Quasi-Non-Hermitian Edge Bursts Induced by Nonuniform Loss

Fuxiang Li; Wei Li; Ze Yang

REVIEW 3 major objections 5 minor 77 references

Quasi-Non-Hermitian Edge Bursts Induced by Nonuniform Loss

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read Spatially nonuniform loss alone can create boundary-localized loss bursts even without the non-Hermitian skin effect, and these bursts grow with the loss gradient.

desk verdict Real numerical observation of a loss-gradient edge burst at zero flux, but the no-NHSE classification is too hand-wavy to take as established. read the letter →

arxiv 2607.19097 v1 pith:FAI2QMET submitted 2026-07-21 quant-ph

Ze Yang , Wei Li , Fuxiang Li This is my paper

classification quant-ph

keywords non-Hermitianskineffectedgeburstnonuniformlossquantumwalkmagneticfluxinverseparticipationratioboundarylocalizationdissipativelattice

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

The paper tries to establish that boundary-localized loss anomalies in non-Hermitian quantum walks do not require the non-Hermitian skin effect (NHSE). In a lattice with magnetic-flux-tunable NHSE and a linearly increasing loss rate, the authors find that at zero flux—where they argue the NHSE is absent—a weak boundary peak in the loss probability still appears and strengthens as the loss gradient grows. They name this the quasi-NHEB and show it is nearly independent of the walker's initial position, unlike the conventional NHSE-driven edge burst. The authors also show that nonuniform loss can suppress the NHSE in the positive-flux regime even when spectral criteria would predict it, indicating that spectral structure and eigenstate localization can decouple under inhomogeneous dissipation. If correct, this separates two physical mechanisms for edge bursts and opens a route to controlling boundary loss through engineered dissipation alone.

What carries the argument

The model is a one-dimensional two-sublattice tight-binding chain with open boundaries, non-Hermiticity from a site-dependent imaginary on-site potential −iγx on the B sublattice, and magnetic flux φ introduced via Peierls phases that control the NHSE direction and strength. The key mechanism the authors invoke is the linearly increasing imaginary potential, which acts as an effective potential barrier that suppresses intercell hopping and biases the walker toward regions of smaller loss, thereby accumulating loss probability near the boundary even without skin-effect eigenstate localization. The inverse participation ratio (IPR) and its mean (MIPR) serve as the operational diagnostics for t

What would settle it

A generalized Brillouin zone computation for the nonuniform-loss model at φ=0 showing that all eigenstates have localization lengths that scale with system size (i.e., a weak NHSE), or an experiment comparing the zero-flux nonuniform-loss case to a zero-flux uniform-loss control that exhibits an equally strong boundary peak, which would show the peak is not caused by the loss gradient.

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Extended reading notes

Core claim

The central claim is that a spatially nonuniform loss profile generates a boundary-localized accumulation of loss probability—termed the quasi-non-Hermitian edge burst (quasi-NHEB)—even in the absence of the non-Hermitian skin effect. At zero magnetic flux, where the authors' MIPR-based analysis indicates no NHSE, the loss probability shows a peak at the boundary whose magnitude increases monotonically with the loss gradient γ, while the peak near the initial position decreases. The quasi-NHEB is distinguished from the conventional NHEB by its broad spatial profile and its near-independence of the initial position x0. When flux is introduced, the interplay between nonuniform loss and the NHS

Load-bearing premise

The claim that the quasi-NHEB occurs in the absence of the NHSE rests on a manually chosen MIPR threshold (≈0.018, specific to L=80) applied to an unspecified subset of eigenstates; if a different subset or threshold were used, the zero-flux case might host a weak skin effect, and the key distinction would dissolve.

Editorial extensions

If this is right

Boundary-localized dissipation anomalies can be engineered purely by shaping the loss profile, without needing nonreciprocal hopping or magnetic flux.
The quasi-NHEB's insensitivity to the initial position provides an experimental fingerprint that separates loss-gradient-driven bursts from skin-effect-driven bursts; measuring P_edge vs x0 distinguishes them.
In systems with nonuniform loss, spectral encirclement criteria for the NHSE can be misleading; eigenstate-based measures such as the MIPR are needed to diagnose skin localization.
The hybrid NHEB regime offers a tunable platform where reversing the magnetic flux can completely switch off the skin effect and its associated edge burst, which could be probed in photonic waveguide arrays or electric circuits with engineered loss.

Reading between the lines

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

If the effective-barrier picture is right, other monotonic loss profiles (e.g., exponential or step-like) should produce quasi-NHEBs with strengths set by the local gradient; this is a direct, testable extension.
The near-independence of P_edge on x0 suggests the quasi-NHEB arises from a local escape process near the boundary rather than from global spectral properties, which could be modelled by a position-dependent decay-rate analysis.
Because the MIPR threshold is size-specific, the claim of NHSE absence at φ=0 should be revisited with a generalized Brillouin zone calculation for the nonuniform-loss model; if a weak NHSE is present at large L, the quasi-NHEB might be a finite-size effect.
The decoupling of spectral and localization properties under inhomogeneous loss may extend to disordered or random loss landscapes, where spectral winding may fail to predict boundary behavior.

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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

Quick take: the paper has a real observation — at zero flux, a boundary loss peak grows with the loss gradient even though the eigenstates look extended — but the way it rules out the NHSE is the weakest link, and the central term 'quasi-NHEB' is only as convincing as that criterion.

What's actually new: the zero-flux case and the asymmetry between φ=-π/2 and φ=π/2 are absent from the earlier papers, which concentrated on φ=-π/2 or uniform loss. The authors reproduce the known φ=-π/2 result, so the numerics are at least consistent. The x0-dependence curves in Fig. 6 are a clean diagnostic: P_edge is flat in x0 for the quasi-NHEB, while the conventional NHEB decays as a power law. That separation is useful and likely to survive scrutiny.

The paper is also honest about the spectral criterion: it explicitly shows the PBC loop enclosing the OBC spectrum at both ±π/2, yet the eigenstate profiles at π/2 are extended. That is a good caution against relying on winding numbers alone.

Soft spots: the NHSE-absence diagnosis rests on a MIPR threshold of 0.018, which the authors admit is specific to L=80, and on a 'subset of eigenstates associated with the NHSE' that is never defined. At φ=π/2 the MIPR is 0.017, essentially equal to the 'no NHSE' value, so the boundary between skin and no-skin is being decided by a number that no one can reproduce. The stress-test worry — that the quasi-NHEB is secretly an imaginary-Stark skin effect — does not land hard for φ=0, because the hopping there is symmetric and a linear loss gradient gives a perfectly natural, NHSE-free explanation. But a GBZ analysis or finite-size scaling of the MIPR would settle it, and the paper has neither.

There is no code, no data, no error bars, and the system sizes (L=80, 150) are modest. The final mechanism — an effective potential barrier from the loss gradient — is heuristic, not derived.

Bottom line: the observed effect is probably real, but the classification is under-supported. The paper deserves a serious referee, and a revision could make the quasi-NHEB a solid addition to the edge-burst literature.

Referee Report

3 major / 5 minor

Summary. The paper studies a one-dimensional non-Hermitian tight-binding model on a two-sublattice ladder with a magnetic flux and a position-dependent imaginary potential. It reports that a boundary-localized loss probability peak persists at zero flux, where the non-Hermitian skin effect (NHSE) is claimed to be absent, and terms this phenomenon a quasi–non-Hermitian edge burst (quasi-NHEB). The authors further classify edge bursts into conventional, hybrid, and quasi types, and use the dependence on the initial position x0 to argue for a distinct bulk–edge scaling relation.

Significance. If the central claim is correct, the paper would establish that spatially nonuniform loss alone can generate boundary-localized loss anomalies without the NHSE, thereby broadening the NHEB framework. The model is concrete and the dynamics are computed directly from the Schrödinger equation, with a check against the previously studied case in Ref. [66]. The proposed x0-dependence as a discriminator between the quasi-NHEB and the conventional NHEB is a useful diagnostic. However, the load-bearing diagnosis of NHSE absence is based on a hand-set MIPR threshold that is acknowledged to be L-specific, and the paper lacks convergence checks and a non-circular boundary-mode criterion. The contribution is potentially significant but currently under-validated.

major comments (3)

[Sec. III, Sec. IV, Fig. 3(a)] The claim that the φ=0 edge burst is NHSE-independent rests on the MIPR≈0.018 threshold, which the paper itself states is 'specific to L=80' and which is applied only to an undefined 'subset of eigenstates associated with the NHSE' (Sec. IV). The PBC/OBC spectral comparison is acknowledged to be inconclusive in Fig. 4: the PBC loop encloses the OBC spectrum for both φ=±π/2, yet only φ=−π/2 shows skin-localized eigenstates. A reliable diagnosis of NHSE absence requires a non-circular criterion, e.g., finite-size scaling of the relevant eigenstate IPRs or a generalized Brillouin zone/winding-number calculation for the nonuniform loss profile. Without this, the distinction between quasi-NHEB and an imaginary-Stark-type NHSE (Ref. [72]) is not established.
[Eq. (4), Sec. V, Fig. 6] The central scaling statement—'P_edge remains nearly independent of x0' for the quasi-NHEB—requires a precise definition of P_edge, which is never given. If P_edge is simply P_x at x=1, it may miss the broad boundary region over which the quasi-NHEB extends; if it is a sum over several sites, the cutoff matters. Because the x0-independence is the key discriminator from the conventional NHEB, the authors should define P_edge explicitly and test the sensitivity of the conclusion to that definition.
[Sec. V, Fig. 6] The paper contains no error bars, convergence checks for the time integral in Eq. (4), or finite-size scaling. The claimed power-law decay of the conventional NHEB and the flatness of P_edge for the quasi-NHEB are asserted from visual inspection. A quantitative fit (with exponents and residuals) and at least one L-scaling test would substantially strengthen the classification.

minor comments (5)

[Sec. IV, after Fig. 3(a)] Typo: 'These difference in the MIPR suggest' should be 'These differences in the MIPR suggest'.
[Sec. IV] The sentence 'an edge burst still emerges, we term this NHSE-independent edge burst a quasi–non-Hermitian edge burst' is a run-on; please rephrase for clarity.
[Sec. IV, Fig. 3(d) inset] The text mentions 'an eigenstate that exhibits a relatively localized probability density near the left side of the chain.' Clarify whether this belongs to the near-real-axis subset or to the vertical imaginary branch, since the later discussion rules it out as the origin of the quasi-NHEB.
[Fig. 4] The axes in the eigenstate density panels (a2) and (b2) are not labeled; adding 'site' and '|ψ|' labels would aid readability.
[Sec. IV] The discussion of Ref. [72] would benefit from a one-sentence summary of the imaginary-Stark NHSE mechanism, so that the reader can appreciate why the present nonuniform-loss case might be related.

Circularity Check

1 steps flagged · score 2.0 of 10

MIPR threshold and eigenstate-subset choice make the 'NHSE absent' label self-referential, but the boundary-peak signal itself is directly simulated and not circular.

self definitional [Sec. III (criterion introduced near Eq. (3)); applied in Sec. IV Fig. 3(a) and Sec. IV text]
"an MIPR value close to 0.018 is taken as a practical criterion for the absence of the NHSE throughout this work. This value is not universal but represents the typical MIPR value observed for the present model in the absence of the NHSE over a wide range of parameter settings, and is specific to the system size L= 80."

The criterion for 'absence of the NHSE' is calibrated on cases already assumed to lack the NHSE, then used to label the φ=0 edge burst as NHSE-independent ('quasi-NHEB'). The MIPR is also evaluated 'only for the subset of eigenstates associated with the NHSE' (Sec. IV), so the no-NHSE verdict is partly fixed by the chosen subset and threshold. This does not manufacture the P_x boundary peak itself (Eq. (4) is independent of MIPR), but it makes the central classification 'quasi-NHEB without NHSE' an internally imposed criterion rather than an externally derived conclusion.

full rationale

The central dynamical result is non-circular: P_x is obtained by numerically solving the non-Hermitian Schrödinger equation and integrating |ψB_x(t)|^2 (Eq. (4)); no fitted parameter is used to produce the boundary peak, and the paper reproduces the known NHEB of Ref. [65] and the nonuniform-loss result of Ref. [66] as external checks. The only circular-adjacent element is the diagnostic used to call the φ=0 burst 'NHSE-independent': the MIPR threshold 0.018 is defined from the model's own assumed no-NHSE regime and is stated to be L=80-specific, while the MIPR is computed on a subset of eigenstates whose selection rule is not specified. This affects interpretation and labeling, and it leaves open the alternative that the quasi-NHEB is an imaginary-Stark skin effect of Ref. [72], but it does not generate the computed quasi-NHEB signal itself. Self-citations are not load-bearing in the derivation, so the score is near zero, with a small penalty for the circularly calibrated classification criterion.

Assumptions & free parameters 1 free parameters · 5 assumptions · 1 invented entities

All claims are numerical; the model is standard. The main non-standard inputs are the linear loss profile and the hand-set MIPR threshold.

free parameters (1)

MIPR NHSE-absence threshold = 0.018 (L=80)
Hand-chosen criterion: 'an MIPR value close to 0.018 is taken as a practical criterion for the absence of the NHSE' (Sec. III); it is model- and size-specific and used throughout to classify NHSE presence/absence.

assumptions (5)

domain assumption Time evolution is governed by the non-Hermitian Schrödinger equation i∂tψ=Hψ with loss encoded as -iγ_x on B sites; P_x=γ_x∫|ψB_x|²dt gives the escape probability.
Standard single-particle loss model; assumes perfect detection at B sites and no re-injection.
standard math Peierls substitution with Landau gauge produces the phase factors in t1 and t3.
Standard minimal-coupling prescription for magnetic flux in tight-binding models.
domain assumption Linear loss profile γ_x=γx with x=1,...,L.
Chosen 'without loss of generality' to model nonuniform loss; the central quasi-NHEB claim depends on this specific monotonic gradient.
domain assumption NHSE presence can be judged from MIPR and eigenstate densities; spectral-loop enclosure is treated as a fallible indicator.
The paper shows the spectral criterion fails for φ=π/2, so the operational definition of NHSE absence is MIPR/subset-based.
domain assumption Open boundary conditions; initial state localized on A-sublattice at x0.
Defines the quantum-walk setup and the boundary where NHEB/quasi-NHEB is measured.

invented entities (1)

quasi-NHEB
purpose: Label for the boundary-localized loss accumulation observed at φ=0 without NHSE
A new named phenomenon category; no independent observable prediction beyond the P_x profiles from which it is defined.

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Cite this review

Pith. "Pith review of Quasi-Non-Hermitian Edge Bursts Induced by Nonuniform Loss." pith.science (2026). https://pith.science/paper/FAI2QMET

@misc{pith2026260719097,
  author       = {Pith},
  title        = {Pith review of: Quasi-Non-Hermitian Edge Bursts Induced by Nonuniform Loss},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FAI2QMET}},
  note         = {Machine review of arXiv:2607.19097}
}

read the original abstract

Non-Hermitian quantum walks on lossy lattices with open boundaries can exhibit an anomalous peak of the loss probability at the boundary, known as the non-Hermitian edge burst (NHEB). This phenomenon has been attributed to the combined effect of the non-Hermitian skin effect (NHSE) and a gapless imaginary spectrum. Here we investigate a class of models in which the NHSE is induced by magnetic flux while the loss is spatially nonuniform. We show that the spatial distribution of loss plays a crucial role in determining the emergence and strength of the NHEB. Notably, even in the absence of the NHSE, a weak boundary accumulation of the loss probability persists. We term this effect a quasi-non-Hermitian edge burst (quasi-NHEB). By analyzing the dependence of the loss probability on the initial position, we further demonstrate that the quasi-NHEB obeys a bulk-edge scaling relation distinct from that of conventional NHEB. Our results show that spatially nonuniform loss alone can generate boundary-localized loss anomalies even without the NHSE, providing new insight into non-Hermitian boundary phenomena and a broader platform for their exploration and potential applications.

Figures

Figures reproduced from arXiv: 2607.19097 by the authors.

**Figure 2.** FIG. 2. (a) For the uniform-loss case, mean inverse participation ratio (MIPR) as a function of magnetic flux [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) For the nonuniform loss model, mean inverse participation ratio (MIPR) as a function of magnetic flux [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. Complex energy spectra of the nonuniform loss [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. Spatial distributions of the loss probability [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6. Dependence of boundary loss probability on the [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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Reference graph

Works this paper leans on

77 extracted references · 2 linked inside Pith

[66]

Yuce and H

C. Yuce and H. Ramezani, Phys. Rev. B107, L140302 (2023)

2023
[72]

H. Lin, J. Pi, Y. Qi, W. Qin, F. Nori, and G.-L. Long, Physical Review Research7, 033150 (2025)

2025
[1]

C. M. Bender and S. Boettcher, Phys. Rev. Lett.80, 5243 (1998)

1998
[2]

C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett.89, 270401 (2002)

2002
[3]

C. M. Bender, M. Berry, and A. Mandilara, Journal of Physics A: Mathematical and General35, L467 (2002)

2002
[4]

Mostafazadeh, Journal of Mathematical Physics43, 205 (2002)

A. Mostafazadeh, Journal of Mathematical Physics43, 205 (2002)

2002
[5]

C. M. Bender, Reports on Progress in Physics70, 947 (2007)

2007
[6]

E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Reviews of Modern Physics93, 015005 (2021)

2021

Show all 77 references

[7]

Ashida, Z

Y. Ashida, Z. Gong, and M. Ueda, Advances in Physics 69, 249 (2020)

2020
[8]

Kawabata, T

K. Kawabata, T. Bessho, and M. Sato, arXiv preprint arXiv:1902.08479 (2019)

1902 arXiv
[10]

W. Rui, Z. Zheng, C. Wang, and Z. Wang, Physical Re- view Letters128, 226401 (2022)

2022
[11]

Matsumoto, K

N. Matsumoto, K. Kawabata, Y. Ashida, S. Furukawa, and M. Ueda, Physical review letters125, 260601 (2020)

2020
[12]

Longhi, Phys

S. Longhi, Phys. Rev. Lett.128, 157601 (2022)

2022
[13]

Xue, M.-R

W.-T. Xue, M.-R. Li, Y.-M. Hu, F. Song, and Z. Wang, Physical Review B103, L241408 (2021)

2021
[14]

Longhi, Phys

S. Longhi, Phys. Rev. B108, 075121 (2023)

2023
[15]

L. Xiao, X. Zhan, Z. Bian, K. Wang, X. Zhang, X. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami,et al., Nature Physics13, 1117 (2017)

2017
[16]

Regensburger, C

A. Regensburger, C. Bersch, M.-A. Miri, G. On- ishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012)

2012
[17]

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljaˇ ci´ c, Nature525, 354 (2015)

2015
[18]

Hodaei, A

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Kha- javikhan, Nature548, 187 (2017)

2017
[19]

Chen, S ¸

W. Chen, S ¸. Kaya ¨Ozdemir, G. Zhao, J. Wiersig, and L. Yang, Nature548, 192 (2017)

2017
[20]

Hatano and D

N. Hatano and D. R. Nelson, Phys. Rev. Lett.77, 570 (1996)

1996
[21]

Hatano and D

N. Hatano and D. R. Nelson, Phys. Rev. B56, 8651 (1997)

1997
[22]

M. S. Rudner and L. S. Levitov, Phys. Rev. Lett.102, 065703 (2009)

2009
[23]

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, Phys. Rev. Lett.115, 040402 (2015)

2015
[24]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett.103, 093902 (2009)

2009
[25]

Feng, Y.-L

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, Nature materials12, 108 (2013)

2013
[26]

El-Ganainy, K

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Na- ture Physics14, 11 (2018)

2018
[27]

St-Jean, V

P. St-Jean, V. Goblot, E. Galopin, A. Lema ˆ ıtre, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, Nature Photonics11, 651 (2017)

2017
[28]

Midya, H

B. Midya, H. Zhao, and L. Feng, Nature communications 9, 2674 (2018)

2018
[29]

X. Wang, Y. Li, X. Hu, R. Gu, Y. Ao, P. Jiang, and Q. Gong, Physical Review A105, 023531 (2022)

2022
[30]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.- F. Chen, Y. Fainman, and A. Scherer, Science333, 729 (2011)

2011
[31]

C. E. R¨ uter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nature physics 6, 192 (2010)

2010
[32]

McDonald and A

A. McDonald and A. A. Clerk, Nature communications 11, 5382 (2020). 10

2020
[33]

C. C. Wanjura, M. Brunelli, and A. Nunnenkamp, Nature communications11, 3149 (2020)

2020
[34]

Parto, C

M. Parto, C. Leefmans, J. Williams, R. M. Gray, and A. Marandi, Light: Science & Applications14, 6 (2025)

2025
[35]

Rotter, Journal of Physics A: Mathematical and The- oretical42, 153001 (2009)

I. Rotter, Journal of Physics A: Mathematical and The- oretical42, 153001 (2009)

2009
[36]

V. M. Martinez Alvarez, J. E. Barrios Vargas, and L. E. F. Foa Torres, Phys. Rev. B97, 121401 (2018)

2018
[37]

S. Yao, F. Song, and Z. Wang, Phys. Rev. Lett.121, 136802 (2018)

2018
[38]

F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Phys. Rev. Lett.121, 026808 (2018)

2018
[39]

C. H. Lee and R. Thomale, Phys. Rev. B99, 201103 (2019)

2019
[40]

Helbig, T

T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. Molenkamp, C. Lee, A. Szameit, M. Gre- iter, and R. Thomale, Nature Physics16, 747 (2020)

2020
[41]

J. C. Budich and E. J. Bergholtz, Phys. Rev. Lett.125, 180403 (2020)

2020
[42]

Longhi, Physical Review Research1, 023013 (2019)

S. Longhi, Physical Review Research1, 023013 (2019)

2019
[43]

L. Xiao, T. Deng, K. Wang, Z. Wang, W. Yi, and P. Xue, Physical Review Letters126, 230402 (2021)

2021
[44]

Weidemann, M

S. Weidemann, M. Kremer, S. Longhi, and A. Szameit, Nature601, 354 (2022)

2022
[45]

D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, Physical review letters124, 056802 (2020)

2020
[46]

F. Song, S. Yao, and Z. Wang, Physical review letters 123, 246801 (2019)

2019
[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, Nature Physics16, 747 (2020)

2020
[48]

L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, Nature Physics16, 761 (2020)

2020
[49]

Ghatak, M

A. Ghatak, M. Brandenbourger, J. Van Wezel, and C. Coulais, Proceedings of the National Academy of Sci- ences117, 29561 (2020)

2020
[50]

Weidemann, M

S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, Science368, 311 (2020)

2020
[51]

T. E. Lee, Physical review letters116, 133903 (2016)

2016
[52]

Yao and Z

S. Yao and Z. Wang, Phys. Rev. Lett.121, 086803 (2018)

2018
[53]

Zhang, Z

K. Zhang, Z. Yang, and C. Fang, Phys. Rev. Lett.125, 126402 (2020)

2020
[54]

Okuma, K

N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Phys. Rev. Lett.124, 086801 (2020)

2020
[55]

H.-Y. Wang, F. Song, and Z. Wang, Phys. Rev. X14, 021011 (2024)

2024
[56]

Z. Yang, K. Zhang, C. Fang, and J. Hu, Phys. Rev. Lett. 125, 226402 (2020)

2020
[57]

W. Li, Z. Sun, Z. Yang, and F. Li, Phys. Rev. B109, 035119 (2024)

2024
[58]

H. Shen, B. Zhen, and L. Fu, Phys. Rev. Lett.120, 146402 (2018)

2018
[59]

Y. Peng, J. Jie, D. Yu, and Y. Wang, Physical Review B 106, L161402 (2022)

2022
[60]

Lu, X.-X

M. Lu, X.-X. Zhang, and M. Franz, Physical review let- ters127, 256402 (2021)

2021
[61]

H. T. Teo, S. Mandal, Y. Long, H. Xue, and B. Zhang, Science Bulletin69, 1667 (2024)

2024
[62]

C.-A. Li, B. Trauzettel, T. Neupert, and S.-B. Zhang, Physical Review Letters131, 116601 (2023)

2023
[63]

C. Xu, Z. Guan, and H. Xu, Physical Review B111, 024201 (2025)

2025
[64]

Longhi, arXiv preprint arXiv:2511.20518 (2025)

S. Longhi, arXiv preprint arXiv:2511.20518 (2025)

2025
[65]

Xue, Y.-M

W.-T. Xue, Y.-M. Hu, F. Song, and Z. Wang, Phys. Rev. Lett.128, 120401 (2022)

2022
[67]

Yuce and H

C. Yuce and H. Ramezani, Phys. Rev. B109, 214301 (2024)

2024
[68]

Sen and C

G. Sen and C. Yuce, Phys. Rev. B111, 174305 (2025)

2025
[69]

Hu, W.-T

Y.-M. Hu, W.-T. Xue, F. Song, and Z. Wang, Physical Review B108, 235422 (2023)

2023
[70]

Zhu, Y.-L

J. Zhu, Y.-L. Mao, H. Chen, K.-X. Yang, L. Li, B. Yang, Z.-D. Li, and J. Fan, Phys. Rev. Lett.132, 203801 (2024)

2024
[71]

Xiao, W.-T

L. Xiao, W.-T. Xue, F. Song, Y.-M. Hu, W. Yi, Z. Wang, and P. Xue, Phys. Rev. Lett.133, 070801 (2024)

2024
[73]

Evers and A

F. Evers and A. D. Mirlin, Reviews of Modern Physics 80, 1355 (2008)

2008
[74]

Zeng and Y

Q.-B. Zeng and Y. Xu, Physical Review Research2, 033052 (2020)

2020
[75]

Lv, T.-C

T. Lv, T.-C. Yi, L. Li, G. Sun, and W.-L. You, Physical Review A105, 013315 (2022)

2022
[76]

Longhi, Phys

S. Longhi, Phys. Rev. B105, 245143 (2022)

2022
[77]

Zhang, Z

K. Zhang, Z. Yang, and C. Fang, Nature communications 13, 2496 (2022)

2022
[78]

Liu and Z

H. Liu and Z. Xu, arXiv preprint arXiv:2605.21034 (2026)

2026 arXiv

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [66]

Yuce and H

C. Yuce and H. Ramezani, Phys. Rev. B107, L140302 (2023)

2023

[2] [72]

H. Lin, J. Pi, Y. Qi, W. Qin, F. Nori, and G.-L. Long, Physical Review Research7, 033150 (2025)

2025

[3] [1]

C. M. Bender and S. Boettcher, Phys. Rev. Lett.80, 5243 (1998)

1998

[4] [2]

C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett.89, 270401 (2002)

2002

[5] [3]

C. M. Bender, M. Berry, and A. Mandilara, Journal of Physics A: Mathematical and General35, L467 (2002)

2002

[6] [4]

Mostafazadeh, Journal of Mathematical Physics43, 205 (2002)

A. Mostafazadeh, Journal of Mathematical Physics43, 205 (2002)

2002

[7] [5]

C. M. Bender, Reports on Progress in Physics70, 947 (2007)

2007

[8] [6]

E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Reviews of Modern Physics93, 015005 (2021)

2021