Controlled probing of localization effects in the non-Hermitian Aubry-Andr\'e model via topolectrical circuits

Dipendu Halder; Saurabh Basu

REVIEW 4 major objections 5 minor 69 references

Controlled probing of localization effects in the non-Hermitian Aubry-Andr\'e model via topolectrical circuits

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The imaginary phase of a quasiperiodic potential is a single control knob that switches a non-Hermitian chain between skin-localized and Anderson-localized states, and the paper shows this switch as a tunable voltage profile in a…

desk verdict A plausible circuit-design paper whose central quantitative claim (the α_c threshold) is quoted from a different geometry and applied inconsistently; the qualitative story likely survives, the numbers need a check. read the letter →

arxiv 2501.04502 v3 pith:XE4IM222 submitted 2025-01-08 cond-mat.dis-nn cond-mat.mes-hallphysics.app-phquant-ph

Dipendu Halder , Saurabh Basu This is my paper · ORCID

classification cond-mat.dis-nncond-mat.mes-hallphysics.app-phquant-ph

keywords non-HermitianAubry-AndrémodelAndersonlocalizationskineffecttopolectricalcircuitsquasiperiodicdisordercomplexphasepotentialnon-reciprocalhoppingtransition

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper is about a single knob that chooses between two ways a wavefunction can be trapped in one dimension: the skin effect, which drives states to a boundary or interface because hopping is non-reciprocal, and Anderson localization, which pins states to disorder-selected sites because of the quasiperiodic potential. The knob is the imaginary phase α of the Aubry-André potential, and the paper claims that crossing α_c = ln|max(t+γ,t−γ)/λ| switches the eigenstates from interface-localized skin states to Anderson-localized states. The same switch is then shown in a topolectrical circuit, a classical network whose Laplacian maps onto the tight-binding Hamiltonian, where the measured voltage profile localizes at the interface below α_c and near the excited node above it. If the claim holds, a classical room-temperature circuit can probe and control the competition between two quantum localization phenomena, and the same parameter also tunes the output amplitude and creates a movable spatial channel for a signal.

What carries the argument

The central object is the complex phase α inside the on-site quasiperiodic potential λ_k = 2λ cos(2πβk + iα), which acts as a tunable imaginary gauge field. It competes with the non-reciprocity γ through the critical line α_c = ln|max(t+γ,t−γ)/λ|, which the paper takes from a Lyapunov-exponent analysis of nonreciprocal quasiperiodic chains: below this value the skin-effect decay length beats the disorder, and above it disorder wins. In the circuit, α is encoded in node-dependent grounded capacitors and resistors, C[k] = −Re(λ_k)/ω_R and R[k] = [Im(λ_k)]^{-1}, and the Laplacian at the resonant frequency ω_R reproduces the tight-binding Hamiltonian; switches S, S1, and S2 select whether non-reciprocity, quasiperiodic disorder, or both are active, which is what makes the voltage profile switch from interface to excitation node.

What would settle it

Compute the inverse participation ratio of the eigenstates of Eq. (1) as a function of α and check whether the skin-to-Anderson transition occurs precisely at α_c = ln|max(t+γ,t−γ)/λ|; if the crossing shifts or broadens, the quantitative control claim fails. In the circuit, sweep α across this value in a 21-node realization and look for the voltage profile to jump from the interface node to the excitation node at the predicted α.

Watch

Extended reading notes

Core claim

The paper's central claim is that the complex phase α of the quasiperiodic potential λ_k = 2λ cos(2πβk + iα) is a control parameter that decides the winner in a non-Hermitian Aubry-André chain with an interface. For α below the critical value α_c = ln|max(t+γ,t−γ)/λ|, the non-reciprocity γ dominates: the eigenstates and the time-evolved density accumulate at the interface, which is the skin effect. For α above α_c, Anderson localization dominates and the states become pinned near disorder-determined sites instead. The authors argue that the same competition appears in a topolectrical circuit whose Laplacian reproduces the Hamiltonian at resonance; the voltage profile is localized at the interface for α < α_c and shifts to the excitation node for α > α_c, with a partially delocalized channel between the two nodes in the crossover. They also report that the tight-binding time evolution under a delta excitation exhibits non-Hermitian jumps between Anderson-localized states for α > α_c, while the circuit driven by a steady sinusoidal current settles into stable localization near the excitation node rather than jumping.

Load-bearing premise

The control claim rests on the assumption that the transition criterion α_c = ln|max(t+γ,t−γ)/λ|, derived for a uniform non-reciprocal quasiperiodic chain, remains exactly valid for the interface-terminated chain of Eq. (1), a step the paper does not rederive.

Editorial extensions

If this is right

Tuning α from below to above α_c switches eigenstate localization in the tight-binding model from the interface to Anderson-localized sites while keeping λ and γ fixed.
In the topolectrical circuit, the same tuning moves the measured voltage profile from the interface node to the excitation node, giving a classical observable for the quantum transition.
In the crossover region the voltage profile is partially delocalized between the excitation node and the interface, forming a spatial channel whose endpoints can be chosen by design.
Increasing α also reduces the output amplitude, so one parameter controls both where the voltage sits and how large it is.
The tight-binding time evolution shows non-Hermitian jumps between Anderson-localized states, whereas the sinusoidally driven circuit does not, so the circuit's dynamics are smoother than the model's.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An implication the authors leave implicit is that the same α-control should transfer to other classical wave platforms with imaginary gauge fields, such as photonic mesh lattices or acoustic lattices, where complex on-site potentials are available.
A direct transfer-matrix derivation of Eq. (2) for the two-domain interface geometry would confirm or correct the quantitative α_c used here; until such a derivation appears, the sharp switching point is an assumption carried over from uniform chains.
A testable extension is to drive the same circuit with a short current pulse instead of a steady sinusoid; if the classical analog remains faithful, the voltage profile should mimic the model's non-Hermitian jumps rather than settling smoothly.
The partially delocalized channel could be characterized quantitatively by measuring two-terminal impedance or transmitted power between the excitation and interface nodes, which would turn the proposed sensing and routing application into a concrete device metric.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things about this paper. First, the genuinely useful piece is the circuit proposal: a topolectrical network with an interface and a complex-phase quasiperiodic potential, where the voltage profile can be steered between interface-localized skin states and excitation-node-localized Anderson states by tuning α. That is a concrete, potentially realizable design, and the LTspice simulations back the qualitative picture. Second, the paper's central quantitative claim—the precise threshold α_c = ln|max(t+γ,t−γ)/λ|—is quoted from a uniform-chain result and applied to a two-domain interface geometry without rederivation, and the paper's own figures don't apply it consistently. That is a load-bearing soft spot, not a cosmetic one.

What the paper does well: the circuit Laplacian derivation in Appendix A is careful, the component values are tied to resonance conditions rather than fitted to localization data, and the authors are honest that only LTspice simulations are shown, not physical measurements. The qualitative competition between NHSE and AL, including the partial delocalization channel between the interface and the excitation node, is a reasonable extension of known work and is clearly presented. The citation pattern is fine; the key prior results (Longhi, Jiang et al., Li et al.) are properly credited.

The soft spots, in proportion: (1) The time-evolution expansion in Eq. (4) assumes the right eigenstates of a non-Hermitian operator form an orthonormal basis, which is not generally true; the paper never introduces left eigenvectors. This is a real technical flaw, but the main qualitative conclusions about NH jumps are standard and probably survive a corrected biorthogonal treatment. (2) The stress-test note is right: for the stated t=0.65, γ=0.35, λ=1 parameters, Eq. (2) gives α_c = ln(1/1) = 0, not the quoted 0.425 for Fig. 2. The paper also quotes 0.54 in Fig. 6 for λ=1.5, and 0.105 in Fig. 1(d) for λ=0.9. These numbers are mutually inconsistent. Either the figures use different implicit parameters or the formula is being misapplied; either way the 'precisely controlled' claim needs a direct numerical check against Eq. (2) for the interface geometry. (3) The all-important transfer of Eq. (2) from a uniform chain to an interface geometry is not justified by any non-Bloch boundary-condition calculation; since the skin states are themselves a boundary-condition effect, the threshold could shift with L0 and interface terms. This is a minor-to-moderate flaw for the qualitative story but a serious one for the quantitative control claim.

Who is this for: condensed-matter experimentalists working on topolectrical circuits, and theorists who want a concrete classical analog of AL-NHSE competition. The paper deserves a serious referee, but the referee should push hard for a consistent parameter set and a derivation or numerical verification of α_c in the interface geometry. I would not cite it for the threshold formula; I would cite it for the circuit design if the numbers get fixed.

Referee Report

4 major / 5 minor

Summary. The manuscript studies a non-Hermitian Aubry-André model with an interface separating two chains of opposite nonreciprocity, and proposes a topolectrical circuit realization. The main claims are: (i) the complex phase α of the quasiperiodic potential controls the competition between Anderson localization and the non-Hermitian skin effect, with a transition at α_c = ln|max(t+γ, t−γ)/λ| (Eq. 2); (ii) time evolution under a single-site excitation exhibits non-Hermitian jumps between skin states and Anderson-localized states; and (iii) an LTspice topolectrical circuit reproduces interface localization of the voltage profile for α < α_c and localization near the excitation node for α > α_c, with a tunable intermediate channel. The paper includes open data and code.

Significance. If the central claims were established, the work would provide a useful and experimentally accessible classical analog of the AL–NHSE competition, with a plausible control knob (the phase α) and a concrete circuit design with explicit component values. The paper is commendable for shipping the simulation data and code (Ref. [69]) and for making falsifiable predictions about the voltage profile. However, the quantitative control claim rests on an imported critical-value formula that is neither rederived for the interface geometry nor applied with consistent parameters, and the time-evolution analysis is built on an invalid orthonormality assumption for non-Hermitian eigenstates. The significance is therefore contingent on fixing these load-bearing issues.

major comments (4)

[Section III, Eq. (4)] Equation (4) assumes that the right eigenstates ψ_q of the non-Hermitian Hamiltonian H form a complete orthonormal basis and sets a_q(0)=⟨ψ_q|Ψ(0)⟩. For a non-Hermitian operator this is not valid: the correct expansion coefficients require the left eigenstates ψ̃_q, namely a_q(0)=⟨ψ̃_q|Ψ(0)⟩/⟨ψ̃_q|ψ_q⟩. Equation (6), the subsequent time evolution in Fig. 2, and the central claim of non-Hermitian jumps all rest on this invalid expansion. The authors should redo the time evolution with a biorthogonal basis or, more robustly, by direct numerical integration of the Schrödinger equation with the norm renormalization in Eqs. (7)–(8), and check whether the reported jumps survive.
[Section II, Eq. (2); Figs. 2, 5, 6] The critical value α_c = ln|max(t+γ, t−γ)/λ| is taken from Ref. [40], which analyzes a uniform nonreciprocal quasiperiodic chain. The Hamiltonian in Eq. (1) instead has two domains with opposite signs of γ and an interface; no derivation or numerical verification is given that the same Lyapunov exponent controls the localization transition in this geometry. The paper's own numbers are also inconsistent with Eq. (2): with the stated TB parameters t=0.65, γ=0.35 and λ=1, Eq. (2) gives α_c=0, not the quoted α_c≈0.425 in Fig. 2; with γ=0, t=0.65 and λ=1 it gives α_c=ln(0.65)<0, not 0.425 in Fig. 5; and for λ=1.5 in Fig. 6, even using the circuit-mapped max(t+γ,t−γ)≈1.53, Eq. (2) gives α_c≈0.02, not 0.54. A definitive numerical determination of the transition for the interface Hamiltonian, for example via the inverse participation ratio or the OBC Lyapunov exponent as a function of α, is needed before the quantitative control claim can be accepted.
[Appendix B, Eq. (B2)] The same orthonormality assumption is applied to the circuit Laplacian: Eq. (B2) assumes V_a^† V_b = δ_ab and projects with V_k^† I(t). However, the Laplacians in Eqs. (A3) and (A4) are non-Hermitian because of the asymmetric off-diagonal couplings (C+C′) vs (C−C′), so their eigenvectors are not generally orthogonal and the projection formula is incorrect. This undermines the claimed correspondence between the circuit voltage response and the time-evolved wavefunction of the tight-binding model. The authors should either use left eigenvectors of the Laplacian or solve the circuit equations directly for the single-source excitation.
[Section IV B and IV D] The identification of the measured voltage profile with eigenstate localization is not quantitatively established. With a single-node current source at the resonant frequency, the steady-state response is governed by the Green's function L(ω_R)^{-1} (with appropriate losses), not directly by the eigenvectors of L. The manuscript relies on RMS values over manually chosen time windows and linear interpolation (Section IV B) and does not provide a direct comparison of the simulated voltage response with the eigenvector profile of the corresponding Laplacian. A calculation or simulation showing that the spatial decay of the voltage response matches the eigenstate localization length is needed to justify the central 'classical analog' claim.

minor comments (5)

[Section II] The text states that 'all parameters in this TB model are in the unit of t', but then quotes t=0.65, γ=0.35 in the figures; please clarify the actual parametrization and consistently state the energy scale for each figure.
[Section IV C] The phrase 'reciprocal NH AA model' is an oxymoron; if S1 is open and the nonreciprocity is removed, the model is the Hermitian (or reciprocal) AA model. Please correct the terminology.
[Section III, after Eq. (6)] The statement that Eq. (6) applies only to systems without boundaries is confusing: the finite chain with open boundaries has well-defined eigenstates that encode the boundaries, and the earlier expansion is in that basis. Please rewrite this passage to state what boundary conditions are actually being imposed.
[Throughout] There are several typographical and formatting issues, including inconsistent spacing in 'Aubry-Andr ´e', 'non-equivalent' (likely 'inequivalent') in the Introduction, and the table formatting in Table I. A careful proofread would improve readability.
[Data availability] The open data and code repository is a strength, but the availability statement should specify whether the LTspice netlists and raw simulation outputs are included alongside the Python codes.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the α_c threshold is imported from external work and the circuit parameters are fixed by resonance conditions, not fitted to the localization output.

full rationale

I walked the derivation chain from Eq. (1) through Eq. (2) and the TEC construction. The central control claim rests on α_c = ln|max(t+γ,t−γ)/λ|, but this formula is quoted from Li et al. (Ref. [40]) and is not derived in this paper; it is an externally imported result, not a quantity fitted to the paper's own localization data. The TEC component values (C, C′, L, Ledge) are fixed by the resonance conditions in Eq. (10) and Eq. (A2) and by the mapping (t±γ) ≡ ω_R(C±C′), rather than by tuning to reproduce the voltage profiles. The voltage-profile localizations are obtained from LTspice simulations and compared with the TB eigenstates; they are not imposed by construction. The only self-citation (Ref. [50]) appears in a list of TEC realization references and is not load-bearing. The apparent inconsistencies among quoted α_c values, e.g., the 0.425 value in Fig. 2 versus Eq. (2) evaluated with the stated TB parameters t=0.65, γ=0.35, λ=1, are numerical/parameter-consistency concerns, not evidence that a parameter was fitted and then renamed a prediction. The transfer of Eq. (2) from a uniform nonreciprocal quasicrystal to the two-domain interface geometry is also a substantive validity assumption, but that is a correctness/evidence caveat, not a circularity: the paper does not define the threshold in terms of the quantity it then predicts, nor does it fit the threshold to the localization output. No circular step meets the required quote-and-reduction standard, so the appropriate finding is no significant circularity.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central claims rest on the borrowed transition criterion Eq. (2), on a false orthonormality assumption for non-Hermitian eigenstates, and on an idealized Laplacian-to-Hamiltonian mapping. No new particles, forces, or dimensions are introduced.

free parameters (2)

time-step dt in norm-renormalized evolution = not specified
Equations (7)-(8) evolve and renormalize the wavefunction at each small interval dt; the jump sequence in Fig. 2(c) can depend on dt, and the paper does not state the value used.
implicit TB parameters for Figs. 2, 5, 6 = not stated
The text fixes t=0.65 and γ=0.35, but the quoted α_c values match neither these nor the circuit-mapped hoppings; the parameter sets for these figures are not specified.

assumptions (3)

domain assumption The transition criterion α_c = ln|max(t+γ,t−γ)/λ| from Li et al. applies unchanged to the two-domain interface Hamiltonian in Eq. (1).
Eq. (2) is cited for a uniform nonreciprocal quasicrystal; the paper uses it for the interface model without rederiving it or checking finite-size and interface corrections.
ad hoc to paper The right eigenvectors of the non-Hermitian H form a complete orthonormal basis, allowing expansion coefficients a_q(0)=⟨ψ_q|Ψ(0)⟩.
Section III, Eq. (4) uses ordinary inner products with right eigenvectors; for non-Hermitian Hamiltonians one needs the biorthogonal left basis, so this assumption is unjustified.
domain assumption Ideal INIC behavior and resonance conditions make the circuit Laplacian exactly replicate the TB Hamiltonian at f_R.
Appendix A assumes ideal op-amps and exact satisfaction of Eq. (A2); real LTspice models include parasitics and finite gain, and the grounded negative-resistor INIC is assumed to add no reciprocal coupling.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Controlled probing of localization effects in the non-Hermitian Aubry-Andr\'e model via topolectrical circuits." pith.science (2026). https://pith.science/paper/XE4IM222

@misc{pith2026250104502,
  author       = {Pith},
  title        = {Pith review of: Controlled probing of localization effects in the non-Hermitian Aubry-Andr\'e model via topolectrical circuits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XE4IM222}},
  note         = {Machine review of arXiv:2501.04502}
}

read the original abstract

Anderson localization and the non-Hermitian skin effect are two distinct confinement phenomena of the eigenfunctions that are driven, respectively, by disorder and nonreciprocity. Understanding their interplay within a unified framework offers valuable insights into the localization properties of low-dimensional systems. To this end, we investigate a non-Hermitian version of the celebrated Aubry-Andr\'e model, which serves as an ideal platform due to its unique self-dual properties and ability to demonstrate a delocalization-localization transition in one dimension. Interestingly, in our setting, the competition between Anderson localization and the skin effect can be precisely controlled via the complex phase of the quasiperiodic disorder. Additionally, by analyzing the time evolution, we demonstrate that quantum jumps between the skin states and the Anderson-localized states occur in the theoretical model. Further, to gain support for our theoretical predictions in an experimental platform, we propose a topolectrical circuit featuring an interface that separates two distinct electrical circuit networks. The voltage profile of the circuit exhibits confinement at the interface, analogous to the skin effect, while the phenomenon of Anderson localization in the circuit can be perceived via a predicted localization behavior near the excitation node, rather than exhibiting sudden non-Hermitian jumps, as observed in the tight-binding framework. This interplay leads to a spatially tunable localization of the output voltage of the circuit. Our findings provide deeper insights into the controlled confinement of the eigenstates of the non-Hermitian Aubry-Andr\'e model by designing analogous features in topolectrical circuits, opening avenues in the fabrication of advanced electronic systems such as highly sensitive sensors and efficient devices for information transfer and communication.

Figures

Figures reproduced from arXiv: 2501.04502 by the authors.

**Figure 2.** FIG. 2. The temporal evolution of the excitation in our model is il [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) Circuit diagram of an INIC is shown. The current entering the INIC from the left(right) side experiences a negative(positive) [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. Colormap of the RMS values of the output VP as functions [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. The RMS values of the output signal, measured over the [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6. (a) The RMS values of the VP are plotted for different values [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7. (a) The NHSE is evident as the [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

69 extracted references · 49 canonical work pages

[69]

The Python codes along with the dataset are available at the fol- lowing link: https://github.com/DIPENDU-PHY/BD14939.git

work page
[40]

S.-Z. Li, E. Cheng, S.-L. Zhu, and Z. Li, Asymmetric trans- fer matrix analysis of lyapunov exponents in one-dimensional nonreciprocal quasicrystals, Phys. Rev. B 110, 134203 (2024)

work page 2024
[1]

P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958)

1958
[2]

Aubry and G

S. Aubry and G. Andr ´e, Analyticity breaking and anderson lo- calization in incommensurate lattices, Ann. Isr. Phys. Soc. 3, 11 133 (1980)

work page 1980
[3]

Ganeshan, K

S. Ganeshan, K. Sun, and S. Das Sarma, Topological zero- energy modes in gapless commensurate aubry-andr ´e-harper models, Phys. Rev. Lett. 110, 180403 (2013)

work page 2013
[4]

Ganeshan, J

S. Ganeshan, J. H. Pixley, and S. Das Sarma, Nearest neighbor tight binding models with an exact mobility edge in one dimen- sion, Phys. Rev. Lett. 114, 146601 (2015)

work page 2015
[5]

De Tomasi, S

G. De Tomasi, S. Bera, J. H. Bardarson, and F. Pollmann, Quan- tum mutual information as a probe for many-body localization, Phys. Rev. Lett. 118, 016804 (2017)

work page 2017
[6]

Lahini, R

Y . Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y . Silberberg, Observation of a localization transition in quasiperiodic photonic lattices, Phys. Rev. Lett. 103, 013901 (2009)

2009

Show all 69 references

[7]

Y . E. Kraus, Y . Lahini, Z. Ringel, M. Verbin, and O. Zilber- berg, Topological states and adiabatic pumping in quasicrystals, Phys. Rev. Lett. 109, 106402 (2012)

2012
[8]

S. Dey, N. R. Das, and S. Ghosh, Exploring unconventional fea- tures of light dynamics in aubrey–andr ´e–harper model based quasi-periodic optical lattices, Optics Communications 506, 127593 (2022)

2022
[9]

Roati, C

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zac- canti, G. Modugno, M. Modugno, and M. Inguscio, Anderson localization of a non-interacting bose–einstein condensate, Na- ture 453, 895 (2008)

2008
[10]

F. A. An, K. Padavi ´c, E. J. Meier, S. Hegde, S. Ganeshan, J. H. Pixley, S. Vishveshwara, and B. Gadway, Interactions and mobility edges: Observing the generalized aubry-andr ´e model, Phys. Rev. Lett. 126, 040603 (2021)

2021
[11]

Li, Y .-Y

H. Li, Y .-Y . Wang, Y .-H. Shi, K. Huang, X. Song, G.-H. Liang, Z.-Y . Mei, B. Zhou, H. Zhang, J.-C. Zhang, S. Chen, S. P. Zhao, Y . Tian, Z.-Y . Yang, Z. Xiang, K. Xu, D. Zheng, and H. Fan, Observation of critical phase transition in a generalized aubry- andr´e-harper model ...

2023
[12]

T. E. Lee, Anomalous edge state in a non-hermitian lattice, Phys. Rev. Lett. 116, 133903 (2016)

2016
[13]

H. Shen, B. Zhen, and L. Fu, Topological band theory for non- hermitian hamiltonians, Phys. Rev. Lett. 120, 146402 (2018)

2018
[14]

Ashida, Z

Y . Ashida, Z. Gong, and M. Ueda, Non-hermitian physics, Ad- vances in Physics 69, 249 (2020)

2020
[15]

E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Excep- tional topology of non-hermitian systems, Rev. Mod. Phys. 93, 015005 (2021)

2021
[16]

Yao and Z

S. Yao and Z. Wang, Edge states and topological invariants of non-hermitian systems, Phys. Rev. Lett. 121, 086803 (2018)

2018
[17]

C. H. Lee and R. Thomale, Anatomy of skin modes and topol- ogy in non-hermitian systems, Phys. Rev. B99, 201103 (2019)

2019
[18]

Okuma, K

N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Topolog- ical origin of non-hermitian skin effects, Phys. Rev. Lett. 124, 086801 (2020)

2020
[19]

D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, Non-hermitian boundary modes and topology, Phys. Rev. Lett. 124, 056802 (2020)

2020
[20]

Zhang, T

X. Zhang, T. Zhang, M. H. Lu, and Y . F. Chen, A review on non-hermitian skin effect, Advances in Physics: X 7, 2109431 (2022)

2022
[21]

Okuma and M

N. Okuma and M. Sato, Non-hermitian topological phenomena: A review, Annual Review of Condensed Matter Physics 14, 83 (2023)

2023
[22]

W. D. Heiss, The physics of exceptional points, Journal of Physics A: Mathematical and Theoretical 45, 444016 (2012)

2012
[23]

Leykam, K

D. Leykam, K. Y . Bliokh, C. Huang, Y . D. Chong, and F. Nori, Edge modes, degeneracies, and topological numbers in non- hermitian systems, Phys. Rev. Lett. 118, 040401 (2017)

2017
[24]

Yokomizo and S

K. Yokomizo and S. Murakami, Non-bloch band theory of non- hermitian systems, Phys. Rev. Lett. 123, 066404 (2019)

2019
[25]

Eichelkraut, R

T. Eichelkraut, R. Heilmann, S. Weimann, S. St ¨utzer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, Mobility transition from ballistic to diffusive transport in non- hermitian lattices, Nature Communications 4, 2533 (2013)

2013
[26]

El-Ganainy, K

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Mus- slimani, S. Rotter, and D. N. Christodoulides, Non-hermitian physics and pt symmetry, Nature Physics 14, 11 (2018)

2018
[27]

A. Wang, Z. Meng, and C. Q. Chen, Non-hermitian topology in static mechanical metamaterials, Science Advances 9 (2023)

2023
[28]

Weidemann, M

S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, Topo- logical funneling of light, Science 368, 311 (2020)

2020
[29]

Fleury, D

R. Fleury, D. Sounas, and A. Al `u, An invisible acoustic sensor based on parity-time symmetry, Nat. Commun. 6, 10.1038/ncomms6905 (2015)

2015 doi
[30]

T. Guo, B. Assouar, B. Vincent, and A. Merkel, Edge states in non-hermitian composite acoustic su schrieffer heeger chains, Journal of Applied Physics 135, 043102 (2024)

2024
[31]

M. Peng, C. Wu, Z. Cui, X. Zhang, Q. Wei, M. Yan, and G. Chen, Acoustic non-hermitian dirac states tuned by flexible designed gain and loss, Applied Physics Letters 125, 193101 (2024)

2024
[32]

Hatano and D

N. Hatano and D. R. Nelson, Localization transitions in non- hermitian quantum mechanics, Phys. Rev. Lett. 77, 570 (1996)

1996
[33]

Longhi, Topological phase transition in non-hermitian qua- sicrystals, Phys

S. Longhi, Topological phase transition in non-hermitian qua- sicrystals, Phys. Rev. Lett. 122, 237601 (2019)

2019
[34]

Schiffer, X.-J

S. Schiffer, X.-J. Liu, H. Hu, and J. Wang, Anderson localiza- tion transition in a robustPT -symmetric phase of a generalized aubry-andr´e model, Phys. Rev. A 103, L011302 (2021)

2021
[35]

Longhi, Phase transitions in a non-hermitian aubry-andr ´e- harper model, Phys

S. Longhi, Phase transitions in a non-hermitian aubry-andr ´e- harper model, Phys. Rev. B 103, 054203 (2021)

2021
[36]

Y . Liu, Q. Zhou, and S. Chen, Localization transition, spec- trum structure, and winding numbers for one-dimensional non- hermitian quasicrystals, Phys. Rev. B 104, 024201 (2021)

2021
[37]

Cai, Localization transitions and winding numbers for non- hermitian aubry-andr´e-harper models with off-diagonal modu- lations, Phys

X. Cai, Localization transitions and winding numbers for non- hermitian aubry-andr´e-harper models with off-diagonal modu- lations, Phys. Rev. B 106, 214207 (2022)

2022
[38]

Gandhi and J

S. Gandhi and J. N. Bandyopadhyay, Topological triple phase transition in non-hermitian quasicrystals with complex asym- metric hopping, Phys. Rev. B 108, 014204 (2023)

2023
[39]

Sun, X.-Y

Y .-M. Sun, X.-Y . Wang, and L.-J. Zhai, Hybrid scaling proper- ties of the localization transition in a non-hermitian disordered aubry-andr´e model, Phys. Rev. B 110, 054202 (2024)

2024
[41]

V . V . Albert, L. I. Glazman, and L. Jiang, Topological properties of linear circuit lattices, Phys. Rev. Lett. 114, 173902 (2015)

2015
[42]

C. H. Lee, S. Imhof, C. Berger, F. Bayer, J. Brehm, L. W. Molenkamp, T. Kiessling, and R. Thomale, Topolectrical cir- cuits, Communications Physics 1, 39 (2018)

2018
[43]

Helbig, T

T. Helbig, T. Hofmann, C. H. Lee, R. Thomale, S. Imhof, L. W. Molenkamp, and T. Kiessling, Band structure engineering and reconstruction in electric circuit networks, Phys. Rev. B 99, 161114 (2019)

2019
[44]

J. Dong, V . Juri ˇci´c, and B. Roy, Topolectric circuits: Theory and construction, Phys. Rev. Res. 3, 023056 (2021)

2021
[45]

S. Guo, G. Pan, J. Huang, R. Huang, F. Zhuang, S. Su, Z. Lin, W. Qiu, and Q. Kan, Realization of the square-root higher- order topology in decorated su–schrieffer–heeger electric cir- cuits, Applied Physics Letters 123, 043102 (2023). 12

2023
[46]

H. Yang, L. Song, Y . Cao, and P. Yan, Circuit realization of topological physics, Physics Reports 1093, 1 (2024), circuit re- alization of topological physics

2024
[47]

Helbig, T

T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, Generalized bulk–boundary correspondence in non-hermitian topolectrical circuits, Nature Physics 16, 747 (2020)

2020
[48]

B. Liu, Y . Li, B. Yang, X. Shen, Y . Yang, Z. H. Hang, and M. Ezawa, Experimental observation of non-hermitian higher- order skin interface states in topological electric circuits, Phys. Rev. Res. 5, 043034 (2023)

2023
[49]

W. Lin, B. Ruan, C. Liu, X. Dai, and Y . Xiang, Evolution of topological extended state in multidimensional non-hermitian topolectrical circuits, Applied Physics Letters 125, 173104 (2024)

2024
[50]

Halder, R

D. Halder, R. Thomale, and S. Basu, Circuit realization of a two-orbital non-hermitian tight-binding chain, Phys. Rev. B 109, 115407 (2024)

2024
[51]

C.-X. Guo, L. Su, Y . Wang, L. Li, J. Wang, X. Ruan, Y . Du, D. Zheng, S. Chen, and H. Hu, Scale-tailored localization and its observation in non-hermitian electrical circuits, Nature Com- munications 15, 9120 (2024)

2024
[52]

S. M. Rafi-Ul-Islam, Z. B. Siu, M. S. H. Razo, and M. B. A. Jalil, Dynamic manipulation of non-hermitian skin effect through frequency in topolectrical circuits (2024), arXiv:2410.16914 [cond-mat.mes-hall]

2024 arXiv
[53]

Jiang, L.-J

H. Jiang, L.-J. Lang, C. Yang, S.-L. Zhu, and S. Chen, Inter- play of non-hermitian skin effects and anderson localization in nonreciprocal quasiperiodic lattices, Phys. Rev. B 100, 054301 (2019)

2019
[54]

Zeng and Y

Q.-B. Zeng and Y . Xu, Winding numbers and generalized mobility edges in non-hermitian systems, Phys. Rev. Res. 2, 033052 (2020)

2020
[55]

X. Xia, K. Huang, S. Wang, and X. Li, Exact mobility edges in the non-hermitian t1−t2 model: Theory and possible exper- imental realizations, Phys. Rev. B 105, 014207 (2022)

2022
[56]

Ganguly and S

S. Ganguly and S. K. Maiti, Electrical analogue of one- dimensional and quasi-one-dimensional aubry–andr ´e–harper lattices, Scientific Reports 13, 13633 (2023)

2023
[57]

H. Wang, W. Zhang, H. Sun, and X. Zhang, Observation of non- abelian anderson localization and transition in topolectrical cir- cuits, Phys. Rev. B 108, 144203 (2023)

2023
[58]

Sun and C

X. Sun and C. Liu, Localization and topological transitions in generalized non-hermitian ssh models, Physics Letters A 482, 129043 (2023)

2023
[59]

Zhang, L

Y . Zhang, L. Su, and S. Chen, Scale-free localization versus anderson localization in unidirectional quasiperiodic lattices, Phys. Rev. B 111, L140201 (2025)

2025
[60]

S. R. Padhi, A. Padhan, S. Banerjee, and T. Mishra, Quasiperi- odic and periodic extended hatano-nelson model: Anomalous complex-real transition and non-hermitian skin effect, Phys. Rev. B 110, 174203 (2024)

2024
[61]

To know more, please visit the following link: https://www.analog.com/en/resources/design-tools-and- calculators/ltspice-simulator.html
[62]

However, we have absorbed a factor of 2 in the QP potential in Eq

In usual notation, the localization-delocalization transition oc- curs at λ = 2t. However, we have absorbed a factor of 2 in the QP potential in Eq. (1)
[63]

Avila, Global theory of one-frequency Schr ¨odinger opera- tors, Acta Mathematica 215, 1 (2015)

A. Avila, Global theory of one-frequency Schr ¨odinger opera- tors, Acta Mathematica 215, 1 (2015)

2015
[64]

Chakrabarty and S

A. Chakrabarty and S. Datta, Skin effect and dynamical delo- calization in non-hermitian quasicrystals with spin-orbit inter- action, Phys. Rev. B 107, 064305 (2023)

2023
[65]

Weidemann, M

S. Weidemann, M. Kremer, S. Longhi, and A. Szameit, Co- existence of dynamical delocalization and spectral localiza- tion through stochastic dissipation, Nature Photonics 15, 576 (2021)

2021
[66]

A. F. Tzortzakakis, K. G. Makris, A. Szameit, and E. N. Economou, Transport and spectral features in non-hermitian open systems, Phys. Rev. Res. 3, 013208 (2021)

2021
[67]

Leventis, K

A. Leventis, K. G. Makris, and E. N. Economou, Non-hermitian jumps in disordered lattices, Phys. Rev. B 106, 064205 (2022)

2022
[68]

F. Y . Wu, Theory of resistor networks: the two-point resis- tance, Journal of Physics A: Mathematical and General 37, 6653 (2004)

2004

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [69]

The Python codes along with the dataset are available at the fol- lowing link: https://github.com/DIPENDU-PHY/BD14939.git

work page

[2] [40]

S.-Z. Li, E. Cheng, S.-L. Zhu, and Z. Li, Asymmetric trans- fer matrix analysis of lyapunov exponents in one-dimensional nonreciprocal quasicrystals, Phys. Rev. B 110, 134203 (2024)

work page 2024

[3] [1]

P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958)

1958

[4] [2]

Aubry and G

S. Aubry and G. Andr ´e, Analyticity breaking and anderson lo- calization in incommensurate lattices, Ann. Isr. Phys. Soc. 3, 11 133 (1980)

work page 1980

[5] [3]

Ganeshan, K

S. Ganeshan, K. Sun, and S. Das Sarma, Topological zero- energy modes in gapless commensurate aubry-andr ´e-harper models, Phys. Rev. Lett. 110, 180403 (2013)

work page 2013

[6] [4]

Ganeshan, J

S. Ganeshan, J. H. Pixley, and S. Das Sarma, Nearest neighbor tight binding models with an exact mobility edge in one dimen- sion, Phys. Rev. Lett. 114, 146601 (2015)

work page 2015

[7] [5]

De Tomasi, S

G. De Tomasi, S. Bera, J. H. Bardarson, and F. Pollmann, Quan- tum mutual information as a probe for many-body localization, Phys. Rev. Lett. 118, 016804 (2017)

work page 2017

[8] [6]

Lahini, R

Y . Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y . Silberberg, Observation of a localization transition in quasiperiodic photonic lattices, Phys. Rev. Lett. 103, 013901 (2009)

2009