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arxiv: 2504.20167 · v3 · submitted 2025-04-28 · 🪐 quant-ph · cond-mat.mes-hall· hep-th· math-ph· math.MP

Emergence of Hermitian topology from non-Hermitian knots

Gaurav Hajong , Ranjan Modak , Bhabani Prasad Mandal This is my paper

Pith reviewed 2026-05-22 17:37 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hallhep-thmath-phmath.MP
keywords non-Hermitian systemsknot topologytopological phase transitionsingular valuesexceptional pointscomplex eigenvaluesHermitian models
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The pith

Treating singular values of a non-Hermitian Hamiltonian as eigenvalues of a Hermitian topological model induces a first-order transition between knot structures in the complex spectrum.

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

The paper shows that when the real singular values of a non-Hermitian Hamiltonian are interpreted as the eigenvalues of a translationally invariant Hermitian model undergoing a topological phase transition, the complex eigenvalues of the original non-Hermitian system undergo a corresponding first-order knot transition between distinct knot structures. This transition occurs without an exceptional point and features abrupt jumps in both the real and imaginary parts of the eigenvalues. The mapping between the two systems remains consistent provided the lattice momentum periodicity is preserved, even though the choice of non-Hermitian Hamiltonian is not unique. The work further establishes that a topological change in the Hermitian model forces a knot change in the non-Hermitian spectrum, while the reverse implication does not hold.

Core claim

If the singular values of an NH Hamiltonian are treated as eigenvalues of prototype translational invariant Hermitian models that undergo a topological phase transition between two distinct topological phases, the complex eigenvalues of the NH Hamiltonian will also undergo a first order knot transition between different knot structures. Unlike the usual knot transition, this transition is not accompanied by an Exceptional point; instead the real and complex parts of the eigenvalues show a discrete jump at the transition point. The connection remains robust when the periodicity in lattice momentum is the same for both models, and a change in the topology of the Hermitian model implies a knot-

What carries the argument

Mapping of singular values of the non-Hermitian Hamiltonian onto eigenvalues of a Hermitian topological model, which drives the knot transition in the complex eigenvalue spectrum.

If this is right

  • The knot transition is first-order and marked by simultaneous jumps in real and imaginary eigenvalue components.
  • No exceptional point appears at the knot transition point.
  • A topological phase change in the Hermitian model forces a change in the non-Hermitian knot structure.
  • The correspondence holds for any non-Hermitian Hamiltonian whose singular values match the Hermitian eigenvalues under identical momentum periodicity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This mapping offers a route to design or predict specific knot topologies in non-Hermitian systems by selecting known Hermitian topological models.
  • Knot transitions could serve as an indirect experimental signature of underlying Hermitian topological changes.
  • The one-way implication suggests that knot topology is more sensitive to Hermitian topology than the reverse.

Load-bearing premise

The singular values of the non-Hermitian Hamiltonian can be matched to the eigenvalues of a Hermitian model that shares the same lattice momentum periodicity.

What would settle it

Finding a knot transition in the complex spectrum that either passes through an exceptional point or lacks discrete jumps in the real and imaginary parts at the critical point would falsify the claimed correspondence.

Figures

Figures reproduced from arXiv: 2504.20167 by Bhabani Prasad Mandal, Gaurav Hajong, Ranjan Modak.

Figure 1
Figure 1. Figure 1: Singular-value spectrum of the Hermitian Hamiltonian HI (k, ω). Panels (a), (b), and (d) show the gapped spectra for ω = 0.5, 1.5, and 34, demonstrating that HI remains gapped for ω ̸= 1. Panel (c) corresponds to ω = 1, where the gap closes at k = π, marking the topological transition between the ν = 0 and ν = 1 phases. IV. RESULTS We now demonstrate our results for both the mod￾els. Note that the Hermitia… view at source ↗
Figure 2
Figure 2. Figure 2: Knot structures of the NH matrix A(ω, k) for Model I with V = σx. Panel (a) shows the unlinked knot structure for ω = 0.5, while panel (b) displays the unknot for ω = 1.5. These two configurations correspond to winding numbers ν = 0 and |ν| = 1, respectively, illustrating the change in knot topology across the transition at ω = 1. We observe an unlink (ω < 1) to unknot (ω > 1) topo￾logical transition with … view at source ↗
Figure 5
Figure 5. Figure 5: Real and imaginary parts of the eigenvalues of A(ω, k) at k = π for Model I. Panels (a) and (c) show the discontinuous jump at ω = 1, indicating the first-order knot transition. Panels (b) and (d) show the behavior near the exceptional point at ω ≃ 34, where the transition occurs without any discontinuity. VI. DISCUSSIONS Our main goal in this manuscript was to establish a connection between the knot topol… view at source ↗
Figure 4
Figure 4. Figure 4: Knot structures of the NH matrix A(ω, k) for Model II with V = σx. Panel (a) shows the unknot configuration for ω = 0.5, while panel (b) displays the Hopf-link structure for ω = 1.5. These correspond to winding numbers |ν| = 1 and |ν| = 2, respectively, illustrating the knot-topology change across the transition at ω = 1. A similar unknot-hopflink knot transition is observed for a different choice of V , w… view at source ↗
Figure 6
Figure 6. Figure 6: Knot structures of A(ω, k) for Model I with the k-dependent matrix V (k). Panel (a) shows the unknot for ω = 0.5, while panel (b) displays the Hopf-link for ω = 1.5. The use of a k-dependent V changes the resulting knot types compared to the traceless, k-independent choices [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Knot structures of A(ω, k) for Model II with the k-dependent matrix V (k). Panel (a) shows the hopflink for ω = 0.5, while panel (b) shows the unknot for ω = 1.5. As in Model I, the k-dependence of V leads to knot configurations distinct from those obtained with traceless, k-independent choices. We observe that the set of knots in Fig. (6) represents the unknot and the Hopf link with |ν| = 1 and |ν| = 2, r… view at source ↗
Figure 8
Figure 8. Figure 8: Eigenvalue spectrum of the unitary matrix V (k), demonstrating that V (k) itself exhibits a knot structure in the complex plane as k is varied [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Knot structures of A′ (ω, k) for Model I, obtained using U with parameter values φ = π 2 , α = π 3 , β = π 6 , and θ = k + π 4 . Panel (a) shows an unlink at ω = 0.5, whereas panel (b) displays an unknot at ω = 1.5. Appendix B: Discontinuity at the transition point Re(E) −2 0 2 Im(E) 0 −1 1 0 1 2 k π ω = 1 E1 E2 (a) −2 Re(E) 0 2 Im(E) −2 0 2 0 1 2 k π ω = 1 E1 E2 (b) [PITH_FULL_IMAGE:figures/full_fig_p011… view at source ↗
Figure 10
Figure 10. Figure 10: Spectral discontinuity of the eigenvalues of A(ω, k) at the transition point ω = 1. Panels (a) and (b) show the discontinuous jump in the eigenvalues at k = π for Model I and Model II, respectively, characterizing the first-order knot transition that occurs without an exceptional point. We have demonstrated that the NH choices of A associated with our Hermitian SSH models exhibit topological transitions a… view at source ↗
Figure 11
Figure 11. Figure 11: Knot structures of A(ω, k) for Model I near the additional NH transition at ω ≃ 34. Panels (a) and (c) show the knot configurations for ω = 30 and ω = 40, respectively, while panel (b) corresponds to ω = 34, where an exceptional point appears at k = π. This NH transition does not coincide with any Hermitian gap closing. In contrast to Fig. (1d) of the main paper, which clearly shows that the configuration… view at source ↗
Figure 12
Figure 12. Figure 12: Knot structures of A(ω, k) for Model II near the additional NH transition at ω ≃ 67. Panels (a) and (c) show the configurations for ω = 60 and ω = 80, respectively, while panel (b) corresponds to ω = 67, where a double exceptional point appears at k = π 2 and k = 3π 2 . This NH transition occurs without a corresponding Hermitian gap closing [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Real and imaginary parts of the eigenvalues of A(ω, k) for Model II. Panels (a) and (c) show the discontinuous jump at ω = 1, indicating the first-order knot transition. Panels (b) and (d) display the smooth behavior near the NH transition at ω ≃ 67, where a double exceptional point occurs without any Hermitian gap closing. Appendix E: Additional examples of V for both models I and II We take up here anot… view at source ↗
Figure 14
Figure 14. Figure 14: Knot structures of A(ω, k) for Model I with the traceless, k-independent choice V = iσz. Panel (a) shows the unlink for ω = 0.5, while panel (b) shows the unknot for ω = 1.5. This demonstrates that the unlink–unknot transition persists for this alternative choice of V . While, for model II, the NH system A with the choice, V = iσz is given by, A =   − ia′√ (3+ω+a′) √ 2(1+e 2ikω+2 cos k) − ia′√ (3+ω−a′)… view at source ↗
Figure 15
Figure 15. Figure 15: Knot structures of A(ω, k) for Model II with the traceless, k-independent choice V = iσz. Panel (a) shows the unknot for ω = 0.5, while panel (b) shows the Hopf-link for ω = 1.5. This confirms that the unknot–hopflink transition of Model II is preserved for this choice of V [PITH_FULL_IMAGE:figures/full_fig_p014_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Knot structures for Model I with V = σy. Panel (a) shows an unknot at ω = 0.5, while panel (b) shows an unlink at ω = 1.5. This choice of V exhibits a reversed knot transition as ω crosses the Hermitian topological transition point. Appendix F: Model I: Effective real-space NH description of A near knot transition In this section, we demonstrate the effective real-space NH description of the A matrix for … view at source ↗
read the original abstract

The non-Hermiticity of the system gives rise to a distinct knot topology in the complex eigenvalue spectrum, which has no counterpart in Hermitian systems. In contrast, the singular values of a non-Hermitian (NH) Hamiltonian are always real by definition, meaning that they can also be interpreted as the eigenvalues of some underlying Hermitian Hamiltonian. In this work, we demonstrate that if the singular values of an NH Hamiltonian are treated as eigenvalues of prototype translational invariant Hermitian models that undergo a topological phase transition between two distinct topological phases, the complex eigenvalues of the NH Hamiltonian will also undergo a {\it{first order knot transition}} between different knot structures. Unlike the usual knot transition, this transition is not accompanied by an Exceptional point (EP); in contrast, the real and complex parts of the eigenvalues of the NH Hamiltonian show a discrete jump at the transition point. We emphasize that the choice of an NH Hamiltonian whose singular values match the eigenvalues of a Hermitian model is not unique. However, our study suggests that this connection between the NH and Hermitian models remains robust as long as the periodicity in lattice momentum is the same for both. Furthermore, we provide an example showing that a change in the topology of the Hermitian model implies a transition in the underlying NH knot topology, but a change in knot topology does not necessarily signal a topological transition in the Hermitian system.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper claims that singular values of a non-Hermitian Hamiltonian can be interpreted as eigenvalues of a translational-invariant Hermitian model undergoing a topological phase transition between distinct phases. Under this mapping, the complex eigenvalues of the NH Hamiltonian undergo a first-order knot transition between different knot structures. This transition occurs without an exceptional point and is instead marked by discrete jumps in the real and imaginary parts of the eigenvalues. The authors note that the NH Hamiltonian choice is non-unique but argue the connection remains robust provided the lattice momentum periodicity matches that of the Hermitian model. An explicit example is given showing that a Hermitian topological transition implies an NH knot transition, but the converse does not necessarily hold.

Significance. If the central mapping holds, the work provides a concrete bridge between Hermitian topological phase transitions and the knot topology of non-Hermitian complex spectra, showing how Hermitian topology can influence NH knot structures via singular values. The explicit example and the explicit acknowledgment that the NH choice is non-unique are positive features that allow the reader to assess the construction directly.

major comments (1)
  1. [Abstract / main construction section] Abstract and the section presenting the main construction: the statement that the NH-Hermitian connection 'remains robust' as long as lattice momentum periodicity is shared is supported only by a single explicit construction and numerical example. No general theorem or argument is supplied showing that knot topology is fixed solely by the singular-value spectrum once periodicity is fixed; other NH realizations sharing the same periodicity could in principle produce matching singular values while yielding different knot transitions or no transition. This is load-bearing for the central claim.
minor comments (2)
  1. Define or reference the precise notion of 'first order knot transition' used here and how the discrete jump in Re/Im parts is quantified (e.g., via a specific order parameter or distance measure between knot invariants).
  2. [Abstract] Clarify the term 'prototype translational invariant Hermitian models' in the abstract; either expand the definition or cite the specific class of models employed in the example.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this important point about the scope of our claims. We address the major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract / main construction section] Abstract and the section presenting the main construction: the statement that the NH-Hermitian connection 'remains robust' as long as lattice momentum periodicity is shared is supported only by a single explicit construction and numerical example. No general theorem or argument is supplied showing that knot topology is fixed solely by the singular-value spectrum once periodicity is fixed; other NH realizations sharing the same periodicity could in principle produce matching singular values while yielding different knot transitions or no transition. This is load-bearing for the central claim.

    Authors: We agree with the referee that the robustness statement is supported only by the explicit construction and numerical example in the paper, and that no general theorem is provided establishing that knot topology is determined solely by the singular-value spectrum for arbitrary NH realizations sharing the same periodicity. In the revised manuscript we will modify the abstract and the main construction section to remove the phrasing that the connection 'remains robust' and instead state that the correspondence is observed in the specific NH Hamiltonians we construct, where singular values are matched to the eigenvalues of the Hermitian model. We will add an explicit caveat noting that other NH realizations with the same periodicity could in principle produce different knot behavior, and that a general proof of uniqueness is beyond the scope of the present work. These changes will accurately reflect the evidential basis of the central claim. revision: yes

Circularity Check

0 steps flagged

No circularity: NH-Hermitian knot mapping rests on explicit example and physical interpretation

full rationale

The paper defines singular values of the NH Hamiltonian as real quantities that can be interpreted as eigenvalues of a separate Hermitian model. It then shows via one explicit construction that a topological phase transition in the Hermitian eigenvalues corresponds to a first-order knot transition in the NH complex eigenvalues, with the connection stated to remain robust under matching lattice-momentum periodicity. This is presented as a demonstration rather than a derivation that reduces the output to the input by construction. No self-definitional equations, fitted parameters renamed as predictions, load-bearing self-citations, or ansatz smuggling appear in the abstract or described chain. The non-uniqueness of the NH choice is explicitly acknowledged, and the result is offered as an observation supported by the example, keeping the derivation self-contained against external definitions of singular values, knot topology, and topological phase transitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard domain assumptions in non-Hermitian and topological physics; no free parameters, new entities, or ad-hoc axioms are introduced beyond the periodicity-matching condition.

axioms (2)
  • domain assumption Singular values of any non-Hermitian Hamiltonian are real and can be interpreted as eigenvalues of an underlying Hermitian Hamiltonian.
    Stated directly in the abstract as a definitional property.
  • domain assumption The periodicity in lattice momentum must be identical for the non-Hermitian and Hermitian models for the connection to remain robust.
    Explicitly emphasized as the condition under which the knot transition follows the Hermitian topological transition.

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Forward citations

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Reference graph

Works this paper leans on

96 extracted references · 96 canonical work pages · cited by 2 Pith papers · 1 internal anchor

  1. [1]

    to hopflink (|ν|= 2) knot transition as graphically shown in Fig. (4). 5 The NH systemA, withV=σ x for this model is represented by the matrix A=   − a′√ (3+ω−a′)√ 2(1+e2ikω+2 cosk) a′√ (3+ω+a′)√ 2(1+e2ikω+2 cosk) √3+ω−a′ √ 2 √3+ω+a′ √ 2   ,(10) where a′ = p 3 +ω 2 + 2 ((2 +ω) cosk+ (1 +ω) cos 2k+ωcos 3k). The knot structures forω <1 amdω >1 for thi...

    work page 2021

  2. [2]

    C. M. Bender and S. Boettcher, Real spectra in non- hermitian hamiltonians havingPTsymmetry, Phys. Rev. Lett.80, 5243 (1998)

    work page 1998

  3. [3]

    C. M. Bender, D. C. Brody, and H. F. Jones, Complex extension of quantum mechanics, Phys. Rev. Lett.89, 270401 (2002)

    work page 2002

  4. [4]

    C. M. Bender, Making sense of non-hermitian hamilto- nians, Reports on Progress in Physics70, 947 (2007), http://arxiv.org/pdf/hep-th/0703096

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  5. [5]

    A. Mostafazadeh, Pseudo-hermitian representation of quantum mechanics, International Journal of Geomet- ric Methods in Modern Physics07, 1191 (2010), https://doi.org/10.1142/S0219887810004816

    work page doi:10.1142/s0219887810004816 2010

  6. [6]

    Mostafazadeh, Pseudo-hermiticity and generalized pt- and cpt-symmetries, Journal of Mathematical Physics 44, 974 (2003)

    A. Mostafazadeh, Pseudo-hermiticity and generalized pt- and cpt-symmetries, Journal of Mathematical Physics 44, 974 (2003)

    work page 2003

  7. [7]

    Mostafazadeh, Krein-space formulation ofPTsym- metry,CPT-inner products, and pseudo-hermiticity, Czechoslovak Journal of Physics56, 10.1007/s10582-006- 0388-8 (2006)

    A. Mostafazadeh, Krein-space formulation ofPTsym- metry,CPT-inner products, and pseudo-hermiticity, Czechoslovak Journal of Physics56, 10.1007/s10582-006- 0388-8 (2006)

    work page doi:10.1007/s10582-006- 2006

  8. [8]

    C.-Y. Ju, A. Miranowicz, G.-Y. Chen, and F. Nori, Non- hermitian hamiltonians and no-go theorems in quantum information, Phys. Rev. A100, 062118 (2019)

    work page 2019

  9. [9]

    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 susceptibility, Phys. Rev. Res.3, 013015 (2021)

    work page 2021

  10. [10]

    Khare and B

    A. Khare and B. P. Mandal, A pt-invariant potential with complex qes eigenvalues, Physics Letters A272, 53 (2000)

    work page 2000

  11. [11]

    Mandal, B

    B. Mandal, B. Mourya, K. Ali, and A. Ghatak, Pt phase transition in a (2+1)-d relativistic system, Annals of Physics363, 185 (2015)

    work page 2015

  12. [12]

    Raval and B

    H. Raval and B. P. Mandal, Deconfinement to confine- ment as pt phase transition, Nuclear Physics B946, 114699 (2019)

    work page 2019

  13. [13]

    B. P. Mandal, B. K. Mourya, and R. K. Yadav, Pt phase transition in higher-dimensional quantum systems, Physics Letters A377, 1043 (2013)

    work page 2013

  14. [14]

    T. Pal, R. Modak, and B. P. Mandal, Parity–time- reversal symmetry-breaking transitions in polymeric sys- tems, Phys. Rev. E111, 014421 (2025)

    work page 2025

  15. [15]

    Peng, S ¸

    B. Peng, S ¸. K.¨Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Parity–time-symmetric whispering-gallery microcavities, Nature Physics10, 394 (2014)

    work page 2014

  16. [16]

    H. Jing, S. K. ¨Ozdemir, X.-Y. L¨ u, J. Zhang, L. Yang, and F. Nori,PT-symmetric phonon laser, Phys. Rev. Lett. 113, 053604 (2014)

    work page 2014

  17. [17]

    J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, Ob- servation of a topological transition in the bulk of a non- hermitian system, Phys. Rev. Lett.115, 040402 (2015)

    work page 2015

  18. [18]

    K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Beam dynamics inPTsymmetric op- tical lattices, Phys. Rev. Lett.100, 103904 (2008)

    work page 2008

  19. [19]

    Regensburger, C

    A. Regensburger, C. Bersch, M.-A. Miri, G. On- ishchukov, D. N. Christodoulides, and U. Peschel, Parity– time synthetic photonic lattices, Nature488, 167 (2012)

    work page 2012

  20. [20]

    L. Feng, R. El-Ganainy, and L. Ge, Non-hermitian pho- tonics based on parity–time symmetry, Nature Photonics 11, 752 (2017)

    work page 2017

  21. [21]

    L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, Single-mode laser by parity-time symmetry breaking, Sci- ence346, 972 (2014)

    work page 2014

  22. [22]

    Shen and L

    H. Shen and L. Fu, Quantum oscillation from in-gap states and a non-hermitian landau level problem, Phys. Rev. Lett.121, 026403 (2018)

    work page 2018

  23. [23]

    Yoshida, R

    T. Yoshida, R. Peters, and N. Kawakami, Non-hermitian perspective of the band structure in heavy-fermion sys- tems, Phys. Rev. B98, 035141 (2018)

    work page 2018

  24. [24]

    Papaj, H

    M. Papaj, H. Isobe, and L. Fu, Nodal arc of disordered dirac fermions and non-hermitian band theory, Phys. Rev. B99, 201107 (2019)

    work page 2019

  25. [25]

    Aravinda, S

    S. Aravinda, S. Banerjee, and R. Modak, Ergodic and mixing quantum channels: From two-qubit to many- body quantum systems, Phys. Rev. A110, 042607 (2024)

    work page 2024

  26. [26]

    H. J. Carmichael, Quantum trajectory theory for cas- caded open systems, Phys. Rev. Lett.70, 2273 (1993)

    work page 1993

  27. [27]

    B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljaˇ ci´ c, Spawning rings of exceptional points out of dirac cones, 7 Nature525, 354 (2015)

    work page 2015

  28. [28]

    Diehl, E

    S. Diehl, E. Rico, M. A. Baranov, and P. Zoller, Topology by dissipation in atomic quantum wires, Nature physics 7, 971 (2011)

    work page 2011

  29. [29]

    Hajong, R

    G. Hajong, R. Modak, and B. P. Mandal, Hellmann- feynman theorem in non-hermitian systems, Phys. Rev. A109, 022227 (2024)

    work page 2024

  30. [30]

    Modak and B

    R. Modak and B. P. Mandal, Eigenstate entanglement entropy in aPT-invariant non-hermitian system, Phys. Rev. A103, 062416 (2021)

    work page 2021

  31. [31]

    Shukla, R

    N. Shukla, R. Modak, and B. P. Mandal, Uncertainty relation for non-hermitian systems, Phys. Rev. A107, 042201 (2023)

    work page 2023

  32. [32]

    Modak and S

    R. Modak and S. Aravinda, Non-hermitian descrip- tion of sharp quantum resetting, arXiv preprint arXiv:2303.03790 (2023)

    work page arXiv 2023

  33. [33]

    Basu-Mallick and B

    B. Basu-Mallick and B. P. Mandal, On an exactly solv- able bn type calogero model with non-hermitian pt in- variant interaction, Physics Letters A284, 231 (2001)

    work page 2001

  34. [34]

    Hasan and B

    M. Hasan and B. P. Mandal, New scattering features in non-hermitian space fractional quantum mechanics, An- nals of Physics396, 371–385 (2018)

    work page 2018

  35. [35]

    Hasan, V

    M. Hasan, V. N. Singh, and B. P. Mandal, Role ofpt- symmetry in understanding hartman effect, Eur. Phys. J. Plus135, 640 (2020)

    work page 2020

  36. [36]

    Ghatak, M

    A. Ghatak, M. Hasan, and B. P. Mandal, A black po- tential for spin less particles, Physics Letters A379, 1326–1336 (2015)

    work page 2015

  37. [37]

    Miri and A

    M.-A. Miri and A. Alu, Exceptional points in optics and photonics, Science363, eaar7709 (2019)

    work page 2019

  38. [38]

    W. D. Heiss, The physics of exceptional points, Journal of Physics A: Mathematical and Theoretical45, 444016 (2012)

    work page 2012

  39. [39]

    Okuma, K

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

    work page 2020

  40. [40]

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

    work page 2020

  41. [41]

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

    work page 2021

  42. [42]

    Yokomizo and S

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

    work page 2019

  43. [43]

    C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with sym- metries, Rev. Mod. Phys.88, 035005 (2016)

    work page 2016

  44. [44]

    Y. Cao, Y. Li, and X. Yang, Non-hermitian bulk- boundary correspondence in a periodically driven system, Phys. Rev. B103, 075126 (2021)

    work page 2021

  45. [45]

    L. Xiao, T. Deng, K. Wang, W. Yi, and P. Xue, Non-hermitian bulk–boundary correspondence in quan- tum dynamics, Nature Physics 10.1038/s41567-020-0836- 6 (2020)

    work page doi:10.1038/s41567-020-0836- 2020

  46. [46]

    Brunelli, C

    M. Brunelli, C. C. Wanjura, and A. Nunnenkamp, Restoration of the non-Hermitian bulk-boundary corre- spondence via topological amplification, SciPost Phys. 15, 173 (2023)

    work page 2023

  47. [47]

    Zhang, Z

    K. Zhang, Z. Yang, and C. Fang, Correspondence be- tween winding numbers and skin modes in non-hermitian systems, Phys. Rev. Lett.125, 126402 (2020)

    work page 2020

  48. [48]

    Manna and B

    S. Manna and B. Roy, Inner skin effects on non- hermitian topological fractals, Communications Physics 6, 10.1038/s42005-023-01130-2 (2023)

    work page doi:10.1038/s42005-023-01130-2 2023

  49. [49]

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

    work page 2018

  50. [50]

    Zhou and J

    H. Zhou and J. Y. Lee, Periodic table for topological bands with non-hermitian symmetries, Phys. Rev. B99, 235112 (2019)

    work page 2019

  51. [51]

    Kawabata, K

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

    work page 2019

  52. [52]

    C. C. Wojcik, X.-Q. Sun, T. c. v. Bzduˇ sek, and S. Fan, Homotopy characterization of non-hermitian hamiltoni- ans, Phys. Rev. B101, 205417 (2020)

    work page 2020

  53. [53]

    Hu and E

    H. Hu and E. Zhao, Knots and non-hermitian bloch bands, Phys. Rev. Lett.126, 010401 (2021)

    work page 2021

  54. [54]

    V. M. Vyas and D. Roy, Topological aspects of peri- odically driven non-hermitian su-schrieffer-heeger model, Phys. Rev. B103, 075441 (2021)

    work page 2021

  55. [55]

    Nehra and D

    R. Nehra and D. Roy, Topology of multipartite non- hermitian one-dimensional systems, Phys. Rev. B105, 195407 (2022)

    work page 2022

  56. [56]

    Nehra and D

    R. Nehra and D. Roy, Anomalous dynamical response of non-hermitian topological phases, Phys. Rev. B109, 094311 (2024)

    work page 2024

  57. [57]

    C. C. Wanjura, M. Brunelli, and A. Nunnenkamp, Cor- respondence between non-hermitian topology and direc- tional amplification in the presence of disorder, Phys. Rev. Lett.127, 213601 (2021)

    work page 2021

  58. [58]

    C. C. Wanjura, M. Brunelli, and A. Nunnenkamp, Topo- logical framework for directional amplification in driven- dissipative cavity arrays, Nature Communications11, 10.1038/s41467-020-16863-9 (2020)

    work page doi:10.1038/s41467-020-16863-9 2020

  59. [59]

    S. Tong, Q. Zhang, L. Qi, G. Li, X. Feng, and C. Qiu, Observation of floquet-bloch braids in non-hermitian spatiotemporal lattices, Phys. Rev. Lett.134, 126603 (2025)

    work page 2025

  60. [60]

    Ryu, J.-H

    J.-W. Ryu, J.-H. Han, C.-H. Yi, M. J. Park, and H. C. Park, Exceptional classifications of non-hermitian systems, Communications Physics 10.1038/s42005-024- 01595-9 (2024)

    work page doi:10.1038/s42005-024- 2024

  61. [61]

    K. Ding, C. Fang, and G. Ma, Non-hermitian topol- ogy and exceptional-point geometries, Nature Reviews Physics 10.1038/s42254-022-00516-5 (2022)

    work page doi:10.1038/s42254-022-00516-5 2022

  62. [62]

    Zhang, G

    X. Zhang, G. Li, Y. Liu, T. Tai, R. Thomale, and C. H. Lee, Tidal surface states as fingerprints of non- hermitian nodal knot metals, Communications Physics 4, 10.1038/s42005-021-00535-1 (2021)

    work page doi:10.1038/s42005-021-00535-1 2021

  63. [63]

    Zhang, Y

    Q. Zhang, Y. Li, H. Sun, X. Liu, L. Zhao, X. Feng, X. Fan, and C. Qiu, Observation of acoustic non- hermitian bloch braids and associated topological phase transitions, Phys. Rev. Lett.130, 017201 (2023)

    work page 2023

  64. [64]

    Yang, C.-K

    Z. Yang, C.-K. Chiu, C. Fang, and J. Hu, Jones polyno- mial and knot transitions in hermitian and non-hermitian topological semimetals, Phys. Rev. Lett.124, 186402 (2020)

    work page 2020

  65. [65]

    Li and R

    Z. Li and R. S. K. Mong, Homotopical characterization of non-hermitian band structures, Phys. Rev. B103, 155129 (2021)

    work page 2021

  66. [66]

    M.-M. Cao, K. Li, W.-D. Zhao, W.-X. Guo, B.-X. Qi, X.- Y. Chang, Z.-C. Zhou, Y. Xu, and L.-M. Duan, Probing complex-energy topology via non-hermitian absorption spectroscopy in a trapped ion simulator, Phys. Rev. Lett. 130, 163001 (2023)

    work page 2023

  67. [67]

    K. Wang, A. Dutt, C. C. Wojcik, and S. Fan, Topological 8 complex-energy braiding of non-hermitian bands, Nature 598, 59 (2021)

    work page 2021

  68. [68]

    Y. S. Patil, J. H¨ oller, P. A. Henry, C. Guria, Y. Zhang, L. Jiang, N. Kralj, N. Read, and J. G. Harris, Mea- suring the knot of non-hermitian degeneracies and non- commuting braids, Nature607, 271 (2022)

    work page 2022

  69. [69]

    Grasedyck, Hierarchical singular value decomposition of tensors, SIAM J

    L. Grasedyck, Hierarchical singular value decomposition of tensors, SIAM J. Matrix Analysis Applications31, 2029 (2010)

    work page 2029

  70. [70]

    Francuz, N

    A. Francuz, N. Schuch, and B. Vanhecke, Stable and effi- cient differentiation of tensor network algorithms, Phys. Rev. Res.7, 013237 (2025)

    work page 2025

  71. [71]

    H. He, Y. Zheng, B. A. Bernevig, and N. Regnault, En- tanglement entropy from tensor network states for stabi- lizer codes, Phys. Rev. B97, 125102 (2018)

    work page 2018

  72. [72]

    Morita, R

    S. Morita, R. Igarashi, H.-H. Zhao, and N. Kawashima, Tensor renormalization group with randomized singular value decomposition, Phys. Rev. E97, 033310 (2018)

    work page 2018

  73. [73]

    J. T. Campbell, Bridging relations between svd in ten- sor networks and common matrix operations in quantum information theory (2024), arXiv:2402.02517 [math.QA]

    work page arXiv 2024

  74. [74]

    T.-P. .-.-.-. Cheng and L.-F. .-.-.-. Li,Gauge Theory of Elementary Particle Physics(Oxford University Press, Oxford, UK, 1984)

    work page 1984

  75. [75]

    B. Zhu, R. Wirth, and H. Hergert, Singular value decom- position and similarity renormalization group evolution of nuclear interactions, Phys. Rev. C104, 044002 (2021)

    work page 2021

  76. [76]

    Datta,Electronic transport in mesoscopic systems (Cambridge university press, 1997)

    S. Datta,Electronic transport in mesoscopic systems (Cambridge university press, 1997)

    work page 1997

  77. [77]

    Prasad, S

    M. Prasad, S. H. Tekur, B. K. Agarwalla, and M. Kulka- rni, Assessment of spectral phases of non-hermitian quan- tum systems through complex and singular values, Phys. Rev. B111, L161408 (2025)

    work page 2025

  78. [78]

    Baggioli, K.-B

    M. Baggioli, K.-B. Huh, H.-S. Jeong, X. Jiang, K.-Y. Kim, and J. F. Pedraza, Singular value decomposition and its blind spot for quantum chaos in non-hermitian sachdev-ye-kitaev models (2025), arXiv:2503.11274 [hep- th]

    work page arXiv 2025

  79. [79]

    Nandy, T

    P. Nandy, T. Pathak, and M. Tezuka, Probing quantum chaos through singular-value correlations in the sparse non-hermitian sachdev-ye-kitaev model, Phys. Rev. B 111, L060201 (2025)

    work page 2025

  80. [80]

    Nandy, T

    P. Nandy, T. Pathak, Z.-Y. Xian, and J. Erdmenger, Krylov space approach to singular value decomposition in non-hermitian systems, Phys. Rev. B111, 064203 (2025)

    work page 2025

Showing first 80 references.