Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems

Abstract

We unveil the geometrical meaning of winding number and utilize it to characterize the topological phases in one-dimensional chiral non-Hermitian systems. While chiral symmetry ensures the winding number of Hermitian systems being integers, it can take half integers for non-Hermitian systems. We give a geometrical interpretation of the half integers by demonstrating that the winding number $\nu$ of a non-Hermitian system is equal to half of the summation of two winding numbers ${\nu }_1$ and ${\nu }_2$ associated with two exceptional points respectively. The winding numbers ${\nu }_1$ and ${\nu }_2$ represent the times of real part of the Hamiltonian in momentum space encircling the exceptional points and can only take integers. We further find that the difference of ${\nu }_1$ and ${\nu }_2$ is related to the second winding number or energy vorticity. By applying our scheme to a non-Hermitian Su-Schrieffer-Heeger model and an extended version of it, we show that the topologically different phases can be well characterized by winding numbers. Furthermore, we demonstrate that the existence of left and right zero-mode edge states is closely related to the winding number ${\nu }_1$ and ${\nu }_2$.

8 pages, 7 figures

References in corpus (12)

• Making Sense of Non-Hermitian Hamiltonians
• The physics of exceptional points
• Non-Hermitian robust edge states in one-dimension: Anomalous localization and eigenspace condensation at exceptional points
• Edge Modes, Degeneracies, and Topological Numbers in Non-Hermitian Systems
• Topological phases in the non-Hermitian Su-Schrieffer-Heeger model
• Weyl Exceptional Rings in a Three-Dimensional Dissipative Cold Atomic Gas
• Topological Transition in a Non-Hermitian Quantum Walk
• Dynamically encircling exceptional points: Exact evolution and polarization state conversion
• PT-Symmetry in Non-Hermitian Su-Schrieffer-Heeger model with complex boundary potentials
• Topological invariance and global Berry phase in non-Hermitian systems
• Topological phase in a non-Hermitian PT symmetric system
• Why does bulk boundary correspondence fail in some non-hermitian topological models

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