Localization−delocalization transitions in non-Hermitian Aharonov−Bohm cages

doi:10.1007/s11467-024-1412-9

Front. Phys.

2024, Vol. 19

Issue (3) : 33211 https://doi.org/10.1007/s11467-024-1412-9

Localization−delocalization transitions in non-Hermitian Aharonov−Bohm cages

Xiang Li, Jin Liu, Tao Liu(

)

School of Physics and Optoelectronics, South China University of Technology, Guangzhou 510640, China

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Abstract

A unique feature of non-Hermitian systems is the extreme sensitivity of the eigenspectrum to boundary conditions with the emergence of the non-Hermitian skin effect (NHSE). A NHSE originates from the point-gap topology of complex eigenspectrum, where an extensive number of eigenstates are anomalously localized at the boundary driven by nonreciprocal dissipation. Two different approaches to create localization are disorder and flat-band spectrum, and their interplay can lead to the anomalous inverse Anderson localization, where the Bernoulli anti-symmetric disorder induces mobility in a full-flat band system in the presence of Aharonov−Bohm (AB) Cage. In this work, we study the localization−delocalization transitions due to the interplay of the point-gap topology, flat band and correlated disorder in the one-dimensional rhombic lattice, where both its Hermitian and non-Hermitian structures show AB cage in the presence of magnetic flux. Although it remains the coexistence of localization and delocalization for the Hermitian rhombic lattice in the presence of the random anti-symmetric disorder, it surprisingly becomes complete delocalization, accompanied by the emergence of NHSE. To further study the effects from the Bernoulli anti-symmetric disorder, we found the similar NHSE due to the interplay of the point-gap topology, correlated disorder and flat bands. Our anomalous localization−delocalization property can be experimentally tested in the classical physical platform, such as electrical circuit.

Keywords non-Hermitian skin effects disorder flat band localization−delocalization transition

Corresponding Author(s): Tao Liu

Issue Date: 24 May 2024

Cite this article:

Xiang Li,Jin Liu,Tao Liu. Localization−delocalization transitions in non-Hermitian Aharonov−Bohm cages[J]. Front. Phys. , 2024, 19(3): 33211.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-024-1412-9
https://academic.hep.com.cn/fop/EN/Y2024/V19/I3/33211

Fig.1 (a) Tight-binding representation of an asymmetric rhombic chain enclosed by a

ϕ

magnetic flux in each plaquette. Each unit cell contains three sublattices indicated by

A

B

and

C

γ ± λ

denote the asymmetric hopping strengths (red and yellow lines with arrows), and

t

is the symmetric hopping strength (black line). Real part (b1) and imaginary part (b2) of single-particle eigenspectrum for

ϕ = 0

. (b3)

R e (E)

vs.

I m (E)

of eigenenergies in complex plane with PBC (blue dots) and OBC (red dots) for

ϕ = 0

. (b4) Probability density distributions

| ψ j | 2

(summed over each unit cell) of eigenstates for their eigenenergies inside point gaps with

| E | ≠ 0

under OBC for

ϕ = 0

, where

| ψ j | 2 = | ψ j, A | 2 + | ψ j, B | 2 +

| ψ j, C | 2

. Real part (c1) and imaginary part (c2) of single-particle eigenspectrum for

ϕ = π

, and its bands are perfect flat, leading to mode localization. The other parameters are

γ / t = 1

and

λ / t = 0.8

Fig.2 (a) Eigenenergies and (b) probability density distributions

| ψ j | 2

(summed over each unit cell) of eigenstates of the Hermitian rhombic lattice, subjected to the random anti-symmetric disorder, under OBCs for

Δ / t = 1

. The red dots indicate the topological boundary states. (c)

| ψ j | 2

for

E = 0

with

Δ / t = 1

. (d) IPR vs.

Δ

. The other parameters for Hermitian conditions are

ϕ = π

γ / t = 1

and

λ / t = 0

Fig.3 The localization and delocalization of the non-Hermitian rhombic lattice subjected to random anti-symmetric disorder

Δ j (B) = − Δ j (C) = Δ j

(

Δ j ∈ [− Δ / 2, Δ / 2]

) for

ϕ = π

. Complex eigenenergies under both OBCs (blue dots) and PBCs (red dots) (a1) for

Δ / t = 1

and

λ / t = γ / t = 1

, (b1) for

Δ / t = 2

and

λ / t = γ / t = 1

, and (c1) for

Δ / t = 2

and

λ / t = γ / t = 5

. The corresponding probability density distributions

| ψ j | 2

(summed over each unit cell) of eigenstates are shown in (a2, b2, c2). (d) mcom as functions of

λ

and

Δ

with

λ = γ

. (e) mcom as functions of

λ

and

Δ

with

γ / t = 1

. The mcom is averaged over 2000 disorder realization with

N = 100

Fig.4 (a) Winding number

w

as functions of

λ

and

Δ

with

γ / t = 1

for the non-Hermitian rhombic lattice subjected to random anti-symmetric disorder. (b) Winding number

w

as functions of

λ

and

Δ

with

λ = γ = 1

for the non-Hermitian rhombic lattice subjected to random anti-symmetric disorder. The results are averaged over

2000

disorder realizations with

N = 200

Fig.5 The localization of the non-Hermitian rhombic lattice subjected to random symmetric disorder

Δ j (B) = Δ j (C) = Δ j

(

Δ j ∈ [− Δ / 2, Δ / 2]

) for

ϕ = π

. (a) Complex eigenenergies under both OBCs (blue dots) and PBCs (red dots) for

Δ / t = 1

, and

λ / t = γ / t = 1

. (b) The corresponding probability density distributions

| ψ j | 2

(summed over each unit cell) of eigenstates under OBCs. (c) mcom as functions of

λ

and

Δ

with

λ = γ

. The mcom is averaged over 2000 disorder realization with

N = 100

Fig.6 Localization and delocalization of the non-Hermitian rhombic lattice subjected to Bernoulli anti-symmetric disorder with

Δ j (B) = − Δ j (C) = Δ j

(

Δ j

randomly takes two values of

± Δ

) for

ϕ = π

. Complex eigenenergies under both OBCs (blue dots) and PBCs (red dots) (a1) for

Δ / t = 1

and

λ / t = γ / t = 0.8

. The corresponding probability density distributions

| ψ j | 2

(summed over each unit cell) of eigenstates are shown in (a2). (b) mcom as functions of

λ

and

Δ

with

λ = γ

. (c) mcom as functions of

λ

and

Δ

with

γ / t = 1

. The mcom is averaged over 2000 disorder realization with

N = 100

Fig.7 Localization of the non-Hermitian rhombic lattice subjected to Bernoulli symmetric disorder with

Δ j (B) = − Δ j (C) = Δ j

(

Δ j

randomly takes two values of

± Δ

) for

ϕ = π

. (a) Complex eigenenergies under both OBCs (blue dots) and PBCs (red dots) for

Δ / t = 1

, and

λ / t = γ / t = 1

. (b) The corresponding probability density distributions

| ψ j | 2

(summed over each unit cell) of eigenstates under OBCs f. (c) mcom as functions of

λ

and

Δ

with

λ = γ

. The mcom is averaged over 2000 disorder realization with

N = 100

Fig.8 (a) Electrical circuit implementation of the model in Eq. (1), corresponding to the lattice structure in (b). The nonreciprocal hopping between nodes

j

and

j + 1

is realized by the negative impedance converters through current inversions (INICs). (c) Details of INIC.

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