Bulk–edge correspondence and stability of multiple edge states of a $\mathcal{PT}$-symmetric non-Hermitian system by using non-unitary quantum walks

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.

Bulk-edge correspondence in nonunitary Floquet systems with chiral symmetry

“…This study systematically clarifies features resulting from chiral symmetry. While the method has been applied to nonunitary Floquet systems [36,47,74,75] based on the analogy to unitary Floquet systems [55], our study gives the microscopic foundation for the validity of the procedure in nonunitary open Floquet systems. We have constructed a model classified into class BDI † or AIII, depending on parameters, with PT symmetry in some situations.…”

“…Equation 24has been employed in nonunitary Floquet systems [36,47,74,75] based on the analogy to unitary Floquet systems in which the same formula is proven to be satisfied [55]. Our derivation gives the microscopic foundation of Eq.…”

“…Since quantum walks are Floquet systems in which time evolves in a discrete manner, topological phases can be different from those which are described by time-independent Hamiltonians. Floquet topological phases of quantum walks have been intensively studied for the last decade [51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69], and topological edge states have been observed in experiments of both closed [70][71][72][73] and open [36,47,74,75] systems. Specifically, much attention has been paid to Floquet systems with chiral symmetry.…”

“…This is because a procedure to calculate topological numbers has been established in the case of chiral symmetric unitary time-evolution operators [55]. On the other hand, regarding open Floquet systems described by nonunitary time-evolution operators, the microscopic foundation for the procedure has not yet been clarified, although it has already been utilized in previous experimental and numerical studies [36,47,74,75]. Also, the enriched symmetries of non-Hermitian open systems [39] have not been discussed so much in nonunitary open Floquet systems.…”

Delocalization of non-Hermitian Quantum Walk on Random Media in One Dimension

“…This is actually related to the P T -symmetric quantum walk defined in Ref. [22,23,24] with a gain-loss operator. The simplest version of the timeevolution operator defined there is given by…”