Non-Hermitian Floquet topological phases in the double-kicked rotor

Jiaxin Pan; Longwen Zhou

REVIEW 2 major objections 5 minor 2 cited by

Non-Hermitian Floquet topological phases in the double-kicked rotor

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Complex kicks turn a kicked rotor into Floquet topological phases, with a pair of winding numbers classifying each phase and a generalized mean chiral displacement measuring them dynamically.

desk verdict A clean analytic extension of non-Hermitian Floquet topology to the double kicked rotor with a dynamical probe that works; the bulk-edge correspondence is the one place where the evidence is thinner than the claim. read the letter →

arxiv 1908.02066 v2 pith:4CZOYNBD submitted 2019-08-06 quant-ph cond-mat.othercond-mat.quant-gas

Longwen Zhou , Jiaxin Pan This is my paper · ORCID

classification quant-phcond-mat.othercond-mat.quant-gas

keywords non-HermitianFloquettopologicalphasesdoublekickedrotorwindingnumbersmeanchiraldisplacementbulk-edgecorrespondencesymmetryperiodicallyquenchedlatticeedgestates

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 paper extends the double kicked rotor into the non-Hmitian regime by allowing the kicking strengths to be complex, and shows that this does not destroy the system's Floquet topology but instead creates a rich set of topological phases. Under the on-resonance condition and with the time delay set so the rotor has two effective bands, each phase is fully classified by a pair of integer winding numbers computed from two symmetric time frames. The paper further shows that these winding numbers can be detected in experiment by measuring a generalized mean chiral displacement of a wave packet. By mapping the rotor to a periodically quenched lattice in real space, the paper obtains topological edge states and demonstrates that a bulk-edge correspondence holds exactly as in the Hermitian limit, with the number of edge states at quasienergies 0 and pi being twice the corresponding winding numbers.

What carries the argument

The load-bearing object is the pair of integer winding numbers (nu_0, nu_pi), defined through complex unit vectors n_alpha(theta) = (n_alpha x, n_alpha y) extracted from the Floquet operators U_1 and U_2 in two symmetric time frames, with nu_alpha = integral(dtheta/2pi)(n_alpha x partial_theta n_alpha y). The quasienergy E(theta) = arccos(cos K_1 cos K_2) enters as the dispersion of the two-band system. A generalized mean chiral displacement C_alpha, defined with a biorthogonal time-evolved operator and a normalization factor, equals nu_alpha/2 in the long-time limit, providing a dynamical probe of the topological invariants. The same Floquet operator is also rewritten as a periodically quenched real-space lattice, whose open-boundary Floquet eigenstates directly expose the topological edge modes and allow the bulk-edge relations n_0 = 2 nu_0 and n_pi = 2 nu_pi to be checked numerically.

What would settle it

Compute the open-boundary Floquet spectrum for the same parameters with a much larger system size (for instance N = 40000) or with a generalized Brillouin-zone (non-Bloch) band theory; if the number of edge modes at quasienergies 0 and pi changes with N or deviates from 2 nu_0 and 2 nu_pi, the bulk-edge correspondence fails. In parallel, initialize the mean chiral displacement at longer evolution times and check that it converges to nu_alpha/2; if normalization biases the long-time average, the dynamical detection scheme would not faithfully extract the winding numbers.

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Extended reading notes

Core claim

The central discovery is that making the kicking strengths of the on-resonance double kicked rotor complex generates non-Hermitian Floquet topological phases protected by chiral symmetry, and that these phases are completely characterized by a pair of integer winding numbers (nu_0, nu_pi) defined through the Floquet operators in two symmetric time frames. The quasienergy dispersion is E(theta) = arccos(cos K_1 cos K_2), and the gap closes when Im E(theta) = 0 and Re E(theta) = 0 or pi, giving explicit analytical conditions for topological phase transitions. A generalized mean chiral displacement, extended to nonunitary evolution via a biorthogonal time-evolution operator and a normalization factor, converges in the long-time limit to half the winding number in each time frame, so the pair (nu_0, nu_pi) is read out as |C_1 + C_2| and |C_1 - C_2|. Mapping the same Floquet operator to a periodically quenched bipartite lattice with open boundaries yields edge states pinned at quasienergies 0 and pi, whose numbers satisfy n_0 = 2 nu_0 and n_pi = 2 nu_pi, confirming the bulk-edge correspondence in the non-Hermitian setting.

Load-bearing premise

The claim that the bulk-edge correspondence holds relies on treating a finite open-boundary lattice with 4000 unit cells as a faithful representation of the infinite kicked rotor, so that non-Hermitian skin effects do not alter the edge-state count in the thermodynamic limit.

Editorial extensions

If this is right

Complex kicking strengths drive a sequence of topological phase transitions, with each transition accompanied by a quantized change of nu_0 or nu_pi by 1, and the system becomes topologically trivial in the large-loss limit.
The generalized mean chiral displacement provides a concrete experimental route to detect non-Hermitian Floquet winding numbers by measuring the shift of a mixed-state wave packet over many driving periods.
The bulk-edge correspondence n_0 = 2 nu_0 and n_pi = 2 nu_pi holds in the non-Hermitian on-resonance double kicked rotor, so the topological edge states at quasienergies 0 and pi are protected by the same chiral symmetry that classifies the bulk phases.
The same machinery of symmetric time frames and normalized chiral displacement should apply to other one-dimensional chiral-symmetric non-Hermitian Floquet systems, including quantum walks and kicked Harper models.
Realizing the complex kicking strengths with lossy optical lattices or photonic waveguides would allow observation of non-Hermitian Floquet topological phases in a driven atomic or photonic system.

Reading between the lines

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

A possible extension is to check whether the bulk-edge correspondence persists when the non-Hermitian skin effect is stronger; if not, the finite-lattice calculation with N = 4000 would need to be replaced by a non-Bloch band theory to count edge states correctly.
The normalization used in the mean chiral displacement may hide transient non-Hermitian dynamics, so experimental probes might need to average over many periods or compare with the unnormalized chiral displacement to distinguish topological contributions from gain-loss artifacts.
A direct generalization to spin-1/2 kicked rotors or kicked Harper models with complex potentials could yield higher winding numbers or anomalous Floquet phases beyond the two-band example treated here.
Because the rotor's momentum lattice is infinite, translating the topological invariants into real-space lattice observables requires the finite-size lattice mapping; experiments with cold atoms or photonic lattices that implement the real-space model would provide a sharper test of the predicted edge-state counts.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here is my take on 1908.02066. It is a real extension of the earlier non-Hermitian Floquet classification into a kicked-rotor setting. The model with complex kicking strengths is new for the DKR; so are the phase diagrams, the generalized MCD, and the numerical edge-state counting. The analytic backbone is honest: Appendix A gives a clean two-band reduction under on-resonance conditions; the gap-closing conditions (14)-(15) and (C6)-(C9) are derived; the MCD derivation in Appendix D is explicit and matches the winding numbers in Figs. 6-7. I would trust the bulk classification.

The soft spot is the bulk-edge correspondence. The relation n0 = 2ν0 and nπ = 2νπ is checked on one finite lattice (N = 4000) for the one-parameter family v1 = v2 = v, and the paper itself flags the skin effect as a known complication and notes a "small deviation" around y5. That is precisely where non-Hermiticity matters. Equation (19) has nonreciprocal inter-cell terms, so the OBC spectrum can in principle differ from Bloch winding predictions. Without finite-size scaling, a non-Bloch calculation, or at least a few other parameter regimes, Eq. (20) is supported but not proven. The text says it was checked elsewhere, but no data is shown. I would call this a moderate gap, not a fatal one. The MCD normalization is introduced ad hoc, but the derivation is explicit and the numerics support it; experimental calibration is a discussion point.

No code or data is released. For a paper whose headline claims are partly numerical, that is a legitimate referee request. The heavy self-citation to Ref. [74] is not a red flag because the classification is indeed from that paper and is published; the authors extend it rather than claim a new classification scheme.

I think this deserves a serious referee. A good referee should ask for data/code and a finite-size scaling check near y5, and should debate whether the skin effect can break the PBC-OBC correspondence in this model. With those additions, the paper would be a solid contribution to the non-Hermitian Floquet literature.

Referee Report

2 major / 5 minor

Summary. The paper introduces a non-Hermitian extension of the on-resonance double kicked rotor, with complex kicking strengths K_j = u_j + i v_j. Under the resonance conditions ℏT = 4π and ℏτ = π, the model is reduced to a two-band Floquet operator U(θ) with a chiral symmetry, and a pair of integer winding numbers (ν0, νπ) is defined from the two symmetric time frames. The authors map out topological phase diagrams for several parameter regimes, propose a generalized mean chiral displacement (MCD) as a dynamical probe of these invariants, and map the system to a periodically quenched lattice to study edge states, claiming the bulk-edge correspondence n0 = 2ν0 and nπ = 2νπ.

Significance. If the results hold, the paper extends non-Hermitian Floquet topological phases to a concrete dynamical kicking model and provides a potentially observable dynamical probe. Strengths include the exact two-band reduction (Appendix A), analytic gap-closing conditions (Eqs. (14)–(15) and (C6)–(C9)), and the apparent internal consistency between the winding-number computations and the MCD numerics in Figs. 6–7. The claims are concrete and falsifiable. However, the two most load-bearing parts—the MCD formula and the bulk-edge correspondence—require additional justification in the non-Hermitian setting.

major comments (2)

[Sec. III C and Appendix D, Eqs. (16) and (17)] The derivation of the MCD is not fully supported as written. Eq. (16) defines Cα(t) as a bare trace without normalization, while Eq. (17) is obtained only after inserting a θ-dependent normalization factor Tr[Ũ†tα U^tα] in Appendix D, described as canceling gain/loss. This changes the observable, and the manuscript does not specify how this normalized quantity is measured experimentally or show that the unnormalized Eq. (16) has the same long-time limit. In addition, the text identifies Ũ as the operator whose left eigenvectors are the right eigenvectors of U, but Appendix D sets Ũ†α = e^{+iE* nα·σ} even though nα is explicitly complex-valued; for Uα = e^{-iE nα·σ}, the adjoint of the transpose (the standard left-eigenvalue operator) would involve n*α, not nα. The trace identities in Eqs. (D9)–(D10) therefore need either a corrected expression for Ũ or an explicit statement that the normalized MCD is being defined rather than derived.
[Sec. III D and Eq. (20)] The bulk-edge correspondence is verified only by counting edge states of the finite open-boundary lattice in Eq. (19) at N = 4000, for the one-parameter family v1 = v2 = v. Because Eq. (19) contains nonreciprocal inter-cell couplings (e.g., the i|n⟩⟨n+1|σ− − i|n+1⟩⟨n|σ+ terms multiplied by complex K1), the non-Hermitian skin effect (Ref. [94]) can in principle decouple the OBC spectrum from the Bloch winding numbers in Eqs. (12)–(13). The manuscript provides no finite-size scaling, no generalized Brillouin zone (non-Bloch) calculation, and the admitted 'small deviation around y5' in Fig. 9 is not analyzed. Without additional evidence that the OBC edge-state count is governed by the PBC winding numbers in the thermodynamic limit, the claim that Eq. (20) holds 'in the same way as in the Hermitian ORDKR' is under-supported at the point where non-Hermiticity is most consequential.

minor comments (5)

[Sec. III B] The text says 'we first analysis the Floquet operator'; this should be 'we first analyze the Floquet operator'.
[Appendix D, Eq. (D3)] The Fourier expansion for |n⟩ is miswritten: it should be |n⟩ = (1/√N) Σ_θ e^{-iθn}|θ⟩, and similarly ⟨n|θ⟩ = e^{iθn}/√N. The current expression has the summation over n on both sides and is not a consistent Fourier transform.
[Eqs. (5)–(8)] The barred quantities K1 and K2 introduced in Eq. (6) are hard to distinguish from the unbarred kicking strengths in the printed notation. Please use explicitly distinct symbols (e.g., ar K_1, ar K_2) throughout to avoid confusion.
[Fig. 9] The claimed 'small deviation around y5 due to finite size effects' is not visible in the figure. Please add an inset or a quantitative measure of the deviation, since this is the only direct evidence about the size dependence of Eq. (20).
[Sec. I and Sec. III C] The paper builds heavily on the classification from the authors' prior work cited as Ref. [74]. Please state explicitly which elements are new here (e.g., the kicked-rotor realization, the specific MCD construction for this model) rather than applications of the general formalism.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: invariants, MCD relation, and bulk-edge check are derived from explicit model definitions; the sole same-author citation is to a published, parameter-free classification.

full rationale

Walking the derivation chain, the NH-ORDKR Floquet operator and its symmetric time frames are constructed explicitly (Eqs. 5-8 and Appendix A), and the winding numbers (12)-(13) are computed from closed-form complex vectors n_alpha in Appendix B rather than fitted to any target quantity. The central MCD claim is a derived identity: Eq. (16) defines a chiral displacement with a mixed initial state, and Appendix D shows, after an explicitly stated normalization that cancels gain/loss, that its long-time average equals nu_alpha/2 with no adjustable parameter; the numerical comparison in Figs. 6-7 evaluates that derived formula against the same analytic n_alpha, which is a validation weakness rather than circular reasoning. The classification via the pair (nu0, nu_pi) cites Ref. [74] by the same first author, but that is a published, parameter-free extension to non-Hermitian Floquet systems and is therefore independent evidence under the review rules. The bulk-edge correspondence n0 = 2 nu0, n_pi = 2 nu_pi is checked by explicit OBC edge-state counting on the mapped lattice Eq. (19); the admitted skin-effect complication and the finite-size deviation around y5 raise correctness concerns, but no equation is redefined to force the result. No fitted input is renamed a prediction, and no central claim reduces by construction to its own input.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The model is specified by four kicking-strength parameters plus the phase delay beta; these are physical inputs scanned in the phase diagrams, not fit parameters. The classification and gap-closing criteria are imported from prior Floquet-topology literature (mainly Ref [74]), and the real-space OBC mapping assumes finite-size truncation is benign. No new physical entities are postulated.

free parameters (1)

Phase delay beta = pi/2
Set to pi/2 in Sec. II to allow nontrivial winding; this is a physically tunable phase, not fitted to data.

assumptions (5)

domain assumption Chiral symmetry Gamma U_alpha Gamma = U_alpha^{-1} holds for both symmetric time frames (Eq. 9).
This is the symmetry that protects the topological phases; it follows from the specific form of the kicks, but its stability for complex K is assumed.
domain assumption The non-Hermitian Floquet topological classification by the pair (nu0, nu_pi) from Ref [74] applies to the NH-ORDKR.
The paper imports the classification scheme and cites Ref [74] rather than rederiving it; this is the framework that defines the phases.
domain assumption Gap-closing condition cos E = +/-1 is the complete criterion for topological phase transitions in the NH-ORDKR.
Used to derive Eqs. (14)-(15) and the numerical gap functions (C8)-(C9); assumes no transition occurs without the quasienergy gap closing.
domain assumption The real-space lattice model with open boundary conditions (N = 4000) faithfully represents the thermodynamic limit for the bulk-edge correspondence check.
Eq. (19) is exact as a lattice representation, but applying OBC by truncating N sites and reading off edge-state counts assumes finite-size effects are small; the paper acknowledges a small deviation near y5.
ad hoc to paper A normalization factor in the mean chiral displacement definition (Appendix D) removes gain/loss effects without altering the winding-number content.
Eq. (D8) inserts a normalization; this choice is motivated by canceling amplitude changes but is not derived from an independent principle.

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Pith. "Pith review of Non-Hermitian Floquet topological phases in the double-kicked rotor." pith.science (2026). https://pith.science/paper/4CZOYNBD

@misc{pith2026190802066,
  author       = {Pith},
  title        = {Pith review of: Non-Hermitian Floquet topological phases in the double-kicked rotor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4CZOYNBD}},
  note         = {Machine review of arXiv:1908.02066}
}

read the original abstract

Dynamical kicking systems possess rich topological structures. In this work, we study Floquet states of matter in a non-Hermitian extension of double kicked rotor model. Under the on-resonance condition, we find various non-Hermitian Floquet topological phases, with each being characterized by a pair of topological winding numbers. A generalized mean chiral displacement is introduced to detect these winding numbers dynamically in two symmetric time frames. Furthermore, by mapping the system to a periodically quenched lattice model, we obtain the topological edge states and unravel the bulk-edge correspondence of the non-Hermitian double kicked rotor. These results uncover the richness of Floquet topological states in non-Hermitian dynamical kicking systems.

Figures

Figures reproduced from arXiv: 1908.02066 by the authors.

**Figure 3.** FIG. 3. The topological phase diagram of NH-ORDKR vs. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** FIG. 4. The topological phase diagram of NH-ORDKR vs. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (4 more)

**Figure 6.** Figure 6: FIG. 6. The MCD [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 7.** Figure 7: FIG. 7. The MCD [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

**Figure 8.** Figure 8: FIG. 8. Floquet spectrum of the NH-ORDKR vs. [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]

**Figure 9.** Figure 9: FIG. 9. Number of edge states at quasienergies [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]

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Forward citations

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Case (i) We ﬁrst consider the case when only one of the kicking strengths (K1 or K2) in Eq. (4) is complex. Under the gapless condition cos[E(θ)] =±1, it can be shown that if v1⁄= 0 and v2 = 0, u1,u 2 and v1 in Eq. (4) satisfy the equation (see Appendix C for derivation details): v1 = u1 nπ arccosh   ±1 cos ( u2 √ 1− n2π2 u2 1 )  , n ∈ Z. (14) Similar...
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Case (i) We ﬁrst consider the case when only one of the kicking strengths (K1 or K2) in Eq. (4) is complex. Under the gapless condition cos[E(θ)] =±1, it can be shown that if v1⁄= 0 and v2 = 0, u1,u 2 and v1 in Eq. (4) satisfy the equation (see Appendix C for derivation details): v1 = u1 nπ arccosh   ±1 cos ( u2 √ 1− n2π2 u2 1 )  , n ∈ Z. (14) Similar...

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Case (ii) In this case, both the two kicking strengthsK1 and K2 take complex values under the constraint that their imaginary parts are equal, i.e., v1 = v2 = v. Using the gapless condition (see Appendix C for more details) and the winding numbers(ν0,νπ), we could then numer- ically characterize the Floquet topological phases of NH- ORDKR at diﬀerent imag...

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Case (iii) In this case, we allow bothK1 and K2 to be complex, with no constraint on their imaginary parts. The re- sulting topological phase diagram versusv1 andv2, with (u1,u 2) = (0 .5π, 5.5π) and (u1,u 2) = (5 .5π, 0.5π) are shown in Figs. 4 and 5, respectively. In each phase diagram, the panels (a) and (b) correspond to the val- ues of winding number...

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