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arXiv:2108.00607 · Physical review. B./Physical review. B

Characterizing bulk-boundary correspondence of one-dimensional non-Hermitian interacting systems by edge entanglement entropy

Weitao Chen, Liangtao Peng, Hantao Lu, Xiancong Lu

Published Feb 14, 2022• 16 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.105.075126

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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.
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.
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 entanglement entropy can be an effective diagnostic tool for probing topological phase transitions. In one-dimensional single particle systems, the periodic driving generates a variety of topological phases and edge modes. In this work, we investigate the topological phase transition of the one-dimensional Floquet Su-Schrieffer-Heeger model using entanglement entropy, and construct the phase diagram based on entanglement entropy. The entanglement entropy exhibits pronounced peaks and follows the logarithmic scaling law at the phase transition points, from which we extract the central charge c=1c=1c=1. We further investigate the entanglement spectrum to accurately distinguish the different topological phases. In addition, the coupling between zero and πππ modes leads to characteristic splittings in the entanglement spectrum, signaling their hybridization under periodic driving. These results remain robust in non-Hermitian regimes and in the presence of next-nearest-neighbor hopping, demonstrating the reliability and universality of entanglement entropy as a diagnostic for topological phase transitions.
We investigate quantum correlations and entanglement in a non-Hermitian quantum system modeled by a 2D stacked Su–Schrieffer–Heeger (SSH) lattice incorporating nonreciprocal hopping amplitudes and balanced on-site gain and loss. Focusing on the interplay between non-Hermiticity, topology, and open-boundary conditions, we analyze how these factors shape bipartite entanglement properties, characterized through the entanglement spectrum and entanglement negativity ( E N ). By examining the complex energy spectrum under open boundary conditions, we identify critical points associated with the closing of complex energy gaps, which significantly influence entanglement structure and signal topological phase transitions. Our findings reveal that boundary-localized non-Bloch modes and nonreciprocal couplings can induce highly tunable and unconventional entanglement features, absent under periodic boundary conditions. These results highlight the breakdown of conventional bulk-boundary correspondence and underscore the utility of entanglement measures in diagnosing non-Hermitian topological phases. Beyond advancing the fundamental understanding of quantum correlations in open systems, our study suggests potential pathways to engineer quantum devices and topological materials where gain, loss, and nonreciprocity are intrinsic elements.
Recent studies of moiré physics have unveiled a wealth of opportunities for significantly advancing the field of quantum phase transitions. However, properties of reentrant phase transitions driven by moiré strength are poorly understood. Here, we investigate the reentrant sequence of phase transitions and the invariant of universality class in moiré-modulated extended Su-Schrieffer-Heeger (SSH) model. For the simplified case with intercell hopping w=0w=0w=0, we analytically derive renormalization relations of Hamiltonian parameters to explain the reentrant phenomenon. For the general case, numerical phase boundaries are calculated in the thermodynamic limit. The bulk boundary correspondence between zero-energy edge modes and entanglement spectrum is revealed from the degeneracy of both quantities. We also address the correspondence between the central charge obtained from entanglement entropy and the change in winding number during the phase transition. Our results shed light on the understanding of universal characteristics and bulk-boundary correspondence for moiré induced reentrant phase transitions in 1D condensed-matter systems.
We investigate the entanglement spectrum of the ground state of the Su-Schrieffer-Heeger-Hubbard model. The topological phases of the model can be identified by degeneracy of the largest eigenvalues of entanglement spectrum. The study of the periodic boundary condition is enough to obtain the phase diagram of the model, without the consideration of the open boundary condition case. By analyzing the states corresponding to the degenerate entanglement spectrum and showing the related edge states in real space, physical interpretation about the relation between bulk-edge correspondence and the entanglement spectrum is presented. The method of the entanglement spectrum can be applicable in studying other topological phases of matter.