Hu Haiping, Zhao Erhai
Department of Physics and Astronomy, George Mason University, Fairfax, Virginia 22030, USA.
Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA.
Phys Rev Lett. 2021 Jan 8;126(1):010401. doi: 10.1103/PhysRevLett.126.010401.
Knots have a twisted history in quantum physics. They were abandoned as failed models of atoms. Only much later was the connection between knot invariants and Wilson loops in topological quantum field theory discovered. Here we show that knots tied by the eigenenergy strings provide a complete topological classification of one-dimensional non-Hermitian (NH) Hamiltonians with separable bands. A Z_{2} knot invariant, the global biorthogonal Berry phase Q as the sum of the Wilson loop eigenphases, is proved to be equal to the permutation parity of the NH bands. We show the transition between two phases characterized by distinct knots occur through exceptional points and come in two types. We further develop an algorithm to construct the corresponding tight-binding NH Hamiltonian for any desired knot, and propose a scheme to probe the knot structure via quantum quench. The theory and algorithm are demonstrated by model Hamiltonians that feature, for example, the Hopf link, the trefoil knot, the figure-8 knot, and the Whitehead link.
纽结在量子物理领域有着曲折的历史。它们曾作为失败的原子模型被摒弃。直到很久以后，人们才发现纽结不变量与拓扑量子场论中的威尔逊圈之间的联系。在此我们表明，由本征能量弦所系的纽结为具有可分离能带的一维非厄米（NH）哈密顿量提供了完整的拓扑分类。一个(Z_{2})纽结不变量，即作为威尔逊圈本征相位之和的全局双正交贝里相位(Q)，被证明等于NH能带的置换奇偶性。我们表明，由不同纽结所表征的两个相之间的转变通过例外点发生，且有两种类型。我们进一步开发了一种算法，用于为任何所需的纽结构造相应的紧束缚NH哈密顿量，并提出了一种通过量子猝灭来探测纽结结构的方案。该理论和算法通过一些模型哈密顿量得到了验证，这些模型哈密顿量具有例如霍普夫链环、三叶纽结、八字纽结和怀特海德链环等特征。
