Pith. sign in

REVIEW 2 major objections 4 minor 78 references

Universality and Dynamical Inequivalence in Isospectral Non-Hermitian Anderson Transitions

T0 review · 2 major / 4 minor · reviewed 2026-07-12 · grok-4.5

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

Pith's one-line read A single non-Hermitian bond on a disordered ring drives the Anderson delocalization transition; static criticality depends only on total imaginary flux, while dynamics do not.

desk verdict Single-bond non-Hermitian flux drives the full Hatano-Nelson LDL transition with identical static criticality but inequivalent dynamics; the exponential bond strength is a real thermodynamic-limit caveat but does not erase the finite-L result. read the letter →

arxiv 2607.07714 v1 pith:KQRULHJG submitted 2026-07-02 cond-mat.mes-hall math-phmath.MP

Aziz Hasan , Anant Vijay Varma , Namit Anand , Sourin Das This is my paper · ORCID
classification cond-mat.mes-hallmath-phmath.MP
keywords non-HermitianAndersontransitionHatano–Nelsonmodelimaginarygaugefluxisospectralfamilylocalization–delocalizationoperatorscramblingsteady-stateentanglementmulti-terminaltransport
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

Anderson localization in one dimension is famously robust: any weak disorder traps every wavefunction. The classic Hatano–Nelson picture shows that spreading non-reciprocal hopping throughout the bulk can overcome that trapping by means of an imaginary gauge flux. This paper shows that the bulk spreading is unnecessary. Concentrating the entire flux onto one boundary bond of a disordered ring is already enough to trigger the same localization–delocalization transition. The authors construct a continuous family of Hamiltonians, all related by complex gauge transformations, that interpolate between the uniform bulk model and the single-bond extreme while keeping the total flux fixed. Every static diagnostic—complex-eigenvalue fraction, inverse-participation ratios, fractal dimensions, winding number—collapses onto the same critical surface controlled solely by that total flux. Yet the same family displays sharply different real-time dynamics: the single-bond member shows rapid operator scrambling, oscillatory wave-packet acceleration, and a double re-entrant steady-state entanglement transition that are absent from the uniform case. The result separates static universality from dynamical inequivalence in non-Hermitian disordered systems and supplies a multi-terminal topological-transport route to realize the effect.

What carries the argument

An exactly isospectral family of non-Hermitian Hamiltonians generated by the similarity transformation S = exp(∑ α_j n_j). The transformation redistributes non-reciprocity while preserving the total imaginary flux Γ = Lγ, thereby collapsing all static critical behavior onto a single gauge-invariant manifold.

What would settle it

Measure the complex-eigenvalue fraction or winding number versus disorder for both the uniform Hatano–Nelson ring and a single-bond ring engineered with the same total flux; if the critical disorder strengths or the extracted exponent ν fail to coincide, the claim of static gauge invariance is false.

Watch

Extended reading notes

Core claim

Extensive bulk non-reciprocity is not required for the non-Hermitian Anderson transition. A single asymmetric boundary bond that carries the entire imaginary gauge flux Γ = Lγ is sufficient to drive the localization–delocalization transition on a disordered ring. All members of the isospectral family generated by complex gauge transformations share identical spectra and identical critical exponents for every static diagnostic; only the total flux matters. Dynamics, however, remain sensitive to the spatial distribution of that flux.

Load-bearing premise

That an exponentially large hopping amplitude on one bond, required to keep the total flux fixed while concentrating it, still counts as a strictly local physical perturbation whose finite-size scaling defines the same universality class as the uniform bulk model.

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.

Referee Report

2 major / 4 minor

Summary. The paper constructs an exactly isospectral family of non-Hermitian disordered ring Hamiltonians H({α_j}) related by the similarity S=exp(∑ α_j n_j) that redistributes imaginary gauge flux while keeping the total flux Γ=Lγ fixed. Static diagnostics of the localization-delocalization transition (fraction of complex eigenvalues f_c, winding number W, biorthogonal IPR and fractal dimension ⟨D_2⟩) are shown to be identical across the family and controlled solely by Γ, with clean finite-size collapses yielding ν=2 and a common critical line in the (w,γ) plane. The single-bond non-Hermitian (SBN) endpoint (all flux on one link) therefore exhibits the same spectral, eigenstate and topological criticality as the uniform Hatano-Nelson model. Dynamics, however, are inequivalent: SBN displays rapid OTOC scrambling, oscillatory center-of-mass acceleration, and a double re-entrant steady-state entanglement transition linked to bimodal localization. An experimental multi-terminal QWZ-BHZ conductance realization is proposed.

Significance. If the thermodynamic-limit interpretation holds, the work cleanly separates static universality (gauge-invariant and fixed by total flux alone) from dynamical inequivalence in non-Hermitian Anderson systems, and shows that bulk delocalization need not require extensive bulk nonreciprocity. The similarity map is rigorous, the SM proofs of invariance of W, IPR and fidelity are explicit, and the finite-size collapses for f_c, W and ⟨D_2⟩ are of high quality. The dynamical distinctions (especially re-entrant entanglement and size-dependent Lieb-Robinson velocity) and the concrete multi-terminal proposal are genuine additions. These results organize a broader class of non-Hermitian disordered models and supply falsifiable dynamical signatures.

major comments (2)
  1. Model Hamiltonian (Eq. 1) and the SBN limit α_j=(j-L)γ: the single-bond amplitude is t_L=e^{Lγ}. For any fixed γ>0 this diverges exponentially with system size. The central claim that 'a single non-Hermitian boundary bond … suffices' and that 'extensive bulk nonreciprocity is not a necessary ingredient' therefore rests on a size-dependent coupling. While the similarity transformation guarantees identical spectra, W, IPR and D_2 for every finite L (SM), the thermodynamic-limit meaning of a 'strictly local' perturbation, and whether the reported scaling collapses (ν=2, w_c≈3.2 at γ=0.1) still define the same universality class when the non-Hermitian strength itself scales with L, is not secured. The SM Lieb-Robinson bound v_LR=4J cosh(Lγ) likewise grows with L. A clear statement of the L oà limit (fixed bond strength vs fixed Γ, or an appropriate scaling of γ) is required for the physical
  2. Dynamics of SBN and End Matter comparison: the Hermitian sector of H_SBN is spatially anisotropic (Eq. 8 and SM acceleration derivation). The oscillatory COM acceleration and the double re-entrant entanglement are traced to this anisotropy plus the boundary current. Because the static criticality is identical by construction, the claim of 'dynamical inequivalence within the same static universality class' is well supported numerically, yet the manuscript should quantify how much of the dynamical distinction survives once the Hermitian anisotropy is removed (the hybrid-SBN model already introduced in the SM). Without that control, it remains possible that part of the reported dynamical separation is an artifact of the Hermitian rather than the non-Hermitian sector.
minor comments (4)
  1. Abstract and p. 5: 'double re-entrant steady state entanglement transition' is clear from Fig. 3(d), but the four-phase sequence (extended/critical/bimodal/localized) should be labeled consistently in the main text and figure captions.
  2. p. 4: 'V on-Neumann' → 'von Neumann'; several other minor typos ('End Matter', 'pbonds', 'la-layer').
  3. Fig. 2 and SM Fig. 11: the spectral window (central 20 %) and Im(E) threshold (10^{-13}) used for f_c and D_2 should be stated once in the main text for reproducibility.
  4. Experimental Realization: the claim that the multi-terminal conductance matrix realizes the disordered isospectral family is plausible for the clean case, but a brief remark on how quenched onsite disorder would be introduced (or post-selected) would strengthen the proposal.

Circularity Check

1 steps flagged · score 2.0 of 10

Static isospectrality and identical critical diagnostics follow by construction from the similarity map; dynamical inequivalence is independently computed and non-circular.

  1. self definitional [Model Hamiltonian and Isospectrality, Eqs. (1)–(3) and following text; SM sections on W, IPR/D2]
    "Consequently, H({αj}) = S H({αj = 0}) S^{-1}, demonstrating that all Hamiltonians sharing the same Γ are related by an exact similarity transformation and therefore possess identical spectra. ... the universal criticality diagnosed by standard global observables (GD) including the fraction of complex eigenvalues (fc), the inverse participation ratio (IPR), fractal dimensions (Dn), ... and the topological winding number (W) remains strictly invariant throughout this manifold."

    Identical spectra (hence fc) are automatic from any similarity transformation. The SM further shows that the same diagonal S leaves det[H(Φ)], biorthogonal IPR and thus D2, and W invariant by direct cancellation. Therefore the claim that static critical behavior is gauge-invariant and controlled solely by total flux is true by construction of the isospectral family, not an independent physical result. Numerics merely confirm the expected invariance.

full rationale

The paper's central static claim (identical spectra, fc, W, IPR/D2, and thus the same LDL critical point/exponents for any spatial distribution of fixed total flux Γ = Lγ) is true by the explicit similarity transformation H({αj}) = S H_HN S^{-1} together with the elementary proofs in the SM that det, biorthogonal IPR and winding number are invariant under diagonal S. This is definitional rather than an independent derivation, but the paper states the construction openly and does not present the static equivalence as a non-trivial prediction. The non-circular content is the demonstration that dynamics (OTOC scrambling, oscillatory COM acceleration, re-entrant SSEE) differ across the family, obtained by direct time evolution of the distinct representatives (SBN vs HN), plus the experimental multi-terminal proposal. No fitted parameters are re-labeled as predictions, no load-bearing self-citation uniqueness theorems appear, and no known empirical pattern is merely renamed. The exponentially large single-bond amplitude is a physical-interpretation issue, not a circularity. Score 2 reflects one mild self-definitional step that is not load-bearing for the dynamical claims.

Assumptions & free parameters 6 free parameters · 5 assumptions · 3 invented entities

The central claim rests on the standard algebraic fact that similarity transformations preserve spectra, on the established 1D Anderson localization paradigm, and on the numerical observation that all static diagnostics collapse when total flux is held fixed. Free parameters are the usual numerical thresholds and the fixed γ = 0.1 used for most scaling plots; the only invented objects are the interpolating family itself and the observed bimodal-localization regime.

free parameters (6)
  • γ (non-Hermiticity strength) = 0.1 (most figures)
    Fixed to 0.1 for the majority of finite-size scaling and dynamical plots; phase diagrams scan γ but the quoted w_c ≈ 3.2 and ν = 2.0 are extracted at this value.
  • w_c (critical disorder) = ≈ 3.2
    Extracted from scale-invariant crossing of f_c and W; quoted as ≈ 3.2 for γ = 0.1.
  • ν (correlation-length exponent) = 2.0
    Assumed equal to 2.0 to produce data collapse of f_c, W and ⟨D_2⟩; not derived analytically.
  • Im(E) threshold for 'complex' = 10^{-13}
    Eigenvalues counted complex only if |Im(E)| ≥ 10^{-13}; controls the fraction f_c.
  • spectral window for diagnostics = central 20 %
    Only central 20 % of the spectrum used for f_c and ⟨D_2⟩; edges are discarded as less sensitive to γ.
  • phase-boundary fit coefficients = 0.005, 0.009, −1.305, 0.865
    Numerical fits γ ≈ 0.005 + 0.009 w^{2} (weak disorder) and γ ≈ −1.305 + 0.865 log w (strong disorder) for f_c/W.
assumptions (5)
  • standard math Similarity transformations S ∈ GL(L,ℂ) leave the spectrum of a matrix invariant.
    Used to prove that every H({α_j}) is isospectral to the Hatano-Nelson Hamiltonian (Eq. 3 and SM).
  • domain assumption In one dimension, arbitrarily weak Hermitian disorder localizes all eigenstates.
    Background fact that makes the non-Hermitian delocalization transition non-trivial (Introduction).
  • domain assumption The biorthogonal inverse-participation ratio and the winding number defined via det[H(Φ)−E_B] are the appropriate diagnostics of localization and topology in non-Hermitian systems.
    Standard in the nH literature; shown to be invariant under the similarity map (SM).
  • domain assumption Free-fermion half-filled Slater determinants and the correlation-matrix formula for von Neumann entropy correctly capture the many-body entanglement dynamics of the non-Hermitian model.
    Used throughout the long-time EE section; equivalent to post-selected jump-free trajectories via the given Lindblad operators.
  • ad hoc to paper A multi-terminal QWZ–BHZ heterostructure with negligible inter-layer tunneling realizes the target non-Hermitian Hamiltonians in the lead-basis conductance matrix.
    Experimental proposal (Fig. 6–7); relies on the specific mass-profile engineering and the Landauer–Büttiker extraction of an effective nH matrix.
invented entities (3)
  • Isospectral family H({α_j}) with total flux Γ = Lγ
    purpose: Continuously interpolates between uniform Hatano-Nelson and single-bond limits while keeping the spectrum fixed.
    Defined by the gauge field {α_j} and the similarity map S; the central object of the paper.
  • Single-bond non-Hermitian (SBN) model
    purpose: Extreme member of the family in which the entire flux sits on the closing bond t_L = e^{Lγ}.
    Obtained by the choice α_j = (j−L)γ; used for all dynamical calculations.
  • Bimodal localization regime
    purpose: Explains the second re-entrance of steady-state entanglement at strong disorder.
    Observed numerically as equal probability density at the nH bond and at the initial site (Fig. 5c); not previously named in the cited literature.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Universality and Dynamical Inequivalence in Isospectral Non-Hermitian Anderson Transitions." pith.science (2026). https://pith.science/paper/KQRULHJG

@misc{pith2026260707714,
  author       = {Pith},
  title        = {Pith review of: Universality and Dynamical Inequivalence in Isospectral Non-Hermitian Anderson Transitions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KQRULHJG}},
  note         = {Machine review of arXiv:2607.07714}
}
read the original abstract

The Hatano Nelson paradigm establishes that extensive bulk nonreciprocity can destabilize Anderson localization via an imaginary gauge flux. Here, we demonstrate that extensive nonreciprocity is not a necessary ingredient: a single non-Hermitian boundary bond in a disordered one-dimensional ring suffices to drive the localization-delocalization transition. More generally, we construct an exactly isospectral family of non-Hermitian Hamiltonians that continuously interpolates between the uniform Hatano Nelson model and the single-bond limit. We show that the universal critical behavior encompassing spectral, eigenstate, and topological diagnostics is gauge invariant and governed solely by the total imaginary gauge flux, regardless of its spatial distribution. Remarkably, despite sharing identical spectra and critical exponents, different configurations within this isospectral family exhibit qualitatively distinct quantum dynamics, establishing a fundamental separation between static and dynamical universality in non-Hermitian systems. Specifically, the single boundary realization features rapid operator scrambling, oscillatory wavepacket acceleration, and a double re-entrant steady state entanglement transition. Finally, we propose an experimentally feasible realization based on multi-terminal topological transport, providing a realistic route toward observing boundary induced non Hermitian criticality and its unconventional dynamical signatures.

Figures

Figures reproduced from arXiv: 2607.07714 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. ( [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The Qi-Wu-Zhang model (left side of the junction, de [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Conductance matrix of the multi-terminal junction for ( [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. OTOCs for different values of [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. ( [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Time evolution of an initially localized wavepacket pre [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Short time evolution of [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

78 extracted references · 4 canonical work pages

  1. [1]

    (d) The steady-state EE,S EE , of the many-particle wavefunction is plotted as a function ofwforL= 50,70

    For a givenw,∆a acc is obtained by averaging the accelerationa(t)over the time interval indicated by the red dashed box in the inset, which corresponds to the period between successive encounters of the wave packet with the nonreciprocal bond and also averaging over1000 disorder realizations.Inset:Time evolution ofa(t)from which the averages are derived. ...

  2. [2]

    QWZ Subsystem.—Under broken time-reversal symme- try (TRS), the system supports a higher topological phase with a total Chern numberC= 2. For the corresponding QWZ HamiltonianH QWZ, the local potentialΓand hopping matri- cesT η are defined as Γ = (2−M)(I s ⊗σ z), T x =− 1 2(Is ⊗σ z)− i 2(Is ⊗σ x), Ty =− 1 2(Is ⊗σ z)− i 2(Is ⊗σ y).(12)

  3. [3]

    BHZ Subsystem.—When TRS is preserved, the cancella- tion of opposing Chern numbers in the spin-up and spin-down channels yields a netC= 0, driving the system into a quan- tum spin Hall phase hosted by helical edge states. ForH BHZ, spin-orbit coupling is introduced vias z along theˆx-direction: Γ = (2−M)(I s ⊗σ z), T x =− 1 2(Is ⊗σ z)− i 2(sz ⊗σ x), Ty =−...

    2025

  4. [4]

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

    1958

  5. [5]

    Hatano and D

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

    1996

  6. [6]

    Hatano and D

    N. Hatano and D. R. Nelson, V ortex pinning and non-hermitian quantum mechanics, Phys. Rev. B56, 8651 (1997)

    1997

  7. [7]

    Hatano and D

    N. Hatano and D. R. Nelson, Non-hermitian delocalization and eigenfunctions, Phys. Rev. B58, 8384 (1998)

    1998

  8. [8]

    Orito and K.-I

    T. Orito and K.-I. Imura, Unusual wave-packet spreading and 8 entanglement dynamics in non-hermitian disordered many- body systems, Phys. Rev. B105, 024303 (2022)

    2022

Show all 78 references
  1. [9]

    Y . C. Hu and T. L. Hughes, Absence of topological insulator phases in non-hermitianpt-symmetric hamiltonians, Phys. Rev. B84, 153101 (2011)

    2011

  2. [10]

    Esaki, M

    K. Esaki, M. Sato, K. Hasebe, and M. Kohmoto, Edge states and topological phases in non-hermitian systems, Phys. Rev. B 84, 205128 (2011)

    2011

  3. [11]

    Schomerus, Topologically protected midgap states in com- plex photonic lattices, Optics Letters38, 1912 (2013)

    H. Schomerus, Topologically protected midgap states in com- plex photonic lattices, Optics Letters38, 1912 (2013)

    1912

  4. [12]

    Malzard, C

    S. Malzard, C. Poli, and H. Schomerus, Topologically protected defect states in open photonic systems with non-hermitian charge-conjugation and parity-time symmetry, Phys. Rev. Lett. 115, 200402 (2015)

    2015

  5. [13]

    San-Jose, J

    P. San-Jose, J. Cayao, E. Prada, and R. Aguado, Majorana bound states from exceptional points in non-topological super- conductors, Scientific Reports6, 10.1038/srep21427 (2016)

    2016 doi

  6. [14]

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

    2016

  7. [15]

    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

  8. [16]

    Xu, S.-T

    Y . Xu, S.-T. Wang, and L.-M. Duan, Weyl exceptional rings in a three-dimensional dissipative cold atomic gas, Phys. Rev. Lett. 118, 045701 (2017)

    2017

  9. [17]

    Kawabata, Y

    K. Kawabata, Y . Ashida, H. Katsura, and M. Ueda, Parity- time-symmetric topological superconductor, Phys. Rev. B98, 085116 (2018)

    2018

  10. [18]

    L. Li, C. H. Lee, and J. Gong, Impurity induced scale-free localization, Communications Physics4, 10.1038/s42005-021- 00547-x (2021)

    2021 doi

  11. [19]

    Liu and S

    Y . Liu and S. Chen, Diagnosis of bulk phase diagram of non- reciprocal topological lattices by impurity modes, Phys. Rev. B 102, 075404 (2020)

    2020

  12. [20]

    Suthar, Boundary-driven many-body phase transitions in a non-hermitian disordered fermionic chain, Phys

    K. Suthar, Boundary-driven many-body phase transitions in a non-hermitian disordered fermionic chain, Phys. Rev. B111, 064202 (2025)

    2025

  13. [21]

    Hamazaki, K

    R. Hamazaki, K. Kawabata, and M. Ueda, Non-hermitian many-body localization, Phys. Rev. Lett.123, 090603 (2019)

    2019

  14. [22]

    Akemann, F

    G. Akemann, F. Balducci, A. Chenu, P. P ¨aßler, F. Roccati, and R. Shir, Two transitions in complex eigenvalue statistics: Her- miticity and integrability breaking, Phys. Rev. Res.7, 013098 (2025)

    2025

  15. [23]

    M ´ak, M

    J. M ´ak, M. J. Bhaseen, and A. Pal, Statics and dynamics of non- hermitian many-body localization, Communications Physics7, 10.1038/s42005-024-01576-y (2024)

    2024 doi

  16. [24]

    P. W. Brouwer, P. G. Silvestrov, and C. W. J. Beenakker, The- ory of directed localization in one dimension, Phys. Rev. B56, R4333(R) (1997)

    1997

  17. [25]

    F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Biorthogonal bulk-boundary correspondence in non-hermitian systems, Phys. Rev. Lett.121, 026808 (2018)

    2018

  18. [26]

    Xiao and C

    Y .-X. Xiao and C. T. Chan, Topology in non-hermitian chern insulators with skin effect, Phys. Rev. B105, 075128 (2022)

    2022

  19. [27]

    Suthar, Y .-C

    K. Suthar, Y .-C. Wang, Y .-P. Huang, H. H. Jen, and J.-S. You, Non-hermitian many-body localization with open boundaries, Phys. Rev. B106, 064208 (2022)

    2022

  20. [28]

    De Tomasi and I

    G. De Tomasi and I. M. Khaymovich, Non-hermitian rosenzweig-porter random-matrix ensemble: Obstruction to the fractal phase, Phys. Rev. B106, 094204 (2022)

    2022

  21. [29]

    Evers and A

    F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys.80, 1355 (2008)

    2008

  22. [30]

    Hentschel and I

    H. Hentschel and I. Procaccia, The infinite number of general- ized dimensions of fractals and strange attractors, Physica D: Nonlinear Phenomena8, 435 (1983)

    1983

  23. [31]

    Grassberger and I

    P. Grassberger and I. Procaccia, Characterization of strange at- tractors, Phys. Rev. Lett.50, 346 (1983)

    1983

  24. [32]

    Y .-T. Tu, I. Jang, P.-Y . Chang, and Y .-C. Tzeng, General prop- erties of fidelity in non-Hermitian quantum systems with PT symmetry, Quantum7, 960 (2023)

    2023

  25. [33]

    Tzeng, C.-Y

    Y .-C. Tzeng, C.-Y . Ju, G.-Y . Chen, and W.-M. Huang, Hunting for the non-hermitian exceptional points with fidelity suscepti- bility, Phys. Rev. Res.3, 013015 (2021)

    2021

  26. [34]

    Sun, J.-C

    G. Sun, J.-C. Tang, and S.-P. Kou, Biorthogonal quantum criticality in non-hermitian many-body systems, Frontiers of Physics17, 33502 (2022)

    2022

  27. [35]

    Jiang, C

    H. Jiang, C. Yang, and S. Chen, Topological invariants and phase diagrams for one-dimensional two-band non-hermitian systems without chiral symmetry, Phys. Rev. A98, 052116 (2018)

    2018

  28. [36]

    Z. Gong, Y . Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Topological phases of non-hermitian systems, Phys. Rev. X8, 031079 (2018)

    2018

  29. [37]

    Yao and Z

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

    2018

  30. [38]

    Kawabata, K

    K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and topology in non-hermitian physics, Phys. Rev. X9, 041015 (2019)

    2019

  31. [39]

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

    2021

  32. [40]

    K. Ding, C. Fang, and G. Ma, Non-hermitian topology and exceptional-point geometries, Nature Reviews Physics4, 745–760 (2022)

    2022

  33. [41]

    Supplemental material, see Supplemental Material for addi- tional derivations and numerical results

  34. [42]

    Ochkan, R

    K. Ochkan, R. Chaturvedi, V . K ¨onye, L. Veyrat, R. Gi- raud, D. Mailly, A. Cavanna, U. Gennser, E. M. Hankiewicz, B. B¨uchner,et al., Non-hermitian topology in a multi-terminal quantum hall device, Nature Physics20, 395 (2024)

    2024

  35. [43]

    L. Yi, E. Steinebronn, A. S. Rashid, N. Samarth, R. El-Ganainy, S. L. Ozdemir, and M. Kayyalha, Non-hermitian dynamics in quantum anomalous hall insulators (2025), arXiv:2510.01125 [cond-mat.mes-hall]

    2025

  36. [44]

    Dalibard, Y

    J. Dalibard, Y . Castin, and K. Mølmer, Wave-function approach to dissipative processes in quantum optics, Phys. Rev. Lett.68, 580 (1992)

    1992

  37. [45]

    Carmichael,An open systems approach to quantum optics: lectures presented at the Universit ´e Libre de Bruxelles October 28 to November 4, 1991(Springer, 1993)

    H. Carmichael,An open systems approach to quantum optics: lectures presented at the Universit ´e Libre de Bruxelles October 28 to November 4, 1991(Springer, 1993)

    1991

  38. [46]

    Chru ´sci´nski, K

    D. Chru ´sci´nski, K. Luoma, J. Piilo, and A. Smirne, How to design quantum-jump trajectories via distinct master equation representations, Quantum6, 835 (2022)

    2022

  39. [47]

    X. Niu, J. Li, S. L. Wu, and X. X. Yi, Effect of quantum jumps on non-hermitian systems, Phys. Rev. A108, 032214 (2023)

    2023

  40. [48]

    Derrida and E

    B. Derrida and E. Gardner, Lyapounov exponent of the one di- mensional Anderson model : weak disorder expansions, Journal de Physique45, 1283 (1984)

    1984

  41. [49]

    Gliozzi, F

    J. Gliozzi, F. Balducci, T. L. Hughes, and G. De Tomasi, Non- hermitian multipole skin effects challenge localization, Physi- cal Review B113, 10.1103/c1jr-zqxg (2026)

    2026 doi

  42. [50]

    Barch, N

    B. Barch, N. Anand, J. Marshall, E. Rieffel, and P. Zanardi, Scrambling and operator entanglement in local non-hermitian quantum systems, Phys. Rev. B108, 134305 (2023)

    2023

  43. [51]

    Maldacena, S

    J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, Journal of High Energy Physics2016, 10.1007/jhep08(2016)106 (2016)

    2016 doi

  44. [52]

    Swingle, Unscrambling the physics of out-of-time-order cor- 9 relators, Nature Physics14, 988 (2018)

    B. Swingle, Unscrambling the physics of out-of-time-order cor- 9 relators, Nature Physics14, 988 (2018)

    2018

  45. [53]

    Xu and B

    S. Xu and B. Swingle, Scrambling dynamics and out-of- time-ordered correlators in quantum many-body systems, PRX Quantum5, 10.1103/prxquantum.5.010201 (2024)

    2024 doi

  46. [54]

    S. H. Shenker and D. Stanford, Black holes and the butterfly effect, Journal of High Energy Physics2014, 1 (2014)

    2014

  47. [55]

    Resta, Quantum-mechanical position operator in extended systems, Phys

    R. Resta, Quantum-mechanical position operator in extended systems, Phys. Rev. Lett.80, 1800 (1998)

    1998

  48. [56]

    Shang and H

    J. Shang and H. Hu, Spreading dynamics in the hatano-nelson model, Phys. Rev. B112, 014205 (2025)

    2025

  49. [57]

    B. M. Manda and V . Achilleos, Crossovers from nonlinear wave-packet acceleration to wave-mixing and self-trapping in the hatano-nelson model, arXiv preprint arXiv:2604.02263 (2026)

    2026

  50. [58]

    Spring, V

    H. Spring, V . K ¨onye, F. A. Gerritsma, I. C. Fulga, and A. R. Akhmerov, Phase transitions of wave packet dynamics in disor- dered non-Hermitian systems, SciPost Phys.16, 120 (2024)

    2024

  51. [59]

    Matsumoto, K

    N. Matsumoto, K. Kawabata, Y . Ashida, S. Furukawa, and M. Ueda, Continuous phase transition without gap closing in non-hermitian quantum many-body systems, Phys. Rev. Lett. 125, 260601 (2020)

    2020

  52. [60]

    Li, Z.-C

    K. Li, Z.-C. Liu, and Y . Xu, Disorder-induced entanglement phase transitions in non-hermitian systems with skin effects (2023), arXiv:2305.12342 [quant-ph]

    2023 arXiv

  53. [61]

    Skinner, J

    B. Skinner, J. Ruhman, and A. Nahum, Measurement-induced phase transitions in the dynamics of entanglement, Phys. Rev. X9, 031009 (2019)

    2019

  54. [62]

    Y . Li, X. Chen, and M. P. A. Fisher, Quantum zeno effect and the many-body entanglement transition, Phys. Rev. B98, 205136 (2018)

    2018

  55. [63]

    A. C. Potter and R. Vasseur, Entanglement dynamics in hybrid quantum circuits, inEntanglement in Spin Chains(Springer In- ternational Publishing, 2022) p. 211–249

    2022

  56. [64]

    Biella and M

    A. Biella and M. Schir ´o, Many-body quantum zeno effect and measurement-induced subradiance transition, Quantum5, 528 (2021)

    2021

  57. [65]

    Turkeshi, A

    X. Turkeshi, A. Biella, R. Fazio, M. Dalmonte, and M. Schir ´o, Measurement-induced entanglement transitions in the quantum ising chain: From infinite to zero clicks, Phys. Rev. B103, 224210 (2021)

    2021

  58. [66]

    Orito and K.-I

    T. Orito and K.-I. Imura, Entanglement dynamics in the many-body hatano-nelson model, Physical Review B108, 10.1103/physrevb.108.214308 (2023)

    2023 doi

  59. [67]

    Z.-C. Liu, K. Li, and Y . Xu, Dynamical transition due to feedback-induced skin effect, Phys. Rev. Lett.133, 090401 (2024)

    2024

  60. [68]

    He and T

    Y . He and T. Ozawa, Anomalous wave-packet dynamics in one- dimensional non-hermitian lattices (2026), arXiv:2512.07484 [physics.optics]

    2026 arXiv

  61. [69]

    Goldberg, H

    A. Goldberg, H. M. Schey, and J. L. Schwartz, Computer- generated motion pictures of one-dimensional quantum- mechanical transmission and reflection phenomena, American Journal of Physics35, 177 (1967)

    1967

  62. [70]

    G. Lee, T. Jin, Y .-X. Wang, A. McDonald, and A. Clerk, En- tanglement phase transition due to reciprocity breaking with- out measurement or postselection, PRX Quantum5, 010313 (2024)

    2024

  63. [71]

    Roccati, Non-hermitian skin effect as an impurity problem, Phys

    F. Roccati, Non-hermitian skin effect as an impurity problem, Phys. Rev. A104, 022215 (2021)

    2021

  64. [72]

    Molignini, O

    P. Molignini, O. Arandes, and E. J. Bergholtz, Anomalous skin effects in disordered systems with a single non-hermitian impu- rity, Phys. Rev. Res.5, 033058 (2023)

    2023

  65. [73]

    Qi, Y .-S

    X.-L. Qi, Y .-S. Wu, and S.-C. Zhang, Topological quantization of the spin hall effect in two-dimensional paramagnetic semi- conductors, Phys. Rev. B74, 085308 (2006)

    2006

  66. [74]

    B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum spin hall effect and topological phase transition in hgte quantum wells, Science314, 1757–1761 (2006)

    2006

  67. [75]

    B ¨uttiker, Four-terminal phase-coherent conductance, Phys

    M. B ¨uttiker, Four-terminal phase-coherent conductance, Phys. Rev. Lett.57, 1761 (1986)

    1986

  68. [76]

    C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal, Kwant: a software package for quantum transport, New Journal of Physics16, 063065 (2014)

    2014

  69. [77]

    Manzano, A short introduction to the lindblad master equa- tion, AIP Advances10, 10.1063/1.5115323 (2020)

    D. Manzano, A short introduction to the lindblad master equa- tion, AIP Advances10, 10.1063/1.5115323 (2020)

    2020 doi

  70. [78]

    Universality and Dynamical Inequivalence in Isospectral Non-Hermitian Anderson Transitions

    H.-P. Breuer and F. Petruccione,The Theory of Open Quantum Systems(Oxford University Press, 2007). END MA TTER As previously established, theGDremain identical across the entire isospectral family. To complement this finding, we include a comparative plot illustrating the dyna...

    2007

Pith tools

Reviewed July 12, 2026 · model on record in the stance chip above.