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arXiv:2406.15564

Entangelment Entropy on Generalized Brillouin Zone

Zhenghao Yang, C.-Y. Lu, Xiancong Lu

Published Jun 21, 2024• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.2406.15564

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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 ccc 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.
Topological phase transitions are found in a variety of systems and were shown to be deeply related with a thermodynamic description through scaling relations. Here we investigate the entanglement entropy, which is a quantity that captures the central charge of a critical model and the thermodynamics of the nonreciprocal Su-Schrieffer-Heeger (SSH) model. Although this model has been widely studied, the thermodynamic properties reveal interesting physics not explored so far. In order to analyze the boundary effects of the model, we use Hill's thermodynamics to split the grand potential in two contributions: the extensive one, related to the bulk, and the subdivision one, related to the boundaries. Then, we derive the thermodynamic entropy for both the edges and the bulk, and the heat capacity for the bulk at the topological phase transitions. The latter is related to the central charge when the underlying theory is a conformal field theory, whereas the first reveals the resilience of the topological edge states to finite temperatures. The phase transition between phases that are adiabatically connected with the Hermitian SSH model display the well-known behavior of systems within the Dirac universality class, but the transition between phases with complex energies shows an unexpected critical behavior that signals the emergence of an imaginary time crystal.
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 phase transitions in the extended Su-Schrieffer-Heeger (SSH) model with next-nearest-neighbor hoppings and an imaginary staggered chemical potential. In the presence of small non-Hermiticity, exceptional points emerge in pairs from the gap-closing momenta near the topological phase boundaries of the Hermitian limit. Utilizing the Krylov spread complexity and entanglement entropy, we analyze two dynamical protocols: (i) preparing the non-Hermitian ground state via a unitary transformation, and (ii) evolving the system under the non-Hermitian Hamiltonian. We show that the spread complexity, and long-time spread complexity as well as entanglement entropy can effectively signal phase transitions in the first and second protocols, respectively. To unravel the detailed structure of the transitions, we introduce the momentum-resolved complexity that identifies the characteristic modes and tracks their evolution with the driving parameter. In the regime where the system possesses a purely imaginary spectrum, we further identify dynamical phases based on the saturation behavior of the spread complexity. The entanglement entropy is also found to exhibit similar saturation behavior, thereby providing a more experimentally accessible probe of the dynamical phases.
Dramatically different from the Hermitian systems, the conventional bulk-boundary correspondence (BBC) is broken in the non-Hermitian systems. In this paper, we use edge entanglement entropy to characterize the topological properties of non-Hermitian Su-Schrieffer-Heeger Hubbard model. For free fermions, we study the scaling behavior of entanglement entropy and demonstrate that the edge entanglement entropy is a good indicator to delimit different phases of non-Hermitian systems. We further generalize the edge entanglement entropy to the non-Hermitian interacting Hubbard chain and obtain the topological phase diagram in the plane of interaction and non-Hermitian hopping amplitudes. It is found that the Hubbard interaction diminishes and weakens the breakdown of the bulk-boundary correspondence, which eventually disappears at some critical value of interaction.
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 PβP_βPβ​ 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 PβP_βPβ​ and the entanglement polarization χχχ of the biorthogonal ground state. Introducing a quasi-reciprocal Hamiltonian H~\tilde{H}H~ that removes the NHSE while preserving bulk topology, we rigorously prove the fundamental identity Pβ≡χ(H~)(mod1)P_β \equiv χ(\tilde{H})\pmod 1Pβ​≡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~\tilde{H}H~ becomes non-local due to the divergence of position variance, we reveal that χ(H~)χ(\tilde{H})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.