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Emergent discrete space-time crystal of Majorana-like quasiparticles in chiral liquid crystals

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper reports that a periodically driven chiral nematic liquid crystal spontaneously organizes into 1+1D and 2+1D discrete space-time crystals, with the internal period doubling relative to the drive through interconversion of…

desk verdict Genuine period-doubling in a Floquet-driven chiral LC, but the Majorana-like mechanism and DSTC classification are overreaches that need revision. read the letter →

arxiv 2507.16977 v1 pith:FXKCI63B submitted 2025-07-22 cond-mat.soft

Hanqing Zhao , Rui Zhang , Ivan I. Smalyukh This is my paper · ORCID
classification cond-mat.soft
keywords discretespace-timecrystalclassicaltimechiralnematicliquidperioddoublingFloquetdrivetopologicalsolitonsdisclinationsMajorana-likequasiparticles
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 reports that a chiral nematic liquid crystal driven by a periodic electrical signal spontaneously forms spatially periodic structures whose optical pattern repeats every two drive periods rather than every one. This is a classical analogue of a discrete space-time crystal, and the authors observe both 1+1-dimensional and 2+1-dimensional versions over wide ranges of temperature, voltage, and drive period. Using simulations, they trace the period-doubling to periodic inter-conversions, creations, and annihilations of topological solitons and disclination lines, which they treat as particle/antiparticle pairs of Majorana-like quasiparticles. The crystals resist random timing jitter and heal their own lattice defects, and their temporal correlation decays as a power law, analogous to a smectic phase. If correct, the work shows that discrete time-translation symmetry breaking is not confined to quantum systems and could be engineered in ordinary soft materials.

What carries the argument

The load-bearing object is the periodic array of topological quasiparticles: singular disclination lines of winding number $\pm 1/2$ terminated by domain-wall solitons, classified by the first homotopy group $\pi_1(\mathbb{S}^2/\mathbb{Z}_2)=\mathbb{Z}_2$. The disclination's local structure is encoded by the twist angle $\beta=\cos^{-1}(\boldsymbol{\tau}\cdot\boldsymbol{\Omega})$ between the line tangent and the rotation vector; as the drive voltage crosses zero, $\beta$ sweeps continuously, morphing domain-wall solitons from one type into another and back, while at the abrupt voltage switch from $+U_{\max}$ to $-U_{\max}$, particle-antiparticle pairs annihilate and regenerate one half spatial period away, which is the step that doubles the period. The mechanism is carried in simulation by a Landau\textendash de Gennes (Ginzburg\textendash Landau) model with flexoelectric and ionic-screening terms, and the comparison to experiment is made through Jones-matrix simulations of the polarized optical micrographs.

What would settle it

Synchronized, time-resolved three-dimensional imaging of the molecular orientation across a full two-period drive cycle should show the $\pm 1/2$ disclination lines smoothly interconverting and the whole array shifting by half a lattice spacing; if the defect cores do not move or transform as simulated, the proposed mechanism for period doubling would be ruled out.

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Extended reading notes

Core claim

The central discovery claim is that a periodically driven chiral nematic liquid crystal spontaneously breaks both spatial and temporal translation symmetry in discrete steps: the emergent pattern of molecular orientation has its own spatial periodicity and returns to itself only after two full drive periods $2T_E$, so the internal clock runs at half the drive frequency. The building blocks are arrays of topological quasiparticles—pairs of $\pm 1/2$ disclination lines joined by $\pi$-rotation domain-wall solitons—whose smooth interconversion between N\'eel and Bloch forms accompanies the voltage crossing zero, and whose annihilation and regeneration at the voltage jump shifts the whole array by half a spatial period $L/2$. Because the shifted array is identical, the system repeats every $2T_E$, producing 1+1D and 2+1D discrete space-time crystals. The paper further claims that the same mechanism underlies a candidate fractional space-time crystal with an internal period near $10T_E/3$ in thicker cells.

Load-bearing premise

The claim rests on the assumption that the periodic patterns seen through the microscope really are the defect structures that the computer simulation shows, rather than some other optical effect.

Editorial extensions

If this is right

  • Period-doubled discrete time crystals appear in a standard, electrically switchable liquid crystal cell, so the phenomenon is accessible in ordinary laboratory soft matter rather than requiring quantum hardware.
  • Both the 1+1D and 2+1D phases occupy wide regions of the temperature\textendash voltage\textendash drive-period phase diagram, with drive periods from about 0.35 s to 1 s and temperatures from 24 \textdegree C to 31 \textdegree C.
  • The crystals survive random temporal perturbations in the drive period up to about $\pm 0.2$ of the mean period and recover from lattice defects, including ones created by a laser tweezer, within tens of drive cycles.
  • The temporal correlation function of the 1+1D crystal decays as a power law with exponent $\eta \approx 0.08$, indicating quasi-long-range temporal order analogous to a smectic phase.
  • A candidate fractional discrete time crystal appears in thicker cells with an internal period near $10T_E/3$, showing that non-integer period multiplication may also be possible in classical systems.

Reading between the lines

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

  • One testable extension: if the topological-quasiparticle mechanism is the real cause of period doubling, then other driven soft or active systems hosting $\pm 1/2$ disclinations, such as active nematics, should also display period-doubled space-time order under periodic stimulation; this prediction goes beyond the paper.
  • The reported tolerance to random timing jitter leaves open how much structured perturbation the crystal can withstand; a natural experiment would be to omit a drive pulse or sweep the drive frequency and measure when the period-doubled order breaks.
  • The fractional candidate near $10T_E/3$ hints at a hierarchy of locked rational periods, so varying the cell thickness-to-pitch ratio $d/p$ might reveal a mode-locking staircase of fractional space-time crystals, a possibility the paper does not explore.
  • Because the experimental evidence is optical retardation rather than direct imaging of the molecular orientation, a natural next step is time-resolved three-dimensional imaging synchronized with the drive to watch the disclination lines transform and shift by $L/2$; the paper leaves this check open.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The manuscript reports experimental observations and numerical modeling of classical discrete space-time crystals (DSTCs) in chiral nematic liquid crystals driven by a sawtooth electrical signal. The authors present polarized optical microscopy (POM) images, space-time plots, and FFT analysis showing a period-doubled response to the external drive, together with phase diagrams over drive period, voltage, temperature, and cell thickness. They interpret the period-doubling as arising from periodic inter-transformations, generation, and annihilation of topological solitons and singular disclinations, which they describe as Majorana-like quasiparticles. They also report robustness to temporal perturbations and spatial defects, long lifetimes, a claim of quasi-long-range temporal order, and a candidate fractional discrete time crystal.

Significance. If the mechanism and order characterization hold, this would be a notable classical analogue of discrete time crystals in a soft-matter system, with a broad parameter range and potential practical relevance. The paper's strengths include directly evidenced period-doubling via FFT and space-time plots, extensive phase diagrams, robustness videos, and qualitative reproduction of the period-doubled state in Landau-de Gennes simulations. However, the load-bearing mechanism claim relies on simulations with unvalidated dimensionless parameters and on an unpublished manuscript for the definition/verification criteria, and the quasi-long-range order claim is not quantitatively established. These issues need to be addressed before the central claims can be considered fully supported.

major comments (3)
  1. [Abstract and 'Majorana-like quasi-particle nature' (pp. 5–7); Methods, LdG modeling (pp. 35–37, Eqs. (4)–(8))] The central claim that 'the phenomenon-enabling period-doubling effect comes from their topological Majorana-like quasiparticle features' is not supported by the evidence presented. The LdG simulations use dimensionless parameters (ζ1=2, ζ2=11, σa=5×10^-5, Umax=2) with no mapping to the experimental 5CB/CTAB cell (d=5–15 μm, p=5 μm, Umax up to 50 V, finite anchoring W=10^-4 J/m^2), and the Majorana-like spinor interpretation is adopted from Ref. 27 without derivation. The paper does not show that the simulated annihilation/generation sequence is robust to parameter variation, so the simulated mechanism cannot be identified as the actual experimental mechanism. This needs either a parameter calibration, a systematic robustness study, or a rephrasing of the mechanism claim.
  2. ['Stability and robustness' section (p. 9) and Ref. 23] The verification criteria for a space-time crystal—spontaneous symmetry breaking in both space and time and robustness against temporal perturbations—are attributed to the authors' unpublished manuscript (Ref. 23). Because Ref. 23 is not available to readers, the paper effectively defines the phenomenon by criteria from an inaccessible source, making the claim that the system 'appears to satisfy' these criteria circular. The criteria should be stated explicitly and justified with published references, or the dependence on Ref. 23 should be removed.
  3. ['Quasi-long-range order' section (pp. 9–10) and Fig. 9] The claim of temporal quasi-long-range order is based on a power-law fit t^-η with η=0.08, but the fitting range, the uncertainty of η, and comparison with alternative fits (e.g., exponential or stretched-exponential) are not provided. Without this information, the power-law decay is not established, and the analogy to smectic order is not quantitatively supported. The same limitation applies to the correlation analysis of the fractional DSTC in Fig. 10e.
minor comments (6)
  1. [Abstract (p. 1)] The phrase 'these classical time crystals comprise particle-like structural features and exists over a wide range' contains a subject-verb agreement error ('exist' is needed).
  2. [p. 8, 'Majorana-like quasi-particle nature' section] The term 'self-free energy' is unusual; it likely should be 'self-energy'.
  3. [Fig. 4k caption (p. 18)] The definition β=cos^-1(τ·Ω) uses τ as the tangent vector and Ω as the rotation vector, but the connection of this definition to the text's description of β∈[0,π] at top/bottom boundaries is not fully explained.
  4. [Methods, 'Materials and sample preparation' (p. 30)] The statement 'the maximum screening ability can be ~10^2V' is unclear; it should specify whether this is a voltage scale, a screening factor, or something else, and the units should be given consistently.
  5. [Fig. 5 (p. 19)] The phase diagrams would be more informative if the number of independent measurements per point and the estimated uncertainty of phase boundaries were stated.
  6. ['Fractional discrete time crystals' (pp. 10–11) and Fig. 10] The fractional period 10TE/3 is presented as a candidate; the text should more explicitly note that the FFT peak at ~0.3fE and correlation peaks at 33TE and 40TE cannot distinguish 10TE/3 from nearby periods such as 3.3TE or 3.4TE without additional analysis.

Circularity Check

1 steps flagged · score 4.0 of 10

One partially load-bearing self-citation in the verification criteria; the central period-doubling observation itself is independent.

  1. self citation load bearing [Results, 'Stability and robustness' section, paragraph beginning 'We examine the rigidity (robustness) of the DSTC...' (near Fig. 6).]
    "The spontaneous symmetry breaking both in space and time and robustness against temporal perturbations are important properties of space-time crystals identified in recent literature23, serving as verification criteria of space-time crystals that our system appears to satisfy."

    Reference 23 is 'Zhao, H. & Smalyukh, I. I., Space-time crystals from particle-like topological solitons (unpublished).' Both authors are also authors of the present paper. The quoted sentence makes this unpublished self-citation the authority for the 'verification criteria of space-time crystals' and then asserts that the system satisfies those criteria. The classification of the observed pattern as a space-time crystal is therefore validated against the authors' own unpublished standard. The period-doubling and robustness measurements themselves are independent observations, so this is a partially load-bearing self-citation for the label rather than a reduction of the data to the model.

full rationale

The central empirical claim—that the optically tracked pattern recurs every 2TE under a TE-periodic Floquet drive—is established by POM imaging, FFT analysis, and trajectory tracking without fitting a model to the data; no fitted parameter is renamed as a prediction. The Landau-de Gennes simulations are presented as an illustration of a mechanism ('we illustrate'), not as a first-principles derivation, and the mismatch between dimensionless simulation parameters and the experimental cell is a correctness/calibration concern, not circularity. The Majorana-like quasiparticle language is an interpretive analogy imported from an external published source (Ref. 27) and from standard defect topology; it does not enter the period-doubling measurement. The only self-referential step is the citation of the authors' own unpublished manuscript (Ref. 23) as the source of the 'verification criteria of space-time crystals.' Since the same criteria are also available from standard DTC literature and the raw observations are independent, this is a partially load-bearing self-citation rather than a circular derivation. Hence score 4.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The central claim rests on the Frank-Oseen and Landau-de Gennes models with a set of dimensionless parameters chosen for the simulations, on the assumption that POM images faithfully reflect director dynamics, and on acceptance of the authors' own unpublished criteria for what constitutes a space-time crystal. No new physical entities with falsifiable predictions are introduced; the Majorana-like quasiparticle is an interpretive label.

free parameters (4)
  • LdG flexoelectric constants zeta1 and zeta2 = zeta1=2, zeta2=11
    Set in the Ginzburg-Landau simulations (Methods) and not measured for the experimental 5CB/cholesterol pelargonate/CTAB mixture; the simulated period-doubling dynamics depend on these values.
  • LdG conductivity anisotropy sigma_a = 5e-5
    Chosen in simulations; no experimental measurement of ionic conductivity anisotropy is provided.
  • LdG dimensionless electric field Umax = 2.0
    Simulation amplitude; the mapping to the experimental 50-90 V sawtooth is not given.
  • Power-law decay exponent eta = 0.08
    Fitted to the measured temporal correlation function G(t) (Fig. 9) to support quasi-long-range order; no error bar or goodness-of-fit is reported.
assumptions (5)
  • domain assumption Frank-Oseen and Landau-de Gennes free energy functionals describe the chiral nematic's elastic, electric, flexoelectric, and ionic screening response.
    Invoked throughout Methods; standard LC continuum theory but approximate for the doped, out-of-equilibrium condition.
  • domain assumption Director dynamics follows the torque balance equation [F]_ni = -gamma * d(ni)/dt and the Ginzburg-Landau equation d(Q)/dt = -Gamma * [delta F/delta Q]_st.
    Methods, numerical modeling section; assumes overdamped relaxational dynamics with a single rotational viscosity.
  • ad hoc to paper The verification criteria for a space-time crystal are the ones stated in Ref 23.
    Ref 23 is the authors' own unpublished manuscript; this makes the classification of the observed phase partially self-referential.
  • ad hoc to paper The director profiles around disclinations transform as spinors following the Majorana equation, as stated in Ref 27.
    The paper imports this to call the defects Majorana-like quasiparticles but does not derive or test the spinor behavior.
  • ad hoc to paper The dimensionless LdG simulation parameters (zeta1=2, zeta2=11, sigma_a=5e-5, Umax=2) represent the experimental system sufficiently for the simulated mechanism to be the actual one.
    No validation of the parameter mapping is provided; the period-doubling result is not tested for sensitivity to these parameters.
invented entities (1)
  • Majorana-like quasiparticles (classical defect states labeled as Majorana particles/antiparticles)
    purpose: To explain the period-doubling as periodic inter-transformation, generation, and annihilation of disclinations and solitons, with the state after one period shifted by half a lattice spacing.
    The underlying disclinations and solitons are observable, but the Majorana-specific character (self-conjugacy, spinor transformation) is not demonstrated experimentally; the label is adopted from Ref 27 without new falsifiable predictions.

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Pith. "Pith review of Emergent discrete space-time crystal of Majorana-like quasiparticles in chiral liquid crystals." pith.science (2026). https://pith.science/paper/FXKCI63B

@misc{pith2026250716977,
  author       = {Pith},
  title        = {Pith review of: Emergent discrete space-time crystal of Majorana-like quasiparticles in chiral liquid crystals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FXKCI63B}},
  note         = {Machine review of arXiv:2507.16977}
}
read the original abstract

Time crystals spontaneously break the time translation symmetry, as recently has been frequently reported in quantum systems. Here we describe the observation of classical analogues of both 1+1-dimensional and 2+1-dimensional discrete space-time crystals in a liquid crystal system driven by a Floquet electrical signal. These classical time crystals comprise particle-like structural features and exists over a wide range of temperatures and electrical driving conditions. The phenomenon-enabling period-doubling effect comes from their topological Majorana-like quasiparticle features, where periodic inter-transformations of co-existing topological solitons and disclinations emerge in response to external stimuli and play pivotal roles. Our discrete space-time crystals exhibit robustness against temporal perturbations and spatial defects, behaving like a time-crystalline analogues of a smectic phase. Our findings show that the simultaneous symmetry breaking in time and space can be a widespread occurrence in numerous open systems, not only in quantum but also in a classical soft matter context.

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Reference graph

Works this paper leans on

51 extracted references · 45 canonical work pages · cited by 1 Pith paper

  1. [1]

    Quantum time crystals

    Wilczek, F. Quantum time crystals. Phys. Rev. Lett. 109, 160401 (2012)

    work page 2012

  2. [2]

    & Wilczek, F

    Shapere, A. & Wilczek, F. Classical time crystals. Phys. Rev. Lett. 109, 160402 (2012)

    work page 2012

  3. [3]

    Time crystals: can diamagnetic currents drive a charge density wave into rotation? Europhys

    Noziè res, P. Time crystals: can diamagnetic currents drive a charge density wave into rotation? Europhys. Lett. 103, 57008 (2013)

    work page 2013

  4. [4]

    Impossibility of spontaneously rotating time crystals: a no-go theorem

    Bruno, P. Impossibility of spontaneously rotating time crystals: a no-go theorem. Phys. Rev. Lett. 111, 070402 (2013)

    work page 2013

  5. [5]

    & Oshikawa, M

    Watanabe, H. & Oshikawa, M. Absence of quantum time crystals. Phys. Rev. Lett. 114, 251603 (2015)

    work page 2015

  6. [6]

    Modeling spontaneous breaking of time-translation symmetry

    Sacha, K. Modeling spontaneous breaking of time-translation symmetry. Phys. Rev. A 91, 033617 (2015)

    work page 2015

  7. [7]

    V., Bauer, B

    Else, D. V., Bauer, B. & Nayak, C. Floquet time crystals. Phys. Rev. Lett. 117, 090402 (2016)

    work page 2016

  8. [8]

    Y., Potter, A

    Yao, N. Y., Potter, A. C., Potirniche, I. -D. & Vishwanath, A. Discrete time crystals: rigidity, criticality, and realizations. Phys. Rev. Lett. 118, 030401 (2017). 9.Zaletel, M. P. et al. Colloquium: quantum and classical discrete time crystals. Rev. Mod. Phys. 95, 031001 (2023)

    work page 2017

Show all 51 references
  1. [10]

    Zhang, J. et al. Observation of a discrete time crystal. Nature 543, 217–220 (2017)

    2017

  2. [11]

    Choi, S. et al. Observation of discrete time-crystalline order in a disordered dipolar many-body system. Nature 543, 221–225 (2017)

    2017

  3. [12]

    Randall, J. et al. Many -body–localized discrete time crystal with a programmable spin -based quantum simulator. Science 374, 1474–1478 (2021)

    2021

  4. [13]

    Mi, X. et al. Time-crystalline eigenstate order on a quantum processor. Nature 601, 531–536 (2022). 27

    2022

  5. [14]

    & Rachel, S

    Frey, P. & Rachel, S. Realization of a discrete time crystal on 57 qubits of a quantum computer. Sci. Adv. 8, eabm7652 (2022)

    2022

  6. [15]

    Keß ler, H. et al. Observation of a dissipative time crystal. Phys. Rev. Lett. 127, 043602 (2021)

    2021

  7. [16]

    B., Maleki, L

    Taheri, H., Matsko, A. B., Maleki, L. & Sacha, K. All-optical dissipative discrete time crystals. Nat. Commun. 13, 848 (2022)

    2022

  8. [17]

    Kyprianidis, A. et al. Observation of a prethermal discrete time crystal. Science 372, 1192– 1196 (2021)

    2021

  9. [18]

    Y., Nayak, C., Balents, L

    Yao, N. Y., Nayak, C., Balents, L. & Zaletel, M. P. Classical discrete time crystals. Nat. Phys. 16, 438–447 (2020)

    2020

  10. [19]

    & Knolle, J

    Pizzi, A., Nunnenkamp, A. & Knolle, J. Classical prethermal phases of matter. Phys. Rev. Lett. 127, 140602 (2021)

    2021

  11. [20]

    & Yao, N

    Ye, B., Machado, F. & Yao, N. Y. Floquet phases of matter via classical prethermalization. Phys. Rev. Lett. 127, 140603 (2021)

    2021

  12. [21]

    Kongkhambut, P. et al. Observation of a continuous time crystal. Science 377, 670–673 (2022)

    2022

  13. [22]

    Liu, T., Ou, J.-Y., MacDonald, K. F. & Zheludev, N. I. Photonic metamaterial analogue of a continuous time crystal. Nat. Phys. 19, 986–991 (2023)

    2023

  14. [23]

    & Smalyukh, I

    Zhao, H. & Smalyukh, I. I. Space -time crystals from particle -like topological solitons . (unpublished)

  15. [24]

    Xu, S. & Wu, C. Space-time crystal and space-time group. Phys. Rev. Lett. 120, 096401 (2018)

    2018

  16. [25]

    Zhang, R. et al. Spatiotemporal control of liquid crystal structure and dynamics through activity patterning. Nat. Mater. 20, 875–882 (2021)

    2021

  17. [26]

    Majorana returns

    Wilczek, F. Majorana returns. Nat. Phys. 5, 614–618 (2009)

    2009

  18. [27]

    Head, L. C. et al. Majorana quasiparticles and topological phases in 3D active nematics. Proc. 28 Natl. Acad. Sci. 121, e2405304121 (2024)

    2024

  19. [28]

    Kauffman, L. H. A construction for Clifford algebras. Preprint at https://doi.org/10.48550/arXiv.2202.13592 (2022)

    2022 doi arXiv

  20. [29]

    & Lorenz, K

    Buka, A. & Lorenz, K. Pattern Formation in Liquid Crystals. (Springer Science & Business Media, 2012)

    2012

  21. [30]

    & Buka, Á

    É ber, N., Salamon, P. & Buka, Á. Electrically induced patterns in nematics and how to avoid them. Liq. Cryst. Rev. 4, 101–134 (2016)

    2016

  22. [31]

    Sohn, H. R. O., Liu, C. D. & Smalyukh, I. I. Schools of skyrmions with electrically tunable elastic interactions. Nat. Commun. 10, 4744 (2019)

    2019

  23. [32]

    de Gennes, P. G. & Prost, J. The Physics of Liquid Crystals. (Clarendon Press,1993)

    1993

  24. [33]

    Lee, T., Trivedi, R. P. & Smalyukh , I. I. Multimodal nonlinear optical polarizing microscopy of long-range molecular order in liquid crystals. Opt. Lett. 35, 3447 (2010)

    2010

  25. [34]

    Yeh, P. & Gu, C. Optics of Liquid Crystal Displays. (Wiley, 2010)

    2010

  26. [35]

    & Žumer, S

    Ravnik, M. & Žumer, S. Landau–de Gennes modelling of nematic liquid crystal colloids. Liq. Cryst. 36, 1201–1214 (2009)

    2009

  27. [36]

    & Alexander, G

    Binysh, J., Kos, Ž., Čopar, S., Ravnik, M. & Alexander, G. P. Three-dimensional active defect loops. Phys. Rev. Lett. 124, 088001 (2020)

    2020

  28. [37]

    Wang, X. et al. Moiré effect enables versatile design of topological defects in nematic liquid crystals. Nat. Commun. 15, 1655 (2024)

    2024

  29. [38]

    Chaikin, P. M. & Lubensky, T. C. Principles of Condensed Matter Physics (Cambridge University Press, 2000)

    2000

  30. [39]

    B., Wu, J.-S

    Zhao, H., Tai, J.-S. B., Wu, J.-S. & Smalyukh, I. I. Liquid crystal defect structures with Mö bius strip topology. Nat. Phys. 19, 451-459 (2023). 29

    2023

  31. [40]

    & Kelly, J

    Cui, M. & Kelly, J. R. Temperature dependence of visco-elastic properties of 5CB. Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. Mol. Cryst. Liq. Cryst. 331, 49–57 (1999)

    1999

  32. [41]

    Landau, L. D. & Lifshitz, E. M. Statistical Physics. (Elsevier, 2013)

    2013

  33. [42]

    Time Crystals

    Sacha, K. Time Crystals. (Springer International Publishing, Cham, 2020)

    2020

  34. [43]

    & Nunnenkamp, A

    Pizzi, A., Knolle, J. & Nunnenkamp, A. Higher -order and fractional discrete time crystals in clean long-range interacting systems. Nat. Commun. 12, 2341 (2021)

    2021

  35. [44]

    Liu, B. et al. Higher -order and fractional discrete time crystals in Floquet -driven Rydberg atoms. Nat. Commun. 15, 9730 (2024)

    2024

  36. [45]

    Meng, C., Wu, J. -S. & Smalyukh, I. I. Topological steering of light by nematic vortices and analogy to cosmic strings. Nat. Mater. 22, 64–72 (2023)

    2023

  37. [46]

    Zhang, Z. et al. Diffusion metamaterials. Nat. Rev. Phys. 5, 218–235 (2023)

    2023

  38. [47]

    Yang, F. et al. Controlling mass and energy diffusion with metamaterials. Rev. Mod. Phys. 96, 015002 (2024)

    2024

  39. [48]

    Tran, L. et al. Change in stripes for cholesteric shells via anchoring in moderation. Phys. Rev. X 7, 041029 (2017)

    2017

  40. [49]

    Chang, S. et al. Electrical tuning of branched flow of light. Nat. Commun. 15, 197 (2024)

    2024

  41. [50]

    Repula, A., Gates, C., Cameron, J. C. & Smalyukh, I. I. Photosynthetically-powered phototactic active nematic liquid crystal fluids and gels. Commun. Mater. 5, 37 (2024). Methods Materials and sample preparation Chiral nematic LCs are prepared by mixing nematic LCs 4-Cyano-4'-...

    2024

  42. [51]

    Ackerman, P. J. & Smalyukh, I. I. Static three-dimensional topological solitons in fluid chiral ferromagnets and colloids. Nat. Mater. 16, 426–432 (2017)

    2017

  43. [52]

    boil out

    Tai, J.-S. B. & Smalyukh, I. I. Three -dimensional crystals of adaptive knots. Science 365, 1449–1453 (2019). Supplementary Video Captions Supplementary Video 1 | Video s showing the 1+1D and 2+1D DSTCs. The POM video for 1+1D DSTC (top left) is obtained for a cell gap d = 5 μ...

    2019

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