Abstract
Abstract Topological phases and the associated multiple edge states are studied for parity and time-reversal ($\mathcal{PT}$)-symmetric non-Hermitian open quantum systems by constructing a non-unitary three-step quantum walk retaining $\mathcal{PT}$ symmetry in one dimension. We show that the non-unitary quantum walk has large topological numbers of the $\mathbb{Z}$ topological phase and numerically confirm that multiple edge states appear as expected from the bulk–edge correspondence. Therefore, the bulk–edge correspondence is valid in this case. Moreover, we study the stability of the multiple edge states against a symmetry-breaking perturbation so that the topological phase is reduced to $\mathbb{Z}_2$ from $\mathbb{Z}$. In this case, we find that the number of edge states does not become one unless a pair of edge states coalesce at an exceptional point. Thereby, this is a new kind of breakdown of the bulk–edge correspondence in non-Hermitian systems. The mechanism of the prolongation of edge states against the symmetry-breaking perturbation is unique to non-Hermitian systems with multiple edge states and anti-linear symmetry. Toward experimental verifications, we propose a procedure to determine the number of multiple edge states from the time evolution of the probability distribution.
Highlights
In closed quantum systems, all observables including Hamiltonians are described by Hermitian operators
We introduce a non-unitary three-step quantum walk with parity and time-reversal (PT) symmetry in one dimension, which can be realized in a quantum optical system
We study the stability of the multiple edge states against a symmetry-breaking perturbation so that the topological phase is reduced to Z2 from Z
Summary
All observables including Hamiltonians are described by Hermitian operators. By using the PT -symmetric non-unitary quantum walk, topological phases and the edge states for open quantum systems have been studied theoretically [24] and experimentally [22]. We focus on the bulk-edge correspondence of PT -symmetric open quantum systems with large topological numbers by using a non-unitary quantum walk. Multiple edge states survive from the perturbation breaking the Z topological phase This is a breakdown of the bulkedge correspondence in non-Hermitian systems, which is different from the previous one [33]. IV, we introduce a time-evolution operator with a symmetry-breaking perturbation so that the topological phase is reduced to Z2 from Z and show the new kind of breakdown of bulk-edge correspondence in nonHermitian systems, which is different from the previous one [33].
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