ℤ𝟐𝟐 Point-Gap Topology of a Non-Unitary Quantum Walk with Time-ReversalSymmetry
Zhiyu Jiang1 and Hideaki Obuse1,212

Department of Applied Physics, Hokkaido University
Institute of Industrial Science, The University of Tokyozhiyu.jiang.t7@elms.hokudai.ac.jp

Introduction
The study of non-Hermitian systems has gained significant attention in recent years due to its potential for novel physicalphenomena and practical applications. Non-Hermitian physics is characterized by systems with non-Hermitian Hamiltonian,which can lead to topological properties different from Hermitian systems such as non-Hermitian skin effects (NHSE) [1].
In the Hatano-Nelson (HN) model exhibiting NHSE, the energy spectrum can be significantly affected by the boundary
conditions due to a non-zero point-gap topological number which is defined by the winding number (∈ ℤ). However, thewinding number vanishes in time-reversal systems consisting of two coupled HN chains with opposite non-reciprocity. Evenin that case, the system can remain NHSE because of the emergence of ℤ2 topological number, which is called ℤ2 nonHermitian skin effect [2]. One promising area of exploration within topology of non-Hermitian physics is the study of nonunitary quantum walks. The standard quantum walks consist of a walker moving on lattices, whose state is defined by unitary
operators with each step. However, the non-unitary quantum walks incorporate gain and loss, resulting in non-Hermitian
Hamiltonian dynamics. Some models of non-unitary quantum walks have been realized theoretically and experimentally bytemporally alternating photon losses [3,4] while there is limited research focused on the topological properties of timereversal symmetric non-unitary quantum walks, which is worth studying to realize ℤ2 NHSE by quantum walks.
In our work [5], we introduce a quantum walk, where the walker traverses two non-unitary quantum-walk chains toachieve non-Hermitian time-reversal symmetry (TRS † ). We demonstrate that for the quantum walk, the quasi-energy
spectrum strongly depends on the boundary conditions at certain energy points that exhibit ℤ2 topological behavior [Fig.1(a)]. Kramers doublets with topologically non-trivial energies are confined to the edges of the chains, indicating theemergence of the ℤ2 NHSE in this quantum walk. Our work highlights the potential for non-unitary quantum walks to serve
as a platform for exploring novel physical phenomena and paves the way for further investigations into non-Hermitiansystems with time-reversal symmetry.

Tables and Figures

FIG. 1. Eigenvalue spectrum of time evolution operator under period boundaries conditions and reflecting boundaryconditions. (a) describes a topological non-trivial quantum walk, while (b) describes a topological trivial quantum walk.
Acknowledgement
Z. J. was supported by JST SPRING (Grant No. JPMJSP2119). This work was supported by JSPS KAKENHI (Grants
No. JP19H00658, No. JP20H01828, No. JP21H01005, No. JP22K03463 and No. JP22H01140)

Reference
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