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Perspective on topological states of non-Hermitian lattices

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Pith's one-line read The paper argues that a non-Hermitian lattice in dimension $d$ can be read as the projection of a Hermitian lattice in dimension $d+1$ plus a boundary condition, making defectiveness a boundary-condition phenomenon.

desk verdict A candid, well-organized Perspective whose main interpretive suggestion is honestly labeled as a conjecture; worth a referee for a review venue, but don't expect a new result. read the letter →

arxiv 1909.00809 v1 pith:MYHA74OE submitted 2019-09-02 cond-mat.mes-hall quant-ph

Luis E. F. Foa Torres This is my paper · ORCID
classification cond-mat.mes-hallquant-ph
keywords non-Hermitianlatticestopologicalstatesdefectivenessexceptionalpointsskineffectbulk-boundarycorrespondencescatteringpictureFloquetsystems
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This Perspective proposes an interpretive key for topological states of non-Hermitian lattices: a non-Hermitian lattice in dimension $d$ can be understood as an effective description of a Hermitian lattice in dimension $d+1$ plus a boundary condition, obtained by projecting the parent's scattering states onto a sub-Hilbert space. On this view the two features that make non-Hermitian topological physics strange—defectiveness (missing eigenvectors) and extreme sensitivity to boundary conditions—are two sides of the same coin: defectiveness is the lack of parent eigenstates compatible with the chosen boundary condition, and the much-studied skin effect is its extreme case. The author draws on a scattering picture, in which gains and losses in the effective Hamiltonian encode where particles enter and leave, and connects the framework to exceptional points, bulk-boundary correspondence, and Floquet systems. A sympathetic reader would care because if the picture is right, many non-Hermitian phenomena stop being isolated curiosities and become boundary-condition constraints inherited from a hidden Hermitian ancestor.

What carries the argument

The load-bearing mechanism is the Hermitian-parent construction: view a non-Hermitian Hamiltonian as the projection, onto a sub-Hilbert space, of a Hermitian parent lattice in one higher dimension together with a boundary condition such as an incidence direction for incoming particles. The boundary condition generates the gains and losses of the effective non-Hermitian child, and defectiveness is the exclusion of parent eigenstates that do not fit that boundary condition. A companion device is the doubled Hamiltonian built from $H$ and $H^\dagger$ as off-diagonal blocks, which restores the information missing from the non-Hermitian child; in the scattering picture the skin effect appears as extreme defectiveness at exceptional points whose order scales with system size.

What would settle it

Pick a canonical non-Hermitian lattice with skin modes, such as the one-dimensional chain with asymmetric hoppings, and try to realize it as the projection of scattering states of a two-dimensional topological ribbon with a chosen incidence direction. If no parent geometry and boundary condition reproduces the skin-mode eigenstates, or if the correspondence holds only at isolated energies that do not cover the whole spectrum, the central claim is falsified. The calculation is concrete: solve the parent scattering problem, project onto the strip, and compare the projected subspace with the child's defective eigenspace.

Watch

Extended reading notes

Core claim

The paper's central claim is that defectiveness—the lack of a full set of linearly independent eigenvectors, the defining non-Hermitian feature at exceptional points—has a concrete physical meaning in a scattering picture. Take a Hermitian lattice with at least one translationally invariant direction, impose a boundary condition such as particles incident from one side, and project the resulting scattering states onto a sub-Hilbert space: the effective Hamiltonian for that subspace is non-Hermitian, with gains and losses that encode the boundary condition. The missing eigenvectors are precisely the parent eigenstates that are incompatible with that boundary condition; they are not absent from the parent, they are excluded by the constraint. The author illustrates this with a ribbon of a two-dimensional topological insulator whose boundary condition selects an edge state at a single edge, and cites the result that real-energy eigenstates of parity-time-symmetric non-Hermitian tight-binding chains correspond to resonant transmission states of a Hermitian parent. The paper also draws the corollary that a non-Hermitian lattice in dimension $d$ is, when a parent exists, the 'shadow' of a Hermitian lattice in dimension $d+1$ plus boundary conditions, and that the link may hold only at discrete resonance energies.

Load-bearing premise

The load-bearing premise is that a non-Hermitian Hamiltonian can always be viewed as the projection of a Hermitian parent Hamiltonian plus a boundary condition; the paper explicitly concedes that no such parent is guaranteed to exist and that, when it does, the link often holds only at discrete resonance energies.

Editorial extensions

If this is right

  • If the scattering picture is right, the non-Hermitian skin effect is a boundary-condition phenomenon: open-boundary eigenstates localize at one edge because the chosen boundary condition excludes the extended states of the parent, so bulk and finite-system spectra can differ even in the thermodynamic limit.
  • Complex spectra force a choice of gap definition, and the choice changes the topology: point gaps allow a single band to wind around a base energy and carry a nonzero winding number, whereas line gaps preserve a more Hermitian-like classification.
  • Bulk-boundary correspondence cannot be assumed; the paper surveys non-Bloch invariants, real-space invariants, Green's-function invariants, and the doubled-Hamiltonian reduction as competing routes to restore it.
  • Floquet systems become a natural testing ground: projecting Floquet space onto a single replica produces effective non-Hermitian-like behavior, and tools such as the doubled Hamiltonian transfer between driven and non-Hermitian problems.
  • Defectiveness at exceptional points can be extreme in pristine lattices; the picture frames higher-order exceptional points and eigenspace condensation as the boundary-condition constraint taken to its limit.

Reading between the lines

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

  • If the shadow picture holds generally, it suggests a constructive search strategy for missing parent lattices: for any anomalous non-Hermitian model, look for a Hermitian lattice in one higher dimension whose scattering subspace, after projection, reproduces the model; this would turn classification questions into a search over parent geometries.
  • The paper's own caveat that the parent link often holds only at resonance energies implies the picture is most secure for effectively single-particle, non-interacting settings; in interacting or time-dependent many-body systems the parent may need to be nonlocal in energy or time, which is a testable limitation rather than a contradiction.
  • A direct experimental test could be built in photonic or acoustic lattices: tune a boundary condition, such as the side from which a waveguide mode enters, and watch whether the number of linearly independent localized modes in the projected system changes exactly as the scattering picture predicts; appearance or disappearance of defectiveness under boundary-condition tuning would confirm the mech
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This Perspective surveys recent work on topological states of non-Hermitian lattices. It reviews possible definitions of a gap in the complex energy plane (point gap and line gap), the meaning of imaginary eigenvalues, the physical origin of gain, loss, and asymmetric couplings, and the role of exceptional points and defectiveness. The paper's central proposal is that a non-Hermitian lattice can be understood as the projection of a Hermitian parent lattice in one higher dimension, with a boundary condition that selects a subset of scattering states; defectiveness is then interpreted as the absence of enough eigenstates compatible with that boundary condition. This idea is illustrated with the Jin-Song construction for PT-symmetric tight-binding chains and with a Chern-ribbon scattering picture, and is connected to the non-Hermitian skin effect. The paper also surveys proposals for non-Hermitian bulk-boundary correspondence and analogies with Floquet systems. It explicitly acknowledges that not every non-Hermitian Hamiltonian admits such a Hermitian parent.

Significance. If the scattering-parent interpretation holds for the intended class of systems, it offers a unifying physical picture of defectiveness and skin effects, and it connects non-Hermitian topology to familiar Hermitian scattering and Floquet concepts. The paper is a useful perspective: it gives a clear account of point-gap versus line-gap notions, summarizes competing routes to bulk-boundary correspondence, and is unusually candid about the main weakness of its own proposal. It does not claim to fit parameters or derive new invariants, and it makes no falsifiable predictions; its contribution is conceptual synthesis. The explicit self-criticism in Section 6 is a strength, because it prevents the interpretive claim from being mistaken for a theorem.

major comments (3)
  1. [Section 6 and Section 6.1] The central interpretive claim that defectiveness 'can be understood as the lack of enough states which are compatible with the boundary condition' (Section 6) is demonstrated only for a restricted class: real-energy eigenstates of PT-symmetric tight-binding models that map to resonant transmission states of a Hermitian parent, as shown in Section 6.1. The footnote to Section 6 concedes that even when a parent exists the effective description is energy dependent and the link is restricted to resonances, and Section 6.2 discusses skin-effect models with complex spectra for which no parent construction is given. Please either restrict the claim to the established class or explicitly promote the general statement to a conjecture and state what evidence would confirm or refute it, for instance a parent construction for an asymmetric-hopping chain with genuinely complex eigenvalues.
  2. [Section 6, dimensionality link] The expectation that 'a non-Hermitian lattice in dimension d is an effective description of an Hermitian lattice in dimension d + 1 plus boundary conditions' is presented as a consequence of the scattering picture, but the supporting references do not establish the existence of a parent for generic models or the preservation of the full spectrum, eigenvectors, and topological invariants under projection. The paper should either mark this dimensionality correspondence as an open conjecture, with a precise statement of the class of non-Hermitian Hamiltonians to which it is intended to apply, or provide a concrete worked example, such as deriving a non-Hermitian SSH chain with non-reciprocal hoppings from a Chern-insulator ribbon parent plus a boundary condition.
  3. [Section 6.1, two-child construction] The argument that each non-Hermitian child is defective at the resonant energy because its time-reversed partner is assigned to the other child is specific to the two-child construction with opposite imaginary potentials. As written, the surrounding text suggests a general mechanism for defectiveness in arbitrary non-Hermitian Hamiltonians. Please label this mechanism as illustrative rather than universal, and clarify that generic defective Hamiltonians may become defective through other mechanisms, such as the higher-order exceptional points discussed in Section 6.2.
minor comments (5)
  1. [Section 6, first paragraph] The dimensionality convention is used inconsistently: the text first maps a d-dimensional lattice to a non-Hermitian lattice in dimension d-1, then states the expectation as a d-dimensional non-Hermitian lattice arising from a d+1-dimensional Hermitian parent. Please fix the notation so the reader can follow the dimensional shift.
  2. [Section 6.2] The phrase 'This anomalous localization was a attributed to the proximity' contains a typo; it should read 'was attributed to the proximity'.
  3. [Figure 4 caption] The phrase 'representing containing an absorbing on-site term' should be reworded, for example 'representing a tight-binding network containing an absorbing on-site term'.
  4. [Section 7] The phrase 'an integral over the the Brillouin zone' contains a duplicated article; it should read 'over the Brillouin zone'.
  5. [Footnote in Section 6] The crucial caveat about the absence of a guarantee that a Hermitian parent exists, and about the energy dependence of the effective description, is important enough to be stated in the main text; consider moving it out of the footnote.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central scattering-parent interpretation is explicitly conditional and rests on external constructions, with no fitted input presented as prediction.

full rationale

This is a Perspective, not a derivation-based paper. The central interpretive claim—defectiveness as a lack of boundary-compatible states and the d-to-(d+1) Hermitian-parent correspondence—is framed as an expectation, with the paper's own Section 6 conceding that 'there is no warranty that a non-hermitian Hamiltonian can always be assimilated to the effective description of a scattering situation' and its footnote noting that even when a parent exists the effective description is energy-dependent and 'the link is restricted to a discrete set of energies (resonances).' The parent-construction procedure is imported from Jin and Song [96,97], which is an external citation rather than a self-citation chain; the author's own prior work ([46,48]) appears only as background for the skin effect and exceptional-point localization and is not used to forbid alternatives or to force a conclusion. No parameter is fitted to data, and no 'prediction' is presented that reduces by construction to an input. The main limitation is therefore one of scope—the scattering-parent picture is demonstrated only for real-energy resonance eigenstates of PT-symmetric chains, not for generic complex-spectrum defective lattices—which is a correctness/scope concern, not circularity. Under the hard rules, a self-contained interpretive paper with no fitted parameters and an explicitly conditional central claim receives score 0.

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

No free parameters are fitted and no new entities are introduced. The paper's interpretive claims rest on two domain assumptions, both explicitly flagged as not universally valid: the existence of a Hermitian parent and the validity of the projection to a lower-dimensional non-Hermitian lattice.

assumptions (2)
  • domain assumption Existence of a Hermitian parent Hamiltonian plus boundary conditions for a non-Hermitian lattice
    Section 6 states this is the main idea and acknowledges 'there is no warranty that a non-hermitian Hamiltonian can always be assimilated to the effective description of a scattering situation.'
  • domain assumption Projection from a translationally invariant lattice under non-equilibrium conditions to a lower-dimensional non-Hermitian lattice
    Section 6 introduces this mapping for the scattering picture and it depends on the existence of the parent and restriction of the Hilbert space.

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Cite this review

Pith. "Pith review of Perspective on topological states of non-Hermitian lattices." pith.science (2026). https://pith.science/paper/MYHA74OE

@misc{pith2026190900809,
  author       = {Pith},
  title        = {Pith review of: Perspective on topological states of non-Hermitian lattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MYHA74OE}},
  note         = {Machine review of arXiv:1909.00809}
}
read the original abstract

The search of topological states in non-Hermitian systems has gained a strong momentum over the last two years climbing to the level of an emergent research front. In this Perspective we give an overview with a focus in connecting this topic to others like Floquet systems. Furthermore, using a simple scattering picture we discuss an interpretation of concepts like the Hamiltonian's defectiveness, i.e. the lack of a full basis of eigenstates, crucial in many discussions of topological phases of non-Hermitian Hamiltonians.

Figures

Figures reproduced from arXiv: 1909.00809 by the authors.

Figure 1
Figure 1. (a) Scheme representing the eigenenergies forming bands (thick segments) in a typical gapped Hermitian system. The red dot marks the Fermi energy. In the non-Hermitian case, the gap can be defined as the prohibition of touching a base energy, also called point gap (b), or the prohibition of touching a line, also called line gap (c). axis, all without touching a base energy set inside the gap (and therefore closing i… view at source ↗
Figure 2
Figure 2. 2 × 2 Hamiltonian with complex onsite terms ωdiff − iγdiff and −ωdiff + iγdiff and reciprocal hopping v exhibiting an exceptional point. (a) and (b) show the evolution of the real and imaginary parts of the eigenvalues, denoted with σ± as a function of the difference in frequencies ωdiff and loss factors γdiff. (c-e) show cuts for different values of γdiff: for γdiff > 0 (c) there is level repulsion in the real part… view at source ↗
Figure 3
Figure 3. (a) Scheme representing a hypothetical Hermitian lattice with translational invariance along the horizontal direction. The red arrow represents the (boundary) condition of particles incoming from the left. (b) When projected on a strip the situation in (a) can also be described by an effective non-Hermitian Hamiltonian (gains and losses stemming from the boundary condition are represented in color). 6. Interpreting … view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: (a) Scheme representing a tight-binding network with an on-site imaginary potential −iγ at one site and an arbitrary network with an Hermitian Hamiltonian represented by Hsub. (b) represents the Hermitian parent of the system in (a) sharing the same eigenstate within t…

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Works this paper leans on

156 extracted references · 40 canonical work pages

  1. [1]

    & Pepper, M

    von Klitzing, K., Dorda, G. & Pepper, M. New Method for High-Accuracy Determination of the Fine- Structure Constant Based on Quantized Hall Resistance. Phys. Rev. Lett. 45, 494–497 (1980). URL http://link.aps.org/doi/10.1103/PhysRevLett.45.494

    doi:10.1103/physrevlett.45.494 1980

  2. [2]

    J., Kohmoto, M., Nightingale, M

    Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall Conductance in a Two- Dimensional Periodic Potential. Phys. Rev. Lett. 49, 405–408 (1982). URL http://link.aps. org/doi/10.1103/PhysRevLett.49.405

    doi:10.1103/physrevlett.49.405 1982

  3. [3]

    Haldane, F. D. M. Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the ”Parity Anomaly”. Phys. Rev. Lett. 61, 2015–2018 (1988). URL http://link. aps.org/doi/10.1103/PhysRevLett.61.2015

    doi:10.1103/physrevlett.61.2015 1988

  4. [4]

    Kane, C. L. & Mele, E. J. Quantum Spin Hall Effect in Graphene. Phys. Rev. Lett. 95, 226801– (2005). URL http://link.aps.org/doi/10.1103/PhysRevLett.95.226801

    doi:10.1103/physrevlett.95.226801 2005

  5. [5]

    A., Hughes, T

    Bernevig, B. A., Hughes, T. L. & Zhang, S.-C. Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells. Science 314, 1757–1761 (2006). URL http://www. sciencemag.org/content/314/5806/1757.abstract

    2006

  6. [6]

    K ¨onig, M. et al. Quantum Spin Hall Insulator State in HgTe Quantum Wells. Science 318, 766–770 (2007). URL http://www.sciencemag.org/content/318/5851/766.abstrac. ††Quote from Edward Abbey’s Desert Solitaire, New York, McGraw-Hill, 1968. Perspective on topological states of non-Hermitian lattices 14

    2007

  7. [7]

    Hsieh, D. et al. A topological Dirac insulator in a quantum spin Hall phase. Nature 452, 970–974 (2008)

    2008

  8. [8]

    Bernevig, B. A. & Hughes, T. L. Topological Insulators and Topological Superconductors (Princeton University Press, 2013)

    2013

Show all 156 references
  1. [9]

    Frank Ortmann, S. O. V ., Stephan Roche & Molenkamp, L. W. (eds.) Topological Insulators: Fundamentals and Perspectives (Wiley, 2015)

    2015

  2. [10]

    K., Oroszl ´any, L

    Asb ´oth, J. K., Oroszl ´any, L. & P ´alyi, A. A Short Course on Topological Insulators , vol. 919 of Lecture Notes in Physics (Springer International Publishing, Cham, 2016). URL http://link. springer.com/10.1007/978-3-319-25607-8

    2016 doi

  3. [11]

    & Vishwanath, A

    Vafek, O. & Vishwanath, A. Dirac Fermions in Solids: From High-Tc Cuprates and Graphene to Topological Insulators and Weyl Semimetals. Annual Review of Condensed Mat- ter Physics 5, 83–112 (2014). URL http://www.annualreviews.org/doi/10.1146/ annurev-conmatphys-031113-133841

    2014

  4. [12]

    Z., Xu, S.-Y ., Belopolski, I

    Hasan, M. Z., Xu, S.-Y ., Belopolski, I. & Huang, S.-M. Discovery of Weyl Fermion Semimetals and Topological Fermi Arc States. Annual Review of Condensed Matter Physics 8, 289–309 (2017). URL https://doi.org/10.1146/annurev-conmatphys-031016-025225

    2017 doi

  5. [13]

    & Aoki, H

    Oka, T. & Aoki, H. Photovoltaic Hall effect in graphene. Phys. Rev. B 79, 081406– (2009). URL http://link.aps.org/doi/10.1103/PhysRevB.79.081406

    2009 doi

  6. [14]

    H., Refael, G

    Lindner, N. H., Refael, G. & Galitski, V . Floquet topological insulator in semiconductor quantum wells. Nat Phys 7, 490–495 (2011). URL http://dx.doi.org/10.1038/nphys1926

    2011 doi

  7. [15]

    McIver, J. W. et al. Light-induced anomalous Hall effect in graphene. arXiv:1811.03522 [cond-mat] (2018). URL http://arxiv.org/abs/1811.03522. ArXiv: 1811.03522

    2018 arXiv

  8. [16]

    Rudner, M. S. & Levitov, L. S. Topological Transition in a Non-Hermitian Quantum Walk. Physical Review Letters 102, 065703 (2009). URL http://link.aps.org/doi/10.1103/ PhysRevLett.102.065703

    2009

  9. [17]

    Diehl, S., Rico, E., Baranov, M. A. & Zoller, P. Topology by dissipation in atomic quantum wires. Nature Physics 7, 971–977 (2011). URL http://www.nature.com/nphys/journal/v7/ n12/full/nphys2106.html

    2011

  10. [18]

    Topological phase in a non-Hermitian symmetric system

    Yuce, C. Topological phase in a non-Hermitian symmetric system. Physics Letters A 379, 1213–1218 (2015). URL http://www.sciencedirect.com/science/article/pii/ S0375960115001681

    2015

  11. [19]

    & Aguado, R

    San-Jose, P., Cayao, J., Prada, E. & Aguado, R. Majorana bound states from exceptional points in non- topological superconductors. Scientific Reports 6, 21427 (2016). URL http://www.nature. com/srep/2016/160211/srep21427/full/srep21427.html

    2016

  12. [20]

    Lee, T. E. Anomalous Edge State in a Non-Hermitian Lattice. Physical Review Letters 116, 133903 (2016). URL http://link.aps.org/doi/10.1103/PhysRevLett.116.133903

    2016 doi

  13. [21]

    Y ., Huang, C., Chong, Y

    Leykam, D., Bliokh, K. Y ., Huang, C., Chong, Y . & Nori, F. Edge Modes, Degeneracies, and Topological Numbers in Non-Hermitian Systems. Physical Review Letters 118, 040401 (2017). URL https: //link.aps.org/doi/10.1103/PhysRevLett.118.040401

    2017 doi

  14. [22]

    Bender, C. M. & Boettcher, S. Real Spectra in Non-Hermitian Hamiltonians Having $\mathsc{P}\mathsc{T}$ Symmetry. Physical Review Letters 80, 5243–5246 (1998). URL https: //link.aps.org/doi/10.1103/PhysRevLett.80.5243

    1998 doi

  15. [23]

    A non-Hermitian Hamilton operator and the physics of open quantum systems

    Rotter, I. A non-Hermitian Hamilton operator and the physics of open quantum systems. Journal of Physics A: Mathematical and Theoretical 42, 153001 (2009)

    2009

  16. [24]

    & Bird, J

    Rotter, I. & Bird, J. P. A review of progress in the physics of open quantum systems: theory and experiment. Reports on Progress in Physics 78, 114001 (2015). URL https://doi.org/10. 1088%2F0034-4885%2F78%2F11%2F114001

    2015

  17. [25]

    Kozii, V . & Fu, L. Non-Hermitian Topological Theory of Finite-Lifetime Quasiparticles: Prediction of Bulk Fermi Arc Due to Exceptional Point. arXiv:1708.05841 [cond-mat] (2017). URL http: //arxiv.org/abs/1708.05841. ArXiv: 1708.05841

    2017 arXiv

  18. [26]

    & Kawakami, N

    Yoshida, T., Peters, R. & Kawakami, N. Non-Hermitian perspective of the band structure in heavy- fermion systems. arXiv:1805.01172 [cond-mat] (2018). URL http://arxiv.org/abs/1805. Perspective on topological states of non-Hermitian lattices 15 01172. ArXiv: 1805.01172

    2018 arXiv

  19. [28]

    Liertzer, M. et al. Pump-Induced Exceptional Points in Lasers. Physical Review Letters 108, 173901 (2012). URL https://link.aps.org/doi/10.1103/PhysRevLett.108.173901

    2012 doi

  20. [29]

    Peng, B. et al. Chiral modes and directional lasing at exceptional points. Proceedings of the National Academy of Sciences 113, 6845–6850 (2016). URL https://www.pnas.org/content/113/ 25/6845

    2016

  21. [30]

    Harari, G. et al. Topological insulator laser: Theory. Science 359, eaar4003 (2018). URL https: //science.sciencemag.org/content/359/6381/eaar4003

    2018

  22. [31]

    Bandres, M. A. et al. Topological insulator laser: Experiments. Science 359, eaar4005 (2018). URL https://science.sciencemag.org/content/359/6381/eaar4005

    2018

  23. [32]

    Zhao, H. et al. Topological hybrid silicon microlasers. Nature Communications 9, 981 (2018). URL https://www.nature.com/articles/s41467-018-03434-2

    2018

  24. [33]

    R ¨uter, C. E. et al. Observation of parity–time symmetry in optics. Nature Physics 6, 192–195 (2010). URL http://www.nature.com/nphys/journal/v6/n3/abs/nphys1515.html

    2010

  25. [34]

    Parity-time symmetry meets photonics: A new twist in non-Hermitian optics

    Longhi, S. Parity-time symmetry meets photonics: A new twist in non-Hermitian optics. EPL 120, 64001 (2017). URL https://epljournal.edpsciences.org/articles/epl/abs/ 2017/24/epl18976/epl18976.html

    2017

  26. [35]

    Feng, L., El-Ganainy, R. & Ge, L. Non-Hermitian photonics based on parity–time symme- try. Nature Photonics 11, 752 (2017). URL https://www.nature.com/articles/ s41566-017-0031-1

    2017

  27. [36]

    Ozawa, T. et al. Topological photonics. Reviews of Modern Physics 91, 015006 (2019). URL https://link.aps.org/doi/10.1103/RevModPhys.91.015006

    2019 doi

  28. [37]

    Y ., Leykam, D., Lein, M

    Bliokh, K. Y ., Leykam, D., Lein, M. & Nori, F. Topological non-Hermitian origin of surface Maxwell waves. Nature Communications 10, 580 (2019). URL https://www.nature.com/articles/ s41467-019-08397-6

    2019

  29. [38]

    Lee, C. H. et al. Topolectrical Circuits. Communications Physics 1, 39 (2018). URL https: //www.nature.com/articles/s42005-018-0035-2

    2018

  30. [39]

    Non-Hermitian higher-order topological states in nonreciprocal and reciprocal systems with their electric-circuit realization

    Ezawa, M. Non-Hermitian higher-order topological states in nonreciprocal and reciprocal systems with their electric-circuit realization. Physical Review B99, 201411 (2019). URL https://link.aps. org/doi/10.1103/PhysRevB.99.201411

    2019 doi

  31. [40]

    Dembowski, C. et al. Experimental Observation of the Topological Structure of Exceptional Points. Physical Review Letters 86, 787–790 (2001). URL https://link.aps.org/doi/10.1103/ PhysRevLett.86.787

    2001

  32. [41]

    Dembowski, C. et al. Encircling an exceptional point. Physical Review E 69, 056216 (2004). URL https://link.aps.org/doi/10.1103/PhysRevE.69.056216

    2004 doi

  33. [42]

    Stehmann, T., Heiss, W. D. & Scholtz, F. G. Observation of exceptional points in electronic circuits. Journal of Physics A: Mathematical and General37, 7813–7819 (2004). URL https://doi.org/ 10.1088%2F0305-4470%2F37%2F31%2F012

    2004

  34. [43]

    Hu, W., Wang, H., Shum, P. P. & Chong, Y . D. Exceptional points in a non-Hermitian topological pump. Physical Review B 95, 184306 (2017). URL https://link.aps.org/doi/10.1103/ PhysRevB.95.184306

    2017

  35. [44]

    & Al `u, A

    Miri, M.-A. & Al `u, A. Exceptional points in optics and photonics. Science 363, eaar7709 (2019). URL https://science.sciencemag.org/content/363/6422/eaar7709

    2019

  36. [45]

    Why does bulk boundary correspondence fail in some non-hermitian topological models

    Xiong, Y . Why does bulk boundary correspondence fail in some non-hermitian topological models. Journal of Physics Communications 2, 035043 (2018). URL https://doi.org/10.1088% 2F2399-6528%2Faab64a

    2018

  37. [48]

    M., Barrios Vargas, J

    Martinez Alvarez, V . M., Barrios Vargas, J. E., Berdakin, M. & Foa Torres, L. E. F. Topological states of non-Hermitian systems. The European Physical Journal Special Topics227, 1295–1308 (2018). URL https://doi.org/10.1140/epjst/e2018-800091-5

    2018 doi

  38. [49]

    Shen, H., Zhen, B. & Fu, L. Topological Band Theory for Non-Hermitian Hamiltonians. Physical Review Letters 120, 146402 (2018). URL https://link.aps.org/doi/10.1103/PhysRevLett. 120.146402

    2018 doi

  39. [50]

    Gong, Z. et al. Topological Phases of Non-Hermitian Systems. Physical Review X 8, 031079 (2018). URL https://link.aps.org/doi/10.1103/PhysRevX.8.031079

    2018 doi

  40. [52]

    & Sato, M

    Kawabata, K., Shiozaki, K., Ueda, M. & Sato, M. Symmetry and Topology in Non-Hermitian Physics. arXiv:1812.09133 [cond-mat, physics:math-ph, physics:physics, physics:quant-ph] (2018). URL http://arxiv.org/abs/1812.09133. ArXiv: 1812.09133

    2018 arXiv

  41. [53]

    Li, L., Lee, C. H. & Gong, J. Geometric classification of non-Hermitian topological systems through the singularity ring. arXiv:1905.04965 [cond-mat, physics:quant-ph] (2019). URL http://arxiv. org/abs/1905.04965. ArXiv: 1905.04965

    2019 arXiv

  42. [54]

    & Wang, Z

    Yao, S., Song, F. & Wang, Z. Non-Hermitian Chern bands and Chern numbers. arXiv:1804.04672 [cond-mat, physics:physics] (2018). URL http://arxiv.org/abs/1804.04672. ArXiv: 1804.04672

    2018 arXiv

  43. [55]

    & Das, T

    Ghatak, A. & Das, T. New topological invariants in non-Hermitian systems. Journal of Physics: Condensed Matter 31, 263001 (2019). URL https://doi.org/10.1088%2F1361-648x% 2Fab11b3

    2019

  44. [56]

    & Chen, S

    Jiang, H., Yang, C. & Chen, S. Topological invariants and phase diagrams for one-dimensional two- band non-Hermitian systems without chiral symmetry. Physical Review A 98, 052116 (2018). URL https://link.aps.org/doi/10.1103/PhysRevA.98.052116

    2018 doi

  45. [57]

    K., Edvardsson, E., Budich, J

    Kunst, F. K., Edvardsson, E., Budich, J. C. & Bergholtz, E. J. Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems. Physical Review Letters 121, 026808 (2018). URL https://link.aps.org/doi/10.1103/PhysRevLett.121.026808

    2018 doi

  46. [58]

    S., Kruchkov, A

    Borgnia, D. S., Kruchkov, A. J. & Slager, R.-J. Non-Hermitian Boundary Modes. arXiv:1902.07217 [cond-mat, physics:quant-ph] (2019). URL http://arxiv.org/abs/1902.07217. ArXiv: 1902.07217

    2019 arXiv

  47. [59]

    Non-Hermitian Quantum Mechanics (2011)

    Moiseyev, N. Non-Hermitian Quantum Mechanics (2011). URL http://www.cambridge.org/us/academic/subjects/physics/ quantum-physics-quantum-information-and-quantum-computation/ non-hermitian-quantum-mechanics

    2011

  48. [60]

    & Martin, T

    Landauer, R. & Martin, T. Barrier interaction time in tunneling. Reviews of Modern Physics66, 217–228 (1994). URL https://link.aps.org/doi/10.1103/RevModPhys.66.217

    1994 doi

  49. [61]

    Ni, X. et al. $\mathcal{PT}$ phase transitions of edge states at $\mathcal{PT}$ symmetric interfaces in non-Hermitian topological insulators. Physical Review B 98, 165129 (2018). URL https: //link.aps.org/doi/10.1103/PhysRevB.98.165129

    2018 doi

  50. [62]

    Edge states at the interface of non-Hermitian systems

    Yuce, C. Edge states at the interface of non-Hermitian systems. Physical Review A 97, 042118 (2018). URL https://link.aps.org/doi/10.1103/PhysRevA.97.042118

    2018 doi

  51. [63]

    Ryu, J.-W. et al. Emergent localized states at the interface of a twofold $ \mathcal{PT}$-symmetric lattice. arXiv:1904.00444 [cond-mat] (2019). URL http://arxiv.org/abs/1904.00444. Perspective on topological states of non-Hermitian lattices 17 ArXiv: 1904.00444

    2019 arXiv

  52. [65]

    H., Li, L

    Lee, C. H., Li, L. & Gong, J. Hybrid Higher-Order Skin-Topological Modes in Nonreciprocal Systems. Physical Review Letters 123, 016805 (2019). URL https://link.aps.org/doi/10.1103/ PhysRevLett.123.016805

    2019

  53. [66]

    & Wigner, E

    von Neuman, J. & Wigner, E. Uber merkw ¨urdige diskrete Eigenwerte. Uber das Verhalten von Eigenwerten bei adiabatischen Prozessen. Physikalische Zeitschrift 30, 467–470 (1929). URL http: //adsabs.harvard.edu/abs/1929PhyZ...30..467V

    1929

  54. [67]

    Perturbation Theory for Linear Operators

    Kato, T. Perturbation Theory for Linear Operators . Classics in Mathematics (Springer-Verlag, Berlin Heidelberg, 1995), 2 edn. URL https://www.springer.com/us/book/9783540586616

    1995

  55. [68]

    Pancharatnam, virtuoso of the Poincar ´e sphere: an appreciation

    Berry, M. Pancharatnam, virtuoso of the Poincar ´e sphere: an appreciation. Current Science 67, 220–223 (1994). URL http://www.jstor.org/stable/24095727

    1994

  56. [69]

    Geometric phase memories (2010)

    Berry, M. Geometric phase memories (2010). URL https://www.nature.com/articles/ nphys1608

    2010

  57. [71]

    Hassan, A. U. et al. Erratum: Chiral state conversion without encircling an exceptional point [Phys. Rev. A 96, 052129 (2017)]. Physical Review A 96, 069908 (2017). URL https://link.aps.org/ doi/10.1103/PhysRevA.96.069908

    2017 doi

  58. [72]

    & Chan, C

    Zhang, X.-L., Wang, S., Hou, B. & Chan, C. Dynamically Encircling Exceptional Points: In situ Control of Encircling Loops and the Role of the Starting Point. Physical Review X 8, 021066 (2018). URL https://link.aps.org/doi/10.1103/PhysRevX.8.021066

    2018 doi

  59. [73]

    Bittner, S. et al. $\mathsc{P}\mathsc{T}$ Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard. Physical Review Letters 108, 024101 (2012). URL https://link.aps. org/doi/10.1103/PhysRevLett.108.024101

    2012 doi

  60. [74]

    Doppler, J. et al. Dynamically encircling an exceptional point for asymmetric mode switching. Nature 537, 76–79 (2016). URL http://www.nature.com/nature/journal/v537/n7618/ full/nature18605.html

    2016

  61. [75]

    Ghosh, S. N. & Chong, Y . D. Exceptional points and asymmetric mode conversion in quasi-guided dual- mode optical waveguides. Scientific Reports 6, 19837 (2016). URL https://www.nature.com/ articles/srep19837

    2016

  62. [76]

    A., Danieli, E

    ´Alvarez, G. A., Danieli, E. P., Levstein, P. R. & Pastawski, H. M. Environmentally induced quantum dynamical phase transition in the spin swapping operation. The Journal of Chemical Physics 124, 194507 (2006). URL https://aip.scitation.org/doi/10.1063/1.2193518

    2006 doi

  63. [77]

    & Harris, J

    Xu, H., Mason, D., Jiang, L. & Harris, J. G. E. Topological energy transfer in an optomechanical system with exceptional points. Nature 537, 80–83 (2016). URL http://www.nature.com/nature/ journal/v537/n7618/full/nature18604.html

    2016

  64. [78]

    Lin, Z. et al. Unidirectional Invisibility Induced by $\mathcal{P}\mathcal{T}$-Symmetric Periodic Structures. Physical Review Letters 106, 213901 (2011). URL https://link.aps.org/doi/ 10.1103/PhysRevLett.106.213901

    2011 doi

  65. [79]

    J., Ma, R.-M., Wang, Y

    Feng, L., Wong, Z. J., Ma, R.-M., Wang, Y . & Zhang, X. Single-mode laser by parity-time symmetry breaking. Science 346, 972–975 (2014). URL https://science.sciencemag. org/content/346/6212/972

    2014

  66. [80]

    Hodaei, H., Miri, M.-A., Heinrich, M., Christodoulides, D. N. & Khajavikhan, M. Parity-time–symmetric microring lasers. Science 346, 975–978 (2014). URL https://science.sciencemag.org/ content/346/6212/975

    2014

  67. [81]

    & Yang, L

    Chen, W., Kaya ¨Ozdemir, Zhao, G., Wiersig, J. & Yang, L. Exceptional points enhance sensing in an optical microcavity. Nature 548, 192–196 (2017). URL https://www.nature.com/ Perspective on topological states of non-Hermitian lattices 18 articles/nature23281

    2017

  68. [82]

    Hodaei, H. et al. Enhanced sensitivity at higher-order exceptional points. Nature 548, 187–191 (2017). URL https://www.nature.com/articles/nature23280

    2017

  69. [83]

    Rechtsman, M. C. Applied physics: Optical sensing gets exceptional. Nature 548, 161–162 (2017). URL https://www.nature.com/articles/548161a

    2017

  70. [84]

    Bird, J. P. & Ochiai, Y . Electron Spin Polarization in Nanoscale Constrictions. Science 303, 1621–1622 (2004). URL https://science.sciencemag.org/content/303/5664/1621

    2004

  71. [85]

    Yoon, Y . et al. Coupling Quantum States through a Continuum: A Mesoscopic Multistate Fano Resonance. Physical Review X 2, 021003 (2012). URL https://link.aps.org/doi/10. 1103/PhysRevX.2.021003

    2012

  72. [86]

    & Zelevinsky, V

    Auerbach, N. & Zelevinsky, V . Super-radiant dynamics, doorways and resonances in nuclei and other open mesoscopic systems. Reports on Progress in Physics 74, 106301 (2011). URL https: //doi.org/10.1088%2F0034-4885%2F74%2F10%2F106301

    2011

  73. [87]

    & Rotter, I

    Eleuch, H. & Rotter, I. Resonances in open quantum systems. Physical Review A 95, 022117 (2017). URL https://link.aps.org/doi/10.1103/PhysRevA.95.022117

    2017 doi

  74. [88]

    Xiao, Y .-X., Zhang, Z.-Q., Hang, Z. H. & Chan, C. T. Anisotropic exceptional points of arbitrary order. arXiv:1903.01737 [cond-mat, physics:quant-ph] (2019). URL http://arxiv.org/abs/1903. 01737. ArXiv: 1903.01737

    2019 arXiv

  75. [89]

    & Yokoyama, T

    Okugawa, R. & Yokoyama, T. Topological exceptional surfaces in non-Hermitian systems with parity- time and parity-particle-hole symmetries. Physical Review B 99, 041202 (2019). URL https: //link.aps.org/doi/10.1103/PhysRevB.99.041202

    2019 doi

  76. [90]

    A., Zyuzin, A

    Moors, K., Zyuzin, A. A., Zyuzin, A. Y ., Tiwari, R. P. & Schmidt, T. L. Disorder-driven exceptional lines and Fermi ribbons in tilted nodal-line semimetals. Physical Review B 99, 041116 (2019). URL https://link.aps.org/doi/10.1103/PhysRevB.99.041116

    2019 doi

  77. [91]

    & Hatsugai, Y

    Yoshida, T. & Hatsugai, Y . Exceptional rings protected by emergent symmetry for mechanical systems. arXiv:1904.10764 [cond-mat] (2019). URL http://arxiv.org/abs/1904.10764. ArXiv: 1904.10764

    2019 arXiv

  78. [92]

    & Hatsugai, Y

    Yoshida, T., Peters, R., Kawakami, N. & Hatsugai, Y . Symmetry-protected exceptional rings in two- dimensional correlated systems with chiral symmetry. Physical Review B 99, 121101 (2019). URL https://link.aps.org/doi/10.1103/PhysRevB.99.121101

    2019 doi

  79. [93]

    Molina, R. A. & Gonz ´alez, J. Surface and 3d Quantum Hall Effects from Engineering of Exceptional Points in Nodal-Line Semimetals. Physical Review Letters 120, 146601 (2018). URL https: //link.aps.org/doi/10.1103/PhysRevLett.120.146601

    2018 doi

  80. [94]

    & Molina, R

    Gonz ´alez, J. & Molina, R. A. Topological protection from exceptional points in Weyl and nodal-line semimetals. Physical Review B 96, 045437 (2017). URL https://link.aps.org/doi/10. 1103/PhysRevB.96.045437

    2017

  81. [95]

    Anomalous parity-time–symmetry transition away from an exceptional point

    Ge, L. Anomalous parity-time–symmetry transition away from an exceptional point. Physical Review A 94, 013837 (2016). URL https://link.aps.org/doi/10.1103/PhysRevA. 94.013837

    2016 doi

  82. [96]

    & Song, Z

    Jin, L. & Song, Z. Physics counterpart of the $ \mathcal{P}\mathcal{T}$ non-Hermitian tight-binding chain. Physical Review A 81, 032109 (2010). URL https://link.aps.org/doi/10.1103/ PhysRevA.81.032109

    2010

  83. [97]

    & Song, Z

    Jin, L. & Song, Z. A physical interpretation for the non-Hermitian Hamiltonian. Journal of Physics A: Mathematical and Theoretical 44, 375304 (2011). URL https://doi.org/10.1088% 2F1751-8113%2F44%2F37%2F375304

    2011

  84. [98]

    & Siegl, P

    Hernandez-Coronado, H., Krej ˇciˇr´ık, D. & Siegl, P. Perfect transmission scattering as a PT- symmetric spectral problem. Physics Letters A 375, 2149–2152 (2011). URL http://www. sciencedirect.com/science/article/pii/S037596011100483X

    2011

  85. [99]

    & Nelson, D

    Hatano, N. & Nelson, D. R. Localization Transitions in Non-Hermitian Quantum Mechanics. Physical Review Letters 77, 570–573 (1996). URL https://link.aps.org/doi/10.1103/ PhysRevLett.77.570. Perspective on topological states of non-Hermitian lattices 19

    1996

  86. [100]

    E., Lahini, Y ., Ringel, Z., Verbin, M

    Kraus, Y . E., Lahini, Y ., Ringel, Z., Verbin, M. & Zilberberg, O. Topological States and Adiabatic Pumping in Quasicrystals. Physical Review Letters 109, 106402 (2012). URL https://link. aps.org/doi/10.1103/PhysRevLett.109.106402

    2012 doi

  87. [101]

    Martinez Alvarez, V . M. & Coutinho-Filho, M. D. Edge states in trimer lattices. Physical Review A 99, 013833 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.99.013833

    2019 doi

  88. [102]

    D’Amato, J. L. & Pastawski, H. M. Conductance of a disordered linear chain including inelastic scattering events. Phys. Rev. B 41, 7411–7420 (1990). URL http://link.aps.org/doi/10.1103/ PhysRevB.41.7411

    1990

  89. [103]

    & Bardarson, J

    Rhim, J.-W., Behrends, J. & Bardarson, J. H. Bulk-boundary correspondence from the intercellular Zak phase. Physical Review B 95, 035421 (2017). URL https://link.aps.org/doi/10.1103/ PhysRevB.95.035421

    2017

  90. [104]

    Rhim, J.-W., Bardarson, J. H. & Slager, R.-J. Unified bulk-boundary correspondence for band insulators. Physical Review B 97, 115143 (2018). URL https://link.aps.org/doi/10. 1103/PhysRevB.97.115143

    2018

  91. [105]

    Ortmann, F., Roche, S., Valenzuela, S. O. & Molenkamp, L. W. (eds.) Topological Insulators: Fundamentals and Perspectives (Wiley, 2015)

    2015

  92. [106]

    Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators (Cambridge University Press, Cambridge, UK, 2018)

    Vanderbilt, D. Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators (Cambridge University Press, Cambridge, UK, 2018). URL https: //doi.org/10.1017/9781316662205

    2018 doi

  93. [107]

    S., Levin, M

    Rudner, M. S., Levin, M. & Levitov, L. S. Survival, decay, and topological protection in non- Hermitian quantum transport. arXiv:1605.07652 [cond-mat] (2016). URL http://arxiv.org/ abs/1605.07652. ArXiv: 1605.07652

    2016 arXiv

  94. [108]

    & Lee, J

    Zhou, H. & Lee, J. Y . Periodic Table for Topological Bands with Non-Hermitian Bernard-LeClair Symmetries. arXiv:1812.10490 [cond-mat, physics:physics, physics:quant-ph] (2018). URL http: //arxiv.org/abs/1812.10490. ArXiv: 1812.10490

    2018 arXiv

  95. [109]

    Ge, Z.-Y . et al. Topological band theory for non-Hermitian systems from a quantum field viewpoint. arXiv:1903.09985 [cond-mat, physics:hep-th, physics:quant-ph] (2019). URL http://arxiv. org/abs/1903.09985. ArXiv: 1903.09985

    2019 arXiv

  96. [110]

    & Hyart, T

    Brzezicki, W. & Hyart, T. Hidden Chern number in one-dimensional non-Hermitian chiral-symmetric systems. arXiv:1908.01553 [cond-mat] (2019). URL http://arxiv.org/abs/1908.01553. ArXiv: 1908.01553

    2019 arXiv

  97. [111]

    P., Ryu, S., Furusaki, A

    Schnyder, A. P., Ryu, S., Furusaki, A. & Ludwig, A. W. W. Classification of topological insulators and superconductors in three spatial dimensions. Physical Review B 78, 195125 (2008). URL https://link.aps.org/doi/10.1103/PhysRevB.78.195125

    2008 doi

  98. [112]

    & Zirnbauer, M

    Altland, A. & Zirnbauer, M. R. Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Physical Review B 55, 1142–1161 (1997). URL https://link.aps.org/ doi/10.1103/PhysRevB.55.1142

    1997 doi

  99. [113]

    & Ueda, M

    Kawabata, K., Higashikawa, S., Gong, Z., Ashida, Y . & Ueda, M. Topological unification of time-reversal and particle-hole symmetries in non-Hermitian physics. Nature Communications 10, 297 (2019). URL https://www.nature.com/articles/s41467-018-08254-y

    2019

  100. [114]

    Pseudo-Hermiticity versus PT-symmetry

    Mostafazadeh, A. Pseudo-Hermiticity versus PT-symmetry. II. A complete characterization of non- Hermitian Hamiltonians with a real spectrum. Journal of Mathematical Physics43, 2814–2816 (2002). URL https://aip.scitation.org/doi/abs/10.1063/1.1461427

    2002 doi

  101. [115]

    Symmetry-protected zero-mode laser with a tunable spatial profile

    Ge, L. Symmetry-protected zero-mode laser with a tunable spatial profile. Physical Review A 95, 023812 (2017). URL https://link.aps.org/doi/10.1103/PhysRevA.95.023812

    2017 doi

  102. [116]

    Rivero, J. D. H. & Ge, L. Chiral symmetry in non-Hermitian systems: product rule, Clifford algebra and pseudo-chirality. arXiv:1903.02231 [cond-mat, physics:physics, physics:quant-ph] (2019). URL http://arxiv.org/abs/1903.02231. ArXiv: 1903.02231

    2019 arXiv

  103. [117]

    Qi, B., Zhang, L. & Ge, L. Defect States Emerging from a Non-Hermitian Flatband of Photonic Zero Modes. Physical Review Letters 120, 093901 (2018). URL https://link.aps.org/doi/10. 1103/PhysRevLett.120.093901. Perspective on topological states of non-Hermitian lattices 20

    2018

  104. [118]

    & Zhai, Y

    Yang, X., Cao, Y . & Zhai, Y . Non-Hermitian Weyl Semimetals: Non-Hermitian Skin Effect and non-Bloch Bulk-Boundary Correspondence. arXiv:1904.02492 [cond-mat] (2019). URL http: //arxiv.org/abs/1904.02492. ArXiv: 1904.02492

    2019 arXiv

  105. [119]

    & Wang, Z

    Song, F., Yao, S. & Wang, Z. Non-Hermitian Topological Invariants in Real Space. arXiv:1905.02211 [cond-mat, physics:physics] (2019). URL http://arxiv.org/abs/1905.02211. ArXiv: 1905.02211

    2019 arXiv

  106. [120]

    & Rosenow, B

    Zirnstein, H.-G., Refael, G. & Rosenow, B. Bulk-boundary correspondence for non-Hermitian Hamiltonians via Green functions. arXiv:1901.11241 [cond-mat, physics:physics, physics:quant-ph] (2019). URL http://arxiv.org/abs/1901.11241. ArXiv: 1901.11241

    2019 arXiv

  107. [121]

    V olovik, G. E. & Yakovenko, V . M. Fractional charge, spin and statistics of solitons in superfluid3he film. Journal of Physics: Condensed Matter 1, 5263–5274 (1989). URL https://doi.org/10. 1088%2F0953-8984%2F1%2F31%2F025

    1989

  108. [122]

    V olovik, G. E. The Universe in a Helium Droplet (Oxford University Press, 2009). URL https: //www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199564842. 001.0001/acprof-9780199564842

    2009

  109. [124]

    Kunst, F. K. & Dwivedi, V . Non-Hermitian systems and topology: A transfer matrix perspective. arXiv:1812.02186 [cond-mat, physics:quant-ph] (2018). URL http://arxiv.org/abs/1812. 02186. ArXiv: 1812.02186

    2018 arXiv

  110. [125]

    & Demler, E

    Kitagawa, T., Oka, T., Brataas, A., Fu, L. & Demler, E. Transport properties of nonequilibrium systems under the application of light: Photoinduced quantum Hall insulators without Landau levels. Phys. Rev. B 84, 235108– (2011). URL http://link.aps.org/doi/10.1103/PhysRevB.84. 235108

    2011 doi

  111. [126]

    & H ¨anggi, P

    Kohler, S., Lehmann, J. & H ¨anggi, P. Driven quantum transport on the nanoscale. Physics Reports 406, 379–443 (2005). URL http://www.sciencedirect.com/science/article/pii/ S0370157304005071

    2005

  112. [127]

    & B ¨uttiker, M

    Moskalets, M. & B ¨uttiker, M. Floquet scattering theory of quantum pumps. Physical Review B 66, 205320 (2002). URL https://link.aps.org/doi/10.1103/PhysRevB.66.205320

    2002 doi

  113. [128]

    & Platero, G

    Gomez-Leon, A. & Platero, G. Floquet-Bloch Theory and Topology in Periodically Driven Lattices. Phys. Rev. Lett. 110, 200403– (2013). URL http://link.aps.org/doi/10.1103/ PhysRevLett.110.200403

    2013

  114. [129]

    M., Usaj, G., Balseiro, C

    Perez-Piskunow, P. M., Usaj, G., Balseiro, C. A. & Foa Torres, L. E. F. Floquet chiral edge states in graphene. Phys. Rev. B 89, 121401(R) (2014). URL http://link.aps.org/doi/10.1103/ PhysRevB.89.121401

    2014

  115. [130]

    H., Steinberg, H., Jarillo-Herrero, P

    Wang, Y . H., Steinberg, H., Jarillo-Herrero, P. & Gedik, N. Observation of Floquet-Bloch States on the Surface of a Topological Insulator. Science 342, 453–457 (2013). URL http://science. sciencemag.org/content/342/6157/453

    2013

  116. [131]

    Kibis, O. V . Metal-insulator transition in graphene induced by circularly polarized photons. Phys. Rev. B 81, 165433– (2010). URL http://link.aps.org/doi/10.1103/PhysRevB.81.165433

    2010 doi

  117. [132]

    L., Pastawski, H

    Calvo, H. L., Pastawski, H. M., Roche, S. & Foa Torres, L. E. F. Tuning laser-induced band gaps in graphene. Appl. Phys. Lett. 98, 232103–3 (2011). URL http://dx.doi.org/10.1063/1. 3597412

    2011 doi

  118. [133]

    Zhou, Y . & Wu, M. W. Optical response of graphene under intense terahertz fields. Phys. Rev. B 83, 245436– (2011). URL http://link.aps.org/doi/10.1103/PhysRevB.83.245436

    2011 doi

  119. [134]

    Sentef, M. et al. Theory of Floquet band formation and local pseudospin textures in pump-probe photoemission of graphene. Nat. Commun. 6, – (2015). URL http://dx.doi.org/10.1038/ ncomms8047

    2015

  120. [135]

    Sie, E. J. et al. Valley-selective optical Stark effect in monolayer WS2. Nature Materials 14, 290–294 (2015). URL https://www.nature.com/articles/nmat4156. Perspective on topological states of non-Hermitian lattices 21

    2015

  121. [136]

    & Schliemann, J

    L ´opez, A., Scholz, A., Santos, B. & Schliemann, J. Photoinduced pseudospin effects in silicene beyond the off-resonant condition. Phys. Rev. B 91, 125105– (2015). URL http://link.aps.org/ doi/10.1103/PhysRevB.91.125105

    2015 doi

  122. [137]

    & Usaj, G

    Huam ´an, A. & Usaj, G. Floquet spectrum and two-terminal conductance of a transition-metal dichalcogenide ribbon under a circularly polarized laser field. Physical Review B 99, 075423 (2019). URL https://link.aps.org/doi/10.1103/PhysRevB.99.075423

    2019 doi

  123. [138]

    & Loss, D

    Klinovaja, J., Stano, P. & Loss, D. Topological Floquet Phases in Driven Coupled Rashba Nanowires. Phys. Rev. Lett. 116, 176401– (2016). URL http://link.aps.org/doi/10. 1103/PhysRevLett.116.176401

    2016

  124. [139]

    Colloquium: Atomic quantum gases in periodically driven optical lattices

    Eckardt, A. Colloquium: Atomic quantum gases in periodically driven optical lattices. Reviews of Modern Physics 89, 011004 (2017). URL https://link.aps.org/doi/10.1103/ RevModPhys.89.011004

    2017

  125. [140]

    & Moessner, R

    D ´ora, B., Cayssol, J., Simon, F. & Moessner, R. Optically Engineering the Topological Properties of a Spin Hall Insulator. Phys. Rev. Lett. 108, 056602– (2012). URL http://link.aps.org/doi/ 10.1103/PhysRevLett.108.056602

    2012 doi

  126. [141]

    & Foa Torres, L

    Dal Lago, V ., Atala, M. & Foa Torres, L. E. F. Floquet topological transitions in a driven one-dimensional topological insulator. Phys. Rev. A92, 023624– (2015). URL http://link.aps.org/doi/10. 1103/PhysRevA.92.023624

    2015

  127. [142]

    & Molina, R

    Gonz ´alez, J. & Molina, R. A. Macroscopic Degeneracy of Zero-Mode Rotating Surface States in 3d Dirac and Weyl Semimetals under Radiation. Phys. Rev. Lett. 116, 156803– (2016). URL http://link.aps.org/doi/10.1103/PhysRevLett.116.156803

    2016 doi

  128. [143]

    & Sondhi, S

    Moessner, R. & Sondhi, S. L. Equilibration and order in quantum Floquet matter. Nature Physics 13, 424–428 (2017). URL https://www.nature.com/articles/nphys4106

    2017

  129. [144]

    Rudner, M. S. & Song, J. C. W. Berryogenesis: self-induced Berry flux and spontaneous non-equilibrium magnetism (2018). URL https://arxiv.org/abs/1807.01708v1

    2018 arXiv

  130. [145]

    Shirley, J. H. Solution of the Schr ¨odinger Equation with a Hamiltonian Periodic in Time. Phys. Rev.138, B979–B987 (1965). URL http://link.aps.org/doi/10.1103/PhysRev.138.B979

    1965 doi

  131. [146]

    Foa Torres, L. E. F., Perez-Piskunow, P. M., Balseiro, C. A. & Usaj, G. Multiterminal Conductance of a Floquet Topological Insulator. Phys. Rev. Lett. 113, 266801 (2014). URL http://journals. aps.org/prl/abstract/10.1103/PhysRevLett.113.266801

    2014 doi

  132. [147]

    S., Lindner, N

    Rudner, M. S., Lindner, N. H., Berg, E. & Levin, M. Anomalous edge states and the bulk-edge correspondence for periodically-driven two dimensional systems. Phys. Rev. X 3, 031005 (2013)

    2013

  133. [148]

    M., Foa Torres, L

    Usaj, G., Perez-Piskunow, P. M., Foa Torres, L. E. F. & Balseiro, C. A. Irradiated graphene as a tunable Floquet topological insulator. Phys. Rev. B 90, 115423 (2014). URL http://link.aps.org/ doi/10.1103/PhysRevB.90.115423

    2014 doi

  134. [149]

    M., Foa Torres, L

    Perez-Piskunow, P. M., Foa Torres, L. E. F. & Usaj, G. Hierarchy of Floquet gaps and edge states for driven honeycomb lattices. Physical Review A 91, 043625 (2015). URL http://link.aps.org/ doi/10.1103/PhysRevA.91.043625

    2015 doi

  135. [150]

    & Harper, F

    Roy, R. & Harper, F. Periodic table for Floquet topological insulators. Physical Review B 96, 155118 (2017). URL https://link.aps.org/doi/10.1103/PhysRevB.96.155118

    2017 doi

  136. [151]

    PT symmetric Floquet topological phase

    Yuce, C. PT symmetric Floquet topological phase. The European Physical Journal D 69, 184 (2015). URL https://link.springer.com/article/10.1140/epjd/e2015-60220-7

    2015 doi

  137. [152]

    & Yuce, C

    Turker, Z., Tombuloglu, S. & Yuce, C. PT symmetric Floquet topological phase in SSH model. Physics Letters A 382, 2013–2016 (2018). URL http://www.sciencedirect.com/science/ article/pii/S0375960118305309

    2018

  138. [153]

    & Gong, J

    Zhou, L. & Gong, J. Non-Hermitian Floquet topological phases with arbitrarily many real-quasienergy edge states. arXiv:1807.00988 [cond-mat, physics:quant-ph] (2018). URL http://arxiv.org/ abs/1807.00988. ArXiv: 1807.00988

    2018 arXiv

  139. [154]

    & Pan, J

    Zhou, L. & Pan, J. Non-Hermitian Floquet topological phases in the double-kicked rotor. arXiv:1908.02066 [cond-mat, physics:quant-ph] (2019). URL http://arxiv.org/abs/1908. 02066. ArXiv: 1908.02066. Perspective on topological states of non-Hermitian lattices 22

    2019 arXiv

  140. [155]

    & Shapiro, B

    Li, H., Kottos, T. & Shapiro, B. Floquet-Network Theory of Nonreciprocal Transport. Phys- ical Review Applied 9, 044031 (2018). URL https://link.aps.org/doi/10.1103/ PhysRevApplied.9.044031

    2018

  141. [156]

    Edvardsson, E., Kunst, F. K. & Bergholtz, E. J. Non-Hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence. Physical Review B 99, 081302 (2019). URL https://link.aps.org/doi/10.1103/PhysRevB.99.081302

    2019 doi

  142. [157]

    Liu, T. et al. Second-Order Topological Phases in Non-Hermitian Systems. Physical Review Letters 122, 076801 (2019). URL https://link.aps.org/doi/10.1103/PhysRevLett.122. 076801

    2019 doi

  143. [158]

    Electric-circuit realization of non-Hermitian higher-order topological systems

    Ezawa, M. Electric-circuit realization of non-Hermitian higher-order topological systems. arXiv:1810.04527 [cond-mat] (2018). URL http://arxiv.org/abs/1810.04527. ArXiv: 1810.04527

    2018 arXiv

  144. [159]

    C., Carlstr ¨om, J., Kunst, F

    Budich, J. C., Carlstr ¨om, J., Kunst, F. K. & Bergholtz, E. J. Symmetry-protected nodal phases in non- Hermitian systems. Physical Review B 99, 041406 (2019). URL https://link.aps.org/doi/ 10.1103/PhysRevB.99.041406

    2019 doi

  145. [160]

    Carlstr ¨om, J., St˚alhammar, M., Budich, J. C. & Bergholtz, E. J. Knotted non-Hermitian metals. Physical Review B 99, 161115 (2019). URL https://link.aps.org/doi/10.1103/PhysRevB. 99.161115

    2019 doi

  146. [161]

    Bergholtz, E. J. & Budich, J. C. Non-Hermitian Weyl physics in topological insulator ferromagnet junctions. Physical Review Research 1, 012003 (2019). URL https://link.aps.org/doi/ 10.1103/PhysRevResearch.1.012003

    2019 doi

  147. [162]

    M., Wang, P., Jin, L

    Yang, X. M., Wang, P., Jin, L. & Song, Z. Visualizing Topology of Real-Energy Gapless Phase Arising from Exceptional Point. arXiv:1905.07109 [cond-mat, physics:physics, physics:quant-ph] (2019). URL http://arxiv.org/abs/1905.07109. ArXiv: 1905.07109

    2019 arXiv

  148. [163]

    Zhou, H. et al. Observation of bulk Fermi arc and polarization half charge from paired exceptional points. Science 359, 1009–1012 (2018). URL https://science.sciencemag.org/content/ 359/6379/1009

    2018

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Reviewed August 14, 2026 · model on record in the stance chip above.