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Non-Hermitian wave-packet dynamics and its realization within a non-Hermitian chiral cavity

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The imaginary part of the Berry curvature acts as an anomalous force on non-Hermitian wave packets, and a two-sign 'complex chirality' sets the direction and dissipation of edge states.

desk verdict A correct-looking central derivation and a clean toy-model test, but the Haldane bulk numerics omit the Berry phase, so the abstract overclaims; still deserves refereeing. read the letter →

arxiv 2501.12163 v1 pith:BJ4EUCM4 submitted 2025-01-21 cond-mat.mes-hall quant-ph

Weicen Dong , Qing-Dong Jiang , Matteo Baggioli This is my paper · ORCID
classification cond-mat.mes-hallquant-ph
keywords non-HermitiantopologysemiclassicalequationsofmotioncomplexBerrycurvaturebiorthogonalphaseHaldanemodelnon-reciprocalhoppingchiralopticalcavityedge-statechirality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to give non-Hermitian topological wave packets a semiclassical description on the same footing as Hermitian ones. Using the biorthogonal complex Berry phase, it derives equations of motion in which the complex Berry curvature produces two effects: a real-part term that bends the real-space velocity and an imaginary-part term that acts as an anomalous force steering the packet in momentum space, alongside a gain-gradient drift. To anchor the formalism, the authors build a non-Hermitian Haldane model with non-reciprocal next-nearest-neighbor hoppings and show that its 'complex chirality'—two signs, one from the real imbalance and one from the imaginary imbalance—separately controls the imaginary and real parts of the Berry curvature, the Chern number, and the direction and dissipation of edge states. Direct numerical wave-packet simulations match the equations, and a non-Hermitian chiral cavity is proposed as a physical platform. If correct, the paper supplies a transport theory for non-Hermitian topological matter and a concrete experimental knob for controlling edge-state behavior.

What carries the argument

The machinery has two parts. First, the biorthogonal Berry connection $A_k = i\langle\psi_L(k)|\nabla_k\psi_R(k)\rangle$, built from left and right eigenstates of the non-Hermitian Hamiltonian, defines a generally complex Berry curvature $B_k = \nabla_k \times A_k$; the paper tracks wave-packet maxima by deforming the $k$-integration contour into the complex plane, which converts the Berry phase into the identity $d(\nabla_k\gamma_k)/dt = F\times B_k$. Second, the 'complex chirality' of the non-Hermitian Haldane model is the sign pair $S_R=\mathrm{sign}(n-m)$ and $S_I=\mathrm{sign}(b-a)$ attached to the real and imaginary imbalances of the non-reciprocal next-nearest-neighbor hoppings $m+ia$ and $n+ib$; in this model $S_I$ fixes the Chern number and edge-state direction, while $S_R$ fixes the sign of the imaginary Berry curvature and the gain/loss asymmetry of edge states.

What would settle it

Take the same strip as in Fig. 5 and compute its open-boundary spectrum using the generalized Brillouin zone (non-Bloch band theory) at large non-reciprocal hopping, rather than the periodic Bloch ansatz. If the number or direction of edge states disagrees with $C_1=-\mathrm{sign}(b-a)$, or if the sign of $\mathrm{Im}(E)$ on the edge modes does not flip with $\mathrm{sign}(n-m)$, the claimed bulk-edge correspondence is false, even though the wave-packet equations (7)-(8) could still hold.

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Extended reading notes

Core claim

Using the biorthogonal Berry connection, the paper derives the semiclassical equations of motion $\dot{\mathbf{r}}_M = \mathrm{Re}(\nabla_{\mathbf{k}} E)|_{\mathbf{k}_M} - \mathrm{Re}(\mathbf{F} \times \mathbf{B})|_{\mathbf{k}_M}$ and $\dot{\mathbf{k}}_M = \mathbf{F} + \Delta^2[\mathrm{Im}(\nabla_{\mathbf{k}} E)|_{\mathbf{k}_M} - \mathrm{Im}(\mathbf{F} \times \mathbf{B})|_{\mathbf{k}_M}]$, where $\mathbf{B}$ is the biorthogonal Berry curvature. The imaginary part of $\mathbf{B}$ therefore produces an anomalous force rather than only an anomalous velocity, and gradients of the imaginary energy push the packet toward gain. In the non-Hermitian Haldane model with non-reciprocal next-nearest-neighbor hoppings $m+ia$ and $n+ib$, the paper finds $C_1 = -\mathrm{sign}(b-a)$ and $\mathrm{sign}(\mathrm{Im}\,B) = \mathrm{sign}(n-m)$, so the sign pair $(S_R,S_I)=(\mathrm{sign}(n-m),\mathrm{sign}(b-a))$ organizes the bulk Chern number, the direction of edge states, and whether edge states amplify or decay. Numerical simulations of both edge and bulk wave packets confirm the derived equations.

Load-bearing premise

The load-bearing premise is that the periodic-lattice Chern number $C_1=-\mathrm{sign}(b-a)$, computed from the biorthogonal Berry curvature, correctly predicts the number and chirality of open-boundary edge states; the authors defer a proof of this non-Hermitian bulk-boundary correspondence, and non-reciprocal systems are exactly where non-Bloch corrections can break such predictions.

Editorial extensions

If this is right

  • A Gaussian wave packet in any non-Hermitian band with complex Berry curvature obeys the new equations of motion, so even a constant external force bends the packet's momentum-space path perpendicular to the force through the term $-\Delta^2\mathrm{Im}(F\times B)$.
  • In the non-Hermitian Haldane model, the Chern number is $C_1 = -\mathrm{sign}(b-a)$, which means the imbalance of the imaginary parts of the next-nearest-neighbor hoppings alone decides whether edge states move clockwise or counterclockwise.
  • The sign of $n-m$, the imbalance of the real parts, fixes the sign of $\mathrm{Im}(B)$ and the sign of the edge-state gain or loss, so one lattice can amplify a chiral edge state moving one way and damp the one moving the other way.
  • For one-dimensional edge transport, the packet's momentum-space center drifts at the rate $\Delta^2\mathrm{Im}(dE/dk)$, toward gain, while its real-space center moves with $\mathrm{Re}(dE/dk)$; the simulations reproduce both motions.
  • The non-Hermitian chiral cavity maps to the model with explicit relations for $m,n,a,b$, so tuning the cavity chirality and loss should control both the Chern number and the edge-state dissipation rate.

Reading between the lines

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

  • If the same peak-tracking derivation is repeated in a Lindblad master equation, the mean position and momentum of an ensemble of trajectories should acquire the same anomalous force $-\Delta^2\mathrm{Im}(F\times B)$ plus stochastic corrections; this would extend the result to genuinely open quantum systems, which the paper does not do.
  • The anomalous force is perpendicular to the external force and grows with the squared packet width $\Delta^2$, so it could be isolated experimentally by sending a wide packet with a tunable drive and measuring transverse drift in momentum space; this test is not proposed in the paper.
  • The cavity mapping implies a direct control protocol: sweeping the cavity polarization angle through the point where $\cos(2\theta)=0$ should flip the Chern number, and switching the cavity from loss to gain should reverse the edge-state dissipation asymmetry; these are corollaries of Eqs. (32)-(35) rather than stated proposals.
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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

4 major / 6 minor

Summary. Using the biorthogonal (complex) Berry connection, the paper derives semiclassical equations of motion for the peak position and momentum of a wave packet in a non-Hermitian topological band. For a Gaussian packet, the claimed EOM are Eq. (7), ṙ_M = Re(∇_k E) − Re(F × B), and Eq. (8), ḳ_M = F + Δ²[Im(∇_k E) − Im(F × B)], where the second Δ² term is a new k-space anomalous force; a non-Gaussian extension is given by Eqs. (13)-(14). The Gaussian result is tested in a toy model with linear complex dispersion and constant complex Berry curvature (Fig. 1). The authors then introduce a non-Hermitian Haldane model with non-reciprocal next-nearest-neighbor hoppings and define a "complex chirality" (S_R = sign(n−m), S_I = sign(b−a)), finding C₁ = −S_I and a sign pattern of Im(B) controlled by S_R. Strip spectra show chiral edge states whose direction tracks S_I and whose gain/loss tracks S_R. Edge and bulk wave-packet simulations are reported, and an optical chiral cavity realization is proposed and mapped to the model parameters.

Significance. If valid, the EOM add a genuinely new term to the semiclassical transport of non-Hermitian bands: an anomalous force −Δ²Im(F × B) in k-space, alongside the anomalous velocity −Re(F × B), and the complex-chirality model provides a concrete two-parameter platform with falsifiable predictions for edge-state direction and dissipation rate. The strengths of the manuscript are the explicit derivation from stated assumptions (no fitting, no free parameters), the toy-model test of Fig. 1 that directly exercises the Berry-curvature terms, the computed (not imposed) Chern numbers, and the independent cavity parameter mapping. The main weaknesses are that the Haldane-model bulk simulation omits the Berry phase entirely, so the topological content of the EOM is not numerically confirmed in the physical model; the non-Gaussian comparison of Fig. 7 is under-specified; and the bulk-boundary correspondence linking C₁ to the edge states is assumed rather than proven for a non-reciprocal non-Hermitian system. These are gaps in verification and in claim strength rather than errors in the core derivation; closing them or qualifying the claims would make the paper a solid contribution.

major comments (4)
  1. [Sec. III.E, Fig. 7, and Abstract] The bulk wave-packet simulation of Sec. III.E (Fig. 7) omits the Berry phase by the authors' own statement ("the Berry phase is not included in the simulation and theoretical calculation due to numerical challenges caused by its gauge dependence"). Since the terms −Re(F × B) and −Δ²Im(F × B) in Eqs. (13)-(14) enter the equations only through the Berry phase via the identity (6), this simulation cannot test the anomalous velocity or the anomalous force; the "good agreement" reported in Fig. 7(i,j) validates only the gain-gradient drift terms Re(∇_k E) and Δ²Im(∇_k E). The only direct check of the Berry-curvature terms is the toy model of Fig. 1, which uses a linear dispersion and a constant, artificial connection A_k = (−Bk_y, 0). The abstract's sentence "Our analytical findings are confirmed by direct numerical simulations" is therefore not supported for the paper's central topological predictions; the authors should either include the Berry phase in the Haldane-model simulation (e.g., by working in a fixed smooth gauge with appropriate branch cuts) or explicitly qualify which findings are confirmed by which figure.
  2. [Sec. II.B, Eq. (14)] Equation (14) is not a closed equation of motion for k_M(t): it determines the time derivative of ∇_k f at the peak, and converting it into a prediction for k_M requires either a prescription for the time evolution of f(k,t,F) or a fixed-shape assumption. The paper does not state which procedure produced the theoretical curves in Fig. 7(i,j), and since the authors themselves find that the initial Gaussian becomes anisotropic and non-Gaussian (Fig. 7(c-e)), a fixed-Gaussian assumption would contradict that observation, whereas inserting the simulated f into Eq. (14) would make the comparison tautological. Please specify the procedure used to generate the theory curves.
  3. [Sec. II.B, Eq. (15)] There is a sign error in the printed form of Eq. (15): with the convention |Ψ(k,t)⟩ = exp[−f(k,t,F)] of Eq. (11), the Gaussian distribution of Eq. (1) corresponds to f_Gaussian = +(k−k̄)²/(2Δ²). With the printed minus sign, exp[−f] is not normalizable, and inserting f_Gaussian into Eq. (14) yields Eq. (8) with the opposite signs on both Δ² terms, so the claimed reduction of Eq. (14) to Eq. (8) is not reproduced as written.
  4. [Sec. III.C and Sec. V] The attribution of the edge-state direction and dissipation rate to the complex chirality (abstract and Sec. III.C) relies on the standard bulk-boundary correspondence between the periodic-boundary Chern number C₁ of Eq. (20) and the open-boundary strip spectra of Fig. 5. For non-Hermitian systems with non-reciprocal hoppings (here n ≠ m), non-Bloch corrections are known to alter this correspondence, and the authors explicitly defer its proof ("a comprehensive investigation of the precise mathematical proof for the novel bulk-boundary correspondence") to future work in Sec. V. The statement in the abstract is therefore stronger than what the manuscript establishes; either compute the non-Bloch (generalized Brillouin zone) topological invariant, or present the edge-state correspondence as an observation for the specific zigzag and armchair strips of Fig. 5 and Appendix B.
minor comments (6)
  1. [Sec. III.A] The sentence "PT symmetry is always broken, hence the eigenvalues of the Hamiltonian are expected to be complex" is inaccurate in the Hermitian limit m = n, a = −b, where the spectrum is real; please write "generically broken away from the Hermitian limit."
  2. [Sec. IV, Eqs. (30)-(35)] The parameter α introduced after Eq. (29) contains the factor (1 + g²γ)/ω_c, and with ω_c = |ω_c|/γ it is complex for generic complex γ, yet Eqs. (30)-(35) treat α Re(γ) and α Im(γ) as real quantities; please clarify the conditions under which the mapping to real m, n, a, b holds.
  3. [Sec. III.C] The phrase "a anticlowise/clockwise moving edge state" contains the typo "anticlowise"; in addition, please spell out the convention by which C₁ = +1 is identified with an anticlockwise edge mode, since this identification is not a universal convention.
  4. [Sec. II.A, Eqs. (5)-(6)] The notation ∇_k γ_k(t)|_{k_M} is ambiguous (derivative of the Berry-phase endpoint integral versus evaluation at the saddle point), and the identity (6), cited to the Hermitian literature, is used at a complex stationary point k_M; please justify its validity for the complex biorthogonal connection and for complex k.
  5. [Sec. I] The statement that a semiclassical EOM for non-Hermitian systems "remains an open question" is not reconciled with Refs. [31-35] and [38], which already treat non-Hermitian wave-packet dynamics; please state explicitly the new ingredients, in particular the biorthogonal connection giving time-independent curvature and the Δ² anomalous force.
  6. [Sec. III.B] The sentence "When S_I ≠ 0, the time-reversal symmetry, H_k = H*_{−k} and B_k = −B_{−k} [7], is explicitly broken..." states the time-reversal identities and then asserts that the symmetry is broken; please rewrite so that the identities are not presented as properties of the actual model.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the semiclassical EOM follow from stated biorthogonal-ansatz assumptions, the Chern/edge-state results are computed from the model rather than imposed, and the numerical caveats are verification gaps rather than circular reductions.

full rationale

The derivation of Eqs. (7)-(8) starts from the Gaussian wave-packet ansatz Eq. (2), the biorthogonal Berry connection A = i⟨ψ_L|∇ψ_R⟩, the saddle-point conditions Eq. (5), and the external identity Eq. (6); no target result is inserted as an input. The Haldane-model statements are obtained by direct computation: C1 = -SI (Eq. (22)) is the outcome of integrating Eq. (21), the signs of Re(B) and Im(B) are read off the numerically computed Berry curvature in Fig. 4, and the edge-state direction and Im(E_k) are read off the strip spectra in Fig. 5, so the complex-chirality correspondence is not fitted. The cavity realization in Sec. IV is a parameter mapping derived from the similarity transformation and Peierls substitution (Eqs. (29)-(35)), not a prediction fed back from the target model; the self-citations [42,43] used for the cavity transformation are not load-bearing for the central EOM or the topological correspondence. The remaining caveats are verification gaps, not circularity: Sec. III E states that "the Berry phase is not included in the simulation and theoretical calculation due to numerical challenges caused by its gauge dependence," so Fig. 7(i,j) tests mainly the drift terms, and Sec. V defers "the precise mathematical proof for the novel bulk-boundary correspondence." These weaken the numerical confirmation and the bulk-edge claim, but no equation in the paper reduces to its own input by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numerical parameters are fitted to the target results. The main load-bearing axioms are the biorthogonal completeness away from exceptional points, the complex generalization of the Berry phase identity, the real-axis peak assumption, the unproved non-Hermitian bulk-boundary correspondence, and the first-order similarity transformation in the cavity section.

assumptions (5)
  • domain assumption Biorthogonal completeness and normalization hold for the non-Hermitian Hamiltonian (langle psiL_i | psiR_j rangle = delta_ij, sum |psiR rangle langle psiL| = I), requiring diagonalizability and no exceptional points in the relevant parameter regime.
    Invoked at the start of Sec. II A after Eq. (1); fails at the exceptional rings shown in Fig. 3, so the derivation applies away from those rings.
  • standard math The identity d(gradient_k gamma_k(t))/dt = F x B (Eq. 6) carries over to the complex biorthogonal Berry connection and complex Berry curvature.
    Used to convert Eq. (5) into Eqs. (7)-(8); taken from Ref. [8] in the Hermitian context and assumed to hold for complex A and B.
  • domain assumption The wave-packet maximum in real space is found at Im k = 0, and |psiR(k)> varies slowly over the Gaussian packet while Delta^2 terms capture the k-space drift.
    Sec. II A, hierarchy of scales and gradient_r Re(g) = 0; underlies the identification of rM and kM as real maxima.
  • domain assumption The Bloch Chern number from biorthogonal Berry curvature in the periodic system determines the edge-state chirality of the open-boundary strip (bulk-boundary correspondence).
    Assumed in Sec. III C to connect C1 to edge states; explicitly flagged as lacking a proof in Sec. V (future studies, precise mathematical proof).
  • domain assumption The similarity transformation U = exp(-i xi p . pi / hbar) decouples light and matter to first order in xi, and the resulting effective Hamiltonian maps exactly onto the tight-binding model with real m, n, a, b.
    Sec. IV and Appendix C; expansion to first order in xi, plus the unstated assumption that alpha is real in Eqs. (30)-(35).

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Pith. "Pith review of Non-Hermitian wave-packet dynamics and its realization within a non-Hermitian chiral cavity." pith.science (2026). https://pith.science/paper/BJ4EUCM4

@misc{pith2026250112163,
  author       = {Pith},
  title        = {Pith review of: Non-Hermitian wave-packet dynamics and its realization within a non-Hermitian chiral cavity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BJ4EUCM4}},
  note         = {Machine review of arXiv:2501.12163}
}
abstract

Topological wave-packet dynamics provide a powerful framework for studying quantum transport in topological materials. However, extending this approach to non-Hermitian quantum systems presents several important challenges, primarily due to ambiguities in defining the Berry phase and the non-unitary evolution of the wave-packets when $\mathcal{P}\mathcal{T}$ symmetry is broken. In this work, we adopt the complex Berry phase definition using the bi-orthogonal formalism and derive the semiclassical equations of motion (EOM) for a wave-packet in a non-Hermitian topological system. Interestingly, we find that the complex Berry curvature introduces both an anomalous velocity and an anomalous force into the semiclassical EOM. To validate the derived EOM, we design a non-Hermitian Haldane model featuring non-reciprocal next-nearest-neighbor (NNN) hopping, where the imbalance in the NNN hopping amplitudes gives rise to an emergent `complex chirality'. We reveal that the real and imaginary components of the complex chirality dictate the signs of both the real and imaginary parts of the complex Berry curvature, as well as the direction and dissipation rate of the edge states. Our analytical findings are confirmed by direct numerical simulations of the wave-packet dynamics. Finally, we suggest a potential experimental realization of this complex Haldane model using a non-Hermitian optical chiral cavity, providing a promising platform for testing our theoretical predictions.

Figures

Figures reproduced from arXiv: 2501.12163 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. by red solid lines. These exceptional rings shrink to degenerated points, K and K′ , when SR = SI = 0. On the other hand, when a ̸= b (right column of the same figure), a topologically protected gap between the two bands appears. B. Berry curvature and Chern number The topological invariant Chern number for the ith band is defined by the integral of the Berry curvature B i k over the first Brillouin zone [28, 46]: C… view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: ). For the ‘anticlockwise’ NNN hopping, ϕij = γα, where α = cos(2θ) g 2 (1+g 2γ) (1+g 2) 2 V0π ωc . From there, we can de￾rive that: t ′ 2 = t2e iα Re(γ) e −α Im(γ) , (30) t ′′ 2 = t2e −iα Re(γ) e α Im(γ) . (31) This implies that the NNN hopping mechanism are now non-r…
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Anomalous Wave-Packet Dynamics in One-Dimensional Non-Hermitian Lattices

    physics.optics 2025-12 accept novelty 6.0 of 10

    In 1D non-Hermitian lattices, gain/loss alone makes wave packets drift in momentum, self-Bloch-oscillate, jump between momentum states even with real spectra, and reflect with positive or negative time shifts.

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Works this paper leans on

53 extracted references · 32 canonical work pages · cited by 1 Pith paper

  1. [1]

    Nash, History of topology pp

    C. Nash, History of topology pp. 359–415 (1999), URL https://www.sciencedirect.com/science/article/ pii/B9780444823755500134

    work page 1999

  2. [2]

    M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010), URL https://link.aps.org/doi/10. 1103/RevModPhys.82.3045

    2010

  3. [3]

    Qi and S.-C

    X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011), URL https://link.aps.org/doi/10. 1103/RevModPhys.83.1057

    work page 2011

  4. [4]

    Karplus and J

    R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 (1954), URL https://link.aps.org/doi/10. 1103/PhysRev.95.1154

    work page 1954

  5. [5]

    Chang and Q

    M.-C. Chang and Q. Niu, Phys. Rev. Lett. 75, 1348 (1995), URL https://link.aps.org/doi/10. 1103/PhysRevLett.75.1348

    work page 1995

  6. [7]

    S. M. Girvin and K. Yang, Modern Condensed Mat- ter Physics (Cambridge University Press, 2019), URL https://doi.org/10.1017/9781316480649

    work page doi:10.1017/9781316480649 2019

  7. [8]

    Y. D. Chong, Phys. Rev. B 81, 052303 (2010), URL https://link.aps.org/doi/10.1103/PhysRevB. 81.052303

    doi:10.1103/physrevb 2010

  8. [9]

    C. M. Bender, Reports on Progress in Physics 70, 947 (2007), URL https://dx.doi.org/10.1088/0034-4885/ 70/6/R03

    doi:10.1088/0034-4885/ 2007

Show all 53 references
  1. [10]

    Z.-J. Li, G. Cardoso, E. J. Bergholtz, and Q.-D. Jiang, 10 Phys. Rev. Res. 6, 043023 (2024), URL https://link. aps.org/doi/10.1103/PhysRevResearch.6.043023

    2024 doi

  2. [11]

    Yang, Q.-D

    F. Yang, Q.-D. Jiang, and E. J. Bergholtz, Phys. Rev. Res. 4, 023160 (2022), URL https://link.aps.org/ doi/10.1103/PhysRevResearch.4.023160

    2022 doi

  3. [12]

    L. Feng, R. El-Ganainy, and L. Ge, Nature Photon- ics 11, 752 (2017), URL https://doi.org/10.1038/ s41566-017-0031-1

    2017

  4. [13]

    Okuma and M

    N. Okuma and M. Sato, Annual Review of Condensed Matter Physics 14, 83 (2023), URL https://doi.org/ 10.1146/annurev-conmatphys-040521-033133

    2023 doi

  5. [14]

    Z. G. Yuto Ashida and M. Ueda, Advances in Physics 69, 249 (2020)

    2020

  6. [15]

    E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Rev. Mod. Phys. 93, 015005 (2021), URL https://link.aps. org/doi/10.1103/RevModPhys.93.015005

    2021 doi

  7. [16]

    Kawabata, K

    K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Phys. Rev. X 9, 041015 (2019), URL https://link.aps.org/ doi/10.1103/PhysRevX.9.041015

    2019 doi

  8. [17]

    Ghatak and T

    A. Ghatak and T. Das, Journal of Physics: Condensed Matter 31, 263001 (2019), URL https://dx.doi.org/ 10.1088/1361-648X/ab11b3

    2019 doi

  9. [18]

    Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Phys. Rev. X 8, 031079 (2018), URL https://link.aps.org/doi/10. 1103/PhysRevX.8.031079

    2018

  10. [19]

    J. Y. Lee, J. Ahn, H. Zhou, and A. Vishwanath, Phys. Rev. Lett. 123, 206404 (2019), URL https://link.aps. org/doi/10.1103/PhysRevLett.123.206404

    2019 doi

  11. [20]

    Leykam, K

    D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, Phys. Rev. Lett. 118, 040401 (2017), URL https://link.aps.org/doi/10.1103/PhysRevLett. 118.040401

    2017 doi

  12. [21]

    S. Yao, F. Song, and Z. Wang, Phys. Rev. Lett. 121, 136802 (2018), URL https://link.aps.org/doi/10. 1103/PhysRevLett.121.136802

    2018

  13. [22]

    V. M. Martinez Alvarez, J. E. Barrios Vargas, M. Berdakin, and L. E. F. Foa Torres, The Eu- ropean Physical Journal Special Topics 227, 1295 (2018), ISSN 1951-6401, URL https://doi.org/10. 1140/epjst/e2018-800091-5

    2018

  14. [23]

    K. Ding, C. Fang, and G. Ma, Nature Reviews Physics 4, 745 (2022), ISSN 2522-5820, URL https://doi.org/ 10.1038/s42254-022-00516-5

    2022 doi

  15. [24]

    Yokomizo and S

    K. Yokomizo and S. Murakami, Phys. Rev. Lett. 123, 066404 (2019), URL https://link.aps.org/doi/10. 1103/PhysRevLett.123.066404

    2019

  16. [25]

    Yao and Z

    S. Yao and Z. Wang, Phys. Rev. Lett. 121, 086803 (2018), URL https://link.aps.org/doi/10. 1103/PhysRevLett.121.086803

    2018

  17. [26]

    J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, Phys. Rev. Lett. 115, 040402 (2015), URL https://link.aps. org/doi/10.1103/PhysRevLett.115.040402

    2015 doi

  18. [27]

    D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, Phys. Rev. Lett. 124, 056802 (2020), URL https://link.aps. org/doi/10.1103/PhysRevLett.124.056802

    2020 doi

  19. [28]

    Esaki, M

    K. Esaki, M. Sato, K. Hasebe, and M. Kohmoto, Phys. Rev. B 84, 205128 (2011), URL https://link.aps.org/ doi/10.1103/PhysRevB.84.205128

    2011 doi

  20. [29]

    Zhang, Z

    K. Zhang, Z. Yang, and C. Fang, Phys. Rev. Lett. 125, 126402 (2020), URL https://link.aps.org/doi/ 10.1103/PhysRevLett.125.126402

    2020 doi

  21. [30]

    Sun, Physica Scripta 48, 393 (1993), URL https: //dx.doi.org/10.1088/0031-8949/48/4/002

    C.-P. Sun, Physica Scripta 48, 393 (1993), URL https: //dx.doi.org/10.1088/0031-8949/48/4/002

    1993 doi

  22. [31]

    Y.-M. R. Hu, E. A. Ostrovskaya, and E. Estrecho, Opt. Mater. Express 14, 664 (2024), URL https://opg. optica.org/ome/abstract.cfm?URI=ome-14-3-664

    2024

  23. [32]

    Y. M. R. Hu, E. A. Ostrovskaya, and E. Estrecho, Role of quantum geometric tensor on wavepacket dynam- ics in two-dimensional non-hermitian systems (2024), 2412.08141, URL https://arxiv.org/abs/2412.08141

    2024 arXiv

  24. [33]

    Silberstein, J

    N. Silberstein, J. Behrends, M. Goldstein, and R. Ilan, Phys. Rev. B 102, 245147 (2020), URL https://link. aps.org/doi/10.1103/PhysRevB.102.245147

    2020 doi

  25. [34]

    Chen Ye, W

    C. Chen Ye, W. L. Vleeshouwers, S. Heatley, V. Grit- sev, and C. Morais Smith, Phys. Rev. Res. 6, 023202 (2024), URL https://link.aps.org/doi/10. 1103/PhysRevResearch.6.023202

    2024

  26. [35]

    Wang, Y.-L

    J.-H. Wang, Y.-L. Tao, and Y. Xu, Chinese Physics Let- ters 39, 010301 (2022), URL https://cpl.iphy.ac.cn/ Y2022/V39/I1/010301

    2022

  27. [36]

    Res´ endiz-V´ azquez, K

    P. Res´ endiz-V´ azquez, K. Tschernig, A. Perez-Leija, K. Busch, and R. d. J. Le´ on-Montiel, Phys. Rev. Res. 2, 013387 (2020), URL https://link.aps.org/doi/10. 1103/PhysRevResearch.2.013387

    2020

  28. [37]

    C. Wu, A. Fan, and S.-D. Liang, AAPPS Bul- letin 32, 39 (2022), URL https://doi.org/10.1007/ s43673-022-00065-0

    2022

  29. [38]

    Muschietti and C

    L. Muschietti and C. Dum, Physics of Fluids B: Plasma Physics 5, 1383 (1993), URL https://doi.org/10. 1063/1.860877

    1993

  30. [39]

    Bylinkin, M

    A. Bylinkin, M. Schnell, M. Autore, F. Calavalle, P. Li, J. Taboada-Guti` errez, S. Liu, J. H. Edgar, F. Casanova, L. E. Hueso, et al., Nature Photonics 15, 197 (2021), URL https://doi.org/10.1038/s41566-020-00725-3

    2021 doi

  31. [40]

    Epstein, D

    I. Epstein, D. Alcaraz, Z. Huang, V.-V. Pusapati, J.- P. Hugonin, A. Kumar, X. M. Deputy, T. Khodkov, T. G. Rappoport, J.-Y. Hong, et al., Science 368, 1219 (2020), URL https://www.science.org/doi/abs/ 10.1126/science.abb1570

    2020 doi

  32. [41]

    B.-S. Song, S. Noda, T. Asano, and Y. Akahane, Na- ture Materials 4, 207 (2005), URL https://doi.org/ 10.1038/nmat1320

    2005 doi

  33. [42]

    Jiang, Angular momentum-dependent spectral shift in chiral vacuum cavities (2023), 2307.14964, URL https://arxiv.org/abs/2307.14964

    Q.-D. Jiang, Angular momentum-dependent spectral shift in chiral vacuum cavities (2023), 2307.14964, URL https://arxiv.org/abs/2307.14964

    2023 arXiv

  34. [43]

    Yang and Q.-D

    L. Yang and Q.-D. Jiang, Emergent haldane model and photon-valley locking in chiral cavities (2024), 2403.11063, URL https://arxiv.org/abs/2403.11063

    2024 arXiv

  35. [44]

    D. C. Brody, Journal of Physics A: Mathematical and Theoretical 47, 035305 (2013), URL https://dx.doi. org/10.1088/1751-8113/47/3/035305

    2013 doi

  36. [45]

    B. A. Bernevig, Topological Insulators and Topological Superconductors (Princeton University Press, Princeton, 2013), ISBN 9781400846733, URL https://doi.org/10. 1515/9781400846733

    2013

  37. [46]

    J. K. Asb´ oth, L. Oroszl´ any, and A. P´ alyi,A short course on topological insulators, vol. 919 (Springer, 2016), URL https://doi.org/10.1007/978-3-319-25607-8

    2016 doi

  38. [47]

    A. Mook, J. Henk, and I. Mertig, Phys. Rev. B 90, 024412 (2014), URL https://link.aps.org/doi/10. 1103/PhysRevB.90.024412

    2014

  39. [48]

    Hatsugai, Phys

    Y. Hatsugai, Phys. Rev. Lett. 71, 3697 (1993), URL https://link.aps.org/doi/10.1103/PhysRevLett.71. 3697

    1993 doi

  40. [49]

    Masuki and Y

    K. Masuki and Y. Ashida, Phys. Rev. B 107, 195104 (2023), URL https://link.aps.org/doi/10. 11 1103/PhysRevB.107.195104

    2023

  41. [50]

    Ashida, A

    Y. Ashida, A. m. c. ˙Imamo˘ glu, and E. Demler, Phys. Rev. Lett. 126, 153603 (2021), URL https://link.aps.org/ doi/10.1103/PhysRevLett.126.153603

    2021 doi

  42. [51]

    Liu, X.-B

    T. Liu, X.-B. Qiang, H.-Z. Lu, and X. C. Xie, National Science Review (2024), ISSN 2053-714X, URL http:// dx.doi.org/10.1093/nsr/nwae334

    2024 doi

  43. [52]

    Cheng, Quantum geometric tensor (fubini-study met- ric) in simple quantum system: A pedagogical introduc- tion (2013), 1012.1337, URL https://arxiv.org/abs/ 1012.1337

    R. Cheng, Quantum geometric tensor (fubini-study met- ric) in simple quantum system: A pedagogical introduc- tion (2013), 1012.1337, URL https://arxiv.org/abs/ 1012.1337

    2013 arXiv

  44. [53]

    M. S. M. de Sousa, A. L. Cruz, and W. Chen, Phys. Rev. B 107, 205133 (2023), URL https://link.aps. org/doi/10.1103/PhysRevB.107.205133

    2023 doi

  45. [54]

    Porlles and W

    D. Porlles and W. Chen, Phys. Rev. B 108, 094508 (2023), URL https://link.aps.org/doi/10. 1103/PhysRevB.108.094508. Appendix A: Quantum metric and fidelity number Let us consider an Hermitian system and define the distance between quantum states in wave vector space, ds2 H = 1...

    2023

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