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arXiv:1502.00443 · Physical Review A

Topological invariance and global Berry phase in non-Hermitian systems

Shi‐Dong Liang, Guang-Yao Huang

Published Jan 17, 2013• 245 citations
arXivHTMLPDF

Abstract

By studying the topological invariance and Berry phase in non-Hermitian systems, we reveal the basic properties of the complex Berry phase and generalize the global Berry phases $Q$ to identify the topological invariance for non-Hermitian systems. We find that $Q$ can identify topological invariance in two kinds of non-Hermitian model, the two-level non-Hermitian Hamiltonian and the bipartite dissipative model. For the bipartite dissipative model, an abrupt change of the Berry phase in the parameter space reveals a quantum phase transition and is related to the exceptional points. These results give the basic relationships between the Berry phase and the quantum and topological phase transitions of non-Hermitian systems.

DOI
10.1103/physreva.87.012118

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By studying the topological invariance and Berry phase in non-Hermitian systems, we reveal the basic properties of the complex Berry phase and generalize the global Berry phases QQQ to identify the topological invariance for non-Hermitian systems. We find that QQQ can identify topological invariance in two kinds of non-Hermitian model, the two-level non-Hermitian Hamiltonian and the bipartite dissipative model. For the bipartite dissipative model, an abrupt change of the Berry phase in the parameter space reveals a quantum phase transition and is related to the exceptional points. These results give the basic relationships between the Berry phase and the quantum and topological phase transitions of non-Hermitian systems.
Abstract We study the complex Berry phases in non-Hermitian systems with parity- and time-reversal   symmetry. We investigate a kind of two-level system with   symmetry. We find that the real part of the the complex Berry phases have two quantized values and they are equal to either 0 or π , which originates from the topology of the Hermitian eigenstates. We also find that if we change the relative parameters of the Hamiltonian from the unbroken-   -symmetry phase to the broken-   -symmetry phase, the imaginary part of the complex Berry phases are divergent at the exceptional points. We exhibit two concrete examples in this work, one is a two-level toys model, which has nontrivial Berry phases; the other is the generalized Su–Schrieffer–Heeger (SSH) model that has physical loss and gain in every sublattice. Our results explicitly demonstrate the relation between complex Berry phases, topology and   -symmetry breaking and enrich the field of the non-Hermitian physics.
As energy dissipation and gain are ubiquitous in the real world, such phenomena demand the generalization of Hermitian methods such as the analysis of topological properties for non-Hermitian systems. However, as non-Hermitian systems typically contain more degrees of freedom, this poses a challenge for analytical approaches to understand their topology and invariants. In this work, we analytically calculate the 2D Zak phase for a 2D non-Hermitian SSH-type Hamiltonian that supports a rich structure and edge currents. Closed-form expressions for eigenstates and divisions of the phase diagram are obtained, including for regions in the phase diagram where different types of exceptional points exist. We use Morse theory to determine the topology of exceptional points in momentum space. Although the band structure breaks down at exceptional points, we show that a specific phase-based topological invariant remains well-defined. Furthermore, our work yields an analytic derivation for counting edge states in the Hermitian limit. These results provide new conceptual and analytical tools for the study of complex topological systems.
We discuss the topology of Bogoliubov excitation bands from a Bose-Einstein condensate in an optical lattice. Since the Bogoliubov equation for a bosonic system is non-Hermitian, complex eigenvalues often appear and induce dynamical instability. As a function of momentum, the onset of appearance and disappearance of complex eigenvalues is an exceptional point (EP), which is a point where the Hamiltonian is not diagonalizable and hence the Berry connection and curvature are ill-defined, preventing defining topological invariants. In this paper, we propose a systematic procedure to avoid EPs in the Brillouin zone by introducing an imaginary part of the momentum. We then define the Berry phase for a one-dimensional bosonic Bogoliubov system. Extending the argument for Hermitian systems, the Berry phase for an inversion-symmetric system is shown to be Z2{\mathbb{Z}}_{2}Z2​. As concrete examples, we numerically investigate two toy models and confirm the bulk-edge correspondence even in the presence of complex eigenvalues. The Z2{\mathbb{Z}}_{2}Z2​ invariant associated with particle-hole symmetry and the winding number for a time-reversal-symmetric system are also discussed.
We study the scaling behavior of the Berry phase in the Yang-Lee edge singularity (YLES) of the non-Hermitian quantum system. A representative model, the one-dimensional quantum Ising model in an imaginary longitudinal field, is selected. For this model, the dissipative phase transition (DPT), accompanying a parity-time (PT) symmetry-breaking phase transition, occurs when the imaginary field changes through the YLES. We find that the real and imaginary parts of the complex Berry phase show anomalies around the critical points of YLES. In the overlapping critical regions constituted by the (0 + 1)D YLES and (1 + 1)D ferromagnetic-paramagnetic phase transition (FPPT), we find that the real and imaginary parts of the Berry phase can be described by both the (0 + 1)D YLES and (1 + 1)D FPPT scaling theory. Our results demonstrate that the complex Berry phase can be used as a universal order parameter for the description of the critical behavior and the phase transition in the non-Hermitian systems.
Topological phases are generally characterized by topological invariants denoted by integer numbers. However, different topological systems often require different topological invariants to measure, and theses definition usually fail at critical points. Therefore, it's challenging to predict what would occur during the transformation between two different topological phases. To address these issues, we propose a general definition based on fidelity and trace distance from quantum information theory: manifold distance (MD). This definition does not rely on the berry connection but rather on the information of the two manifolds - their ground state wave functions. Thus, it can measure different topological systems (including traditional band topology models, non-Hermitian systems, and gapless systems, etc.) and exhibit some universal laws during the transformation between two topological phases. Our research demonstrates for different topological manifolds, the change rate (first-order derivative) or susceptibility (second-order derivative) of MD exhibit various divergent behaviors near the critical points. Compared to the strange correlator, which could be used as a diagnosis for short-range entangled states in 1D and 2D, MD is more universal and could be applied to non-Hermitian systems and long-range entangled states. For subsequent studies, we expect the method to be generalized to real-space or non-lattice models, in order to facilitate the study of a wider range of physical platforms such as open systems and many-body localization.