Detecting bulk and edge exceptional points in non-Hermitian systems through generalized Petermann factors

doi:10.1007/s11467-023-1337-8

Front. Phys.

2024, Vol. 19

Issue (2) : 23201 https://doi.org/10.1007/s11467-023-1337-8

RESEARCH ARTICLE

Detecting bulk and edge exceptional points in non-Hermitian systems through generalized Petermann factors

Yue-Yu Zou, Yao Zhou, Li-Mei Chen, Peng Ye(

)

School of Physics, State Key Laboratory of Optoelectronic Materials and Technologies, and Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, Sun Yat-sen University, Guangzhou 510275, China

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Abstract

Non-orthogonality in non-Hermitian quantum systems gives rise to tremendous exotic quantum phenomena, which can be fundamentally traced back to non-unitarity. In this paper, we introduce an interesting quantity (denoted as $η$ ) as a new variant of the Petermann factor to directly and efficiently measure non-unitarity and the associated non-Hermitian physics. By tuning the model parameters of underlying non-Hermitian systems, we find that the discontinuity of both $η$ and its first-order derivative (denoted as $∂ η$ ) pronouncedly captures rich physics that is fundamentally caused by non-unitarity. More concretely, in the 1D non-Hermitian topological systems, two mutually orthogonal edge states that are respectively localized on two boundaries become non-orthogonal in the vicinity of discontinuity of $η$ as a function of the model parameter, which is dubbed “edge state transition”. Through theoretical analysis, we identify that the appearance of edge state transition indicates the existence of exceptional points (EPs) in topological edge states. Regarding the discontinuity of $∂ η$ , we investigate a two-level non-Hermitian model and establish a connection between the points of discontinuity of $∂ η$ and EPs of bulk states. By studying this connection in more general lattice models, we find that some models have discontinuity of $∂ η$ , implying the existence of EPs in bulk states.

Keywords non-Hermitian Su−Schrieffer−Heeger (SSH) model exceptional point

Corresponding Author(s): Peng Ye

Issue Date: 07 October 2023

Cite this article:

Yue-Yu Zou,Yao Zhou,Li-Mei Chen, et al. Detecting bulk and edge exceptional points in non-Hermitian systems through generalized Petermann factors[J]. Front. Phys. , 2024, 19(2): 23201.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-023-1337-8
https://academic.hep.com.cn/fop/EN/Y2024/V19/I2/23201

Discontinuity	EPs	Model

$η$	Topological edge states	I
$∂ η$	Bulk states	II, III, IV

Tab.1 The correspondence between the location of EPs and the discontinuity of

η

and its derivative

∂ η

in the models I−IV.

Fig.1 (a) Real part of energy spectrum of the model-I (2) as a function of

t 1

. The topological phase transition occurs at

t 1 = t 22 − g 2 ≈ 0.99

. (b) and (c) respectively show the quantity

η

and its derivative as a function of

t 1

. The discontinuity of

η

and the local maximum of

η

appear respectively at

t 1 ≈ 0.15

and

t 1 ≈ 0.99

. (d) and (e) respectively demonstrate two distributions of edge states at

t 1 = 0.133

and

t 1 = 0.333

near the discontinuity point

t 1 = 0.15

of the quantity

η

. Here,

t 2 = 1, g = 0.1

, the length of the system (2)

L = 150

Fig.2 (a) and (b) respectively represent the quantity

η

and its derivative as a function of the parameter

γ

in the two-level model (3).

Fig.3 (a) and (b) respectively show the quantity

η

and its derivative as a function of the potential strength

V

in the model-III (4) [42]. Here,

J R = 1, J L = 0.5

Fig.4 (a) is the energy spectrum of the Hamiltonian (5), (b) and (c) are respectively the quantity

η

and its derivative with the wave vector

k

varying. Here

(w, u, v) = (0.7, 0.5, 0.8)

Fig. A1 The location of discontinuity point of

η

with the size of the model-I increasing. Here

g = 0.1

t 2 = 1

Fig. C1 (a) Energy spectrum of the model (C-1) with open boundary condition as a function of the parameter

g

. (b) and (c) respectively show the quantity

η

and its derivative as a function of parameter

g

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