Observation of impurity-induced scale-free localization in a disordered non-Hermitian electrical circuit

doi:10.15302/frontphys.2025.014203

Front. Phys.

2025, Vol. 20

Issue (1) : 14203 https://doi.org/10.15302/frontphys.2025.014203

Observation of impurity-induced scale-free localization in a disordered non-Hermitian electrical circuit

Hao Wang, Jin Liu, Tao Liu(

), Wenbo Ju(

)

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

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Abstract

One of unique features of non-Hermitian systems is the extreme sensitive to their boundary conditions, e.g., the emergence of non-Hermitian skin effect (NHSE) under the open boundary conditions, where most of bulk states become localized at the boundaries. In the presence of impurities, the scale-free localization can appear, which is qualitatively distinct from the NHSE. Here, we experimentally design a disordered non-Hermitian electrical circuits in the presence of a single non-Hermitian impurity and the nonreciprocal hopping. We observe the anomalous scale-free accumulation of eigenstates, opposite to the bulk hopping direction. The experimental results open the door to further explore the anomalous skin effects in non-Hermitian electrical circuits.

Keywords non-Hermitian scale-free localization electrical circuit non-Hermitian skin effect

Corresponding Author(s): Tao Liu,Wenbo Ju

Just Accepted Date: 02 August 2024 Issue Date: 20 September 2024

Cite this article:

Hao Wang,Jin Liu,Tao Liu, et al. Observation of impurity-induced scale-free localization in a disordered non-Hermitian electrical circuit[J]. Front. Phys. , 2025, 20(1): 14203.

URL:

https://academic.hep.com.cn/fop/EN/10.15302/frontphys.2025.014203
https://academic.hep.com.cn/fop/EN/Y2025/V20/I1/14203

Fig.1 (a) Schematic of HN model in the presence of onsite disorder and a single non-Hermitian impurity.

t ± γ

denotes the nonreciprocal hopping strength,

ν ± δ

is the nonreciprocal hopping strength between the first and last sites, severing as a single non-Hermitian impurity, and

V n ∈ [− V, V]

is the random onsite potential with

V = 0.05

. (b) Phase diagram of the model as functions of

t

and

ν

with

t = γ

and

ν = δ ≠ 0

. (c) Electrical circuit implementation of the model. The nodes are interconnected by INIC and capacitors in parallel, achieving nonreciprocal hopping. (d) Photographne of the experimental circuit board.

Fig.2 Simulated results for the scale-free localization in the electrical circuit. Frequency-resolved voltage distribution

V n (ω)

excited by the alternating current (AC) at the different node

n

(a) for

C 1 = 9.4

nF and

C ν = 47

nF, and (b) for

C 1 = 22

nF and

C ν = 2.2

nF. (c, d) The corresponding normalized spatial distribution

Φ n

of the voltage at different normalized node indices

(n − 1) / (N − 1)

for different lattice size

N

, where the node index is mapped to the range

[0, 1]

. (e, f) Localization length

ξ

(blue dots) of bulk modes as a function of the lattice size

N

. The black dashed line denotes a linear fit to

ξ

Fig.3 Experimentally measured voltages of the admittance under chirp signal excitation. Frequency-resolved voltage distribution (a) for

C 1 = 9.4

nF and

C ν = 47

nF, and (b) for

C 1 = 22

nF and

C ν = 2.2

nF, respectively. (c, d) The corresponding spatial distribution of the voltage at the peak frequency, indicating the left-localized and right-localized states.

Fig.4 Experimentally measured scale-free localization in electrical circuit. (a, b) Normalized spatial distribution

Φ n

of the voltage at different normalized node index

(n − 1) / (N − 1)

for the different lattice size

N

, where the node index is mapped to the range

[0, 1]

. Here, (a) for

C 1 = 9.4

nF and

C ν = 47

nF, and (b) for

C 1 = 22

nF and

C ν = 2.2

nF. (c, d) Localization length

ξ

(blue dots) of bulk modes as a function of the lattice size

N

. The black dashed line denotes a linear fit to

ξ

. The measured eigenvalues of the admittance (e) for

C 1 = 9.4

nF and

C ν = 47

nF, and (f) for

C 1 = 22

nF and

C ν = 2.2

nF.

Fig.A1 (a) Experimental circuit board diagram containing eleven unit cells for each board. Multiple boards can be connected to create a longer chain. The first node and the last node are connected by the external wires acting as the single impurity. The nonreciprocal hopping between nodes

n

and

n + 1

is realized by the negative impedance converters through current inversions (INICs) in (b), where INIC consists of capacitor, resistor and operational amplifier. (c) Circuit board of each unit cell, where the red dashed curve indicates the INIC.

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