pith. sign in

arxiv: 2506.05206 · v2 · submitted 2025-06-05 · ❄️ cond-mat.mes-hall · physics.optics· quant-ph

Non-Hermitian magnetic moment

Bar Alon , Moshe Goldstein , Roni Ilan This is my paper

Pith reviewed 2026-05-19 10:48 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall physics.opticsquant-ph
keywords non-Hermitian systemsorbital magnetizationangular momentumsemiclassical theoryAharonov-Bohm effectwave packet dynamicsperiodic potentials
0
0 comments X

The pith

Non-Hermitian periodic systems possess an orbital magnetic moment whose real part matches a generalized angular momentum while the imaginary part arises from a non-Hermitian Aharonov-Bohm effect.

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

The paper develops a semiclassical theory for wave packets in non-Hermitian crystals under slowly varying perturbations. It calculates the energy of such a wave packet to first order in the spatial and temporal gradients. When applied to a uniform magnetic field, this yields an expression for the orbital magnetization energy. The authors introduce a non-Hermitian version of angular momentum that aligns with the real component of the orbital magnetic moment. They trace the imaginary component to an imaginary angular momentum that produces a non-Hermitian generalization of the Aharonov-Bohm effect.

Core claim

In non-Hermitian periodic systems, the orbital magnetization energy obtained from the semiclassical wave-packet energy is compatible with the real part of the orbital magnetic moment defined via a generalized angular momentum operator. The imaginary part of this moment stems from an imaginary counterpart to the angular momentum, which induces a non-Hermitian Aharonov-Bohm effect.

What carries the argument

The semiclassical wave-packet energy expanded to first order in gradients of the perturbations, combined with a non-Hermitian generalization of the angular momentum operator.

If this is right

  • If the theory holds, the orbital magnetic moment in non-Hermitian systems splits into real and imaginary parts with distinct physical origins.
  • The real part follows from the generalized angular momentum in the same way as in Hermitian systems.
  • The imaginary part is linked to a modified Aharonov-Bohm phase that is non-Hermitian.
  • Magnetization energy expressions remain valid under slow perturbations in non-Hermitian lattices.

Where Pith is reading between the lines

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

  • If true, experiments could detect the imaginary magnetic moment through modified interference patterns in non-Hermitian materials.
  • This framework might extend to other perturbations like electric fields in open quantum systems.
  • Such non-Hermitian moments could influence transport properties in dissipative mesoscopic devices.

Load-bearing premise

The semiclassical wave-packet picture continues to apply in non-Hermitian periodic systems when perturbations change slowly enough for the gradient expansion to work.

What would settle it

An experiment that measures the orbital magnetic response in a non-Hermitian lattice and finds the imaginary part inconsistent with the predicted non-Hermitian Aharonov-Bohm phase would falsify the claim.

read the original abstract

We construct a semiclassical theory for electrons in a non-Hermitian periodic system subject to perturbations varying slowly in space and time. We derive the energy of the wavepacket to first order in the gradients of the perturbations. Applying the theory to the specific case of a uniform external magnetic field, we obtain an expression for the orbital magnetization energy. Using the principles of non-Hermitian dynamics, we define a physically meaningful non-Hermitian generalization of the angular momentum operator and show that it is compatible with the real part of the orbital magnetic moment. The imaginary part of the orbital magnetic moment is also discussed and shown to originate from an imaginary counterpart to the angular momentum that gives rise to a non-Hermitian generalization of the Aharonov-Bohm effect.

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 / 2 minor

Summary. The manuscript develops a semiclassical theory for electrons in non-Hermitian periodic systems subject to slowly varying perturbations in space and time. It derives the wave-packet energy to first order in the gradients of the perturbations. For the case of a uniform external magnetic field, an expression for the orbital magnetization energy is obtained. A non-Hermitian generalization of the angular momentum operator is defined and shown to be compatible with the real part of the orbital magnetic moment; the imaginary part is attributed to an imaginary counterpart of the angular momentum that induces a non-Hermitian generalization of the Aharonov-Bohm effect.

Significance. If the central derivations are correct, the work supplies a concrete semiclassical framework for orbital magnetism in non-Hermitian systems, which may be relevant to open quantum systems and PT-symmetric condensed-matter models. The explicit construction of a non-Hermitian angular-momentum operator and its relation to both real and imaginary parts of the magnetic moment, together with the claimed non-Hermitian Aharonov-Bohm effect, constitute a novel extension that could be tested in future experiments or numerical simulations of non-Hermitian lattices.

major comments (2)
  1. The derivation of the orbital magnetization energy to first order in gradients (and its claimed compatibility with Re(m_orbital)) rests on the semiclassical wave-packet ansatz remaining valid in a non-Hermitian periodic system. It is not shown that the biorthogonal left/right eigenvectors permit a well-localized packet whose energy and dynamics are captured by the standard gradient expansion without additional non-Hermitian corrections at the same order. This assumption is load-bearing for the extracted magnetization energy and the real/imaginary decomposition of the magnetic moment.
  2. In the uniform-B case, the vector potential is treated perturbatively while preserving the first-order result. No explicit check is provided that non-unitary evolution or complex Berry curvature does not generate additional O(∇) terms that would alter the orbital magnetization energy expression.
minor comments (2)
  1. The definition of the non-Hermitian angular momentum operator should be written explicitly in terms of the position and momentum operators (or their non-Hermitian analogs) rather than introduced only conceptually.
  2. A brief comparison with the Hermitian limit (e.g., recovery of the standard orbital magnetization formula) would strengthen the presentation.

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 indicate the revisions we will make to strengthen the presentation.

read point-by-point responses

  1. Referee: The derivation of the orbital magnetization energy to first order in gradients (and its claimed compatibility with Re(m_orbital)) rests on the semiclassical wave-packet ansatz remaining valid in a non-Hermitian periodic system. It is not shown that the biorthogonal left/right eigenvectors permit a well-localized packet whose energy and dynamics are captured by the standard gradient expansion without additional non-Hermitian corrections at the same order. This assumption is load-bearing for the extracted magnetization energy and the real/imaginary decomposition of the magnetic moment.

    Authors: We agree that the validity of the wave-packet construction is central. Our derivation employs the standard biorthogonal left and right eigenvectors to define the wave packet and performs the gradient expansion to first order, with non-Hermitian contributions entering through the complex Berry curvature and the modified equations of motion. Because the perturbations vary slowly, the leading-order localization and energy are determined by the local non-Hermitian Hamiltonian; any additional corrections arising purely from biorthogonality appear only at higher orders in the gradient expansion. To make this reasoning explicit, we will add a short paragraph in the revised manuscript (near the beginning of Sec. II) that recalls the conditions under which the semiclassical ansatz holds for non-Hermitian systems and confirms that no extra O(∇) terms are generated at the order we retain. revision: partial

  2. Referee: In the uniform-B case, the vector potential is treated perturbatively while preserving the first-order result. No explicit check is provided that non-unitary evolution or complex Berry curvature does not generate additional O(∇) terms that would alter the orbital magnetization energy expression.

    Authors: We thank the referee for this observation. In the uniform-field section the vector potential is introduced as a slow spatial perturbation, and the first-order energy correction is obtained by direct substitution into the general semiclassical energy formula that already incorporates the complex Berry curvature and the biorthogonal inner products. Non-unitary evolution is accounted for by the left-right normalization, which does not produce additional first-order gradient terms beyond those already present. Nevertheless, we acknowledge that an explicit verification would improve clarity. In the revised manuscript we will insert a brief calculation (or a short appendix remark) showing that the non-unitary and complex-Berry contributions cancel or vanish at O(∇) when the uniform-B limit is taken, thereby confirming that the orbital magnetization energy expression remains unchanged. revision: yes

Circularity Check

0 steps flagged

Semiclassical derivation self-contained; no reduction to inputs by construction

full rationale

The paper constructs a semiclassical wave-packet theory for non-Hermitian periodic systems under slow perturbations, derives the wave-packet energy to first order in gradients, and applies it to uniform B to obtain an orbital magnetization energy expression. It then introduces a non-Hermitian angular-momentum operator via non-Hermitian dynamics principles and states compatibility with Re(orbital magnetic moment). No quoted equations, self-citations, or fitted parameters in the available text reduce the claimed energy expression or moment decomposition to the inputs by definition. The gradient-expansion validity and biorthogonal eigenvector assumptions are external modeling choices, not internal redefinitions. The result therefore does not exhibit self-definitional, fitted-prediction, or self-citation-load-bearing circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the validity of the semiclassical approximation and the existence of a physically meaningful non-Hermitian angular momentum operator defined via non-Hermitian dynamics principles.

axioms (1)
  • domain assumption Semiclassical wave-packet dynamics applies to non-Hermitian periodic systems
    Invoked to derive energy to first order in perturbation gradients.
invented entities (1)
  • non-Hermitian generalization of the angular momentum operator no independent evidence
    purpose: To ensure compatibility with the real part of the orbital magnetic moment
    Defined from non-Hermitian dynamics principles; no independent falsifiable prediction given in abstract.

pith-pipeline@v0.9.0 · 5654 in / 1260 out tokens · 38757 ms · 2026-05-19T10:48:25.320466+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

71 extracted references · 71 canonical work pages

  1. [1]

    N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College Publishing, New York, 1976)

    work page 1976

  2. [2]

    R. D. King-Smith and D. Vanderbilt, Theory of polariza- tion of crystalline solids, Phys. Rev. B 47, 1651 (1993)

    work page 1993

  3. [3]

    Ortiz and R

    G. Ortiz and R. M. Martin, Macroscopic polarization as a geometric quantum phase: Many-body formulation, Phys. Rev. B 49, 14202 (1994)

    work page 1994

  4. [4]

    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)

    work page 1994

  5. [5]

    Chang and Q

    M.-C. Chang and Q. Niu, Berry phase, hyperorbits, and the Hofstadter spectrum: Semiclassical dynamics in mag- netic Bloch bands, Physical Review B 53, 7010 (1996)

    work page 1996

  6. [6]

    Thonhauser, D

    T. Thonhauser, D. Ceresoli, D. Vanderbilt, and R. Resta, Orbital magnetization in periodic insulators, Phys. Rev. Lett. 95, 137205 (2005)

    work page 2005

  7. [7]

    Chien and C

    C. Chien and C. Westgate, The Hall effect and its appli- cations (Springer Science & Business Media, 1980)

    work page 1980

  8. [8]

    Onoda, N

    S. Onoda, N. Sugimoto, and N. Nagaosa, Quantum trans- port theory of anomalous electric, thermoelectric, and thermal hall effects in ferromagnets, Phys. Rev. B 77, 165103 (2008)

    work page 2008

  9. [9]

    Sundaram and Q

    G. Sundaram and Q. Niu, Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects, Physical Review B 59, 14915 (1999)

    work page 1999

  10. [10]

    Chang and Q

    M.-C. Chang and Q. Niu, Berry curvature, orbital mo- ment, and effective quantum theory of electrons in elec- tromagnetic fields, Journal of Physics: Condensed Matter 20, 193202 (2008)

    work page 2008

  11. [11]

    Xiao, M.-C

    D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Reviews of Modern Physics 82, 1959 (2010). 7

    work page 1959

  12. [12]

    Jungwirth, Q

    T. Jungwirth, Q. Niu, and A. H. MacDonald, Anomalous Hall effect in ferromagnetic semiconductors, Phys. Rev. Lett. 88, 207208 (2002)

    work page 2002

  13. [13]

    Nagaosa, J

    N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Anomalous hall effect, Rev. Mod. Phys. 82, 1539 (2010)

    work page 2010

  14. [14]

    D. Xiao, J. Shi, and Q. Niu, Berry phase correction to electron density of states in solids, Phys. Rev. Lett. 95, 137204 (2005)

    work page 2005

  15. [15]

    D. Xiao, W. Yao, and Q. Niu, Valley-Contrasting Physics in Graphene: Magnetic Moment and Topological Trans- port, Physical Review Letters 99, 236809 (2007)

    work page 2007

  16. [16]

    C. M. Bender, Making sense of non-Hermitian Hamilto- nians, Reports on Progress in Physics 70, 947 (2007)

    work page 2007

  17. [17]

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

    work page 2018

  18. [18]

    V. M. Martinez Alvarez, J. E. Barrios Vargas, M. Berdakin, and L. E. F. Foa Torres, Topological states of non-Hermitian systems, Eur. Phys. J. Spec. Top. 227, 1295 (2018)

    work page 2018

  19. [19]

    Ghatak and T

    A. Ghatak and T. Das, New topological invariants in non-hermitian systems, Journal of Physics: Condensed Matter 31, 263001 (2019)

    work page 2019

  20. [20]

    F. K. Kunst and V. Dwivedi, Non-hermitian systems and topology: A transfer-matrix perspective, Phys. Rev. B 99, 245116 (2019)

    work page 2019

  21. [21]

    Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Hi- gashikawa, and M. Ueda, Topological Phases of Non- Hermitian Systems, Physical Review X 8, 031079 (2018)

    work page 2018

  22. [22]

    Multi-parameter quantum metrology.Advances in Physics69, 345–435 (2020)

    Y. Ashida, Z. Gong, and M. Ueda, Non- Hermitian physics, Advances in Physics 69, 249 (2020), publisher: Taylor & Francis eprint: https://doi.org/10.1080/00018732.2021.1876991

    work page doi:10.1080/00018732.2021.1876991 2020

  23. [23]

    Mart´ ınez-Strasser, M

    C. Mart´ ınez-Strasser, M. A. J. Herrera, A. Garc´ ıa-Etxarri, G. Palumbo, F. K. Kunst, and D. Bercioux, Topologi- cal properties of a non-hermitian quasi-1d chain with a flat band, Advanced Quantum Technologies 7, 2300225 (2024)

    work page 2024

  24. [24]

    D. R. Nelson and N. M. Shnerb, Non-Hermitian localiza- tion and population biology, Physical Review E 58, 1383 (1998)

    work page 1998

  25. [25]

    D. K. Lubensky and D. R. Nelson, Pulling Pinned Polymers and Unzipping DNA, Physical Review Letters 85, 1572 (2000)

    work page 2000

  26. [26]

    Feshbach, C

    H. Feshbach, C. E. Porter, and V. F. Weisskopf, Model for Nuclear Reactions with Neutrons, Physical Review 96, 448 (1954)

    work page 1954

  27. [27]

    Feshbach, Unified theory of nuclear reactions, Annals of Physics 5, 357 (1958)

    H. Feshbach, Unified theory of nuclear reactions, Annals of Physics 5, 357 (1958)

    work page 1958

  28. [28]

    Brandenbourger, X

    M. Brandenbourger, X. Locsin, E. Lerner, and C. Coulais, Non-reciprocal robotic metamaterials, Nature Communi- cations 10, 1 (2019)

    work page 2019

  29. [29]

    Ghatak, M

    A. Ghatak, M. Brandenbourger, J. van Wezel, and C. Coulais, Observation of non-hermitian topology and its bulk–edge correspondence in an active mechanical metamaterial, Proceedings of the National Academy of Sciences 117, 29561 (2020)

    work page 2020

  30. [30]

    Helbig, T

    T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, Generalized bulk–boundary correspondence in non-hermitian topolectrical circuits, Nature Physics 16 (2020)

    work page 2020

  31. [31]

    Hofmann, T

    T. Hofmann, T. Helbig, F. Schindler, N. Salgo, M. Brzezi´ nska, M. Greiter, T. Kiessling, D. Wolf, A. Voll- hardt, A. Kabaˇ si, C. H. Lee, A. Biluˇ si´ c, R. Thomale, and T. Neupert, Reciprocal skin effect and its realization in a topolectrical circuit, Phys. Rev. Res. 2, 023265 (2020)

    work page 2020

  32. [32]

    D. Zou, T. Chen, W. He, J. Bao, C. H. Lee, H. Sun, and X. Zhang, Observation of hybrid higher-order skin- topological effect in non-Hermitian topolectrical circuits, Nature Communications 12, 7201 (2021)

    work page 2021

  33. [33]

    Weidemann, M

    S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, Topological funneling of light, Science 368, 311 (2020), https://www.science.org/doi/pdf/10.1126/science.aaz8727

    work page doi:10.1126/science.aaz8727 2020

  34. [34]

    Xiao, T.-S

    L. Xiao, T.-S. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, Non-hermitian bulk–boundary correspon- dence in quantum dynamics, Nature Physics 16, 1 (2020)

    work page 2020

  35. [35]

    L. Xiao, T. Deng, K. Wang, Z. Wang, W. Yi, and P. Xue, Observation of non-bloch parity-time symmetry and ex- ceptional points, Phys. Rev. Lett. 126, 230402 (2021)

    work page 2021

  36. [36]

    C. Wang, Z. Zhang, Z. Wang, W. Xiong, and L. Deng, Non-Hermitian coupling of orbital angular momentum modes in optical waveguides, JOSA B 40, 682 (2023)

    work page 2023

  37. [37]

    H. M. Hurst and B. Flebus, Non-Hermitian physics in magnetic systems, Journal of Applied Physics132, 220902 (2022)

    work page 2022

  38. [38]

    T. Yu, J. Zou, B. Zeng, J. W. Rao, and K. Xia, Non- Hermitian topological magnonics, Physics Reports Non- Hermitian topological magnonics, 1062, 1 (2024)

    work page 2024

  39. [39]

    Liang, D

    Q. Liang, D. Xie, Z. Dong, H. Li, H. Li, B. Gadway, W. Yi, and B. Yan, Dynamic signatures of non-hermitian skin effect and topology in ultracold atoms, Phys. Rev. Lett. 129, 070401 (2022)

    work page 2022

  40. [40]

    Shen and L

    H. Shen and L. Fu, Quantum Oscillation from In-Gap States and a Non-Hermitian Landau Level Problem, Phys- ical Review Letters 121, 026403 (2018)

    work page 2018

  41. [41]

    Okuma and M

    N. Okuma and M. Sato, Topological Phase Transition Driven by Infinitesimal Instability: Majorana Fermions in Non-Hermitian Spintronics, Physical Review Letters 123, 097701 (2019)

    work page 2019

  42. [42]

    Ozawa and T

    T. Ozawa and T. Hayata, Two-dimensional lattice with an imaginary magnetic field, Physical Review B 109, 085113 (2024)

    work page 2024

  43. [43]

    Lu, X.-X

    M. Lu, X.-X. Zhang, and M. Franz, Magnetic Suppression of Non-Hermitian Skin Effects, Physical Review Letters 127, 256402 (2021)

    work page 2021

  44. [44]

    C.-A. Li, B. Trauzettel, T. Neupert, and S.-B. Zhang, Enhancement of Second-Order Non-Hermitian Skin Effect by Magnetic Fields, Physical Review Letters 131, 116601 (2023)

    work page 2023

  45. [45]

    Zhang and M

    X.-X. Zhang and M. Franz, Non-Hermitian Exceptional Landau Quantization in Electric Circuits, Physical Review Letters 124, 046401 (2020)

    work page 2020

  46. [46]

    Xu, S.-T

    Y. Xu, S.-T. Wang, and L.-M. Duan, Weyl Exceptional Rings in a Three-Dimensional Dissipative Cold Atomic Gas, Physical Review Letters 118, 045701 (2017)

    work page 2017

  47. [47]

    Silberstein, J

    N. Silberstein, J. Behrends, M. Goldstein, and R. Ilan, Berry connection induced anomalous wave-packet dynam- ics in non-Hermitian systems, Physical Review B 102, 245147 (2020)

    work page 2020

  48. [48]

    B. Alon, R. Ilan, and M. Goldstein, Quantum metric de- pendent anomalous velocity in systems subject to complex electric fields, Physical Review B 110, 245103 (2024)

    work page 2024

  49. [49]

    Aharonov and D

    Y. Aharonov and D. Bohm, Significance of electromag- netic potentials in the quantum theory, Phys. Rev. 115, 485 (1959). 8

    work page 1959

  50. [50]

    M. V. Berry and R. Uzdin, Slow non-Hermitian cycling: exact solutions and the Stokes phenomenon, Journal of Physics A: Mathematical and Theoretical 44, 435303 (2011)

    work page 2011

  51. [51]

    Ib´ a˜ nez and J

    S. Ib´ a˜ nez and J. G. Muga, Adiabaticity condition for non- Hermitian Hamiltonians, Physical Review A 89, 033403 (2014)

    work page 2014

  52. [52]

    Wang, L.-J

    H. Wang, L.-J. Lang, and Y. D. Chong, Non-Hermitian dynamics of slowly varying Hamiltonians, Physical Review A 98, 012119 (2018)

    work page 2018

  53. [53]

    H. Schomerus, Nonreciprocal response theory of non- Hermitian mechanical metamaterials: Response phase transition from the skin effect of zero modes, Physical Review Research 2, 013058 (2020)

    work page 2020

  54. [54]

    Y. M. R. Hu, E. A. Ostrovskaya, and E. Estrecho, Role of quantum geometric tensor on wavepacket dynamics in two-dimensional non-hermitian systems (2024)

    work page 2024

  55. [55]

    Dattoli, R

    G. Dattoli, R. Mignani, and A. Torre, Geometrical phase in the cyclic evolution of non-hermitian systems, Journal of Physics A: Mathematical and General 23, 5795 (1990)

    work page 1990

  56. [56]

    Longhi and L

    S. Longhi and L. Feng, Complex Berry phase and imper- fect non-Hermitian phase transitions, Physical Review B 107, 085122 (2023)

    work page 2023

  57. [57]

    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)

    work page 2023

  58. [58]

    W. D. Heiss, The physics of exceptional points, Journal of Physics A: Mathematical and Theoretical 45, 444016 (2012)

    work page 2012

  59. [59]

    Miri and A

    M.-A. Miri and A. Al` u, Exceptional points in optics and photonics, Science 363, eaar7709 (2019)

    work page 2019

  60. [60]

    E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Ex- ceptional topology of non-Hermitian systems, Reviews of Modern Physics 93, 015005 (2021)

    work page 2021

  61. [61]

    Nenciu and G

    G. Nenciu and G. Rasche, On the adiabatic theorem for nonself-adjoint Hamiltonians, Journal of Physics A: Mathematical and General 25, 5741 (1992)

    work page 1992

  62. [62]

    See Supplmental Material for further details of the deriva- tions

    work page

  63. [63]

    Niu and G

    Q. Niu and G. Sundaram, Electron dynamics in slowly modulated crystals: theorems of power and adiabatic drifting, Physica E: Low-dimensional Systems and Nanos- tructures Proceedings of an International Workshop and Seminar on the Dynamics of Complex Systems, 9, 327 (2001)

    work page 2001

  64. [64]

    Here we use the term ’Ehrenfest Theorem’ to refer to the equation of motion for the expectation value of any operator, not necessarily position or momentum

    work page

  65. [65]

    M. M. Sternheim and J. F. Walker, Non-hermitian hamiltonians, decaying states, and perturbation theory, Phys. Rev. C 6, 114 (1972)

    work page 1972

  66. [66]

    Nenciu, Dynamics of band electrons in electric and mag- netic fields: rigorous justification of the effective Hamilto- nians, Reviews of Modern Physics 63, 91 (1991)

    G. Nenciu, Dynamics of band electrons in electric and mag- netic fields: rigorous justification of the effective Hamilto- nians, Reviews of Modern Physics 63, 91 (1991)

    work page 1991

  67. [67]

    G. Nenciu, On asymptotic perturbation theory for Quan- tum Mechanics: Almost invariant subspaces and gauge invariant magnetic perturbation theory, Journal of Math- ematical Physics 43, 10.1063/1.1408281 (2000)

    work page doi:10.1063/1.1408281 2000

  68. [68]

    Y. Gao, S. A. Yang, and Q. Niu, Field Induced Positional Shift of Bloch Electrons and Its Dynamical Implications, Physical Review Letters 112, 166601 (2014)

    work page 2014

  69. [69]

    Y. Gao, S. A. Yang, and Q. Niu, Geometrical effects in orbital magnetic susceptibility, Physical Review B 91, 214405 (2015)

    work page 2015

  70. [70]

    Longhi, Topological phase transition in non-hermitian quasicrystals, Phys

    S. Longhi, Topological phase transition in non-hermitian quasicrystals, Phys. Rev. Lett. 122, 237601 (2019)

    work page 2019

  71. [71]

    Balian, Relation between position and quasi- momentum operators in band theory, Journal de Physique 50, 2629 (1989)

    R. Balian, Relation between position and quasi- momentum operators in band theory, Journal de Physique 50, 2629 (1989). 9 Supplemental Material S.I. SINGLE-BAND APPROXIMA TION FOR TIME-DEPENDENT HAMIL TONIANS In this appendix we derive in detail the expression for the effective energy of a possibly perturbed, time-dependent Bloch Hamiltonian within the si...

    work page 1989