Generalized Berry phase for a bosonic Bogoliubov system with exceptional points
Generalized Berry phase for a bosonic Bogoliubov system with exceptional points
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DOI：
10.1103/physreva.101.013625
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发表时间：
2019-04
影响因子：
2.9
通讯作者：
Terumichi Ohashi;S. Kobayashi;Y. Kawaguchi
中科院分区：
文献类型：
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作者：
Terumichi Ohashi;S. Kobayashi;Y. Kawaguchi
讨论了光学晶格中玻色-爱因斯坦凝聚体的波哥留波夫激发带的拓扑结构。由于玻色子系统的Bogoliubov方程是非厄米方程，因此经常出现复特征值并引起动力学不稳定。作为动量的函数，复特征值出现和消失的起始点是一个例外点（EP），这是一个哈密顿量不可对角的点，因此Berry连接和曲率是不定义的，无法定义拓扑不变量。在本文中，我们提出了一个系统的程序，以避免在布里渊区通过引入虚部的动量EPs。然后我们定义了一维玻色子Bogoliubov系统的Berry相。推广厄米系统的论证，反转对称系统的Berry相位为${\mathbb{Z}}_{2}$。作为具体的例子，我们对两个玩具模型进行了数值研究，并证实了即使在存在复特征值的情况下，体积-边缘的对应关系。讨论了与粒子-空穴对称有关的${\mathbb{Z}}_{2}$不变量和时间逆对称系统的圈数。
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 ${\mathbb{Z}}_{2}$. As concrete examples, we numerically investigate two toy models and confirm the bulk-edge correspondence even in the presence of complex eigenvalues. The ${\mathbb{Z}}_{2}$ invariant associated with particle-hole symmetry and the winding number for a time-reversal-symmetric system are also discussed.