based on a GBZ transformation, a quasi-reciprocal many-body Hamiltonian can be constructed which, under periodic boundary conditions (PBC), captures the physics of the original non-Hermitian model under open boundary conditions (OBC). Using exact diagonalization (ED), we determine the phase diagram for the quasi-reciprocal many-body Hamiltonian by computing the Zak phase and the structure factor of the charge-density-wave (CDW) phase. We further investigate the entanglement properties and find that the degeneracy of the low-lying entanglement spectrum characterizes each phase in the phase diagram. These findings demonstrate that the topological properties in interacting non-Hermitian system is encoded in the entanglement spectrum of the quasi-reciprocal model. Our work establishes a route to studying many-body non-Hermitian physics within the GBZ formalism.
We investigate the entanglement properties of non-Hermitian Su-Schrieffer-Heeger (SSH) model from the perspective of the Generalized Brillouin Zone (GBZ). The non-Bloch entanglement entropy is defined on a quasi-reciprocal lattice, obtained by performing an ordinary Fourier transformation on the non-Bloch Hamiltonian. We demonstrate that the broken bulk-boundary correspondence is recovered in terms of the non-Bloch entanglement entropy. When the GBZ is circular, we show that the non-Bloch entanglement entropy is well-defined (real and positive-definite) in large parameter regions, except close to the exceptional points (EPs). In the critical region, we found that each Fermi point contributes precisely 1 to the central charge
c of the logarithmic scaling. At the EP, the central charge becomes negative due to the presence of the exceptional bound state. For the case of non-circular GBZ, long-range hopping emerges in the quasi-reciprocal lattice, and the von Neumann entropy on the GBZ is no longer real. However, the non-Bloch edge entanglement entropy remains real, which serves as a reliable topological indicator and respects the bulk-boundary correspondence. We compute the topological phase diagram, and reveal the critical behavior along the exceptional phase boundaries.
The physical interpretation of entanglement entropy in non-Hermitian systems remains unclear. The authors construct here an artificial lattice by performing a Fourier transformation on the non-Bloch Hamiltonian. Interestingly, when the generalized Brillouin zone (GBZ) is circular, the authors demonstrate that the entanglement entropy on the artificial lattices is well defined (real and positive definite). For noncircular GBZ, the non-Bloch edge entanglement entropy remains real and quantized, serving as a reliable topological indicator. The non-Bloch entanglement entropy opens new possibilities for studying many-body non-Hermitian physics.
The discovery of topological phases has ushered in a new era of condensed matter physics and revealed a variety of natural and artificial materials. They obey the bulk-boundary correspondence (BBC), which guarantees the emergence of boundary states with nonzero topological invariants in the bulk. Widespread attention has been paid to extending topological phases to nonlinear and non-Hermitian systems. However, the BBC and topological invariants of non-Hermitian nonlinear systems remain largely unexplored. Here, we establish a complete BBC and topological characterization of the topological phases in a class of non-Hermitian nonlinear-eigenvalue systems by introducing an auxiliary system. We restore the BBC broken by non-Hermiticity via employing the generalized Brillouin zone on the auxiliary system. Remarkably, we discover that the interplay between non-Hermiticity and nonlinearity creates an exotic complex-band topological phase that coexists with the real-band topological phase. Our results enrich the family of nonlinear topological phases and lay a foundation for exploring novel topological physics in metamaterial systems.
The non-Hermitian skin effect (NHSE) is a hallmark of non-Hermitian system, yet its generalized Brillouin zone (GBZ) description is restricted to periodic systems. We develop a site-resolved theory via a local scaling transformation (LST), introducing local twisting
Tn to quantify metric operator
ξ nontriviality. This elucidates the NHSE's origin and uncovers the generalized multiple-channel skin effect (MCSE). Exploiting
Tn's translational independence, we define the Zahlen-Brillouin Zone (ZBZ), extending non-Hermitian band theory to nonperiodic and disordered lattices. Furthermore, we unify the
ξ with GBZ Riemannian geometry, establishing the metric and state correspondence (MSC) as the principle for real-space localization. With a global skin index
Γ for phase transitions, our results provide a universal paradigm for non-Hermitian physics in both crystalline and amorphous media.
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
Pβ 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
Pβ and the entanglement polarization
χ of the biorthogonal ground state. Introducing a quasi-reciprocal Hamiltonian
H~ that removes the NHSE while preserving bulk topology, we rigorously prove the fundamental identity
Pβ≡χ(H~)(mod1) 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
H~ becomes non-local due to the divergence of position variance, we reveal that
χ(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.
Non-Hermitian Hamiltonians enrich quantum physics by extending conventional phase diagrams, enabling novel topological phenomena, and realizing exceptional points with potential applications in quantum sensing. Here, we present an experimental photonic platform capable of simulating a non-unitary quantum walk generated by a peculiar type of non-Hermitian Hamiltonian, largely unexplored in the literature. The novelty of this platform lies in its direct access to the reciprocal space, which enables us to scan the quasi-momentum across the entire Brillouin zone and thus achieve a precise tomographic reconstruction of the underlying non-Hermitian Hamiltonian, indicated by the comparison between theoretical predictions and experimental measurements. From the inferred Hamiltonian, it is possible to retrieve complex-valued band structures, resolve exceptional points in momentum space, and detect the associated parity-time symmetry breaking through eigenvector coalescence. Our results, presented entirely in quasi-momentum space, represent a substantial shift in perspective in the study of non-Hermitian phenomena.