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Physical review. B./Physical review. B

Entanglement entropy on generalized Brillouin zone

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

Published Dec 11, 2024• 7 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.110.235127

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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.
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 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 formulate quantum geometry for non-Hermitian systems under open boundary conditions. By defining quantum-geometric quantities in both real-space and non-Bloch representations, we establish a unified framework beyond conventional Bloch band theory. Our central result is an exact equivalence between the real-space integrated quantum metric and a non-Bloch integrated quantum metric defined on the generalized Brillouin zone. We further introduce localized non-Bloch Wannier functions in the presence of the non-Hermitian skin effect and show that the non-Bloch integrated quantum metric gives the gauge-invariant part of their spread functional. These results establish quantum geometry as a natural framework for characterizing open-boundary non-Hermitian band structures and the localization properties encoded in skin modes.
Entanglement entropy is a powerful tool in characterizing universal features in quantum many-body systems. In quantum chaotic Hermitian systems, typical eigenstates have near maximal entanglement with very small fluctuations. Here, we show that for Hamiltonians displaying non-Hermitian many-body quantum chaos, modeled by the Ginibre ensemble, the entanglement entropy of typical eigenstates is greatly suppressed. The entropy does not grow with the Hilbert space dimension for sufficiently large systems, and the fluctuations are of equal order. We derive the novel entanglement spectrum that has infinite support in the complex plane and strong energy dependence. We provide evidence of universality, and similar behavior is found in the non-Hermitian Sachdev-Ye-Kitaev model, indicating the general applicability of the Ginibre ensemble to dissipative many-body quantum chaos.
Entanglement entropy serves as a fundamental diagnostic for quantum many-body correlations, distinguishing quantum entanglement from classical stochastic fluctuations. This review systematically explores the scaling behaviors of entanglement entropy, beginning with the foundational area law in Hermitian systems and progressing to the exotic regimes of non-Hermitian physics. We focus on the mathematical apparatus of the Fisher-Hartwig theorem in fermionic chains, the biorthogonal formalism in non-unitary systems, and the emergence of negative entanglement entropy driven by exceptional boundary states and non-unitary conformal field theories.
Non-Bloch band theory recovers the open-boundary spectrum of a non-Hermitian chain from its Bloch Hamiltonian evaluated along the generalised Brillouin zone rather than the unit circle. We develop its finite-size counterpart for banded Toeplitz matrices with real open-boundary spectrum. Reality of the spectrum forces the generalised Brillouin zone to be parametrised by an analytic polar curve, so the known sufficient condition for a real spectrum is also necessary under some mild non-degeneracy conditions. At finite size nnn, the eigenvalues of the original and the symmetrised Bloch Hamiltonians are given by samples at nnn quantised momenta from the same limiting band, although separated by a phase mismatch of order 1/n1/n1/n which we obtain in closed form. Iterating this correction KKK times yields a quantisation rule on the generalised Brillouin zone reproducing the finite spectrum with an error O(n−K−1)\mathcal{O}(n^{-K-1})O(n−K−1) for any fixed KKK. At finite size, the generalised Brillouin zone itself is unchanged and only its parametrisation depends on nnn.