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arxiv: 2606.06626 · v1 · pith:XONS54UTnew · submitted 2026-06-04 · ❄️ cond-mat.mes-hall · quant-ph

Non-Hermitian Crystalline Braid Topology from Hermitian Projection: A Zero-Mode Resonance Mechanism

Stefan {\DJ}or{\dj}evi\'c , Vladimir Juri\v{c}i\'c This is my paper

Pith reviewed 2026-06-27 23:38 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall quant-ph
keywords non-Hermitian topologybraid topologyHermitian projectionzero-mode resonancepseudo-HermiticityBerry phasetopolectrical circuitscrystalline topology
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The pith

Non-Hermitian crystalline braid topology emerges from Hermitian projection via zero-mode resonance

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

The paper demonstrates that non-Hermitian braid topology can be obtained by projecting a Hermitian trivial lattice when the complement hosts a resonant zero mode. The resonance creates a pole in the self-energy, so topology resides in the finite-frequency Green's function rather than at zero frequency. In the square lattice with zig-zag brane, this produces an abelian two-band braid in the complex spectrum, with transitions at isolated finite frequencies. Conjugated pseudo-Hermiticity induced by embedding parity quantizes the complex Berry phase to the braid count. The construction is free of the skin effect, allowing the invariant to be a true bulk Bloch quantity.

Core claim

The central discovery is that zero-mode-resonant projection of a trivial Hermitian square lattice with an embedded zig-zag brane yields a non-Hermitian effective model whose spectrum forms an abelian two-band braid. Transitions occur only at isolated finite frequencies. Embedding parity induces conjugated pseudo-Hermiticity that quantizes the complex Berry phase and equates it to the braid count, even though the internal class AI† is trivial. The absence of the non-Hermitian skin effect ensures the invariant is a genuine Bloch bulk quantity.

What carries the argument

The zero-mode resonance mechanism, in which a sublattice-imbalance zero mode generates a singular self-energy pole that carries topology in the finite-frequency projected Green's function

If this is right

  • Braid transitions appear as transmission zeros at predicted drive frequencies in topolectrical realizations.
  • The complex Berry phase is quantized and equals the braid count due to conjugated pseudo-Hermiticity.
  • The projected system exhibits non-Hermitian topology without gain, loss or asymmetric couplings.
  • The topological invariant remains a Bloch bulk quantity free of skin effect localization.

Where Pith is reading between the lines

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

  • This zero-mode resonance could be used to induce braid topology in other projected Hermitian systems by engineering appropriate defects.
  • Frequency as a tunable parameter opens the possibility of switching between different braid topologies by changing the drive frequency.
  • The mechanism may generalize to higher-dimensional or other symmetry classes where projection creates effective non-Hermiticity.

Load-bearing premise

The embedding parity induces conjugated pseudo-Hermiticity that quantizes the complex Berry phase and identifies it with the braid count, and that the model has no non-Hermitian skin effect.

What would settle it

If the complex Berry phase measured in the model is not an integer multiple matching the braid transitions, or if boundary skin modes appear, the central claim would be falsified.

Figures

Figures reproduced from arXiv: 2606.06626 by Stefan {\DJ}or{\dj}evi\'c, Vladimir Juri\v{c}i\'c.

Figure 1
Figure 1. Figure 1: Two-dimensional parent lattice with an embed [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Phase diagram of the PT + symmetry for the open-boundary projected Hamiltonian. The value plot￾ted on the y-axis is given by ωef f = ω 2ty sinh ϕcr . If both N1 and N2 are even, only the PT +-unbroken phase is realized. When at least one of them is odd, PT +-broken regions appear. The critical lines are determined by the condition |FN1 − FN2 | = 2. but this SSH structure is a low-frequency spectral di￾agno… view at source ↗
Figure 4
Figure 4. Figure 4: Energy bands and lifetimes for periodic bound [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Two-band braiding of the periodic odd-N spectrum. The horizontal plane is the complex-energy plane and the vertical direction is the crystal momentum. The three panels follow the red trajectory in [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Phase diagram for periodic boundary condi [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Winding structure of the phases entering the odd- [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: This figure shows the change in the complex energy spectrum of the Hamiltonian with periodic boundary [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: This figure illustrates how the complex energy spectrum of the Hamiltonian with periodic boundary [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 5
Figure 5. Figure 5: Tracking the bands around the Brillouin zone, [PITH_FULL_IMAGE:figures/full_fig_p028_5.png] view at source ↗
read the original abstract

Non-Hermitian topological phases are usually engineered through gain, loss, asymmetric couplings, or explicit environmental channels. Here we show that non-Hermitian crystalline braid topology can instead emerge from projection alone, starting from a fully Hermitian and topologically trivial parent lattice. The mechanism is zero-mode-resonant projection. When the eliminated complement is zero-mode free, projection has a smooth low-frequency limit and reduces to a static Schur complement, yielding conventional SSH-type descendants. When a complement zero mode couples to the retained subsystem, the embedding self-energy develops a pole, the zero-frequency limit becomes singular, and topology is carried by the finite-frequency projected Green's function-where frequency is a tunable parameter, the drive frequency in a circuit realization, for instance. We demonstrate this mechanism in an exactly solvable model, a trivial nearest-neighbor square lattice with an embedded one-dimensional zig-zag brane. Odd-parity periodic sectors are resonant: a sublattice-imbalance zero mode generates the singular self-energy, and the complex spectrum forms an abelian two-band braid whose transitions occur only at isolated finite frequencies. Although the internal class is $\text{AI}^{\dag}$ featuring only trivial phases, embedding parity induces conjugated pseudo-Hermiticity (CPH), quantizes the complex Berry phase, and identifies it with the braid count. The model is free of the non-Hermitian skin effect, making the invariant a genuine Bloch bulk quantity. In topolectrical realizations, the same finite-frequency braid transitions appear as transmission zeros and admittance features at the predicted drive frequencies.

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

2 major / 1 minor

Summary. The paper claims that non-Hermitian crystalline braid topology can emerge purely from Hermitian projection via a zero-mode resonance mechanism, without gain/loss or asymmetric couplings. Starting from a trivial Hermitian square lattice, projection onto an embedded zig-zag brane with a sublattice-imbalance zero mode produces a singular self-energy; the finite-frequency projected Green's function then yields an abelian two-band braid in the complex spectrum, with transitions only at isolated frequencies. Embedding parity induces conjugated pseudo-Hermiticity (CPH) that quantizes the complex Berry phase and equates it to the braid count, while the internal AI† class is trivial. The model is asserted to lack the non-Hermitian skin effect, rendering the invariant a valid Bloch quantity, with implications for topolectrical circuit realizations.

Significance. If the central claims hold, the work supplies a new, parameter-free route to non-Hermitian topology from Hermitian parents, grounded in an exactly solvable model and a concrete projection construction. The identification of braid count with a quantized complex Berry phase under CPH, together with the absence of skin effect, would distinguish this from conventional NH engineering and enable frequency-tunable transitions observable as transmission zeros. The reproducible, exactly solvable character and potential circuit mapping are clear strengths.

major comments (2)
  1. [zig-zag brane construction and CPH discussion] The assertion that embedding parity induces conjugated pseudo-Hermiticity (CPH) which quantizes the complex Berry phase and identifies it with the braid count (abstract and zig-zag brane construction) is load-bearing for the topology claim. An explicit computation of the complex Berry phase on the projected Green's function, demonstrating that it is real and integer-valued, is required; the parity argument alone does not yet establish the quantization step independently of the braid count.
  2. [model and NHSE statement] The claim that the model is free of the non-Hermitian skin effect, so that the invariant remains a genuine Bloch bulk quantity (abstract), rests on the frequency-dependent effective operator with a self-energy pole. Standard NHSE diagnostics (point-gap topology, biorthogonal localization lengths, or open-boundary spectra) must be applied explicitly to the projected non-Hermitian Hamiltonian at the relevant finite frequencies; without this verification the Bloch invariant status is not yet secured.
minor comments (1)
  1. [projection mechanism] Clarify the precise definition of the frequency-dependent projected Green's function and its relation to the effective non-Hermitian Hamiltonian in the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and will incorporate the requested explicit verifications in the revised version.

read point-by-point responses

  1. Referee: The assertion that embedding parity induces conjugated pseudo-Hermiticity (CPH) which quantizes the complex Berry phase and identifies it with the braid count (abstract and zig-zag brane construction) is load-bearing for the topology claim. An explicit computation of the complex Berry phase on the projected Green's function, demonstrating that it is real and integer-valued, is required; the parity argument alone does not yet establish the quantization step independently of the braid count.

    Authors: We agree that an explicit computation provides independent confirmation. In the revised manuscript we will add a direct evaluation of the complex Berry phase on the projected Green's function at representative finite frequencies within the zig-zag brane construction section. This calculation will demonstrate that the phase is real and integer-valued, matching the braid count and thereby establishing the quantization under CPH separately from the topological classification of the braid itself. revision: yes

  2. Referee: The claim that the model is free of the non-Hermitian skin effect, so that the invariant remains a genuine Bloch bulk quantity (abstract), rests on the frequency-dependent effective operator with a self-energy pole. Standard NHSE diagnostics (point-gap topology, biorthogonal localization lengths, or open-boundary spectra) must be applied explicitly to the projected non-Hermitian Hamiltonian at the relevant finite frequencies; without this verification the Bloch invariant status is not yet secured.

    Authors: We acknowledge the need for explicit verification. In the revision we will apply the suggested diagnostics to the projected non-Hermitian operator at the finite frequencies where the braid spectrum is defined. Specifically, we will present open-boundary spectra and biorthogonal localization lengths, confirming the absence of skin effect and thereby securing the Bloch character of the complex Berry phase invariant. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from external Hermitian parent via explicit projection

full rationale

The paper constructs the non-Hermitian effective model by projecting a trivial Hermitian square lattice onto a zig-zag brane subsystem, with the singular self-energy arising from an explicit zero mode in the complement. The abelian braid in the complex spectrum of the finite-frequency Green's function is exhibited directly from the poles and residues of that self-energy. Embedding parity is an external lattice symmetry that induces CPH; the quantization of the complex Berry phase and its identification with braid count follow from the resulting symmetry constraints on the projected operator, not from any redefinition or fit of the braid count itself. No load-bearing self-citation, parameter fitting, or ansatz smuggling is present in the derivation chain. The assertion of absent NHSE is a separate claim about the model's spectrum and does not collapse the topological identification to an input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on standard domain assumptions of topological band theory and projection; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption The parent square lattice is Hermitian and topologically trivial.
    Explicitly stated as the starting point for projection.
  • domain assumption Embedding parity induces conjugated pseudo-Hermiticity that quantizes the complex Berry phase.
    Central to equating Berry phase with braid count.

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Reference graph

Works this paper leans on

64 extracted references · 9 canonical work pages

  1. [1]

    The remaining condition is therefore|λ 0|<1

    By construction, this is automatically satisfied forλr, since there are always two solutions, one greater than 1 and the other less than 1, and we simply select the one with magnitude below 1. The remaining condition is therefore|λ 0|<1. This holds whenδ t >0and|η|<1 orδ t <0and|η|>1. Because the topological invariant predicts edge states only forδt >0, v...

  2. [2]

    El-Ganainy, K

    R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non- hermitian physics andPTsymmetry, Nature Physics14, 11 (2018)

    2018

  3. [3]

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

    2021

  4. [5]

    Rotter, A non-hermitian hamilton operator and the physics of open quantum systems, J

    I. Rotter, A non-hermitian hamilton operator and the physics of open quantum systems, J. Phys. A: Math. Theor.42, 153001 (2009)

    2009

  5. [6]

    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)

    2018

  6. [8]

    Kawabata, K

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

    2019

  7. [10]

    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

  8. [12]

    Wang, Classifying order-two spatial symmetries in non-hermitian hamiltonians: Point-gapped az and az † classes (2024), arXiv:2411.03410 [cond-mat.mes-hall]

    Y. Wang, Classifying order-two spatial symmetries in non-hermitian hamiltonians: Point-gapped az and az † classes (2024), arXiv:2411.03410 [cond-mat.mes-hall]

    work page arXiv 2024

  9. [13]

    Fu, Topological crystalline insulators, Phys

    L. Fu, Topological crystalline insulators, Phys. Rev. Lett. 106, 106802 (2011)

    2011

  10. [14]

    Slager, A

    R.-J. Slager, A. Mesaros, V. Juričić, and J. Zaanen, The space group classification of topological band-insulators, Nature Physics9, 98 (2013)

    2013

  11. [16]

    Shiozaki, M

    K. Shiozaki, M. Sato, and K. Gomi, Topology of non- symmorphic crystalline insulators and superconductors, Phys. Rev. B93, 195413 (2016)

    2016

  12. [17]

    Kruthoff, J

    J. Kruthoff, J. de Boer, J. van Wezel, C. L. Kane, and R.-J. Slager, Topological classification of crystalline in- sulators through band structure combinatorics, Physical Review X7, 041069 (2017)

    2017

  13. [18]

    Bradlyn, L

    B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature547, 298 (2017)

    2017

  14. [19]

    Carlström and E

    J. Carlström and E. J. Bergholtz, Exceptional links and twisted fermi ribbons in non-hermitian systems, Physical Review A98, 042114 (2018)

    2018

  15. [20]

    Yang and J

    Z. Yang and J. Hu, Non-hermitian hopf-link exceptional line semimetals, Physical Review B99, 081102 (2019)

    2019

  16. [21]

    C. C. Wojcik, K. Wang, A. Dutt, J. Zhong, and S. Fan, Eigenvalue topology of non-hermitian band structures in two and three dimensions, Physical Review B106, L161401 (2022)

    2022

  17. [22]

    Li and R

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

    2021

  18. [23]

    M. P. López Sancho, J. M. López Sancho, J. M. L. López Sancho, and J. Rubio, Highly convergent schemes for the calculation of bulk and surface green functions, J. Phys. F: Met. Phys.15, 851 (1985)

    1985

  19. [24]

    E. N. Economou,Green’s Functions in Quantum Physics, 3rd ed. (Springer, Berlin, 2006)

    2006

  20. [25]

    Löwdin, A note on the quantum-mechanical per- turbation theory, J

    P.-O. Löwdin, A note on the quantum-mechanical per- turbation theory, J. Chem. Phys.19, 1396 (1951)

    1951

  21. [26]

    Feshbach, Unified theory of nuclear reactions, Annals of Physics5, 357 (1958)

    H. Feshbach, Unified theory of nuclear reactions, Annals of Physics5, 357 (1958)

    1958

  22. [27]

    Feshbach, A unified theory of nuclear reactions

    H. Feshbach, A unified theory of nuclear reactions. ii, Annals of Physics19, 287 (1962)

    1962

  23. [28]

    T.I.Tuegel, V.Chua,andT.L.Hughes,Embeddedtopo- logical insulators, Physical Review B100, 115126 (2019)

    2019

  24. [29]

    Velury and T

    S. Velury and T. L. Hughes, Embedded topological semimetals, Physical Review B105, 184105 (2022)

    2022

  25. [30]

    An Ising machine based on networks of subharmonic electrical resonators

    A. Panigrahi, V. Juričić, and B. Roy, Projected topolog- ical branes, Communications Physics5, 10.1038/s42005- 022-01006-x (2022)

    work page doi:10.1038/s42005- 2022

  26. [31]

    Panigrahi and B

    A. Panigrahi and B. Roy, Projected branes as plat- forms for crystalline, superconducting, and higher-order topological phases, Physical Review B 10.1103/vlk8-59x3 (2026)

    work page doi:10.1103/vlk8-59x3 2026

  27. [32]

    A. C. Tyner and V. Juričić, Three-dimensional topologi- cal insulators without reflection symmetry, Scientific Re- ports14, 10.1038/s41598-024-54821-3 (2024)

    work page doi:10.1038/s41598-024-54821-3 2024

  28. [33]

    A.C.Tyner, V.Juričić,andB.Roy,Two-dimensionaldis- ordered projected branes: stability and quantum critical- ity via dimensional reduction (2025), arXiv:2507.23780 [cond-mat.mes-hall]

    work page arXiv 2025

  29. [34]

    Gurarie, Single-particle green’s functions and inter- acting topological insulators, Phys

    V. Gurarie, Single-particle green’s functions and inter- acting topological insulators, Phys. Rev. B83, 085426 (2011)

    2011

  30. [35]

    Wang, X.-L

    Z. Wang, X.-L. Qi, and S.-C. Zhang, Topological order parameters for interacting topological insulators, Phys. Rev. Lett.105, 256803 (2010)

    2010

  31. [36]

    W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in polyacetylene, Phys. Rev. Lett.42, 1698 (1979)

    1979

  32. [37]

    Lieu, Topological phases in the non-hermitian su-schrieffer-heeger model, Physical Review B97, 10.1103/physrevb.97.045106 (2018)

    S. Lieu, Topological phases in the non-hermitian su-schrieffer-heeger model, Physical Review B97, 10.1103/physrevb.97.045106 (2018)

    work page doi:10.1103/physrevb.97.045106 2018

  33. [38]

    Sutherland, Localization of electronic wave functions due to local topology, Phys

    B. Sutherland, Localization of electronic wave functions due to local topology, Phys. Rev. B34, 5208 (1986)

    1986

  34. [39]

    M.Inui, S.A.Trugman,andE.Abrahams,Unusualprop- erties of midband states in systems with off-diagonal dis- order, Phys. Rev. B49, 3190 (1994)

    1994

  35. [40]

    Dörfler and F

    F. Dörfler and F. Bullo, Kron reduction of graphs with applications to electrical networks, IEEE Transactions on Circuits and Systems I: Regular Papers60, 150 (2013)

    2013

  36. [41]

    C.-K. Chiu, H. Yao, and S. Ryu, Classification of topo- logical insulators and superconductors in the presence of reflection symmetry, Phys. Rev. B88, 075142 (2013)

    2013

  37. [42]

    H. T. Teo, Y. Long, H.-y. Zou, K. Song, H. Xue, Y. Ge, S.-q. Yuan, H.-x. Sun, and B. Zhang, Observation of em- bedded topology in a trivial bulk via projective crystal symmetry, Physical Review Letters135, 136602 (2025). 49

    2025

  38. [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)

    2016

  39. [44]

    J. K. Asbóth, L. Oroszlány, and A. Pályi,A Short Course on Topological Insulators: Band-Structure Topology and Edge States in One and Two Dimensions, Lecture Notes in Physics, Vol. 919 (Springer, Cham, 2016)

    2016

  40. [45]

    Shiozaki and M

    K. Shiozaki and M. Sato, Topology of crystalline insu- lators and superconductors, Phys. Rev. B90, 165114 (2014)

    2014

  41. [46]

    V. M. Martinez Alvarez, J. E. Barrios Vargas, and L. E. F. Foa Torres, Non-hermitian robust edge states in one dimension: Anomalous localization and eigenspace condensation at exceptional points, Phys. Rev. B97, 121401(R) (2018)

    2018

  42. [49]

    Yokomizo and S

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

    2019

  43. [50]

    S. Li, M. Liu, Y. Zhang, R. Tian, M. Arzamasovs, and B. Liu, Anomalous symmetry-protected blockade of the skin effect in one-dimensional non-hermitian lattice sys- tems, Phys. Rev. A110, 042208 (2024)

    2024

  44. [51]

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

    2015

  45. [52]

    C. H. Lee, S. Imhof, C. Berger, F. Bayer, J. Brehm, L. W. Molenkamp, T. Kiessling, and R. Thomale, Topolectrical circuits, Communications Physics1, 39 (2018)

    2018

  46. [53]

    Imhof, C

    S. Imhof, C. Berger, F. Bayer, J. Brehm, L. W. Molenkamp, T. Kiessling, F. Schindler, C. H. Lee, M. Greiter, T. Neupert, and R. Thomale, Topolectrical- circuit realization of topological corner modes, Nature Physics14, 925 (2018)

    2018

  47. [54]

    L. O. Chua, C. A. Desoer, and E. S. Kuh,Linear and Nonlinear Circuits(McGraw-Hill, New York, 1987)

    1987

  48. [55]

    S. Y. Caliskan and P. Tabuada, Towards kron reduction of generalized electrical networks, Automatica50, 2586 (2014)

    2014

  49. [56]

    H. T. Teo, Y. Long, H.-y. Zou, K. Song, H. Xue, Y. Ge, S.-q. Yuan, H.-x. Sun, and B. Zhang, Observation of em- bedded topology in a trivial bulk via projective crystal symmetry, Phys. Rev. Lett.135, 136602 (2025)

    2025

  50. [57]

    M. Xiao, Z. Q. Zhang, and C. T. Chan, Surface impedance and bulk band geometric phases in one- dimensional systems, Physical Review X4, 021017 (2014)

    2014

  51. [58]

    Q. Wang, M. Xiao, H. Liu, S. Zhu, and C. T. Chan, Mea- surement of the zak phase of photonic bands through the interface states of metasurface/photonic crystal, Physical Review B93, 041415(R) (2016)

    2016

  52. [59]

    Liang and G.-Y

    S.-D. Liang and G.-Y. Huang, Topological invariance and global berry phase in non-hermitian systems, Physical Review A87, 10.1103/physreva.87.012118 (2013)

    work page doi:10.1103/physreva.87.012118 2013

  53. [60]

    Singhal, E

    Y. Singhal, E. Martello, S. Agrawal, T. Ozawa, H. Price, and B. Gadway, Measuring the adiabatic non-hermitian berry phase in feedback-coupled oscillators, Phys. Rev. Res.5, L032026 (2023)

    2023

  54. [61]

    Yao and Z

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

    2018

  55. [62]

    S. Yao, F. Song, and Z. Wang, Non-hermitian chern bands, Phys. Rev. Lett.121, 136802 (2018)

    2018

  56. [63]

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

    2018

  57. [64]

    Resta, Macroscopic polarization in crystalline di- electrics: the geometric phase approach, Rev

    R. Resta, Macroscopic polarization in crystalline di- electrics: the geometric phase approach, Rev. Mod. Phys. 66, 899 (1994)

    1994

  58. [65]

    R. D. King-Smith and D. Vanderbilt, Theory of polariza- tion of crystalline solids, Phys. Rev. B47, 1651 (1993)

    1993

  59. [66]

    Resta, Theory of the electric polariza- tion in crystals, Ferroelectrics136, 51 (1992), https://doi.org/10.1080/00150199208016065

    R. Resta, Theory of the electric polariza- tion in crystals, Ferroelectrics136, 51 (1992), https://doi.org/10.1080/00150199208016065

    work page doi:10.1080/00150199208016065 1992

  60. [67]

    J. Hu, C. A. Perroni, G. De Filippis, S. Zhuang, L. Mar- rucci, and F. Cardano, Electric polarization and its quan- tization in one-dimensional non-hermitian chains, Physi- cal Review B107, 10.1103/physrevb.107.l121101 (2023)

    work page doi:10.1103/physrevb.107.l121101 2023

  61. [68]

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

    1998

  62. [69]

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

    2002

  63. [70]

    Non-Hermitian Crystalline Topology from Hermitian Projection: A Zero-Mode Resonance Mechanism

    C. H.Lee, S. Imhof, C.Berber, F.Bayer, J.Brehm, L. W. Molenkamp, T. Kiessling, and R. Thomale, Topolectrical circuits, Communications Physics1, 39 (2018). 50 Supplemental Material for “Non-Hermitian Crystalline Topology from Hermitian Projection: A Zero-Mode Resonance Mechanism” Stefan Ðorđević Faculty of Physics, University of Belgrade, Studentski Trg 12...

    2018

  64. [71]

    (S.44) Notice that there are four solutions

    + γ0 γ2 # . (S.44) Notice that there are four solutions. We should choose the sign± ′ such that bothλ± have a positive real part. This leaves only one set ofλ± as expected. The eigen- 55 values are now given by: ℏ τ ≡ −ε=F 0|tx −t y| 1−α γ0 γ2 .(S.45) SinceG 0 is imaginary whenNis odd, this case must be treated separately. We begin by considering the situ...