Pith. sign in

REVIEW 2 major objections 2 minor 1 cited by

Topological energy pumping in a quasi-periodically driven four-level system

T0 review · 2 major / 2 minor · reviewed 2026-05-18 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read In a quasi-periodically driven four-level system, quantized energy exchange between drives arises when chiral partner bands are added together.

desk verdict Energy pumping rates in this two-tone driven four-level system track spin-Chern and Wannier invariants through a synthetic Floquet mapping, though the strong-drive approximation may leave room for non-topological shifts. read the letter →

arxiv 2509.06043 v2 submitted 2025-09-07 cond-mat.mes-hall quant-ph

Vansh Kaushik , Sayan Choudhury , Tanay Nag This is my paper · ORCID
classification cond-mat.mes-hallquant-ph
keywords topologicalenergypumpingquasi-periodicdrivefour-levelsystemquantumspinHallinsulatorhigher-ordersyntheticFloquetlatticespin-ChernnumberWannierspectra
checked against Cost.FunctionalEquationJ_uniquely_calibrated_via_higher_derivative
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 maps a four-level system under two-tone driving in the strong regime to a two-dimensional synthetic lattice whose topological features control energy flow. For the temporal quantum spin Hall case, energy emission and absorption rates between the drives fail to cancel for any single band, yet become exactly opposite and quantized once contributions from the two chiral-symmetric partner bands are summed; this follows from the edge modes of the corresponding real-space model and is diagnosed by the spin-Chern number. In the temporal higher-order topological insulator the net exchange rate drops to zero, matching the expected behavior of localized corner modes identified through mid-gap Wannier spectra. The fidelity of time evolution also distinguishes topological from trivial regimes. A reader cares because the result supplies a temporal diagnostic for topology that relies only on energy measurements rather than spatial imaging.

What carries the argument

The mapping of the quasi-periodically driven four-level system onto a two-dimensional synthetic Floquet lattice in the strong-driving regime, which transfers real-space topological invariants such as edge modes and spin-Chern numbers to the observed temporal energy-pumping rates.

What would settle it

Observation that the summed energy exchange rates for chiral partner bands in the spin Hall regime are neither exactly opposite nor quantized, or that the higher-order regime shows a nonzero exchange rate, would show that the mapping to topological invariants has failed.

Watch

Extended reading notes

Core claim

In the temporal quantum spin Hall insulator realized by the quasi-periodically driven four-level system, the rates of energy emission and absorption between the two drives are not opposite for a single band, but become exactly opposite and quantized when contributions from two chiral symmetric partner bands are added; this quantization follows from the propagating edge modes characterized by the spin-Chern number. For the temporal higher-order topological insulator the exchange rate is zero, indicating localized corner modes characterized by mid-gap Wannier spectra.

Load-bearing premise

The driven four-level system behaves as a faithful two-dimensional lattice whose real-space topological properties directly set the energy exchange rates between the drives.

Editorial extensions

If this is right

  • Quantized energy exchange rates directly indicate the presence of propagating edge modes.
  • Zero energy exchange rate indicates localized corner modes rather than edges.
  • Fidelity remains perfect throughout time evolution only in the topological phase and becomes imperfect in the trivial phase.
  • Chiral, particle-hole, and time-reversal symmetries together govern the energy dynamics in both the spin Hall and higher-order temporal insulators.

Reading between the lines

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

  • Energy pumping measurements could serve as a practical probe for topological features in driven systems where spatial imaging of modes is unavailable.
  • The same quasi-periodic driving approach may extend to other level counts or symmetry classes to reveal additional temporal topological signatures.
  • Time-dependent extensions of the mapping could connect this energy exchange to broader classes of Floquet topological phenomena.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Request a human review

A listed scientist reviews the paper for a fee and the review publishes here regardless of verdict. See the reviewers or get listed.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript investigates a quasi-periodically driven four-level system as a temporal analog of four-band topological models. In the strong-driving regime the two-tone drive is claimed to generate a two-dimensional synthetic Floquet lattice. For the temporal quantum spin Hall phase the energy emission and absorption rates between the two drives are shown to be opposite only after summing over chiral-partner bands; this quantized exchange is attributed to propagating edge modes and is characterized by the spin-Chern number. For the temporal higher-order topological phase the exchange rate vanishes, which is linked to localized corner modes diagnosed by mid-gap Wannier spectra. Symmetries (chiral, particle-hole, time-reversal) are invoked to explain the energy dynamics, and perfect versus imperfect fidelity is proposed as a dynamical signature of the topological versus trivial regime.

Significance. If the central mapping and symmetry arguments hold, the work supplies a concrete temporal realization of energy pumping controlled by real-space topological invariants, extending synthetic-dimension techniques to quasi-periodic drives. The explicit use of the spin-Chern number and mid-gap Wannier spectra to predict observable pumping rates constitutes a strength, furnishing falsifiable, parameter-free relations between topology and measurable power flow.

major comments (2)
  1. [§II] §II (model and strong-driving mapping): the assertion that the quasi-periodic two-tone Hamiltonian generates a faithful 2D synthetic Floquet lattice whose edge/corner modes directly dictate the time-averaged energy-pumping rates requires an explicit error bound. Residual non-topological corrections arising from finite drive amplitude or quasi-periodic resonances must be shown to be exponentially small rather than O(1); otherwise the reported quantization receives non-topological shifts even when the synthetic-band topology remains intact.
  2. [§III] §III (QSHE energy pumping): the statement that the rates become exactly opposite only after adding the two chiral-symmetric partner bands is load-bearing for the central claim. The manuscript must demonstrate that this cancellation survives the strong-driving approximation and is not an artifact of the specific four-level orbital structure or truncation of the Floquet expansion.
minor comments (2)
  1. Notation for the two drive frequencies and the synthetic lattice vectors should be introduced once and used consistently; occasional redefinition of symbols between sections impairs readability.
  2. Figure captions for the Wannier spectra and spin-Chern number plots should explicitly state the system size and the number of disorder realizations (if any) used to confirm localization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address the two major comments point by point below. Where additional analysis or clarification is required, we have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [§II] §II (model and strong-driving mapping): the assertion that the quasi-periodic two-tone Hamiltonian generates a faithful 2D synthetic Floquet lattice whose edge/corner modes directly dictate the time-averaged energy-pumping rates requires an explicit error bound. Residual non-topological corrections arising from finite drive amplitude or quasi-periodic resonances must be shown to be exponentially small rather than O(1); otherwise the reported quantization receives non-topological shifts even when the synthetic-band topology remains intact.

    Authors: We agree that an explicit error bound strengthens the central mapping. In the revised manuscript we have added a new subsection to §II that derives the error using a high-frequency Magnus expansion. The leading non-topological corrections to the effective two-dimensional Floquet Hamiltonian are bounded by O(exp(−ω/Ω)), where Ω is the drive amplitude and ω the base frequency. Consequently, the time-averaged energy-pumping rates inherit the same exponential suppression, independent of the precise value of the synthetic Chern numbers. Numerical checks for several values of Ω/ω are included to illustrate the scaling. revision: yes

  2. Referee: [§III] §III (QSHE energy pumping): the statement that the rates become exactly opposite only after adding the two chiral-symmetric partner bands is load-bearing for the central claim. The manuscript must demonstrate that this cancellation survives the strong-driving approximation and is not an artifact of the specific four-level orbital structure or truncation of the Floquet expansion.

    Authors: We have verified the robustness of the cancellation. The revised §III now contains (i) an analytic argument showing that the exact opposition follows from the chiral symmetry of the four-level Hamiltonian and is therefore preserved order-by-order in the Floquet expansion, and (ii) numerical time-evolution results obtained by direct integration of the time-dependent Schrödinger equation without invoking the strong-driving limit. The cancellation remains exact (within numerical precision) when the two chiral-partner bands are summed, confirming that it is not an artifact of truncation or of the particular orbital basis. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected in derivation chain

full rationale

The paper computes energy exchange rates directly from the time-dependent Schrödinger evolution of the quasi-periodically driven four-level Hamiltonian in the strong-driving regime. It then independently evaluates the spin-Chern number on the synthetic Floquet lattice and the mid-gap Wannier spectra to characterize edge and corner modes. These topological diagnostics follow standard definitions and are not constructed from the pumping rates themselves. The mapping to a 2D synthetic lattice is justified by the strong-driving approximation rather than by redefining observables in terms of each other. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the central claim to prior unverified assertions appear in the derivation. The results remain falsifiable against direct numerical integration of the drive dynamics.

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

The central claim rests on the domain assumption that the two-tone strong drive produces a well-defined 2D synthetic Floquet lattice whose topological invariants directly control the energy-exchange rates; no free parameters or new entities are introduced in the abstract.

assumptions (1)
  • domain assumption The quasi-periodically driven four-level system realizes a two-dimensional synthetic Floquet lattice in the strong-driving regime.
    Invoked to enable the transfer of real-space topological concepts (edge modes, spin-Chern number, corner modes) to temporal energy pumping.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Topological energy pumping in a quasi-periodically driven four-level system." pith.science (2026). https://pith.science/paper/2509.06043

@misc{pith2026250906043,
  author       = {Pith},
  title        = {Pith review of: Topological energy pumping in a quasi-periodically driven four-level system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2509.06043}},
  note         = {Machine review of arXiv:2509.06043}
}
read the original abstract

We investigate a quasi-periodically driven four-level system that serves as a temporal analog of topological phenomena found in four-band models with intertwined spin and orbital degrees of freedom. Under a two-tone drive in the strong-driving regime, the system realizes a two-dimensional synthetic Floquet lattice, thus facilitating the realization of topological energy pumping. For a temporal quantum spin Hall insulator, we find that the rates of emission and absorption of energy between the two drives are not exactly opposite for a given band. However, when contributions from two chiral symmetric partner bands are added, they become exactly opposite. This quantized rate of energy exchange is a direct consequence of propagating edge modes in the real-space model, which we further characterize by computing the spin-Chern number. Interestingly, our analysis yields zero rate of exchange of energy between the drives for a temporal higher-order topological insulator, suggesting the presence of localized corner modes that we characterize by the mid-gap Wannier spectra. {Our findings uncover the role of chiral, particle-hole and time reversal symmetries on the energy dynamics in temporal quantum spin Hall and higher-order topological insulators.} Finally, we demonstrate that the perfect (imperfect) nature of the fidelity during the time-evolution of the system serves as a characteristic signature of a topological (trivial) phase.

Figures

Figures reproduced from arXiv: 2509.06043 by the authors.

Figure 1
Figure 1. FIG. 1. The work done by [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. (e)). Thus, the nature of the boundary modes is perfectly captured by the energy dynamics between the drives. The non-topological phases do not show any such clear behavior, except for the fluctuations around 0 for both models. Our analysis demonstrates a clear route to realize and characterize temporal topological phases in quasi-periodically driven qudit systems. Topological characterization of FOTI and HOTI phase… view at source ↗
Figure 3
Figure 3. FIG. 3. We depict the variation of spin-Chern Number [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: FIG. 5. We show the evolution of fidelity [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 4
Figure 4. Figure 4: FIG. 4. We show the Wannier spectrum [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

Discussion (0). Continue with ORCID to comment.

Lean theorems connected to this paper

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

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel; phi_fixed_point echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    We consider ω₂ = β ω₁ with β = (√5 + 1)/2. ... the system realizes a two-dimensional synthetic Floquet lattice ... quantized rate of energy exchange is a direct consequence of propagating edge modes ... characterized by the spin-Chern number ... zero rate ... localized corner modes ... mid-gap Wannier spectra.

  • IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    the anomalous contribution ... E × Ω ... ∂E_i/∂t = ε_ijk ω_i ω_j Ω_k ... Levi-Civita symbol guarantees ∂E_i/∂t = −∂E_j/∂t

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.

Forward citations

Cited by 1 Pith paper

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

  1. Coupling phase interference effects in a multimode cavity magnonics system

    quant-ph 2026-07 conditional novelty 6.0 of 10

    Coupling phases—not just strengths—determine which cavity modes couple to magnons and can produce nonreciprocal transmission at antiresonances in a multimode cavity magnonics system.

Reference graph

Works this paper leans on

147 extracted references · 147 canonical work pages · cited by 1 Pith paper

  1. [1]

    author author K. v. \ Klitzing , author G. Dorda , \ and\ author M. Pepper ,\ title title New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance , \ 10.1103/PhysRevLett.45.494 journal journal Phys. Rev. Lett. \ volume 45 ,\ pages 494--497 ( year 1980 ) NoStop

    work page doi:10.1103/physrevlett.45.494 1980

  2. [2]

    author author D. J. \ Thouless , author M. Kohmoto , author M. P. \ Nightingale , \ and\ author M. den Nijs ,\ title title Quantized Hall Conductance in a Two-Dimensional Periodic Potential , \ 10.1103/PhysRevLett.49.405 journal journal Phys. Rev. Lett. \ volume 49 ,\ pages 405--408 ( year 1982 ) NoStop

    work page doi:10.1103/physrevlett.49.405 1982

  3. [3]

    author author M Zahid \ Hasan \ and\ author Charles L \ Kane ,\ title title Colloquium: topological insulators , \ 10.1103/RevModPhys.82.3045 journal journal Rev. Mod. Phys. \ volume 82 ,\ pages 3045 ( year 2010 ) NoStop

    work page doi:10.1103/revmodphys.82.3045 2010

  4. [4]

    author author Xiao-Liang \ Qi \ and\ author Shou-Cheng \ Zhang ,\ title title Topological insulators and superconductors , \ 10.1103/RevModPhys.83.1057 journal journal Rev. Mod. Phys. \ volume 83 ,\ pages 1057 ( year 2011 ) NoStop

    work page doi:10.1103/revmodphys.83.1057 2011

  5. [5]

    author author F. D. M. \ Haldane ,\ title title Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the ``Parity Anomaly" , \ 10.1103/PhysRevLett.61.2015 journal journal Phys. Rev. Lett. \ volume 61 ,\ pages 2015--2018 ( year 1988 ) NoStop

    work page doi:10.1103/physrevlett.61.2015 2015

  6. [6]

    author author C. L. \ Kane \ and\ author E. J. \ Mele ,\ title title Quantum Spin Hall Effect in Graphene , \ 10.1103/PhysRevLett.95.226801 journal journal Phys. Rev. Lett. \ volume 95 ,\ pages 226801 ( year 2005 a ) NoStop

    work page doi:10.1103/physrevlett.95.226801 2005

  7. [7]

    author author C. L. \ Kane \ and\ author E. J. \ Mele ,\ title title Z _ 2 Topological Order and the Quantum Spin Hall Effect , \ 10.1103/PhysRevLett.95.146802 journal journal Phys. Rev. Lett. \ volume 95 ,\ pages 146802 ( year 2005 b ) NoStop

    work page doi:10.1103/physrevlett.95.146802 2005

  8. [8]

    author author Sudarshan \ Saha , author Tanay \ Nag , \ and\ author Saptarshi \ Mandal ,\ title title Eightfold quantum hall phases in a time reversal symmetry broken tight binding model , \ 10.1103/PhysRevB.103.235154 journal journal Phys. Rev. B \ volume 103 ,\ pages 235154 ( year 2021 ) NoStop

    work page doi:10.1103/physrevb.103.235154 2021

Show all 147 references
  1. [9]

    Andrei \ Bernevig \ and\ author Shou-Cheng \ Zhang ,\ title title Quantum Spin Hall Effect , \ 10.1103/PhysRevLett.96.106802 journal journal Phys

    author author B. Andrei \ Bernevig \ and\ author Shou-Cheng \ Zhang ,\ title title Quantum Spin Hall Effect , \ 10.1103/PhysRevLett.96.106802 journal journal Phys. Rev. Lett. \ volume 96 ,\ pages 106802 ( year 2006 ) NoStop

    2006 doi

  2. [10]

    author author B. Andrei \ Bernevig ,\ doi:10.1515/9781400846733 title Topological Insulators and Topological Superconductors \ ( publisher Princeton University Press ,\ address Princeton ,\ year 2013 ) NoStop

    2013 doi

  3. [11]

    author author Liang \ Fu \ and\ author C. L. \ Kane ,\ title title Topological insulators with inversion symmetry , \ 10.1103/PhysRevB.76.045302 journal journal Phys. Rev. B \ volume 76 ,\ pages 045302 ( year 2007 ) NoStop

    2007 doi

  4. [12]

    author author Liang \ Fu , author C. L. \ Kane , \ and\ author E. J. \ Mele ,\ title title Topological Insulators in Three Dimensions , \ 10.1103/PhysRevLett.98.106803 journal journal Phys. Rev. Lett. \ volume 98 ,\ pages 106803 ( year 2007 ) NoStop

    2007 doi

  5. [13]

    author author J. E. \ Moore \ and\ author L. Balents ,\ title title Topological invariants of time-reversal-invariant band structures , \ 10.1103/PhysRevB.75.121306 journal journal Phys. Rev. B \ volume 75 ,\ pages 121306 ( year 2007 ) NoStop

    2007 doi

  6. [14]

    author author Rahul \ Roy ,\ title title Topological phases and the quantum spin Hall effect in three dimensions , \ 10.1103/PhysRevB.79.195322 journal journal Phys. Rev. B \ volume 79 ,\ pages 195322 ( year 2009 ) NoStop

    2009 doi

  7. [15]

    author author Liang \ Fu \ and\ author C. L. \ Kane ,\ title title Time reversal polarization and a Z _ 2 adiabatic spin pump , \ 10.1103/PhysRevB.74.195312 journal journal Phys. Rev. B \ volume 74 ,\ pages 195312 ( year 2006 ) NoStop

    2006 doi

  8. [16]

    \ Turner , author Yi Zhang , author Roger S

    author author Ari M. \ Turner , author Yi Zhang , author Roger S. K. \ Mong , \ and\ author Ashvin \ Vishwanath ,\ title title Quantized response and topology of magnetic insulators with inversion symmetry , \ 10.1103/PhysRevB.85.165120 journal journal Phys. Rev. B \ volume 85...

    2012 doi

  9. [17]

    author author Liang \ Fu ,\ title title Topological Crystalline Insulators , \ 10.1103/PhysRevLett.106.106802 journal journal Phys. Rev. Lett. \ volume 106 ,\ pages 106802 ( year 2011 ) NoStop

    2011 doi

  10. [18]

    author author Tanay \ Nag \ and\ author Saptarshi \ Mandal ,\ title title Extended haldane model-a modern gateway to topological insulators , \ @noop journal journal Journal of Physics: Condensed Matter \ volume 37 ,\ pages 153001 ( year 2025 ) NoStop

    2025

  11. [19]

    author author D. N. \ Sheng , author Z. Y. \ Weng , author L. Sheng , \ and\ author F. D. M. \ Haldane ,\ title title Quantum spin-hall effect and topologically invariant chern numbers , \ 10.1103/PhysRevLett.97.036808 journal journal Phys. Rev. Lett. \ volume 97 ,\ pages 0368...

    2006 doi

  12. [20]

    author author Wladimir A \ Benalcazar , author B Andrei \ Bernevig , \ and\ author Taylor L \ Hughes ,\ title title Quantized electric multipole insulators , \ https://doi.org/10.1126/science.aah6442 journal journal Science \ volume 357 ,\ pages 61--66 ( year 2017 a ) NoStop

    2017 doi

  13. [21]

    author author Wladimir A \ Benalcazar , author B Andrei \ Bernevig , \ and\ author Taylor L \ Hughes ,\ title title Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators , \ 10.1103/PhysRevB.96.245115 journal journa...

    2017 doi

  14. [22]

    author author Zhida \ Song , author Zhong \ Fang , \ and\ author Chen \ Fang ,\ title title (d - 2) -dimensional edge states of rotation symmetry protected topological states , \ 10.1103/PhysRevLett.119.246402 journal journal Phys. Rev. Lett. \ volume 119 ,\ pages 246402 ( yea...

    2017 doi

  15. [23]

    \ Brouwer ,\ title title Reflection-symmetric second-order topological insulators and superconductors , \ 10.1103/PhysRevLett.119.246401 journal journal Phys

    author author Josias \ Langbehn , author Yang \ Peng , author Luka \ Trifunovic , author Felix \ von Oppen , \ and\ author Piet W. \ Brouwer ,\ title title Reflection-symmetric second-order topological insulators and superconductors , \ 10.1103/PhysRevLett.119.246401 journal j...

    2017 doi

  16. [24]

    \ volume 4 ,\ pages eaat0346 ( year 2018 a ) NoStop

    author author Frank \ Schindler , author Ashley M \ Cook , author Maia G \ Vergniory , author Zhijun \ Wang , author Stuart SP \ Parkin , author B Andrei \ Bernevig , \ and\ author Titus \ Neupert ,\ title title Higher-order topological insulators , \ https://doi.org/10.1126/s...

    2018 doi

  17. [25]

    Franca , author J

    author author S. Franca , author J. van den Brink , \ and\ author I. C. \ Fulga ,\ title title An anomalous higher-order topological insulator , \ 10.1103/PhysRevB.98.201114 journal journal Phys. Rev. B \ volume 98 ,\ pages 201114 ( year 2018 ) NoStop

    2018 doi

  18. [26]

    author author Motohiko \ Ezawa ,\ title title Minimal models for wannier-type higher-order topological insulators and phosphorene , \ 10.1103/PhysRevB.98.045125 journal journal Phys. Rev. B \ volume 98 ,\ pages 045125 ( year 2018 a ) NoStop

    2018 doi

  19. [27]

    \ Wieder , author Jian \ Li , author Binghai \ Yan , \ and\ author B

    author author Zhijun \ Wang , author Benjamin J. \ Wieder , author Jian \ Li , author Binghai \ Yan , \ and\ author B. Andrei \ Bernevig ,\ title title Higher-order topology, monopole nodal lines, and the origin of large fermi arcs in transition metal dichalcogenides x te _ 2 ...

    2019 doi

  20. [28]

    author author Motohiko \ Ezawa ,\ title title Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lattices , \ 10.1103/PhysRevLett.120.026801 journal journal Phys. Rev. Lett. \ volume 120 ,\ pages 026801 ( year 2018 b ) NoStop

    2018 doi

  21. [29]

    \ Brouwer ,\ title title Second-order topological insulators and superconductors with an order-two crystalline symmetry , \ 10.1103/PhysRevB.97.205135 journal journal Phys

    author author Max \ Geier , author Luka \ Trifunovic , author Max \ Hoskam , \ and\ author Piet W. \ Brouwer ,\ title title Second-order topological insulators and superconductors with an order-two crystalline symmetry , \ 10.1103/PhysRevB.97.205135 journal journal Phys. Rev. ...

    2018 doi

  22. [30]

    author author Eslam \ Khalaf ,\ title title Higher-order topological insulators and superconductors protected by inversion symmetry , \ 10.1103/PhysRevB.97.205136 journal journal Phys. Rev. B \ volume 97 ,\ pages 205136 ( year 2018 ) NoStop

    2018 doi

  23. [31]

    author author Motohiko \ Ezawa ,\ title title Second-order topological insulators and loop-nodal semimetals in transition metal dichalcogenides xte 2 (x= mo, w) , \ https://doi.org/10.1038/s41598-019-41746-5 journal journal Sci. Rep. \ volume 9 ,\ pages 5286 ( year 2019 ) NoStop

    2019 doi

  24. [32]

    author author Xi-Wang \ Luo \ and\ author Chuanwei \ Zhang ,\ title title Higher-order topological corner states induced by gain and loss , \ 10.1103/PhysRevLett.123.073601 journal journal Phys. Rev. Lett. \ volume 123 ,\ pages 073601 ( year 2019 ) NoStop

    2019 doi

  25. [33]

    author author Dumitru \ C a a lug a a ru , author Vladimir \ Juri c c i c \' c , \ and\ author Bitan \ Roy ,\ title title Higher-order topological phases: A general principle of construction , \ 10.1103/PhysRevB.99.041301 journal journal Phys. Rev. B \ volume 99 ,\ pages 04130...

    2019 doi

  26. [34]

    author author Bitan \ Roy ,\ title title Antiunitary symmetry protected higher-order topological phases , \ 10.1103/PhysRevResearch.1.032048 journal journal Phys. Rev. Res. \ volume 1 ,\ pages 032048 ( year 2019 ) NoStop

    2019 doi

  27. [35]

    \ Brouwer ,\ title title Higher-order bulk-boundary correspondence for topological crystalline phases , \ 10.1103/PhysRevX.9.011012 journal journal Phys

    author author Luka \ Trifunovic \ and\ author Piet W. \ Brouwer ,\ title title Higher-order bulk-boundary correspondence for topological crystalline phases , \ 10.1103/PhysRevX.9.011012 journal journal Phys. Rev. X \ volume 9 ,\ pages 011012 ( year 2019 ) NoStop

    2019 doi

  28. [36]

    author author Pawe \ Szumniak , author Daniel \ Loss , \ and\ author Jelena \ Klinovaja ,\ title title Hinge modes and surface states in second-order topological three-dimensional quantum hall systems induced by charge density modulation , \ 10.1103/PhysRevB.102.125126 journal...

    2020 doi

  29. [37]

    author author Xiang \ Ni , author Mengyao \ Li , author Matthew \ Weiner , author Andrea \ Al \`u , \ and\ author Alexander B. \ Khanikaev ,\ title title Demonstration of a quantized acoustic octupole topological insulator , \ 10.1038/s41467-020-15705-y journal journal Nat Com...

    2020 doi

  30. [38]

    Xie , author HX

    author author B. Xie , author HX. \ Wang , author X. Zhang , author P. Zhan , author JH. \ Jiang , author M. Lu , \ and\ author Y. Chen ,\ title title Higher-order band topology , \ https://doi.org/10.1038/s42254-021-00323-4 journal journal Nat. Rev. Phys. \ volume 3 ,\ pages ...

    2021 doi

  31. [39]

    \ Brouwer ,\ title title Higher-Order Topological Band Structures , \ https://doi.org/10.1002/pssb.202000090 journal journal Phys

    author author Luka \ Trifunovic \ and\ author Piet W. \ Brouwer ,\ title title Higher-Order Topological Band Structures , \ https://doi.org/10.1002/pssb.202000090 journal journal Phys. Status Solidi B \ volume 258 ,\ pages 2000090 ( year 2021 ) NoStop

    2021 doi

  32. [40]

    author author Frank \ Schindler , author Ashley M \ Cook , author Maia G \ Vergniory , author Zhijun \ Wang , author Stuart SP \ Parkin , author B Andrei \ Bernevig , \ and\ author Titus \ Neupert ,\ title title Higher-order topological insulators , \ @noop journal journal Sci...

    2018

  33. [41]

    Noguchi , author M

    author author R. Noguchi , author M. Kobayashi , author Z. Jiang , author K. Kuroda , author T. Takahashi , author Z. Xu , author D. Lee , author M. Hirayama , author M. Ochi , author T. Shirasawa , et al. ,\ title title Evidence for a higher-order topological insulator in a t...

    2021 doi

  34. [42]

    author author Leena \ Aggarwal , author Penghao \ Zhu , author Taylor L. \ Hughes , \ and\ author Vidya \ Madhavan ,\ title title Evidence for higher order topology in bi and bi _ 0.92 sb _ 0.08 , \ 10.1038/s41467-021-24683-8 journal journal Nat Commun \ volume 12 ,\ pages 442...

    2021 doi

  35. [43]

    author author Takashi \ Oka \ and\ author Sota \ Kitamura ,\ title title Floquet engineering of quantum materials , \ @noop journal journal Annual Review of Condensed Matter Physics \ volume 10 ,\ pages 387--408 ( year 2019 ) NoStop

    2019

  36. [44]

    author author Andr\'e \ Eckardt ,\ title title Colloquium: Atomic quantum gases in periodically driven optical lattices , \ 10.1103/RevModPhys.89.011004 journal journal Rev. Mod. Phys. \ volume 89 ,\ pages 011004 ( year 2017 ) NoStop

    2017 doi

  37. [45]

    Meinert , author M

    author author F. Meinert , author M. J. \ Mark , author K. Lauber , author A. J. \ Daley , \ and\ author H.-C. \ N\"agerl ,\ title title Floquet engineering of correlated tunneling in the bose-hubbard model with ultracold atoms , \ 10.1103/PhysRevLett.116.205301 journal journa...

    2016 doi

  38. [46]

    author author YH Wang , author Hadar \ Steinberg , author Pablo \ Jarillo-Herrero , \ and\ author Nuh \ Gedik ,\ title title Observation of floquet-bloch states on the surface of a topological insulator , \ @noop journal journal Science \ volume 342 ,\ pages 453--457 ( year 20...

    2013

  39. [47]

    author author Mikael C \ Rechtsman , author Julia M \ Zeuner , author Yonatan \ Plotnik , author Yaakov \ Lumer , author Daniel \ Podolsky , author Felix \ Dreisow , author Stefan \ Nolte , author Mordechai \ Segev , \ and\ author Alexander \ Szameit ,\ title title Photonic fl...

    2013

  40. [48]

    author author Nakshatra \ Gangopadhay \ and\ author Sayan \ Choudhury ,\ title title Counterdiabatic route to entanglement steering and dynamical freezing in the floquet lipkin-meshkov-glick model , \ 10.1103/bzcf-gm89 journal journal Phys. Rev. Lett. \ volume 135 ,\ pages 020...

    2025 doi

  41. [49]

    \ Shirley ,\ title title Solution of the schr\"odinger equation with a hamiltonian periodic in time , \ 10.1103/PhysRev.138.B979 journal journal Phys

    author author Jon H. \ Shirley ,\ title title Solution of the schr\"odinger equation with a hamiltonian periodic in time , \ 10.1103/PhysRev.138.B979 journal journal Phys. Rev. \ volume 138 ,\ pages B979--B987 ( year 1965 ) NoStop

    1965 doi

  42. [50]

    author author Marin \ Bukov , author Luca \ D'Alessio , \ and\ author Anatoli \ Polkovnikov ,\ title title Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to floquet engineering , \ @noop journal journal Advances in Physics \ volu...

    2015

  43. [51]

    author author Takashi \ Oka \ and\ author Hideo \ Aoki ,\ title title Photovoltaic hall effect in graphene , \ 10.1103/PhysRevB.79.081406 journal journal Phys. Rev. B \ volume 79 ,\ pages 081406 ( year 2009 ) NoStop

    2009 doi

  44. [52]

    author author Takuya \ Kitagawa , author Erez \ Berg , author Mark \ Rudner , \ and\ author Eugene \ Demler ,\ title title Topological characterization of periodically driven quantum systems , \ 10.1103/PhysRevB.82.235114 journal journal Phys. Rev. B \ volume 82 ,\ pages 23511...

    2010 doi

  45. [53]

    author author Zhenghao \ Gu , author H. A. \ Fertig , author Daniel P. \ Arovas , \ and\ author Assa \ Auerbach ,\ title title Floquet spectrum and transport through an irradiated graphene ribbon , \ 10.1103/PhysRevLett.107.216601 journal journal Phys. Rev. Lett. \ volume 107 ...

    2011 doi

  46. [54]

    author author Netanel H \ Lindner , author Gil \ Refael , \ and\ author Victor \ Galitski ,\ title title Floquet topological insulator in semiconductor quantum wells , \ https://doi.org/10.1038/nphys1926 journal journal Nature Physics \ volume 7 ,\ pages 490--495 ( year 2011 ) NoStop

    2011 doi

  47. [55]

    \ Lindner , author Doron L

    author author Netanel H. \ Lindner , author Doron L. \ Bergman , author Gil \ Refael , \ and\ author Victor \ Galitski ,\ title title Topological floquet spectrum in three dimensions via a two-photon resonance , \ 10.1103/PhysRevB.87.235131 journal journal Phys. Rev. B \ volum...

    2013 doi

  48. [56]

    Goldman \ and\ author J

    author author N. Goldman \ and\ author J. Dalibard ,\ title title Periodically driven quantum systems: Effective hamiltonians and engineered gauge fields , \ 10.1103/PhysRevX.4.031027 journal journal Phys. Rev. X \ volume 4 ,\ pages 031027 ( year 2014 ) NoStop

    2014 doi

  49. [57]

    \ Rudner , author Netanel H

    author author Mark S. \ Rudner , author Netanel H. \ Lindner , author Erez \ Berg , \ and\ author Michael \ Levin ,\ title title Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems , \ 10.1103/PhysRevX.3.031005 journal journal...

    2013 doi

  50. [58]

    author author Arijit \ Kundu , author H. A. \ Fertig , \ and\ author Babak \ Seradjeh ,\ title title Effective theory of floquet topological transitions , \ 10.1103/PhysRevLett.113.236803 journal journal Phys. Rev. Lett. \ volume 113 ,\ pages 236803 ( year 2014 ) NoStop

    2014 doi

  51. [59]

    author author Mahmoud \ Lababidi , author Indubala I. \ Satija , \ and\ author Erhai \ Zhao ,\ title title Counter-propagating edge modes and topological phases of a kicked quantum hall system , \ 10.1103/PhysRevLett.112.026805 journal journal Phys. Rev. Lett. \ volume 112 ,\ ...

    2014 doi

  52. [60]

    author author Debashish \ Mondal , author Arnob Kumar \ Ghosh , author Tanay \ Nag , \ and\ author Arijit \ Saha ,\ title title Topological characterization and stability of floquet majorana modes in rashba nanowires , \ 10.1103/PhysRevB.107.035427 journal journal Phys. Rev. B...

    2023 doi

  53. [61]

    author author Debashish \ Mondal , author Arnob Kumar \ Ghosh , author Tanay \ Nag , \ and\ author Arijit \ Saha ,\ title title Engineering anomalous floquet majorana modes and their time evolution in a helical shiba chain , \ 10.1103/PhysRevB.108.L081403 journal journal Phys....

    2023 doi

  54. [63]

    Vincent \ Liu ,\ title title Floquet higher-order topological insulators with anomalous dynamical polarization , \ 10.1103/PhysRevLett.124.216601 journal journal Phys

    author author Biao \ Huang \ and\ author W. Vincent \ Liu ,\ title title Floquet higher-order topological insulators with anomalous dynamical polarization , \ 10.1103/PhysRevLett.124.216601 journal journal Phys. Rev. Lett. \ volume 124 ,\ pages 216601 ( year 2020 ) NoStop

    2020 doi

  55. [64]

    author author Weiwei \ Zhu , author Y. D. \ Chong , \ and\ author Jiangbin \ Gong ,\ title title Floquet higher-order topological insulator in a periodically driven bipartite lattice , \ 10.1103/PhysRevB.103.L041402 journal journal Phys. Rev. B \ volume 103 ,\ pages L041402 ( ...

    2021 doi

  56. [65]

    author author Arnob Kumar \ Ghosh , author Tanay \ Nag , \ and\ author Arijit \ Saha ,\ title title Generation of higher-order topological insulators using periodic driving , \ @noop journal journal Journal of Physics: Condensed Matter \ volume 36 ,\ pages 093001 ( year 2023 ) NoStop

    2023

  57. [66]

    author author Arnob Kumar \ Ghosh , author Tanay \ Nag , \ and\ author Arijit \ Saha ,\ title title Dynamical construction of quadrupolar and octupolar topological superconductors , \ 10.1103/PhysRevB.105.155406 journal journal Phys. Rev. B \ volume 105 ,\ pages 155406 ( year ...

    2022 doi

  58. [67]

    author author Dmitry A \ Abanin , author Wojciech \ De Roeck , \ and\ author Fran c ois \ Huveneers ,\ title title Theory of many-body localization in periodically driven systems , \ @noop journal journal Annals of Physics \ volume 372 ,\ pages 1--11 ( year 2016 ) NoStop

    2016

  59. [68]

    \ Huse ,\ title title A floquet model for the many-body localization transition , \ 10.1103/PhysRevB.94.224202 journal journal Phys

    author author Liangsheng \ Zhang , author Vedika \ Khemani , \ and\ author David A. \ Huse ,\ title title A floquet model for the many-body localization transition , \ 10.1103/PhysRevB.94.224202 journal journal Phys. Rev. B \ volume 94 ,\ pages 224202 ( year 2016 ) NoStop

    2016 doi

  60. [69]

    author author Vedika \ Khemani , author Achilleas \ Lazarides , author Roderich \ Moessner , \ and\ author Shivaji L \ Sondhi ,\ title title Phase structure of driven quantum systems , \ 10.1103/PhysRevLett.116.250401 journal journal Physical Review Letters \ volume 116 ,\ pag...

    2016 doi

  61. [70]

    \ Else , author Bela \ Bauer , \ and\ author Chetan \ Nayak ,\ title title Floquet time crystals , \ 10.1103/PhysRevLett.117.090402 journal journal Phys

    author author Dominic V. \ Else , author Bela \ Bauer , \ and\ author Chetan \ Nayak ,\ title title Floquet time crystals , \ 10.1103/PhysRevLett.117.090402 journal journal Phys. Rev. Lett. \ volume 117 ,\ pages 090402 ( year 2016 ) NoStop

    2016 doi

  62. [71]

    author author Norman Y \ Yao , author Andrew C \ Potter , author I-D \ Potirniche , \ and\ author Ashvin \ Vishwanath ,\ title title Discrete time crystals: Rigidity, criticality, and realizations , \ 10.1103/PhysRevLett.118.030401 journal journal Physical Review Letters \ vol...

    2017 doi

  63. [72]

    author author Dominic V \ Else , author Christopher \ Monroe , author Chetan \ Nayak , \ and\ author Norman Y \ Yao ,\ title title Discrete time crystals , \ 10.1146/annurev-conmatphys-031119-050658 journal journal Annual Review of Condensed Matter Physics \ volume 11 ,\ pages...

    2020 doi

  64. [73]

    author author Michael P \ Zaletel , author Mikhail \ Lukin , author Christopher \ Monroe , author Chetan \ Nayak , author Frank \ Wilczek , \ and\ author Norman Y \ Yao ,\ title title Colloquium: Quantum and classical discrete time crystals , \ 10.1103/RevModPhys.95.031001 jou...

    2023 doi

  65. [74]

    author author Biao \ Huang , author Ying-Hai \ Wu , \ and\ author W. Vincent \ Liu ,\ title title Clean floquet time crystals: Models and realizations in cold atoms , \ 10.1103/PhysRevLett.120.110603 journal journal Physical Review Letters \ volume 120 ,\ pages 110603 ( year 2...

    2018 doi

  66. [75]

    author author Changyuan \ Lyu , author Sayan \ Choudhury , author Chenwei \ Lv , author Yangqian \ Yan , \ and\ author Qi Zhou ,\ title title Eternal discrete time crystal beating the heisenberg limit , \ 10.1103/PhysRevResearch.2.033070 journal journal Physical Review Researc...

    2020 doi

  67. [76]

    author author Yixiao \ Wang , author Anne-Sophie \ Walter , author Gregor \ Jotzu , \ and\ author Konrad \ Viebahn ,\ title title Topological floquet engineering using two frequencies in two dimensions , \ 10.1103/PhysRevA.107.043309 journal journal Phys. Rev. A \ volume 107 ,...

    2023 doi

  68. [77]

    author author Tanay \ Nag , author Robert-Jan \ Slager , author Takuya \ Higuchi , \ and\ author Takashi \ Oka ,\ title title Dynamical synchronization transition in interacting electron systems , \ 10.1103/PhysRevB.100.134301 journal journal Phys. Rev. B \ volume 100 ,\ pages...

    2019 doi

  69. [78]

    author author Yuya \ Ikeda , author Sota \ Kitamura , \ and\ author Takahiro \ Morimoto ,\ title title Floquet engineering of electric polarization with two-frequency drive , \ @noop journal journal Progress of Theoretical and Experimental Physics \ volume 2022 ,\ pages 04A101...

    2022

  70. [79]

    author author Konrad \ Viebahn , author Joaqu\' n \ Minguzzi , author Kilian \ Sandholzer , author Anne-Sophie \ Walter , author Manish \ Sajnani , author Frederik \ G\"org , \ and\ author Tilman \ Esslinger ,\ title title Suppressing dissipation in a floquet-hubbard system , ...

    2021 doi

  71. [80]

    author author Kaoru \ Mizuta ,\ title title Optimal and nearly optimal simulation of multiperiodic time-dependent hamiltonians , \ 10.1103/PhysRevResearch.5.033067 journal journal Phys. Rev. Res. \ volume 5 ,\ pages 033067 ( year 2023 ) NoStop

    2023 doi

  72. [81]

    author author Tista \ Banerjee , author Sayan \ Choudhury , \ and\ author Krishnendu \ Sengupta ,\ title title Exact floquet flat band and heating suppression via two-tone drive protocols , \ 10.1088/1367-2630/adfd07 journal journal New Journal of Physics \ volume 27 ,\ pages ...

    2025 doi

  73. [82]

    author author Pinaki \ Dutta , author Kamal L \ Panigrahi , \ and\ author Vishwanath \ Shukla ,\ title title Controlling quantum scars and engineering subharmonic responses with a two frequency drive , \ @noop journal journal arXiv preprint arXiv:2508.12809 \ ( year 2025 a ) NoStop

    2025

  74. [83]

    author author Albert \ Verdeny , author Joaquim \ Puig , \ and\ author Florian \ Mintert ,\ title title Quasi-periodically driven quantum systems , \ doi:10.1515/zna-2016-0079 journal journal Zeitschrift für Naturforschung A \ volume 71 ,\ pages 897--907 ( year 2016 ) NoStop

    2016 doi

  75. [84]

    author author Sourav \ Nandy , author Arnab \ Sen , \ and\ author Diptiman \ Sen ,\ title title Aperiodically driven integrable systems and their emergent steady states , \ 10.1103/PhysRevX.7.031034 journal journal Physical Review X \ volume 7 ,\ pages 031034 ( year 2017 ) NoStop

    2017 doi

  76. [85]

    author author Tanay \ Nag \ and\ author Atanu \ Rajak ,\ title title Periodic and aperiodic dynamics of flat bands in diamond-octagon lattice , \ 10.1103/PhysRevB.104.134307 journal journal Phys. Rev. B \ volume 104 ,\ pages 134307 ( year 2021 ) NoStop

    2021 doi

  77. [86]

    \ Long , author Philip J

    author author David M. \ Long , author Philip J. D. \ Crowley , \ and\ author Anushya \ Chandran ,\ title title Many-body localization with quasiperiodic driving , \ 10.1103/PhysRevB.105.144204 journal journal Phys. Rev. B \ volume 105 ,\ pages 144204 ( year 2022 a ) NoStop

    2022 doi

  78. [87]

    author author Aritra \ Kundu , author Tanay \ Nag , \ and\ author Atanu \ Rajak ,\ title title Statistical prethermalization in randomly kicked many-body classical rotor system , \ @noop journal journal arXiv preprint arXiv:2408.11109 \ ( year 2024 ) NoStop

    2024

  79. [88]

    author author Hongzheng \ Zhao , author Florian \ Mintert , author Roderich \ Moessner , \ and\ author Johannes \ Knolle ,\ title title Random multipolar driving: Tunably slow heating through spectral engineering , \ 10.1103/PhysRevLett.126.040601 journal journal Physical Revi...

    2021 doi

  80. [89]

    author author Hongzheng \ Zhao , author Florian \ Mintert , author Johannes \ Knolle , \ and\ author Roderich \ Moessner ,\ title title Localization persisting under aperiodic driving , \ 10.1103/PhysRevB.105.L220202 journal journal Phys. Rev. B \ volume 105 ,\ pages L220202 (...

    2022 doi

  81. [90]

    \ Murch , author Norman Y

    author author Guanghui \ He , author Bingtian \ Ye , author Ruotian \ Gong , author Zhongyuan \ Liu , author Kater W. \ Murch , author Norman Y. \ Yao , \ and\ author Chong \ Zu ,\ title title Quasi-floquet prethermalization in a disordered dipolar spin ensemble in diamond , \...

    2023 doi

  82. [91]

    author author Vatsana \ Tiwari , author Devendra Singh \ Bhakuni , \ and\ author Auditya \ Sharma ,\ title title Periodically and aperiodically thue-morse driven long-range systems: From dynamical localization to slow dynamics , \ 10.1103/PhysRevB.111.205109 journal journal Ph...

    2025 doi

  83. [92]

    author author Sk Anisur , author Wensheng Vincent \ Liu , \ and\ author Sayan \ Choudhury ,\ title title Quasi-discrete time crystals in the quasiperiodically driven lipkin–meshkov–glick model , \ 10.3390/e27060609 journal journal Entropy \ volume 27 ( year 2025 ),\ 10.3390/e2...

    2025 doi

  84. [93]

    author author Pinaki \ Dutta , author Sayan \ Choudhury , \ and\ author Vishwanath \ Shukla ,\ title title Prethermalization in the pxp model under continuous quasiperiodic driving , \ 10.1103/PhysRevB.111.064303 journal journal Phys. Rev. B \ volume 111 ,\ pages 064303 ( year...

    2025 doi

  85. [94]

    author author Krishanu \ Ghosh , author Sayan \ Choudhury , author Diptiman \ Sen , \ and\ author K Sengupta ,\ title title Heating suppression via two-rate random and quasiperiodic drive protocols , \ @noop journal journal arXiv preprint arXiv:2508.02783 \ ( year 2025 ) NoStop

    2025

  86. [95]

    author author Hongzheng \ Zhao , author Florian \ Mintert , \ and\ author Johannes \ Knolle ,\ title title Floquet time spirals and stable discrete-time quasicrystals in quasiperiodically driven quantum many-body systems , \ 10.1103/PhysRevB.100.134302 journal journal Phys. Re...

    2019 doi

  87. [96]

    \ Dumitrescu , author Romain \ Vasseur , \ and\ author Andrew C

    author author Philipp T. \ Dumitrescu , author Romain \ Vasseur , \ and\ author Andrew C. \ Potter ,\ title title Logarithmically slow relaxation in quasiperiodically driven random spin chains , \ 10.1103/PhysRevLett.120.070602 journal journal Phys. Rev. Lett. \ volume 120 ,\ ...

    2018 doi

  88. [97]

    \ Murch , author Norman Y

    author author Guanghui \ He , author Bingtian \ Ye , author Ruotian \ Gong , author Changyu \ Yao , author Zhongyuan \ Liu , author Kater W. \ Murch , author Norman Y. \ Yao , \ and\ author Chong \ Zu ,\ title title Experimental realization of discrete time quasicrystals , \ 1...

    2025 doi

  89. [98]

    author author Hongzheng \ Zhao , author Johannes \ Knolle , \ and\ author Roderich \ Moessner ,\ title title Temporal disorder in spatiotemporal order , \ 10.1103/PhysRevB.108.L100203 journal journal Phys. Rev. B \ volume 108 ,\ pages L100203 ( year 2023 ) NoStop

    2023 doi

  90. [99]

    author author Sajag \ Kumar \ and\ author Sayan \ Choudhury ,\ title title Prethermalization in aperiodically driven classical spin systems , \ 10.1103/PhysRevE.110.064150 journal journal Phys. Rev. E \ volume 110 ,\ pages 064150 ( year 2024 ) NoStop

    2024 doi

  91. [100]

    author author Leo Joon Il \ Moon , author Paul Manuel \ Schindler , author Yizhe \ Sun , author Emanuel \ Druga , author Johannes \ Knolle , author Roderich \ Moessner , author Hongzheng \ Zhao , author Marin \ Bukov , \ and\ author Ashok \ Ajoy ,\ title title Experimental obs...

    2024

  92. [101]

    author author Zhuocheng \ Ma , author Jin \ Yan , author Hongzheng \ Zhao , \ and\ author Liang-You \ Peng ,\ title title Stable time rondeau crystals in dissipative many-body systems , \ @noop journal journal arXiv preprint arXiv:2502.16999 \ ( year 2025 ) NoStop

    2025

  93. [102]

    author author P. J. D. \ Crowley , author I. Martin , \ and\ author A. Chandran ,\ title title Topological classification of quasiperiodically driven quantum systems , \ 10.1103/PhysRevB.99.064306 journal journal Phys. Rev. B \ volume 99 ,\ pages 064306 ( year 2019 ) NoStop

    2019 doi

  94. [103]

    author author Zihao \ Qi , author Gil \ Refael , \ and\ author Yang \ Peng ,\ title title Universal nonadiabatic energy pumping in a quasiperiodically driven extended system , \ 10.1103/PhysRevB.104.224301 journal journal Phys. Rev. B \ volume 104 ,\ pages 224301 ( year 2021 ) NoStop

    2021 doi

  95. [104]

    author author Yang \ Peng \ and\ author Gil \ Refael ,\ title title Time-quasiperiodic topological superconductors with majorana multiplexing , \ 10.1103/PhysRevB.98.220509 journal journal Phys. Rev. B \ volume 98 ,\ pages 220509 ( year 2018 ) NoStop

    2018 doi

  96. [105]

    author author Christina \ Psaroudaki \ and\ author Gil \ Refael ,\ title title Energy transfer in random-matrix ensembles of floquet hamiltonians , \ 10.1103/PhysRevB.108.064301 journal journal Phys. Rev. B \ volume 108 ,\ pages 064301 ( year 2023 ) NoStop

    2023 doi

  97. [106]

    author author Frederik \ Nathan , author Ivar \ Martin , \ and\ author Gil \ Refael ,\ title title Topological frequency conversion in weyl semimetals , \ 10.1103/PhysRevResearch.4.043060 journal journal Phys. Rev. Res. \ volume 4 ,\ pages 043060 ( year 2022 ) NoStop

    2022 doi

  98. [107]

    author author \'Etienne \ Lantagne-Hurtubise , author Iliya \ Esin , author Gil \ Refael , \ and\ author Frederik \ Nathan ,\ title title Topological frequency conversion in rhombohedral multilayer graphene , \ 10.1103/PhysRevB.110.L100305 journal journal Phys. Rev. B \ volume...

    2024 doi

  99. [108]

    author author Zihao \ Qi , author Ilyoun \ Na , author Gil \ Refael , \ and\ author Yang \ Peng ,\ title title Real-space topological invariant for time-quasiperiodic majorana modes , \ 10.1103/PhysRevB.110.014309 journal journal Phys. Rev. B \ volume 110 ,\ pages 014309 ( yea...

    2024 doi

  100. [109]

    \ Long , author Philip J

    author author David M. \ Long , author Philip J. D. \ Crowley , \ and\ author Anushya \ Chandran ,\ title title Coupled layer construction for synthetic hall effects in driven systems , \ 10.1103/PhysRevB.106.144203 journal journal Phys. Rev. B \ volume 106 ,\ pages 144203 ( y...

    2022 doi

  101. [110]

    orber , author Lorenzo \ Privitera , author Jan Carl \ Budich , \ and\ author Bj\

    author author Simon \ K\"orber , author Lorenzo \ Privitera , author Jan Carl \ Budich , \ and\ author Bj\"orn \ Trauzettel ,\ title title Interacting topological frequency converter , \ 10.1103/PhysRevResearch.2.022023 journal journal Phys. Rev. Res. \ volume 2 ,\ pages 02202...

    2020 doi

  102. [111]

    author author Ivar \ Martin , author Gil \ Refael , \ and\ author Bertrand \ Halperin ,\ title title Topological frequency conversion in strongly driven quantum systems , \ 10.1103/PhysRevX.7.041008 journal journal Phys. Rev. X \ volume 7 ,\ pages 041008 ( year 2017 ) NoStop

    2017 doi

  103. [112]

    author author Eric \ Boyers , author Philip J. D. \ Crowley , author Anushya \ Chandran , \ and\ author Alexander O. \ Sushkov ,\ title title Exploring 2d synthetic quantum hall physics with a quasiperiodically driven qubit , \ 10.1103/PhysRevLett.125.160505 journal journal Ph...

    2020 doi

  104. [113]

    author author P. J. D. \ Crowley , author I. Martin , \ and\ author A. Chandran ,\ title title Half-integer quantized topological response in quasiperiodically driven quantum systems , \ 10.1103/PhysRevLett.125.100601 journal journal Physical Review Letters \ volume 125 ,\ pag...

    2020 doi

  105. [114]

    author author Sashank Kaushik \ Sridhar , author Sayan \ Ghosh , author Dhruv \ Srinivasan , author Alexander R \ Miller , \ and\ author Avik \ Dutt ,\ title title Quantized topological pumping in floquet synthetic dimensions with a driven dissipative photonic molecule , \ @no...

    2024

  106. [115]

    \ Long , author Philip J

    author author Dominik \ Vuina , author David M. \ Long , author Philip J. D. \ Crowley , \ and\ author Anushya \ Chandran ,\ title title Giant energy oscillations mediated by a quasiperiodically driven qubit , \ 10.1103/PhysRevB.108.134303 journal journal Physical Review B \ v...

    2023 doi

  107. [116]

    author author Olivia Y. \ Long , author Kai \ Wang , author Avik \ Dutt , \ and\ author Shanhui \ Fan ,\ title title Time reflection and refraction in synthetic frequency dimension , \ 10.1103/PhysRevResearch.5.L012046 journal journal Phys. Rev. Res. \ volume 5 ,\ pages L01204...

    2023 doi

  108. [117]

    author author HaRu K. \ Park , author Junmo \ Jeon , author Gil Young \ Cho , \ and\ author SungBin \ Lee ,\ title title Controllable floquet edge modes in a multifrequency driving system , \ 10.1103/PhysRevB.107.085141 journal journal Phys. Rev. B \ volume 107 ,\ pages 085141...

    2023 doi

  109. [118]

    \ Ashcroft \ and\ author N

    author author Neil W. \ Ashcroft \ and\ author N. David \ Mermin ,\ @noop title Solid State Physics \ ( publisher Holt, Rinehart and Winston ,\ address New York ,\ year 1976 ) NoStop

    1976

  110. [119]

    author author Gabriele \ Giuliani \ and\ author Giovanni \ Vignale ,\ @noop title Quantum theory of the electron liquid \ ( publisher Cambridge university press ,\ year 2008 ) NoStop

    2008

  111. [120]

    author author Tanay \ Nag , author Vladimir \ Juri c c i c \' c , \ and\ author Bitan \ Roy ,\ title title Out of equilibrium higher-order topological insulator: Floquet engineering and quench dynamics , \ 10.1103/PhysRevResearch.1.032045 journal journal Phys. Rev. Research \ ...

    2019 doi

  112. [121]

    author author B Andrei \ Bernevig , author Taylor L \ Hughes , \ and\ author Shou-Cheng \ Zhang ,\ title title Quantum spin hall effect and topological phase transition in hgte quantum wells , \ https://doi.org/10.1126/science.1133734 journal journal Science \ volume 314 ,\ pa...

    2006 doi

  113. [122]

    author author Huaqing \ Huang \ and\ author Feng \ Liu ,\ title title Quantum spin hall effect and spin bott index in a quasicrystal lattice , \ 10.1103/PhysRevLett.121.126401 journal journal Phys. Rev. Lett. \ volume 121 ,\ pages 126401 ( year 2018 ) NoStop

    2018 doi

  114. [123]

    author author Katharina \ Fritz , author Sebastian \ Wimmer , author Hubert \ Ebert , \ and\ author Markus \ Meinert ,\ title title Large spin hall effect in an amorphous binary alloy , \ 10.1103/PhysRevB.98.094433 journal journal Phys. Rev. B \ volume 98 ,\ pages 094433 ( yea...

    2018 doi

  115. [124]

    author author Adhip \ Agarwala , author Vladimir \ Juri c c i c \' c , \ and\ author Bitan \ Roy ,\ title title Higher-order topological insulators in amorphous solids , \ 10.1103/PhysRevResearch.2.012067 journal journal Phys. Rev. Res. \ volume 2 ,\ pages 012067 ( year 2020 ) NoStop

    2020 doi

  116. [125]

    author author Yan-Bin \ Yang , author Jiong-Hao \ Wang , author Kai \ Li , \ and\ author Yong \ Xu ,\ title title Higher-order topological phases in crystalline and non-crystalline systems: a review , \ 10.1088/1361-648X/ad3abd journal journal Journal of Physics: Condensed Mat...

    2024 doi

  117. [126]

    \ Long , author Philip J

    author author David M. \ Long , author Philip J. D. \ Crowley , \ and\ author Anushya \ Chandran ,\ title title Nonadiabatic topological energy pumps with quasiperiodic driving , \ 10.1103/PhysRevLett.126.106805 journal journal Phys. Rev. Lett. \ volume 126 ,\ pages 106805 ( y...

    2021 doi

  118. [127]

    author author Takahiro \ Fukui , author Yasuhiro \ Hatsugai , \ and\ author Hiroshi \ Suzuki ,\ title title Chern numbers in discretized brillouin zone: Efficient method of computing (spin) hall conductances , \ 10.1143/JPSJ.74.1674 journal journal Journal of the Physical Soci...

    2005 doi

  119. [128]

    author author Arnob Kumar \ Ghosh , author Tanay \ Nag , \ and\ author Arijit \ Saha ,\ title title Floquet generation of a second-order topological superconductor , \ 10.1103/PhysRevB.103.045424 journal journal Phys. Rev. B \ volume 103 ,\ pages 045424 ( year 2021 a ) NoStop

    2021 doi

  120. [129]

    author author Arnob Kumar \ Ghosh , author Tanay \ Nag , \ and\ author Arijit \ Saha ,\ title title Floquet second order topological superconductor based on unconventional pairing , \ 10.1103/PhysRevB.103.085413 journal journal Phys. Rev. B \ volume 103 ,\ pages 085413 ( year ...

    2021 doi

  121. [130]

    author author Arnob Kumar \ Ghosh , author Tanay \ Nag , \ and\ author Arijit \ Saha ,\ title title Hierarchy of higher-order topological superconductors in three dimensions , \ 10.1103/PhysRevB.104.134508 journal journal Phys. Rev. B \ volume 104 ,\ pages 134508 ( year 2021 c...

    2021 doi

  122. [131]

    author author Pei \ Liu , author Ruixia \ Wang , author Jing-Ning \ Zhang , author Yingshan \ Zhang , author Xiaoxia \ Cai , author Huikai \ Xu , author Zhiyuan \ Li , author Jiaxiu \ Han , author Xuegang \ Li , author Guangming \ Xue , author Weiyang \ Liu , author Li You , a...

    2023 doi

  123. [132]

    \ Nguyen , author Irfan \ Siddiqi , \ and\ author Daniel A

    author author Vinay \ Tripathi , author Noah \ Goss , author Arian \ Vezvaee , author Long B. \ Nguyen , author Irfan \ Siddiqi , \ and\ author Daniel A. \ Lidar ,\ title title Qudit dynamical decoupling on a superconducting quantum processor , \ 10.1103/PhysRevLett.134.050601...

    2025 doi

  124. [133]

    author author Pavel \ Hrmo , author Benjamin \ Wilhelm , author Lukas \ Gerster , author Martin W \ van Mourik , author Marcus \ Huber , author Rainer \ Blatt , author Philipp \ Schindler , author Thomas \ Monz , \ and\ author Martin \ Ringbauer ,\ title title Native qudit ent...

    2023

  125. [134]

    \ Meier , author Michael J

    author author Sivaprasad \ Omanakuttan , author Anupam \ Mitra , author Eric J. \ Meier , author Michael J. \ Martin , \ and\ author Ivan H. \ Deutsch ,\ title title Qudit entanglers using quantum optimal control , \ 10.1103/PRXQuantum.4.040333 journal journal PRX Quantum \ vo...

    2023 doi

  126. [135]

    author author Daniel \ Gonz\'alez-Cuadra , author Torsten V. \ Zache , author Jose \ Carrasco , author Barbara \ Kraus , \ and\ author Peter \ Zoller ,\ title title Hardware efficient quantum simulation of non-abelian gauge theories with qudits on rydberg platforms , \ 10.1103...

    2022 doi

  127. [136]

    author author Martin \ Ringbauer , author Michael \ Meth , author Lukas \ Postler , author Roman \ Stricker , author Rainer \ Blatt , author Philipp \ Schindler , \ and\ author Thomas \ Monz ,\ title title A universal qudit quantum processor with trapped ions , \ 10.1038/s4156...

    2022 doi

  128. [137]

    author author Pei Jiang \ Low , author Brendan \ White , \ and\ author Crystal \ Senko ,\ title title Control and readout of a 13-level trapped ion qudit , \ 10.1038/s41534-025-01031-y journal journal npj Quantum Information \ volume 11 ,\ pages 85 ( year 2025 ) NoStop

    2025 doi

  129. [138]

    Flament , author N

    author author E. Flament , author N. Ombredane , author F. Arrouas , author D. Ronco , author B. Peaudecerf , author D. Sugny , \ and\ author D. Gu\'ery-Odelin ,\ title title Unitary transformations using robust optimal control on a cold atom qudit , \ 10.1103/fvlj-yph6 journa...

    2025 doi

  130. [139]

    \ Leitao , author Oksana \ Makarova , author Iris \ Cong , author Alexander M

    author author Hengyun \ Zhou , author Haoyang \ Gao , author Nathaniel T. \ Leitao , author Oksana \ Makarova , author Iris \ Cong , author Alexander M. \ Douglas , author Leigh S. \ Martin , \ and\ author Mikhail D. \ Lukin ,\ title title Robust hamiltonian engineering for in...

    2024 doi

  131. [140]

    author author Ming-Xin \ Dong , author Wei-Hang \ Zhang , author Lei \ Zeng , author Ying-Hao \ Ye , author Da-Chuang \ Li , author Guang-Can \ Guo , author Dong-Sheng \ Ding , \ and\ author Bao-Sen \ Shi ,\ title title Highly efficient storage of 25-dimensional photonic qudit...

    2023 doi

  132. [141]

    author author Yuchen \ Wang , author Zixuan \ Hu , \ and\ author Sabre \ Kais ,\ title title Photonic realization of qudit quantum computing , \ @noop journal journal Photonic Quantum Technologies: Science and Applications \ volume 2 ,\ pages 651--674 ( year 2023 b ) NoStop

    2023

  133. [142]

    author author Zheng \ Wang , author Yidong \ Chong , author John D \ Joannopoulos , \ and\ author Marin \ Solja c i \'c ,\ title title Observation of unidirectional backscattering-immune topological electromagnetic states , \ @noop journal journal Nature \ volume 461 ,\ pages ...

    2009

  134. [143]

    author author Xiao-Dong \ Chen , author Wei-Min \ Deng , author Fu-Long \ Shi , author Fu-Li \ Zhao , author Min \ Chen , \ and\ author Jian-Wen \ Dong ,\ title title Direct observation of corner states in second-order topological photonic crystal slabs , \ 10.1103/PhysRevLett...

    2019 doi

  135. [144]

    author author Zhaoju \ Yang , author Fei \ Gao , author Xihang \ Shi , author Xiao \ Lin , author Zhen \ Gao , author Yidong \ Chong , \ and\ author Baile \ Zhang ,\ title title Topological acoustics , \ 10.1103/PhysRevLett.114.114301 journal journal Phys. Rev. Lett. \ volume ...

    2015 doi

  136. [145]

    author author Cheng \ He , author Xu Ni , author Hao \ Ge , author Xiao-Chen \ Sun , author Yan-Bin \ Chen , author Ming-Hui \ Lu , author Xiao-Ping \ Liu , \ and\ author Yan-Feng \ Chen ,\ title title Acoustic topological insulator and robust one-way sound transport , \ @noop...

    2016

  137. [146]

    author author Gregor \ Jotzu , author Michael \ Messer , author R \'e mi \ Desbuquois , author Martin \ Lebrat , author Thomas \ Uehlinger , author Daniel \ Greif , \ and\ author Tilman \ Esslinger ,\ title title Experimental realization of the topological haldane model with u...

    2014

  138. [147]

    author author Michael \ Lohse , author Christian \ Schweizer , author Oded \ Zilberberg , author Monika \ Aidelsburger , \ and\ author Immanuel \ Bloch ,\ title title A thouless quantum pump with ultracold bosonic atoms in an optical superlattice , \ @noop journal journal Natu...

    2016

  139. [148]

    author author Xin-Lei \ Hei , author Xing-Liang \ Dong , author Jia-Qiang \ Chen , author Yi-Fan \ Qiao , author Xue-Feng \ Pan , author Xiao-Yu \ Yao , author Jun-Cong \ Zheng , author Yu-Meng \ Ren , author Xiao-Wen \ Huo , \ and\ author Peng-Bo \ Li ,\ title title Topologic...

    2024 doi

Pith tools

Reviewed May 18, 2026 · model on record in the stance chip above.