Parity-dependent skin effects and topological properties in the multilayer nonreciprocal Su−Schrieffer−Heeger structures

doi:10.1007/s11467-023-1350-y

Front. Phys.

2024, Vol. 19

Issue (3) : 33204 https://doi.org/10.1007/s11467-023-1350-y

RESEARCH ARTICLE

Parity-dependent skin effects and topological properties in the multilayer nonreciprocal Su−Schrieffer−Heeger structures

Jia-Rui Li¹, Cui Jiang², Han Su¹, Di Qi¹, Lian-Lian Zhang¹, Wei-Jiang Gong¹(

)

¹. College of Sciences, Northeastern University, Shenyang 110819, China
². Basic Department, Shenyang Institute of Engineering, Shenyang 110136, China

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Abstract

We concentrate on the skin effects and topological properties in the multilayer non-Hermitian Su−Schrieffer−Heeger (SSH) structure, by taking into account the nonreciprocal couplings between the different sublattices in the unit cells. Following the detailed demonstration of the theoretical method, we find that in this system, the skin effects and topological phase transitions induced by nonreciprocal couplings display the apparent parity effect, following the increase of the layer number of this SSH structure. On the one hand, the skin effect is determined by the parity of the layer number of this SSH system, as well as the parity of the band index of the bulk states. On the other hand, for the topological edge modes, such an interesting parity effect can also be observed clearly. Next, when the parameter disorders are taken into account, the zero-energy edge modes in the odd-layer structures tend to be more robust, whereas the other edge modes are easy to be destroyed. In view of these results, it can be ascertained that the findings in this work promote to understand the influences of nonreciprocal couplings on the skin effects and topological properties in the multilayer SSH lattices.

Keywords multilayer SSH lattice nonreciprocal couplings band structure skin effect

Corresponding Author(s): Wei-Jiang Gong

About author:

Peng Lei and Charity Ngina Mwangi contributed equally to this work.

Issue Date: 10 November 2023

Cite this article:

Jia-Rui Li,Cui Jiang,Han Su, et al. Parity-dependent skin effects and topological properties in the multilayer nonreciprocal Su−Schrieffer−Heeger structures[J]. Front. Phys. , 2024, 19(3): 33204.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-023-1350-y
https://academic.hep.com.cn/fop/EN/Y2024/V19/I3/33204

Fig.1 (a) Schematic diagram of the multilayer nonreciprocal SSH structure. The red and blue dots represent the sub-lattices

A

and

B

. The green and purple arrow lines indicate the intracell hopping terms

t 1 L

and

t 1 R

, and black lines stand for the intercell hopping term

t 2

. In addition, the yellow lines denote the interlayer hopping coefficient

t 3

, and

m

means the number of layers. (b, c) Spectra of Im(

E

)−Re(

E

) in the cases of

M = 1

and

M = 2

, respectively. The red circles and blue stars mean the spectra corresponding to the PBC and OBC cases, respectively. Other parameters are

t 1

= 0.4,

t 2

t 3

= 1.0, and

γ = 0.5

Tab.1 Matrix forms of the

T R S †

P H S †

, and

C S

symmetries in the cases of

M = 1

M = 2

, and

M = 3

, respectively. The matrix dimension of all three types of symmetric operators is equal to

d = M ∗ 2 N

Fig.2 Spectra of

| β (E) |

with the change of

t 1

, under the condition of

t 2 = 1.0

when

γ = 0.5

(a, c, e) and

γ = 1.5

(b, d, f). To be specific, (a) and (b) describe the result of

E = ± t 3

for

M = 2

. (c) and (d) indicate the case of

E = 0

when

M = 3

. (e) and (f) correspond to the case of

E = 2 t 3

M = 3

. Different color lines describe the different roots

| β (E) |

Fig.3 (a−d) OBC energy spectra with the change of

t 1

, where (a, b)

γ = 0.5

, and (c, d)

γ = 1.5

. The left column is the real part of eigenenergies, and the right column is the corresponding imaginary part. (e, f) Spectra of

| E |

as a function of

t 1

when

γ = 0.5

and

γ = 1.5

, respectively. Blue lines are related to the non-Hermitian multilayer nonreciprocal SSH structure, and black lines correspond to the Hermitian system after similarity transformation. Parameters are taken to be

t 2 = t 3 = 1.0

Fig.4 Eigenenergy spectra and wavefunction probability density distribution under OBC. The colors represent the values of

I P R

, where (a)

γ = 0.5

and

t 1 = 0.4

, (b)

γ = 1.5

and

t 1 = 1.4

. The right column corresponds to the wavefunction probability density distribution of edge modes and bulk states. Other parameters are set to be

t 2 = t 3 = 1.0

Fig.5 (a, b) OBC energy spectra with the change of

t 1

, where

γ = 0.5

. The left column is the real part of the eigenenergies, and the right column is the corresponding imaginary part. (c, d) Spectra of

| E |

as a function of

t 1

when

γ = 0.5

and

γ = 1.5

, respectively. Blue lines for the non-Hermitian multilayer nonreciprocal SSH structure, and black lines denote the corresponding Hermitian system after similarity transformation. (e, f) Eigenenergy and probability density spectra. The right column represents the localization probability density of edge modes. Relevant parameters are set as (a)

t 1 = 0.4

and

γ = 0.5

, and (b)

t 1 = 1.4

and

γ = 1.5

. The others are taken to be

t 2 = t 3 = 1.0

Fig.6 Eigenenergy spectra in

M = 3

, for the case of

γ = 0.5

. The colors represent the values of IPR. (i−ix) Probability density spectra of these three-classes bulk bands. (i−iii) represent the localization probability density of

E = ± 2.41 ± 0.30 i

E = ± 2.38 ± 0.18 i

, and

E = ± 2.36

, respectively; (iv−vi) shows the probability density in the cases of

E = ± 1.0 ± 0.30 i

E = ± 0.97 ± 0.18 i

, and

E = ± 0.95

, respectively; (vii−ix) stand for the probability density of

E = ± 0.42 ± 0.30 i

E = ± 0.45 ± 0.18 i

, and

E = ± 0.47

, respectively. Relevant parameters are taken to be

t 1 = 0.4

t 2 = t 3 = 1.0

Fig.7 (a−c) Eigenenergy and probability density spectra for

M = 4

in the case of

t 3 = 1.5

. (a, b) Real and imaginary parts of energy. (c) Eigenenergy spectra of

t 1 = 0.4

. (i, ii) indicate the probability density spectra with two types of edge modes. (d−f) Eigenenergy and probability density spectra under the situation of

M = 5

with

t 3 = 2.0

. (d, e) Real and imaginary parts of energy. (f) Eigenenergy of

t 1 = 0.4

. (i−iii) describe the probability density spectra with three types of edge modes. Other parameters are

t 2 = 1.0

and

γ = 0.5

Fig.8 Eigenenergy spectra for

M = 4

, where

γ = 0.5

. The colors represent the values of

I P R

. (i−xii) Probability density spectra of these four-classes bulk bands. Specifically, (i−iii) denote the localization probability density of

E = ± 3.45 ± 0.30 i

E = ± 3.40 ± 0.18 i

and

E = ± 3.38

, respectively; (iv−vi) respectively correspond to the cases of

E = ± 1.92 ± 0.30 i

E = ± 1.90 ± 0.18 i

and

E = ± 1.88

; (vii−ix) indicate the cases of

E = ± 1.43 ± 0.30 i

E = ± 1.46 ± 0.18 i

and

E = ± 1.48

; (x−xii) indicate the cases of

E = ± 0.07 ± 0.30 i

E = ± 0.04 ± 0.18 i

and

E = ± 0.02

, respectively. Other parameters are taken to be

t 1 = 0.4

t 2 = 1.0

, and

t 3 = 1.5

Fig.9 Eigenenergy spectra of

M = 5

, under the condition of

γ = 0.5

. The colors represent the values of

I P R

. (i−xv) Probability density spectra of these five-classes bulk bands. To be specific, (i−iii) are the localization probability density of

E = ± 4.46 ± 0.30 i

E = ± 4.43 ± 0.18 i

, and

E = ± 4.41

, respectively; (iv−vi) show the wavefunction probability density of

E = ± 3.00 ± 0.30 i

E = ± 2.97 ± 0.18 i

, and

E = ± 2.95

; (vii−ix) mean the localization probability density of

E = ± 2.47 ± 0.30 i

E = ± 2.50 ± 0.18 i

, and

E = ± 2.52

, respectively; (x−xii) stand for the cases of

E = ± 1.00 ± 0.30 i

E = ± 1.03 ± 0.18 i

, and

E = ± 1.05

; (xiii−xv) indicate the probability density spectra of

E = ± 0.99 ± 0.30 i

E = ± 0.97 ± 0.18 i

, and

E = ± 0.95

. Relevant parameters are taken to be

t 1 = 0.4

t 2 = 1.0

t 3 = 2.0

Tab.2 Summary of non-Hermitian skin effect (NHSE), the number of edge modes (Pair), whether it is a zero-energy mode, robustness to disorder, and locality of bulk states and edge modes in this five types of structures. In the bulk bands, we present the probability density spectra of

I m (E) = 0

. The lightness of the color indicates the magnitude of the probability density in the “Localization” column of bulk and edge modes.

Fig.10 (a−d) Energy spectra influenced by the different-strength disorder. The disorder is applied according to the manner of

t 2 ′ = t 2 + d w n

with

w n ∈ [− 1.0, 1.0]

and

d

is the disorder parameter. The parameters are set as (a) and (b)

t 1 = 0.4

γ = 0.5

M = 2

, (c) and (d)

t 1 = 0.4

γ = 0.5

M = 3

. The spectra are the average over 100 disorder results. Zoomed-in view of edge modes of

M = 2

in (a) and zero-modes of

M = 3

in (b), respectively.

Fig.11 (a−d) Energy spectra influenced by the different-strength disorder. The disorder is applied according to the manner of

t 3 ′ = t 3 + d w n

with

w n ∈ [− 1.0, 1.0]

and

d

is the disorder parameter. The parameters are set as (a) and (b)

t 1 = 0.4

γ = 0.5

M = 2

, (c) and (d)

t 1 = 0.4

γ = 0.5

M = 3

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