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

Domain: biology
Published: 2026-03-24
Editorial outcome: major_revision

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 exist over a wide range of temperatures and electrical driving conditions. The phenomenon-enabling period-doubling effect in 1+1-dimensional discrete space-time crystals 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. 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.

Introduction
Time crystals, originally proposed by Wilczek a decade ago1,2, have captured the interest of numerous researchers who are fascinated by the search of these new “crystals”. In analogy to the conventional space crystals, the Wilczek’s original proposal suggested that time translation symmetry can be spontaneously broken in the lowest energy states of closed many-body systems, both in a quantum and classical manner1,2. Unfortunately, a series of no-go theorems demonstrate that these closed systems are prohibited by nature at equilibrium3–5. However, in non-equilibrium situations with Floquet external drives, it is possible to break time translation symmetry discretely, where the period of the system's dynamics is an integer multiple of that of the external drive (the integer must be larger than one), resulting in what is called a discrete (Floquet) time crystal6–9. The discrete time crystals were first observed in systems of nuclear spins and trapped ions10,11, and recently many-body localization discrete time crystals were observed in quantum simulators and processors12–14, which use the many-body localization to prevent the system from a thermalized fate. In addition to the many-body localization, discrete time crystals can be made to persist for a long time by incorporating dissipation15,16 or introducing a prethermal state17, which are also predicted for classical systems18–20. Indeed, the so-called "continuous time crystals" (a different type of time crystals with spontaneous symmetry breaking driven by a constant external source) have been recently observed in both quantum and classical systems, including in liquid crystals (LCs)21–23. While quantum mechanical discrete time crystalline effects received a great deal of attention, their long-awaited discovery in classical systems may be equally important from both fundamental and applied perspectives24,25.

Here we report the observation of classical discrete space-time crystals (DSTCs) in a chiral nematic LC system widely known for its widespread technological use. Electrical switching of LCs is at the heart of the modern LC-enabled trillion-dollar industries, including information displays and electro-optic devices, but the possibility of discrete-time-crystal emergence in these soft matter systems was never analysed. By applying a Floquet electrical signal to a chiral nematic LC sandwiched between parallel electrodes, we find that both the spatial and temporal symmetries of the emergent LC’s structure revealed by optical images can be broken discretely and spontaneously under well-defined experimental conditions, and the internal temporal periodicity of the system doubles in relation to the external drive. Both 1+1 dimensional (1+1D) and 2+1 dimensional (2+1D) discrete space-time crystals are observed, with the time crystallization phases depending on temperature and external driving parameters, as illustrated by constructing a phase diagram. Utilizing computer simulations, we illustrate that the period-doubling effect is intimately related to the inter-transformations, generations, and annihilations of coexisting topological solitons and singular disclinations. Remarkably, the different states of these topological objects can be viewed as the particle and anti-particle states of the observed Majorana-like quasiparticles (a classical analogue of Majorana particles26–28) forming our 1+1D space-time crystals. We verify the rigidity (robustness) of the time crystals against temporal perturbation and spatial defects, with the DSTC phase maintaining order locally for a remarkably long time. Moreover, a potential candidate for a fractional discrete space-time crystal is observed when changing the sample thickness. Our findings may lead to a new paradigm of time-crystalline LC meta matter, with potential fundamental science impacts and technological utility.

Results
Emergent space-time crystal in a chiral nematic medium. A typical studied sample is prepared by sandwiching a chiral nematic LC between two electrically conductive transparent substrates (Fig. 1), where the LC is further doped with ionic substances29–31. In response to a Floquet electrical signal (Fig. 1a,b and Fig. 2), the confined LC can spontaneously form spatially periodic configurations (Supplementary Movie 1), which are captured by a camera of an optical microscope system (Fig. 1a).

Because our system consists of a chiral nematic LC with ion doping under surface confinement with perpendicular boundary conditions, which is incompatible with the uniform twist of the chiral nematic phase, it is important to examine its response to different electrical signals. Using the experimental setup shown in Fig. 2a, we measure the light transmission over a relatively large area (~1 mm2) captured by the detector. The system responds within a few milliseconds after turning the voltage on (Fig. 2b,c), whereas the relaxation back to the initial state is a slower process when the voltage is turned off (Fig. 2b,d). As we apply different sawtooth electrical signals, we observe complex responses (Fig. 2e,f), where the relaxation process occurs on the timescales of ~0.1-1s.

The polarized light interference pattern exhibits clear spatial periodicity under Floquet sawtooth electrical signal (Fig. 1d and Fig. 2f), which can be reproduced by numerical simulations of the director configurations based on both Frank−Oseen and Landau−de Gennes free energy models (Figs. 3,4, Supplementary Fig. S1 and Methods)32,33 and the subsequent modelling of polarized light propagation through such a system with the Jones-matrix method34. While the spatiotemporal crystallization in light polarization (and its colour) patterns is apparent from imaging, it also reveals similar behaviour in the spatiotemporal response of the chiral nematic LC medium. By analysing the interference-colour signal within the camera-captured micrographs of these regions over time (Fig. 1e and Supplementary Movie 1), we find that the spatial pattern recurs every two periods of the external Floquet drive (Fig. 1f). This behaviour is often called the “period-doubling” phenomenon7–9, indicating that the time translation symmetry is broken discretely. The near-neighbour correlations are antiferromagnetic-like both in time and in space, similar to the case of previous theoretical studies18–20 for different systems. Furthermore, the discrete time crystallization phase can be observed for different chemical chiral LC substances (Methods).

We have identified three distinct configurations of the 2+1D DSTCs (Fig. 1g-i). Notably, configuration 1 (Fig. 1g) and configuration 2 (Fig. 1h and Fig. 3g-k) are observed under the same conditions (to be detailed later), while configuration 3 (Fig. 1i) is typically observed only for higher d/p ratios, where d is the cell thickness and p is the helicoidal pitch of the chiral nematic LC. Although these three configurations of 2+1D classical discrete time crystals can have different spatial lattice periodicities that are comparable to the helicoidal pitch, along the temporal axis they feature the period-doubled response to the external drive (Fig. 1g-i). To characterize the period-doubling phenomenon within long times, we track the coloured regions (Methods), which correspond to different localized structures of director field n(r). Figure 1j shows positions-vs-time trajectories of the centre of the blue regions, overlaid on top of the images, revealing that these regions in odd/even lines of the lattice move in opposite directions within each drive period (anti-ferromagnetic-like), which is also evident by tracking the average blue signals (Fig. 1k). Furthermore, the individual (Fig. 1l) and collective spatial displacements (Fig. 1m) take place while maintaining the period-doubling behaviour.

Majorana-like quasi-particle nature of building blocks of time crystals. Our computer simulations based on the Ginzburg−Landau equation accounting for dielectric, flexoelectric, and screening-charge effects (Methods) reproduce the experimental observations of the sample's nonsingular initial state at zero field, as well as the topological quasiparticles and the period-doubling phenomenon in the DSTC states with both nonsingular domain wall solitons and singular line defects (Fig. 4a-f and Supplementary Movie 2)35. When the initial state (Fig. 3) transforms into the 1+1D DSTC state (Fig. 4), the symmetry breaking becomes different during the time crystal formation over tens of electrical driving periods. This transition takes place while complying with the sample's perpendicular boundary conditions for n(r) on confining substrates (Methods). The period-doubling phenomenon within the 1+1D DSTC is directly observed in the snapshots of the director field and the scalar order parameter's spatial distribution (Fig. 4c,f). To understand the underlying mechanism, we thoroughly analyse the dynamic process, finding the fascinating inter-transformations, generations, and annihilations of coexisting topological solitons and singular disclinations (Fig. 4g-o). The singular disclination structures, as well as the fragments of solitonic nonsingular walls between them, are the building blocks of the 1+1D time crystal, which can be treated as Majorana-like quasiparticle and respective anti-particles (Fig. 4k)26–28, because the studied disclination profiles in their cross- sections smoothly transform as spinors following the Majorana equation27. The solitonic walls exhibit temporal evolutions that comply with the transformations of the singular line defects at their ends (Fig. 4g-i).

In the time crystal phase, the particle-antiparticle inter-transformation occurs when the external voltage U smoothly changes from negative to positive (Fig. 4g-j). The director field around the translationally invariant disclination line region can be quantified by the characteristic twist angle β (β ∈ [0, π])36–38, whereas the core (centre) of the disclination line is a singularity in n(r), within which the director orientations cannot be defined. When the amplitude of instantaneous negative voltage U is relatively large, β equals π for the top and 0 for the bottom disclinations (Fig. 4g), respectively, which means that n(r) confined to the plane perpendicular to disclination lines (with the effective order parameter space being 𝕊1/ℤ2, the one sphere - or circle - with diametrically opposite points identified, for the director confined to the cross-sections). These defect lines can be referred to as wedge disclinations39 with winding numbers −1/2 and +1/2 defined in the two-dimensional cross-sections orthogonal to the defect lines (Fig. 4j), respectively.39 The 2D winding numbers ±1/2 of disclinations relate to the accumulated angle of n(r) rotation as one circumnavigates the disclination core once, divided by 360°, with the positive sign corresponding to the counterclockwise rotation, matching that of circumnavigation, and negative sign referring to the clockwise case. However, once the director is allowed to rotate out of the 2D plane orthogonal to the defect lines, becoming 3D in nature, the initial −1/2 and +1/2 structures become topologically the same. While the −1/2 and +1/2 cross-sectional structures would correspond to topologically distinct defect states in a purely 2D system, they are topologically analogous and deformable to one another in our system because 3D rotations of director are allowed. Indeed, these structures morph one to another over time via smooth director rotations (Fig. 4g-i), so that they can be treated as Majorana-like quasiparticles and respective anti-quasiparticles. These defect lines within our LC's 3D space are the topologically nontrivial elements of the first homotopy group 𝜋1(𝕊2/ℤ2)= ℤ2, where 𝜋1 refers to the measuring circle around the defect line and the director's order parameter space is 𝕊2/ℤ2, the two-sphere with diametrically opposite points identified.

Between the two disclination regions, the −π-rotation Néel domain wall solitons (with +/− defined by counterclockwise/clockwise rotation when traversing the topological soliton from left to right) are spatially embedded (Fig. 4g). The domain wall solitons are also labelled as elements of the first homotopy group 40, 𝜋1(𝕊2/ℤ2)= ℤ2, and are also allowed to smoothly inter-transform to have 2D cross-sections of different types (Néel or Bloch type) while here being terminated by the two disclination lines. Thus, both singular and solitonic 𝜋1(𝕊2/ℤ2)= ℤ2 structures morph between geometrically different but topologically the same states under electric driving—akin to deforming a donut into a coffee mug without changing the topological properties (Fig. 4g-i). As the voltage U increases from negative values to zero (Fig. 4h), the angle β characterizing both wall-terminating disclinations smoothly changes to π/2 (corresponding to pure twist winding), and the initial −π-rotation Néel domain wall solitons (with bend-splay structures) transform to Bloch domain wall solitons (with pure twist structures). As the voltage U further increases to U > 0 (Fig. 4i), β changes to 0 for the top and π for the bottom defect lines, and the Bloch domain wall solitons transform to the +π-rotation Néel domain wall solitons. The transition is continuous during the linear voltage change, and the positions of the disclinations and solitons do not shift spatially, just morph geometrically.

The array of topological quasiparticles formed by solitonic walls terminated on singular defect lines maintains its topological nature (Fig. 4i) until U suddenly changes from the positive maximum value to the negative maximum value (Fig. 4l), at which point topological particle-antiparticle pairs annihilate (Supplementary Fig. S2a) and re-generate (Supplementary Fig. S2b), while the locations of the emerged quasiparticles are found synchronously shifted by half a spatial period (L/2) away from the previous ones. The origin of quasiparticles’ annihilation and regeneration refers to the discontinuity of external drive: the voltage changes dramatically, but the director field cannot deform correspondingly fast. After the generation, compared to the structures of the director field one temporal period (TE) before (Fig. 4g-i,m-o), the disclinations and topological solitons are the same, with an L/2 shift, so that the system adopts the exact same configuration every 2TE, resulting in the period-doubling phenomenon. In the experiments and simulations alike (Fig. 4d,e), even when the optical images appear almost homogeneously dark under a high electric field between crossed polarizers, the defects and director deformations do not fully disappear, as evident from the enhanced contrast POM images that reveal effective "memory" of the periodic deformations that allows for correlating positions of these topological objects within 2TE.

Serving as building blocks of time crystals, the self-free energy of a quasiparticle unit (Fig. 4a, including a pair of Majorana-like singular defect quasiparticles and domain wall solitons in-between them) relative to the ground state in a passive LC is ~ 1.3×103 kBT/μm (kB is the Boltzmann constant and T is the temperature), as the system is powered by the external drive to overcome the energy gap, consistent with the anticipated existence of Majorana-like quasiparticles in active and driven nematic LC systems out of equilibrium26,27. Moreover, the elasticity-mediated energy of the interaction between neighbouring units is ~10 kBT with a 10% compression or stretching calculated from Landau–de Gennes free energy (Fig. 4b), further indicating that these topological quasiparticles behave as particle-like objects, which can collectively form a crystal. In addition to reproducing the topological time-crystalline character, by changing the voltage amplitude in the computer simulations, the Fourier analysis of the time-dependent n(r,t) reveals the transitions of period-doubling time-crystalline states to other dynamic states (Supplementary Figs. S3,S4), which are consistent with experimental results, as elaborated in the next section below.

Stability and robustness. The stability range of emergent phase of discrete time crystallization depends on the cell thickness, LC's helicoidal pitch, temperature, voltage amplitude Umax, and external drive period TE. To construct the phase diagram, we employ samples of different cell thickness but the same LC pitch while controlling temperature, Umax, and TE (Fig. 5a-d). We find five distinct phases (Fig. 5e-i) within explored ranges of these parameters: a time-symmetry-unbroken phase (Fig. 5e, Supplementary Movie 3) with the temporal periodicity of LC system being the same as the external drive, a disordered phase (Fig. 5f, Supplementary Movie 3) with a disordered response to the external drive, and a phase co-existence region (Fig. 5i, Supplementary Movie 3), where both the 1+1D DSTC (Fig. 5g) and 2+1D DSTC (Fig. 5h) phases can co-exist. For a thin cell, we observe both the 1+1D and 2+1D DSTC phases (Supplementary Movie 4) depending on TE (ranging from 0.35s to 1s) and temperature (ranging from 24°C to 31°C). This indicates that the DSTC is quite robust within a broad range of parameters as the maximum drive period can be ~ 3 times longer than the minimum one, whereas the intrinsic elastic and dielectric properties of the LC can undergo a ~40% change with such temperature variations correlating with changes of the LC’s scalar order parameter32,41. The temporal range of TE in DSTC phases is on the same timescale as the response time of chiral LCs to a sawtooth electrical signal (Fig. 2e-f), which further demonstrates the importance of the coupling between the electrical drive and the LC director field that is structurally complex even in its initial state at no applied voltages (Fig. 3). Interestingly, we find that the phase diagrams in the coordinates of temperature and TE exhibit an unexpected resemblance: increasing the temperature yields behaviour similar to that of decreasing TE. This is because when TE is small, the director field fails to maintain its topological structure as the external electric field changes too rapidly; similarly, as the temperature increases, the elastic and dielectric constants decrease, decreasing the interaction energy between the quasiparticles, so the system cannot maintain its topological structure either. For a thick cell, the DSTC phase also exists over a broad range (Supplementary Movie 4).

We examine the rigidity (robustness) of the DSTC starting from its formation, where the DSTC spontaneously “boils out” from the disordered state (Fig. 6a,b and Supplementary Movie 5), resembling previous theoretical findings18, which indicates the spontaneous symmetry breaking both in space and time. Within the DSTC phase, the DSTC region grows as the disordered region shrinks (Fig. 6c) because the disordered state is “unstable” relative to the ordered configuration. We further examine the rigidity by adding random temporal perturbations ΔT to the external drive period at each period, finding that both the 1+1D and 2+1D DSTC phases are robust under such temporal perturbations (Fig. 6d,e and Supplementary Movie 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.

Lattice defects and long-range order. Just like conventional space crystals often have defects39, various crystal lattice imperfections (like dislocations, vacancies and self-interstitial points) can emerge in DSTCs (Fig. 7 and Supplementary Movie 7). However, because of the rigidity of the time crystals, the discrete space-time crystallization phase tends to recover from such lattice imperfections after tens of drives (Fig. 7a). Additionally, lattice defects can be introduced by manipulating the LC with a focused infrared beam of laser tweezers (Methods). The laser beam (Supplementary Movie 8) forms defects in space coordinates of the DSTC, which also tend to disappear with time (Fig. 7b). These results motivate us to experimentally measure how long our classical DSTC (Fig. 8 and Supplementary Movie 9) can show the period-doubling phenomenon, revealing that the 2+1D DSTCs maintain period-doubling locally (within the camera captured region) for hours (~104 drives) and 1+1D DSTCs for about 24 hours (~105 drives). To verify the local temporal correlation, we define the temporal correlation function G(t) = ⟨Φ(t)Φ(0)⟩−⟨Φ(t)⟩⟨Φ(0)⟩, where the brackets indicate an average and Φ is the signals captured by the camera (Methods). It’s well-known that for a smectic phase, the spatial correlation function G(r) decays in a power-law manner depending on the distance r, where the power-law index should be small (as shown in Supplementary Fig. S5). Interestingly, an exponential decay (Fig. 9 and Supplementary Fig. S6) on t could fit the temporal correlation function of 1+1D DSTC, with both the spatial and temporal fluctuations tending to eventually destroy the corresponding correlation in the long ranges9,18,42–44 (Supplementary Fig. S7-9). Achieving extended correlations therefore requires enhancing system controls that preclude issues like thermally- or vibrations-induced drift. Numerically, by introducing a white-noise stochastic perturbation fnoise(t), with an auto-correlation <fnoise(t)fnoise(t')>=2ηTeffδ(t-t') (where η is the friction coefficient and Teff is the effective temperature of the thermal perturbation), we observe that the order of the system decreases dramatically as the effective temperature increases (Supplementary Fig. S10). More defects emerge in the bulk, and the lifetime (correlation time) decreases exponentially with increasing Teff, suggesting a (thermally) activated time crystal9,18,43. Under conditions around the phase transition (Fig. 5), the lifetime of the DSTC can be very short due to frequent occurrences of line dislocations (Fig. 7c), causing emergence of the disordered states within only ten-to-hundred drives.

Potential candidates of fractional discrete time crystals. The emergent spatiotemporal response of chiral nematic LCs under the Floquet drive can give origins to plentiful other phenomena, such as, the quasi-hexagonal lattices seen in the spatial coordinates (Fig. 10a and Supplementary Movie 10) when d/p > 3, which potentially can be a fractional discrete time crystal19,45–47 (Fig. 10). The intrinsic time periodicity is not an integer multiple of the external drive (Fig. 10c and Supplementary Fig. S11), and the FFT analysis (Fig. 10d) reveals a peak around 10TE/3 (f ≈ 0.3fE). The strong noise around the peak in FFT analysis originates from the emergence of lattice imperfections (such as 5-7 defects, Fig. 10b) in spatial coordinates. We observe 15 lattice imperfections within a region of ~10 hexagonal units over 200 external drives. In addition, the temporal correlation function (Fig. 10e) further suggests the internal temporal periodicity is approximately 33TE/10 or 10TE/3 (40TE/12). Interestingly, fractional discrete time crystals of similar fractional numbers (~3.3 and 100/29) have been observed in two different quantum systems46,47. Whether they share common universal mechanisms remains an open question.

Methods
Materials and sample preparation
Chiral nematic LCs are prepared by mixing nematic LCs 4-Cyano-4'-pentylbiphenyl (5CB, EM Chemicals) or E7 (Shijiazhuang Chengzhi Yonghua Display Material Co.) with a small amount of a left-handed chiral additive, cholesterol pelargonate (Sigma-Aldrich). The helicoidal pitch, p, of the mixtures is fixed to be 5 μm and controlled by the concentration, 𝐶dopant, of the chiral additive with known helical twisting power ℎhtp, according to the relation 𝑝 = 1/(ℎhtp ⋅𝐶dopant), where the helical twisting power ℎhtp = 6.25 μm−1 for the cholesterol pelargonate. The chiral nematic LCs are further doped by ~0.1 wt% of a cationic surfactant hexadecyltrimethylammonium bromide (CTAB, Sigma-Aldrich) in order to boost LC's conductance29–31, the maximum screening ability can be ~102V; we present results for the chiral nematic LCs based on 5CB unless specified otherwise.

LC cells are assembled from indium-tin-oxide (ITO)-coated glass slides or coverslips treated with polyimide SE5661 (Nissan Chemicals) to obtain strong perpendicular (homeotropic) boundary conditions on their inner surfaces without any pre-patterning. The polyimide is applied to the substrates by spin-coating at 2,700 rpm for 30 s followed by baking (5 min at 90 °C and then 1 h at 180 °C). Then, the two ITO-coated glass slides or coverslips are glued together with optical adhesive (NOA 63, Norland Products) and the LC cell gap thickness is defined by silica spheres as spacers between two substrates to be 5-15 μm. Metal wires are attached to ITO and connected to a data acquisition board (NIDAQ-6363, National Instruments) with a signal amplifier (Model 7600, Krohn-Hite) for electrical control. Additionally, we use custom-created Matlab codes controlling the data acquisition board, connected to a computer for modulation of the voltage output. The amplitude of the sawtooth wave is Umax (U ranges from - Umax to + Umax) and the temporal periodicity TE = 0.5 s (unless specified differently).

Optical microscopy and laser tweezers
POM is a method that utilizes polarized light to image the birefringent materials with two crossed polarizers. The sample is illuminated by a wide spectrum of visible light from a lamp, and the light first passes through a polarizer, becoming linearly polarized. Because of the spatially varying optical phase retardation patterns produced by the LC with complex structure of director driven by the field, the linearly polarized illumination light transforms into patterns of generally elliptically polarized light with different polarization ellipse’s major axis orientations. These polarized-light spatiotemporal periodic patterns can be vividly revealed by inserting an additional first-order full-wave retardation plate (Fig. 1c), where addition and subtraction of the phase retardations due to the LC and the accessory plate converts the spatial variations of light's polarization ellipse orientations into that of first- and second-order polarized interference colours. By measuring the polarized light interference patterns, we find that the spatial spacings of configuration 1 and configuration 2 2+1D DSTCs are 8.2 μm and 11.6 μm, respectively, in a sample with a cell thickness of 10 μm. For configuration 3, the spatial spacing is 18.4 μm in a sample with a cell thickness of 15 μm.

POM images and movies are obtained with a multi-modal imaging setup built around an IX-81 Olympus inverted microscope, which is also integrated with an ytterbium-doped fibre laser (YLR-10-1064, IPG Photonics, operating at 1,064nm). All presented POM snapshots and movies are captured with charge-coupled-device cameras (Grasshopper, Point Grey Research). Olympus objectives 100×, 40×, 20× and 10× with numerical apertures of 1.4, 0.75, 0.4, and 0.4, respectively, are used.

Non-contact manipulation by laser tweezers is achieved using a tightly focused 1064 nm laser beam at powers of less than 20 mW. For this, we utilize the Ytterbium-doped fibre laser and a phase-only spatial light modulator (P512-1064, Boulder Nonlinear Systems) integrated into a holographic laser tweezers setup40,55,56. The laser tweezers are integrated with the three-dimensional nonlinear optical imaging setup described below, enabling the simultaneous optical control and nondestructive imaging of the LC structures.

Three-dimensional nonlinear optical imaging of quasi-static director configurations
Three-dimensional nonlinear optical imaging of the LC structures is key to understanding many physical phenomena in LCs33. For time-crystalline structures, ideally, the temporal evolution of director field configurations should be probed as well. While doing this is challenging for our time crystals that have the 3D field configuration changing completely within a fraction of a second, our 3D imaging can still provide valuable insights based on imaging quasi-static configurations from which or into which the time-crystalline structures evolve. Our 3D imaging is performed using the three-photon excitation fluorescence polarizing microscopy setup built around the IX-81 Olympus inverted optical microscope33. We use a Ti-Sapphire oscillator (Chameleon Ultra II; Coherent) operating at 870 nm with 140-fs pulses at an 80 MHz repetition rate, as the source of the linearly polarized laser excitation light. An oil-immersion 100× objective (NA = 1.4) is used to collect the fluorescence signals, which are detected by a photomultiplier tube (H5784-20, Hamamatsu) after a 417/60-nm bandpass filter. The LC molecules are excited via the three-photon absorption process and the signal intensity scales ∝ cos6𝛽0, where 𝛽0 is the angle between the linear polarization direction of the excitation light and the LC director. Polarization states of the excitation (as shown in Fig. 3e,f) are controlled by a half-wave plate. When n(r) is nearly parallel to the linear polarization of the laser beam, the large cos𝛽0 corresponds to the strong three-photon excitation fluorescence polarizing microscopy signal intensity. Computer simulations of the three-photon excitation fluorescence polarizing microscopy images are also based on this dependence of the signal intensity.

Numerical modeling of quasi-static structures based on the Frank−Oseen free energy functional
While the full modelling of observed time-crystalline structures is challenging, helpful insights can be obtained by 3D imaging and energy-minimization-based modelling of quasi-static structures that the time crystalline patterns evolve from or lead to. For chiral nematic LCs, the energy cost of spatial deformations of the director field n(r) can be expressed by the Frank−Oseen free energy functional57:
𝐹elastic = ∫d3𝐫{𝐾11/2 (∇⋅𝐧)2 + 𝐾22/2 [𝐧⋅(∇×𝐧)+ 2π/𝑝]2 + 𝐾33/2 [𝐧×(∇×𝐧)]2}, (1)
where the Frank elastic constants 𝐾11, 𝐾22 and 𝐾33 determine the energy costs of splay, twist and bend deformations, respectively. The surface energy is
𝐹surface = −∫d2𝐫 (W/2) (𝐧 ⋅𝐧0)2, (2)
where W is the surface anchoring strength and n0 is the director's easy axis orientation at the surface, which is perpendicular to the substrate. When external electric field E is applied, dielectric response of the LC yields an additional dielectric coupling term, so that the free energy is supplemented by the following term:
𝐹electric = −(𝜀0Δ𝜀/2) ∫d3𝐫(𝐄⋅𝐧)2, (3)
where 𝜀0 is the vacuum permittivity, and Δε is the dielectric anisotropy of the LC. The total free energy 𝐹 = 𝐹elastic +𝐹surface +𝐹electric.

To computer-simulate the spontaneously formed spatial configuration of DSTCs, we assume that these patterns can emerge under locally relatively weak electric field since the ions can screen the electric field applied to substrates. Inspired by the corresponding 3D imaging insights (Fig. 3a-d), we set the translationally invariant undulated configuration as initial configurations, where the 1+1D DSTC-related quasi-static spatial configurations emerge as local or global minima of F, and a relaxation routine based on the variational method is used to identify an energy-minimizing n(r) configuration. Applying this method, at each iteration of the numerical simulation n(r) is updated using a formula derived from the Euler-Lagrange equation, 𝐧new𝑖 = 𝐧old𝑖 − MSTS [𝐹/𝐧𝑖], where subscript i denotes spatial coordinates, [𝐹/𝐧𝑖] denotes the functional derivative of F with respect to n, and MSTS is the maximum stable time step of the minimization routine, determined by the elastic constants and the spacing of the computational grid. The end-of-the-relaxation condition is identified by monitoring the change in the spatially averaged functional derivatives in consecutive iterations. When the free energy change approaches zero, it signifies proximity of the system to a steady state, and the relaxation routine comes to a halt, yielding the equilibrium or metastable structure for given conditions.

When we apply a time-dependent voltage to our samples, a viscous torque associated with rotational viscosity 𝛾1 opposes the fast rotation of n within the LC in response to the competing electric and elastic torques. The system tends to evolve towards the energy-minimizing configuration, even though it may never approach one due to the changing voltage amplitude. The resulting director dynamics is then governed by a torque balance equation, [𝐹/𝐧𝑖] = −𝛾1(𝜕𝐧𝑖/𝜕𝑡), from which we can obtain the temporal evolution n(t) towards the equilibrium, and the time interval equals 𝛾1 MSTS for each iteration.

When we set the 1+1D DSTC spatial structures as initial configurations and apply the time-dependent Floquet electrical field (sawtooth wave, maximum amplitude Umax = 0.2V and temporal periodicity TE = 0.5s), the 2+1D DSTC (configuration 2) spatial configurations spontaneously emerge (Fig. 3i-k). The system does not exhibit a period-doubling phenomenon over every two periods, as the driving voltage is small, thus showing a time-symmetry-unbroken behavior. The spatial discretization is performed on large 3D square-periodic 80×80×40 grids, and the spatial derivatives are calculated using finite-difference methods with the second-order accuracy. For all simulations, the following parameters are used if not specified: d/p = 2, 𝐾11 = 6.4×10−12 N, 𝐾22 = 3×10−12 N, 𝐾33 = 10×10−12 N, 𝑊 = 10−4 J m−2 and 𝛾1 = 77 mPas. While this approach captures many fine features of quasi-static configurations seen in experiments for some parts of phase diagrams, it does not yield the dynamic features of the time-crystalline structures, this is because the numerical simulation based on Frank-Oseen free energy cannot simulate disclinations or low scalar order parameter regions35,40, for which we adopt a different approach described below.

Numerical modeling based on the Landau−de Gennes free energy functional
The quasi-static spatial structures of 1+1D DSTC (Fig. 3a-d, Supplementary Fig. S1) can be revealed by the Landau−de Gennes free energy functional, in which flexoelectric terms and ion-induced electric field are incorporated. The static or quasi-static spatial structures (Fig. 3) obtained via vectorial or tensorial modeling can be used as initial states of the tensor-based numerical simulations to model the dynamic behaviours. Only tensorial modelling is capable of capturing the details of time crystalline structural evolution because of the singular nature of the observed defect lines.

The tensor order parameter is defined as Q = 𝑆(𝐧𝐧−𝐈/3), where 𝑆 is the LC's scalar order parameter and 𝐈 is the identity matrix, and the energy cost of the spatial deformations of Q(r) can be expressed as:
𝐹elasticLdG = ∫d3𝐫{𝐿1/2 (𝜕𝑄𝑖𝑗/𝜕𝑥𝑘)(𝜕𝑄𝑖𝑗/𝜕𝑥𝑘) + 𝐿2/2 (𝜕𝑄𝑖𝑗/𝜕𝑥𝑗)(𝜕𝑄𝑖𝑘/𝜕𝑥𝑘) + 𝐿3/2 (𝜕𝑄𝑖𝑗/𝜕𝑥𝑘)(𝜕𝑄𝑖𝑘/𝜕𝑥𝑗) + 𝐿6/2 𝑄𝑖𝑗(𝜕𝑄𝑘𝑙/𝜕𝑥𝑖)(𝜕𝑄𝑘𝑙/𝜕𝑥𝑗) + (4π/𝑝) 𝐿4 𝜀𝑖𝑘𝑙 𝑄𝑖𝑗(𝜕𝑄𝑙𝑗/𝜕𝑥𝑘)}, (4)
where 𝜀𝑖𝑘𝑙 is the Levi-Civita symbol and 𝐿𝑖’s are the elasticity parameters. In addition, the Landau−de Gennes free energy functional includes thermotropic terms that describe the nematic-isotropic transition of the LC:
𝐹thermotropicLdG = ∫d3𝐫{𝐴/2 (1−𝑈LdG/3)𝑄𝑖𝑗𝑄𝑗𝑖 − (𝐴𝑈LdG/3)𝑄𝑖𝑗𝑄𝑗𝑘𝑄𝑘𝑖 + (𝐴𝑈LdG/4) (𝑄𝑖𝑗𝑄𝑗𝑖)2}, (5)
where 𝐴 and 𝑈LdG are the nematic material parameters. When an external electric field E is applied, the total electric field E in the LC is a superposition of Eexternal and Eion, where Eion is the ion-induced electric field, and can be calculated by the Poisson equation29:
∇∙(𝜺∙𝑬+𝑷flexo) = 𝜌el,(6)
where 𝜺 is dielectric tensor, 𝑷flexo is polarization field due to flexoelectric, and 𝜌el is the ionic charge satisfying ∇∙(𝝈∙𝑬) = −𝜕𝑡𝜌el, with 𝝈 being the conductivity tensor. The dielectric and conductivity tensor are related to the Q-tensor via 𝜺 = 𝜀̅𝐈+𝜀a𝑸 and 𝝈 = 𝜎̅𝐈+σa𝑸, where 𝜀̅ and 𝜎̅ are the mean dielectric and conductivity constants, respectively, and 𝜀a and σa are the dielectric anisotropy and conductivity anisotropy, respectively.

The free energy is supplemented by the following electric coupling terms:
𝐹electricLdG = ∫d3𝐫{−(𝜀0/2) 𝜀̅ 𝐸𝑖2 − (𝜀0/2) 𝜀a 𝑄𝑖𝑗 𝐸𝑖𝐸𝑗 +𝜁1 (𝜕𝑄𝑖𝑗/𝜕𝑥𝑖) 𝐸𝑗 +𝜁2 𝑄𝑗𝑘 (𝜕𝑄𝑖𝑗/𝜕𝑥𝑘) 𝐸𝑖}, (7)
where 𝜁𝑖’s are the flexoelectric constants. The first two terms describe dielectric coupling between Q and the electric field E, and the last two terms describe the flexoelectric effect.

The surface free energy describing the surface anchoring at the substrates reads
𝐹surfaceLdG = −∫d2𝐫 (𝑊/2) (𝑄𝑖𝑗 −𝑄𝑖𝑗(0))2, (8)
where 𝑄𝑖𝑗(0) defines the preferred orientation and order of LC at the surfaces, corresponding to perpendicular boundary conditions in our experiments. The total Landau−de Gennes free energy 𝐹 = 𝐹elasticLdG +𝐹thermotropicLdG +𝐹electricLdG +𝐹surfaceLdG. The evolution of the system follows the Ginzburg−Landau equation35:
𝜕𝑡𝐐 = −Γ[𝛿𝐹/𝛿𝐐]st,
where […]st is a symmetric and traceless operator and the relaxation coefficient Γ is determined by the rotational viscosity 𝛾1 via Γ = 2𝑆02/𝛾1 with 𝑆0 being the constant equilibrium bulk order parameter (𝑆0 = 1/4 + 3/4 √1− 8/3𝑈LdG). Note that the above free energy is time-dependent because of the time-varying external electric field 𝑬external.

By applying a constant external electric field, the ions play the similar role to screen the external field as in the Frank−Oseen free energy method, and the results are the same as the quasi-static field configuration shown in Fig. 3. When applying a large sawtooth external electric field, we observe that both director field in the bulk and ion-induced electric field exhibit the period-doubling phenomenon locally. In the simulation, a stripe-like periodic pattern appears periodically overtime (Fig. 4a,c). After one external temporal period TE, the pattern shifts by a half spatial period. Thus, the full temporal period of the simulated director field corresponds to 2TE. The Fourier analysis of the director field shows a clear signal of the period-doubling phenomenon (Supplementary Fig. S3). When we increase the amplitude of the electric field, the period-doubling phenomenon transitions to a disordered phase, as the Fourier spectrum of the director field spans a broad range of frequencies. As we decrease the amplitude further, the temporal periodicity of the local director field is the same as TE, which is in agreement with the experimental results (Fig. 5).

The numerical model parameters are set to be the following: 𝑈LdG = 5 leading to 𝑆0 ≅ 0.76, 𝐿1 = 1.0, 𝐿2 = 𝐿3 = 𝐿6 = 0, 𝑝 = 75, 𝐴 = 1, 𝜀̅= 1, 𝜀a = 1, 𝜎̅ = 1×10−4, σa = 5×10−5, 𝜁1 = 2, 𝜁2 = 11, and Umax has a maximum magnitude of 2.0. The simulation box size is chosen to be [𝐿𝑥,𝐿𝑧] = [300,150] such that channel height to pitch ratio is 𝐿𝑧/𝑝 = 2. Infinite homeotropic anchoring condition with 𝑊 = ∞ is applied for the two confining substrates, and the periodic boundary conditions are assumed along the 𝑥-direction.

Our tensorial modelling captures the change of the symmetry breaking during the transition from the initial state at zero field to the dynamic state of the 1+1D DSTC. Under homeotropic boundary conditions, the system is topologically neutral as the initial state consists of alternating + π and –π domain-wall solitons in the x-y plane (Fig. 3a,b). In the 1+1D DSTC phase, pairs of disclination lines can be annihilated (Fig. 4l), which also indicates an overall topologically neutral state, dictating that the topological objects' invariants self-compensate. Both in simulations and experiments, we observe that the initial state transforms into the 1+1D DSTC state within tens of driving periods, marked by the change of symmetry breaking upon the time crystal formation.

Simulation of polarizing optical micrographs
The POMs are simulated for the studied structure by means of the Jones-matrix method34. We first split the cell into 40 thin sublayers along the z direction. Then we calculate the Jones matrix for each pixel in each sublayer by identifying the local optical axis and ordinary and extraordinary modes' phase retardation for the light traversing the LC medium. The optical axis is determined by the direction of the local average molecular orientation, while the phase retardation originates from LC's optical anisotropy. We obtain the Jones matrix for the whole LC cell by sequentially multiplying Jones matrices corresponding to each sublayer, and a first-order full-wave retardation plate is included and also described by a Jones matrix. The simulated single-wavelength POM is obtained as the respective component of the product of the ensuing Jones matrix and Jones vectors describing polarizers. To properly reproduce the coloured features in POMs seen in experiments, we generate images separately for three different wavelengths spanning the entire visible spectrum (450, 550, and 650 nm) and then superimpose them, according to light source intensities at corresponding wavelengths.

Calculation of the correlation functions
To calculate the spatial, temporal and spatiotemporal correlation functions, we directly use the data from movie frames. For 1+1D DSTC, the Фi,j(τ) represents the signal at column i, row j at time τ, and we set j and r are along the direction of spatial spacing, i is perpendicular to the spatial spacing. The spatial correlation function G(r)~ ∑𝜏 ∑𝑖,𝑗 Φi,j(𝜏)Φi,j+r(𝜏), the temporal correlation function G(t)~ ∑𝜏 ∑𝑖,𝑗 Φi,j(𝜏)Φi,j(𝜏+t) and the spatiotemporal correlation function G(r,t)~∑𝜏 ∑𝑖,𝑗 Φi,j(𝜏)Φi,j+r(𝜏+t), with background subtraction and normalization applied. The sum over t for Supplementary Fig. S8b is over 1500 drives, and the sum over r is over 200 μm.

Tracking of quasi-particle-like regions
The displacement trajectories of quasi-particle-like regions in Fig.1j,l,m are analysed using freeware (ImageJ) from National Institutes of Health. We first convert the movies into grayscale and retain only the frames (when the 2+1D DSTC pattern appears) with a time interval TE. Since the quasi-particle-like regions can be recognized as particles in the software, the center position of each quasi-particle-like region, together with its time information, can be directly obtained using the software’s tracking functions. We also perform manual tracking, and the results are consistent with those obtained from the automatic tracking.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors report on the experimental demonstration of the non-equilibrium system, presented by the electrically driven\nliquid crystal, that spontaneously breaks the temporal symmetry of the drive. In other words, a subharmonic response to the\ndriving frequency is observed. The work is very well presented, and the physical content of the observed oscillations is an\nexcellent presentation of the non-linear system with parametric response. However, I have difficulty calling this subharmonic\nresponse a Time Crystal. As discussed, for example, by R. E. Goldstein in Physics Today N. 9, p32-38 (2018), this effect can\nbe related to Faraday instability and is known in non-linear physics. Including the classical systems in the time crystals\nrealm is unjustified for the discussed system.\nIt might be that time crystal is a useful concept in quantum mechanics, where the equations are linear, and the issue of\nheating under external forcing is nontrivial. However, for systems like the one presented here, the concepts of classical non-\nlinear physics appear to be adequate.\nFinally, the studied system presents a novel platform for studying non-linear phenomena. However, I would suggest\nresubmitting the paper to a more specialized journal.", "reviewer_2": "I reviewed the paper entitled “Observation of discrete space-time crystals in a classical soft matter system” from H. Zhao and\ncoauthors where they demonstrated a time and spatial control of a specific liquid crystal by applying surface alignment and\nperiodic electrical drive. The overall space-time periodic order has been observed via polarizing optical microscopy with a\nlambda plate, enhancing the static birefringence to observe the periodic driven orientation. The spatial periodicity seems to\nbe comparable to the chiral pitch while the temporal periodicity is doubled within a period of the electrical drive.\nThe paper is overall well written and the several videos available in supplementary materials are certainly helping the\nvisualization of what the authors define a classical time-space crystal. The periodic control over space and time of a large\nsurface in a thin LC cells is certainly extremely interesting, exploiting as far as I can understand a right combination between\nelectrical spatial modulation, chiral pitch, elastic constant and temperature. The chosen LC mixture moreover appears to be\nrobust to several (hours up to 24 hours) voltage-cycles using high to very high fields amplitudes (150V-25/ 5-10 micron~ 300-25 kV/cm is certainly a high field drive) most probably tanks to the ionic doping which avoid screening effects at the\nelectrodes (only explained in the methods at line 474) and ensure local “weak” electric fields – quite in contrast with the\nactual values used in the experiment.\nDespite the exhaustive description of the phenomenon as it appears and which conditions allows different symmetries to\ntake place, in space and time, the physical explanation detailing the overall effect is somehow not clear. This would certainly\nhelp the reader to understand many aspects of the work here presented.\nFor example, in lines 121-122, the authors refer to “LC’s complex response to the external drive depending on TE and\nelastic and dielectric constants’ which is quite vague.\nI completely miss an elucidation about the origin of the doubling period. The closest the authors come to explaining the\nmechanism is by saying “These DSTCs can be described as comprising arrays of spatially and temporarily localized\nsolitonic domain walls interacting with each other within the overall out-of-equilibrium setting” which is only appearing at line\n167, and left quite unclear. This non-clarity comes quite unexpected given the fact that the authors are presenting 2\nmodels which matches the observed effects. There should be an intuitive explanation for example for the torque balance equation\nonly mentioned in the methods at line 492, or about the apparently important role of the ion-induced electric field introduced\nin the LdG free energy functional. Why the Frank-Oseen functional fails in reproducing the doubling period while the LdG\ndoes? This is not clear to me, and this part should find a place in the main text too in my opinion.\nIn the same spirit, looking at the extended data Fig. 5, why as the voltage amplitude increases the noisier the response of the\nsystem? What’s the correlation between voltage amplitude and the LC frequency response? More generally, how do the\ndetails of the driving voltage impact the behavior? Does it only work for uni-polar saw-tooth signals?\nWhat’s the role of the anchoring at the surfaces? Only strong homeotropic conditions will allow to form such time-space\npattern?\nMore in detail:\n• Line 61: the authors claim to boost the conductance of their LC materials by ionic doping, but do not comment any further.\nDoes the conductivity contribute to the pattern formation and doubling of periodicity observed? How does the period\ndoubling scale with ionic doping? And would it lead to another dimension in the phase diagrams presented (fig 2)? How do\nthe ionic currents expected at these low modulation frequencies play into the underlying mechanism for the period doubling?\n• Lines 64 – 72: I would suggest to shorten at least in the main text the discussion about the Polarizing Microscope and\nmove the long discussion in the methods, also because is quite a standard method.\n• Line 74: The authors confuse the optic axis of a crystal with the optical axis (geometrical center of an optical system).\n• Line 99: what does it mean ‘while maintaining a good order’? It is quite vague.\n• Lines 136-137: what the authors mean with higher dimensional space-time coordinates?\n• Lines 158-160: I am confused. In the first place because this sentence is located just at the end before basically the\nconclusions where the last words are: ” …which potentially could be related to the fact that the DSTC phase is not\nobserved.”, and in the second place because I do not understand which is the phase under doubt: is the 1+1D compared to 2\n+1D? How many defects prevents one to define this structure a time/space crystal?", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "In this work, the authors consider the dynamics of a liquid crystal system under a periodic modulation of the electric field.\nDepending on the details of the drive (and thickness) they observe a variety of spatial-temporal ordering, in the orientation of\nthe liquid crystal molecules. These patterns are measured through the polarization of the light polarization, which is mapped\nonto a color scheme.\nWhile the observations in the paper are intriguing and study time crystalline behavior in a novel setting, I find that the\nmanuscript does not address which aspects of the physical setup and protocol enable the observations of the robust period\ndoubling behavior. As a result, it is difficult to relate the observations in the current manuscript with other time crystalline\nbehavior and how the authors' observations can be generalized to other settings.\nFor example, while the authors emphasize the observation of the time crystalline behavior in a soft matter system, it is\nunclear what aspect of the soft matter system are crucial for the observation. Is it the director nature of the liquid crystal? Is it\nthe chiral nature of the interactions? Or the strong coupling to a finite temperature bath?\nGiven that the authors have access to an effective model that accurately captures the dynamics, I think it is important to\naddress some of these questions before I can support the publication in Nature Communications.\nBelow I expand upon these and other questions (as well as some broad comments), whose answers would greatly improve\nthe manuscript.\n# Questions:\n1) In the abstract, the authors note that the system behaves \"like a time-crystal analogues of a **smectic** phase\". However\nthe observed time crystalline behaviors corresponds to a period doubling behavior -- in what sense is that related to the\nsmectic phase? Is it because of the spatial ordering?\n2) The authors introduce two different free energy models to capture the observed phenomena. From my understanding of\ndiscussion in the Methods section, only the Landau-de Gennes approach is able to capture the dynamical observations. Can\nthe authors expand on the differences between the two approaches? Besides a more complex order parameter (built upon\nthe same director as in the Frank-Oseen approach), what physical content is present in the second free energy that enables\nthe better modeling of the system? At the same time, aspect is captured by the Frank-Oseen approach that is not captured by\nthe Landau-de Gennes one?\n3) The observations correspond to a simultaneous breaking of the spatial and time translation symmetry. Are the observed\nspatial symmetry breaking patterns known in previous literature? Are they a requirement for the observation of the temporal\nordering?\nI ask because there are classes of time crystals that arise from the presence of different symmetry broken states, with the\ndrive serving to toggle the system between the different stable configurations and thus induce a period doubling behavior. Is\nthis an example of one such time crystal or are the spatial patterns only stable in the presence of the period doubling\nbehavior? If not, are these spatial patterns metastable?\n4) The authors discuss the presence of different \"types\" of DSTCs. Is there a precise distinction between the different types\nof DSTCs? While the visual patterns are slightly different, they appear to have the same symmetry properties.\n1) Focusing on Fig, 1h, it is actually unclear that the system is exhibiting a period-doubling response--in particular, all the\nsnapshots appear to have the same pattern. Why is that a DSTC?\n5) What happens for larger TE in Fig2a? And smaller temperature for 2b?\n6) Looking through the videos, it seems like the patterns described in the paper only emerge for very brief moments, with the\nsystem mostly looking homogeneous.\n6.1) Do these patterns occur at the max/min of the voltage or near the zero voltage point?\nI think that emphasizing this point in the main text is important.\n6.2) How does the system preserve a \"memory\" of the previous pattern even when looking homogeneous?\n7) What sets the orientation of the checkerboard pattern in the 2+1D DSTC? Naively, the system has a rotation symmetry\naround the z-axis, yet the patterns appear to form along the picture axis. Is this some feature of the boundary conditions of\nthe experiments that are not made explicit.\n8) I was unable to parse the statement \"While defect regions tend to disappear in the space coordinates, they still\noccasionally emerge in higher-dimensional space-time coordinates.\". Can the authors clarify the meaning of higher-\ndimensional in this context?\n9) \"Φ is the signals [sic] captured by the quasi-long-range the camera.\" In this sentence, what constitutes the signal?\nNaively, there are multiple colors being displayed, so how is that information mapped onto a scalar quantity? Why is such\nquantity not used when displaying the period doubling behavior in Fig 1k, and instead the position of the middle region is\nprefered?\n10) What is the meaning of the quasi-long-range order in this context? While the system displays a U(1) symmetry (for a\nrotation of the order parameter around the z axis), the behavior in the time direction has a discrete symmetry, so what causes\nthe power-law decay in time? Is it inherited from the spatial decay of correlations?\n11) What is the FT response of the hexagonal order? What is the ratio between the drive frequency and the response\nfrequency?\nIt would be very surprising if there there is a non-frational ratio, yet I couldn't find additional data\non this observation.\n13) How is the \"relative noise\" quantified in the context of the comment in the discussion? The fact that the system is\ncoupled to a bath at some temperature suggests that there is indeed a large source of noise that is much bigger than the\nprethermal regime of an isolated quantum system? Or is this a comment with regards to the coherence time of modern\nquantum platforms?\n14) I'm confused how to compatibilize two claims of the paper: (i) that the DSTC can have day long lifetime (Extended Data\nFig 7), and (ii) that you observe power-law decay of the spatial temporal correlations (Extended Data Fig 8). These two\nobservations seem to be at odd since one seems compatible with very stable long-lived two point correlation function, where\nthe later suggests that there is no sharp way of defining a long lifetime for the behavior being explored. Can the authors\nclarify what they are claiming in this context?\n15) As a non-expert in the physics of liquid crystal, I was confused by the description of the measurement scheme. Why do\ndifferent colors get shifted differently by the same pattern of LC directors and how is that related to the average director along\nthe beam path?\n# Comments:\n1) I found the lack of discussion of the periodic drive makes the discussion of the DSTC behavior harder to parse. For\nexample, the authors only note the shape of the profile in the figure and never the main text, and do not address how the\nshape of the profile was chosen and how much it impacts the observations.\nWhile the authors have a mean-field model that is able to capture the observed dynamics, it remains unclear what feature of\nthe system/drive is able to generate the time crystalline behavior. As such it is hard to understand the origin and nature of the\nobserved phenomena.\n2) For 1j, the markers might benefit from having two colors that differ more -- i.e. green and orange. This can be combined\nwith different markers for the two parity of periods.\n3) Because the processing for the \"center\" of the blue region is somewhat obscure, I might suggest the authors append their\ndata with the value/color of the image as a function of time -- e.g. near the black points of 1g, it is clear that the data fluctuates\nbetween blue and purple with a sub-harmonic response."}, "author_response": "Nature Communications manuscript NCOMMS-24-59238A\nEmergent discrete space-time crystal of Majorana-like quasiparticles in a chiral\nliquid crystals\nH. Zhao, R. Zhang and I. I. Smalyukh\nSummary of Changes and Responses to Requests/Remarks\nAuthors:\nWe appreciate the consideration of our manuscript. We thank the Referees for their positive reviews\nand cogent comments. We are pleased that Referees report that our work “ is very well presented, and\nthe physical content of the observed oscillations is an excellent presentation …” and our studied system\n“presents a novel platform for studying non-linear phenomena” (Referee #1), that our paper “ is overall\nwell written”, evaluate our studies “define a classical time-space crystal” and “… is certainly extremely\ninteresting” (Referee #2) and that our observations are “are intriguing and study time crystalline\nbehavior in a novel setting …” (Referee #4). In accordance with the comments and suggestions put\nforward by the referees, we have made clarifications and significant revisions, which are detailed point-\nby-point below. These revisions fully address all remarks and suggestions. We also feel that they\nimprove the accessibility of the paper significantly. We would like to thank the Referees for the\nthoughtful valuable comments that helped direct these improvements.\nChanges to Account for the Referees’ Suggestions.\nReport of Referee 1:\nThe authors report on the experimental demonstration of the non-equilibrium system, presented by the\nelectrically driven liquid crystal, that spontaneously breaks the temporal symmetry of the drive. In\nother words, a subharmonic response to the driving frequency is observed. The work is very well\npresented, and the physical content of the observed oscillations is an excellent presentation of the non-\nlinear system with parametric response.\nAuthors:\nWe thank the Referee #1 for the above positive remarks, as well as for the helpful comments.\nReferee 1:\nHowever, I have difficulty calling this subharmonic response a Time Crystal. As discussed, for\nexample, by R. E. Goldstein in Physics Today N. 9, p32-38 (2018), this effect can\nbe related to Faraday instability and is known in non-linear physics. Including the classical systems in the time\ncrystals realm is unjustified for the discussed system.\nIt might be that time crystal is a useful concept in quantum mechanics, where the equations are\nlinear, and the issue of heating under external forcing is nontrivial. However, for systems like the one\npresented here, the concepts of classical non-linear physics appear to be adequate.\nAuthors:\nWe thank the Referee for these comments. As mentioned in the suggested paper, the same issue\nof Physics Today includes a follow-up paper by Norman Y. Yao and Chetan Nayak, which discusses\n1\nwhy time crystals differ from previous studies, including classical time crystals. Indeed, the \"time\ncrystal\" referred to in the original paper should be classified as a many-body-localized discrete\n(Floquet) time crystal, which, as the Referee points out, exhibits nontrivial properties under external\ndriving and avoids thermalization due to many-body localization. However, as the field has rapidly\nevolved, researchers used different mechanisms to avoid heating (thermalization), such as introducing\ndissipation (dissipative time crystals) or preparing the system in a prethermal state (prethermal time\ncrystals, which can exhibit exponentially long lifetime of the time crystal phase). Although these\nfindings were initially based on quantum systems, classical counterparts of theoretical prediction\nhave been identified in many cases. For example, friction and noise have been used to realize\n\"classical discrete time crystals\" (Ref. 18), and prethermal states have been used to study \"classical\nprethermal discrete time crystals\" (Refs. 19 and 20). These systems, described by classical nonlinear\nphysics, not only provide new ways to study and understand subharmonic behavior but also bridge\nthe gap between quantum and classical systems.\nAdditionally in the revised paper, we carefully studied the quasi-hexagonal lattice pattern and found\nthat its response exhibits a fractional relationship (~10/3) to the external drive (Fig. 10). Interestingly,\nsimilar fractional responses (0.3 and 100/33) have recently been observed in two quantum systems: a\nnumerical simulation of a spin-1/2 system and an experimental study on Rydberg atoms. The\nobservation of such close fractional responses across different systems raises the question of whether\nthere is a general underlying mechanism (like period doubling arising from ℤ symmetry or ℤ\n2\nsymmetry). We believe that aligning these systems under the same framework will help distinguish\ntheir differences and identify commonalities, advancing our understanding of \"time crystals.\"\nFurthermore, our system may differ from Faraday instability. Systems like Faraday waves are described\nby the Mathieu equation, which treats the system as a single-body or continuum medium. In contrast,\nour system is challenging to describe using a single-body equation: utilizing computer simulations, we\nillustrate that the period-doubling effect is intimately related to the inter-transformations, creations and\nannihilations of coexisting topological solitons and singular disclinations. Remarkably, the different\nstates of these topological objects can be viewed as the particle and anti-particle states of the observed\nMajorana-like quasiparticles (a classical analogue of Majorana particles that was recently discussed in\nthe context of active nematic liquid crystal systems [26-28]) forming our 1+1D space-time crystals\n(Fig. 4 and Supplementary Fig. S1).\nWe thank the Referee for these comments again, which prompted us to carefully reconsider the\ndescription and the use of terminology in the rapidly evolving research field of time crystalline\nphenomena. In the revised manuscript, overall, we have carefully checked terminology and softened\nclaims.\nReferee 1:\nFinally, the studied system presents a novel platform for studying non-linear phenomena. However, I\nwould suggest resubmitting the paper to a more specialized journal.\nAuthors: We appreciate Referee #1 noting that \"the studied system presents a novel platform for\nstudying non-linear phenomena.\" As shown in the manuscript, our system not only serves as a platform\nfor observing discrete time symmetry breaking but also provides a unique opportunity to study spatio-\ntemporal defects and classical Majorana-like quasiparticles, which have not been elaborately studied\nin previous work. Our system also exhibits similarities to both classical time crystal systems (e.g., the\n\"boil out\" effect in Ref. 18, anti-ferromagnetic order in space and time as in Refs. 18–20, etc.) and\nquantum time crystal systems (e.g., similar fractional responses, etc.) that have not been reported in a\n2\nclassical system context, to the best of our knowledge.\nAdditionally, our system has potential real-world applications. Liquid crystals are technologically\nimportant materials, with the LC-based global industry estimated to approach 1 trillion USD annually.\nElectrically controlled LCs are at the heart of modern display technologies, where the refresh rate of\nscreens ranges from 60 Hz to hundreds of Hz. Could discrete time symmetry breaking occur under\ncertain conditions, and could these phenomena play a role in technological applications? The transition\nfrom purely fundamental curiosity to technological utility can lead to further growth of research efforts\nstudying these phenomena. Moreover, we are pleased to share a good news: following a careful review,\nour university’s Technology Transfer Office has expressed strong interest in discrete space-time\ncrystals within liquid crystalline systems & filed our patent application with the United States Patent\nand Trademark Office, making our work ready for public disclosure.\nTherefore, we believe our study is of very broad interest for the scientific, technological, and\nengineering communities.\nReport of Referee 2:\nI reviewed the paper entitled “Observation of discrete space-time crystals in a classical soft matter\nsystem” from H. Zhao and coauthors where they demonstrated a time and spatial control of a\nspecific liquid crystal by applying surface alignment and periodic electrical drive. The overall space-\ntime periodic order has been observed via polarizing optical microscopy with a lambda plate,\nenhancing the static birefringence to observe the periodic driven orientation. The spatial periodicity\nseems to be comparable to the chiral pitch while the temporal periodicity is doubled within a period\nof the electrical drive.\nThe paper is overall well written and the several videos available in supplementary materials are\ncertainly helping the visualization of what the authors define a classical time-space crystal. The\nperiodic control over space and time of a large surface in a thin LC cells is certainly extremely\ninteresting, exploiting as far as I can understand a right combination between electrical spatial\nmodulation, chiral pitch, elastic constant and temperature.\nAuthors:\nWe thank the Referee #2 for this very thoughtful and positive report noting that our paper “ is overall\nwell written”, evaluate our studies “define a classical time-space crystal” and “… is certainly extremely\ninteresting”, as well as for the helpful and detailed suggestions. We have accounted for all suggestions,\nas detailed below, which helped us to further improve the manuscript.\nReferee 2:\nThe chosen LC mixture moreover appears to be robust to several (hours up to 24 hours) voltage-cycles\nusing high to very high fields amplitudes (150V-25/ 5-10 micron~ 300-25 kV/cm is certainly a high\nfield drive) most probably tanks to the ionic doping which avoid screening effects at the electrodes\n(only explained in the methods at line 474) and ensure local “weak” electric fields – quite in contrast with the\nactual values used in the experiment.\nAuthors: We thank the Referee for this comment. As an estimate, the maximum screening voltage\nU_screen~0.001d2ρe/mεε , where 0.001 refers to the weight concentration 0.1%, m is the mass of\nr 0\nsingle molecule of CTAB, d is the cell thickness, ρ is the density of LC, e is the charge of electron and\n3\nε is relative permittivity of the LC. Considering that d is 5-10μm, ρ is 1.03kg/m3 and ε is in the order\nr r\nof 102 at 2Hz, the maximum ability of U_screen is about 30-120V. When combined with the screening\neffect, the actual electric field applied across the nematic bulk can be one to two orders of magnitude\nsmaller than the maximum external drive field. Additionally, as suggested by Referee 3, we have added\na new figure (Fig. 2f) showing that the most obvious spatial patterns emerge when the amplitude of the\nexternal drive voltage is close to zero. As the external voltage increases, the LC pattern tends to be\nuniform under crossed polarizers but preserves “memory cores” (Fig. 4e, can be seen with the help of\nan enhanced contrast POM images) related to the mechanism of period-doubling detailed in the revised\nmanuscript.\nReferee 2:\nDespite the exhaustive description of the phenomenon as it appears and which conditions allows\ndifferent symmetries to take place, in space and time, the physical explanation detailing the overall\neffect is somehow not clear. This would certainly help the reader to understand many aspects of the\nwork here presented.\nAuthors: We thank the Referee for this helpful remark. We have detailed the mechanism of the\noverall effect in the responses to the comments below, as well as in the revised manuscript, we\nprovide a deeper description of the phenomena observed in this work.\nReferee 2:\nFor example, in lines 121-122, the authors refer to “LC’s complex response to the external drive\ndepending on TE and elastic and dielectric constants’ which is quite vague.\nAuthors: We thank the Referee for this comment. In the revised manuscript, we have clarified the\ncorresponding sentences and provided a more detailed and precise explanation to address the\nvagueness problem.\nReferee 2:\nI completely miss an elucidation about the origin of the doubling period. The closest the authors\ncome to explaining the mechanism is by saying “These DSTCs can be described as comprising\narrays of spatially and temporarily localized solitonic domain walls interacting with each other\nwithin the overall out-of-equilibrium setting” which is only appearing at line 167, and left quite\nunclear. This non-clarity comes quite unexpected given the fact that the authors are presenting 2\nmodels which matches the observed effects. There should be an intuitive explanation for example for\nthe torque balance equation only mentioned in the methods at line 492, or about the apparently\nimportant role of the ion-induced electric field introduced in the LdG free energy functional. Why the\nFrank-Oseen functional fails in reproducing the doubling period while the LdG does? This is not\nclear to me, and this part should find a place in the main text too in my opinion.\nAuthors: We thank the Referee for these helpful questions. The Frank–Oseen free energy based\nmodel describes very well the initial states when only nonsingular solitonic structures are present, but\nit cannot describe the states with singular half-integer disclinations that emerge during the voltage\ndriving scheme, which ultimately yields the period doubling effect. Utilizing LdG free energy\nfunctional based simulations, we reveal that the period doubling effect is intimately related to the\ninter-transformations, generations, and annihilations of coexisting topological solitons and singular\ndisclinations (Fig. 4g–o). The +1/2 and −1/2 singular disclination structures are the building blocks\nof 1+1D time crystals, which can be treated as a Majorana-like quasiparticle and its anti-particle (Fig.\n4\n4k) [26-28], because the disclination profiles smoothly transform as spinors following the Majorana\nequation (see Ref. 27 for details). In the time crystal phase, the particle-antiparticle inter-transformation occurs when the\nexternal voltage U smoothly changes from negative to positive (Fig.\n4g-i). The director field around the disclination region can be characterized by the characteristic twist\nangle β (β ∈ [0, π])[36,37]. When U < 0, β equals to π at the top and 0 at the bottom, indicating a\n−1/2 and +1/2 wedge disclination[38] in the two-dimensional cross-sections orthogonal to the defect\nlines, respectively. The winding number ±1/2 of disclinations relates to the accumulated angle of n(r)\nrotation as one circumnavigates the disclination core once, divided by 360°, with the positive sign\ncorresponding to the counterclockwise rotation matching that of circumnavigation and negative sign\nreferring to the opposite clockwise case. Between the two disclination regions, the −π-rotation Néel\ndomain wall solitons (with +/− defined by counterclockwise/clockwise rotation when traversing the\ntopological soliton from left to right) are spatially embedded (Fig. 4g). The domain wall solitons are\nlabelled as elements of the first homotopy group[39], 𝜋 (𝕊2/ℤ )= ℤ , and allowed to smoothly inter-\n1 2 2\ntransform to have 2D cross-sections of different types while here being terminated by two\ndisclinations 𝜋 (𝕊2/ℤ )= ℤ also inter-transforming between geometrically different but\n1 2 2\ntopologically the same states. As the voltage U increases from negative values to zero (Fig. 4h), the\nangle β characterizing both wall-terminating disclinations smoothly changes to π/2 (corresponding to\npure twist winding), and the initial −π-rotation Néel domain wall solitons (bend-splay structures)\ntransform to Bloch domain wall solitons (with pure twist structures). As the voltage U further\nincreases to U > 0 (Fig. 4i), β changes to 0 at the top and π at the bottom, and the Bloch domain wall\nsolitons transform to the +π-rotation Néel domain wall solitons. The transition is continuous during\nthe linear voltage change, and the positions of the disclinations and solitons do not shift spatially.The\narray of topological quasiparticles formed by solitonic walls terminated on singular defect lines\nmaintains its topological nature (Fig. 4i) until U suddenly changes from the positive maximum value\nto the negative maximum value (Fig. 4l), at which point topological particle-antiparticle pairs\nannihilate (Supplementary Fig. S1a) and re-generate (Supplementary Fig. S1b), while the locations of\nthe emerged quasiparticles are found synchronously shifted by half a spatial period (L/2) away from\nthe previous ones. After the generation, compared to the structures of the director field one temporal\nperiod (T ) before (Fig. 4g-i,m-o), the disclinations and topological solitons are the same, with an L/2\nE\nshift, so that the system adopts the exact same configuration every 2T , resulting in the period-\nE\ndoubling phenomenon. In the experiments and simulations alike (Fig. 4d,e), even when the optical\nimages appear almost homogeneously dark under a high electric field between crossed polarizers, the\ndefects and director deformations do not fully disappear, as evident from the enhanced contrast POM\nimages that reveal effective \"memory\" of the periodic deformations that allows for correlating\npositions of these topological objects within 2T .\nE\nThe details of the topological transformations make it especially clear why the Frank–Oseen\nfunctional based numerical modeling fails to reproduce the period-doubling behavior: this is because,\nunlike the LdG free energy functional, it cannot simulate disclinations and reduced scalar order\nparameter regions (Refs. 35 and 39). However, the LdG free energy functional is computationally\ntime-consuming for simulating large-scale structures. Therefore, we use the Frank–Oseen free energy\nto computationally obtain fully nonsingular quasi-static configurations in spatiotemporal diagram\nregions, which can be directly compared in detail to experimental POM and 3PEF-PM images.\nWe thank the Referee again for all these insightful questions. In the revised manuscript, we have\ndetailed the methodology and explanations.\nReferee 2:\nnoisier the response of the system? What’s the correlation between voltage amplitude and the LC\nfrequency response?\nAuthors: We thank the Referee for these questions. In the simulation, we find that when the system\nis near the period doubling point or when the voltage is low enough, the noise is small. Otherwise, the\nnoise is relatively large. In the revised manuscript, we add more results as we change the voltage\namplitude to clearly show this effect (Supplementary Fig. S2).\nReferee 2:\nMore generally, how do the details of the driving voltage impact the behavior? Does it only work for\nuni-polar saw-tooth signals?\nAuthors: We thank the Referee for these questions. As shown in the phase diagrams in Fig. 5, the\namplitude and temporal periodicity of the driving voltage are important control parameters to affect\nthe time crystal behavior, and larger temporal perturbations of the driving field can also destroy the\ntime crystal phase (Fig. 6d,e). Regarding the waveform shape of the external drive, we have not\nobserved a clear time crystal behavior with other waveform shape (triangle, square, and sinusoid\nwave), but we do not exclude the possibility for proper conditions and material parameters to be\nfound for this to occur. Related to the period-doubling mechanism described above, the transitions of\ntopological solitons and disclination regions could be a key point to design a different waveform.\nReferee 2:\nWhat’s the role of the anchoring at the surfaces? Only strong homeotropic conditions will allow to\nform such time-space pattern?\nAuthors: We have tried planar conditions, but we did not observe time crystal behavior under these\nconditions. For homeotropic conditions with a weak anchoring strength, it is harder to observe the\ntime crystal phase either, with the relatively strong homeotropic anchoring based on described\npreparation seemingly being ideal. We thank the Referee for these questions and the opportunity to\nprovide this information.\nReferee 2:\nMore in detail:\n• Line 61: the authors claim to boost the conductance of their LC materials by ionic doping, but do\nnot comment any further. Does the conductivity contribute to the pattern formation and doubling of\nperiodicity observed? How does the period doubling scale with ionic doping? And would it lead to\nanother dimension in the phase diagrams presented (fig 2)? How do the ionic currents expected at\nthese low modulation frequencies play into the underlying mechanism for the period doubling?\nAuthors: We thank the Referee for these questions. In experiments, we find that the DSTC phase\noccurs when ion doping is between 0.05-0.1 wt%. We tested pure LC without ionic doping and\nobserved the disordered and the time-symmetry-unbroken states. With higher ion doping (>0.2 wt%),\nno time crystal behavior is observed either. We observed the same behavior in simulations, when\nslightly increase or decrease ion concentrations (Supplementary Fig. S3a), the DSTC preserves.\nHowever, without ion doping or with too high ion doping concentration, there is no DSTC phase. In\naddition, we also plot the ionic current from simulations, we find that the ionic current also has a\nspatial/orientation distribution dependence (Supplementary Fig. S3b), which relates to the topological\nsolitons and disclinations in the system coupled with medium's anisotropy.\n• Lines 64 – 72: I would suggest to shorten at least in the main text the discussion about the\nPolarizing Microscope and\nmove the long discussion in the methods, also because is quite a standard method.\nAuthors: We thank the referee for this suggestion. We have shortened and moved the corresponding\ndescription to the methods part.\nReferee 2:\n• Line 74: The authors confuse the optic axis of a crystal with the optical axis (geometrical center of\nan optical system).\nAuthors: We thank the referee for this helpful remark, which we accounted for.\nReferee 2:\n• Line 99: what does it mean ‘while maintaining a good order’? It is quite vague.\nAuthors: We thank the referee for this question. In the revised manuscript, we have modified it to\n\"while maintaining the period-doubling behaviour\" to provide a clearer and more precise description.\nReferee 2:\n• Lines 136-137: what the authors mean with higher dimensional space-time coordinates?\nAuthors: We thank the Referee for this question. Illustrating this as an example, in spatial\ncoordinates, a defect in the xy-plane may appear at t=t and disappear at t=t . If we consider only the\n0 1\ntwo-dimensional xy-plane, the defect eventually vanishes. However, when we include the time\ncoordinate in an xy-t frame (referred to as higher-dimensional space-time coordinates compared to\nthe xy frame), the temporal axis (though intrinsically different from spatial axes) behaves similarly to\nthe z-axis. In this extended frame, the defect does not simply \"appear\" or \"disappear\"; instead, it\nemerges at specific positions along the t-coordinate. We thank the Referee again for this remark.\nSince the related discussion is not central to the DSTC behaviours and may cause confusion, in the\nrevised manuscript, we removed this sentence to ensure the clarity of the defect descriptions.\nReferee 2:\n• Lines 158-160: I am confused. In the first place because this sentence is located just at the end\nbefore basically the conclusions where the last words are: ” …which potentially could be related to\nthe fact that the DSTC phase is not observed.”, and in the second place because I do not understand\nwhich is the phase under doubt: is the 1+1D compared to 2 +1D? How many defects prevents one to\ndefine this structure a time/space crystal?\nAuthors: We thank the Referee for this helpful remark. From the video, we observe that the quasi-\nhexagonal lattice patterns do not recur every two or three periods in response to the external drive. By\ncarefully analyzing the relationship using Fourier Transform and temporal correlation functions, we\nfind that the internal temporal periodicity is approximately 10/3 times the external temporal periodicity,\nindicating a fractional discrete time crystal. Regarding defects, we observe 15 defects within a region\n(including 10 hexagonal units) over 200 external drives. This explains the strong noise around the peak\nin the FT spectrum and the rapid decay of the temporal correlation function compared to the case shown\nin Fig. 9. In the revised manuscript, we have added a new figure (Fig. 10) and provided a detailed\ndescription of this phenomenon. We thank the referee 2 again for this thoughtful report, which has\nsignificantly enhanced the presentation and clarity of our findings.\nReport of Referee 3:\nReviewer #3 (Remarks to the Author):\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part\nof the Nature Communications initiative to facilitate training in peer review and to provide\nappropriate recognition for Early Career Researchers who co-review manuscripts.\nReport of Referee 4:\nReviewer #4 (Remarks to the Author):\nIn this work, the authors consider the dynamics of a liquid crystal system under a periodic\nmodulation of the electric field. Depending on the details of the drive (and thickness) they observe a\nvariety of spatial-temporal ordering, in the orientation of the liquid crystal molecules. These patterns\nare measured through the polarization of the light polarization, which is mapped onto a color\nscheme.\nWhile the observations in the paper are intriguing and study time crystalline behavior in a novel\nsetting, I find that the manuscript does not address which aspects of the physical setup and protocol\nenable the observations of the robust period doubling behavior. As a result, it is difficult to relate the\nobservations in the current manuscript with other time crystalline behavior and how the authors'\nobservations can be generalized to other settings.\nFor example, while the authors emphasize the observation of the time crystalline behavior in a soft\nmatter system, it is unclear what aspect of the soft matter system are crucial for the observation. Is it\nthe director nature of the liquid crystal? Is it the chiral nature of the interactions? Or the strong\ncoupling to a finite temperature bath?\nGiven that the authors have access to an effective model that accurately captures the dynamics, I\nthink it is important to address some of these questions before I can support the publication in Nature\nCommunications.\nAuthors:\nWe thank the Referee #4 for this very thoughtful and detailed report saying that our observations “are\nintriguing and study time crystalline behavior in a novel setting …”, as well as appreciate the helpful\nsuggestions.\nIn the revised manuscript, we have expanded the discussion on the connections, similarities,\nand commonalities between our space-time crystals and other time crystals. We have also further\nelaborated on the unique advantages of utilizing the particle like topological solitons and disclinations\nwithin the liquid crystal platform for realizing time crystals. While we observe it as an emergent\nphenomenon in a very specific type of soft matter system, our work demonstrates that such effects can\nemerge in a classical setting, which is a result of general significance, and we foresee that they will\nlead to future discoveries of similar effects in other types of liquid crystals and other types of\n8\nsoft/classical matter.\nRegarding the discussion of generalization to other time crystalline behavior, we would first\nlike to thank Referee 4 for these insightful questions (Questions 11 & 13), which inspired and guided\nus to investigate the emergence of temporally fractional response periods of the spatially quasi-\nhexagonal lattice. Surprisingly, although our system is classical, the fractional periods we observed are\nclose to those reported in a theoretical study and an experimental study on quantum systems, which we\nwill describe in detail later. Additionally, we have discovered that our system exhibits intrinsic and\nprofound connections to other time crystals in several aspects. For example, we observed the \"boil out\"\nphenomenon—where space-time crystals emerge from disorder (Ref. 18). Our system also provides a\nvalidation of space-time group theory (Ref. 24). Furthermore, the role of anti-ferromagnetic order in\nboth space and time, and other features highlights more significant connection (Refs. 18-20). These\nfindings not only help bridge the gap between classical and quantum time crystals but also establish a\nbroader connection to soft matter and other classical systems.\nIn terms of the advantages of utilizing the liquid crystal platform, we show that the observed\nphenomena arise from a combination of intrinsic soft matter properties, particle-like behavior,\ntopological features, and other factors. The many-body interactions between particle-like solitonic and\nsingular defect objects are of key importance. The strong (relative to thermal energy) elasticity-\nmediated interactions between quasi-particles ensure that the system remains relatively stable even in\nthe presence of temporal perturbations and spatial defects (Fig. 4a,b). As a result, once the time crystal\nphase emerges under suitable conditions, we can often observe the long lifetime of time crystallization.\nRegarding the mechanism of time crystal formation in our system, we have further elaborated on the\nmechanism in the revised manuscript using a 1+1D space-time crystal as an example (Fig. 4). We find\nduring each drive, the inter-transformations, creations/generations, and annihilations of coexisting\ntopological solitons and singular disclinations play a crucial role, which will be detailed later in the\nresponse to Question #2&6. Remarkably, the different states of these topological objects can be viewed\nas the particle and anti-particle states of the observed Majorana-like quasiparticles (a classical analogue\nof Majorana particles that were recently studied in active nematic LC systems [26-28]). These particle-\nlike structures in liquid crystal systems also show potential realizations in other soft matter systems.\nFurthermore, the spatial scale of the liquid crystal system that forms space-time crystals ranges from\nmicrometers to millimeters. This allows us to directly observe the time crystal under a microscope. In\ncontrast, many previous studies on discrete time crystals in quantum systems were limited to either\nmeasuring on-site period-doubling signals in systems with fewer than tens of spins or qubits or\ndetecting global signals such as total magnetization. Our liquid crystal system, however, enables the\nanalyses of large-scale spatiotemporal structural relationships.\nWe anticipate that soft matter media will allow for discoveries of many discrete, continuous, and\nfractional time and space-time crystals of different dimensionalities.\nWe would like to thank the Referee 4 again for these insightful and helpful questions and comments.\nWe have accounted for all suggestions, as detailed below, which helped us to further improve the\nmanuscript.\nReferee 4:\nBelow I expand upon these and other questions (as well as some broad comments), whose answers\nwould greatly improve the manuscript.\n# Questions:\n9\n1) In the abstract, the authors note that the system behaves \"like a time-crystal analogues of a\n**smectic** phase\". However the observed time crystalline behaviors corresponds to a period\ndoubling behavior -- in what sense is that related to the smectic phase? Is it because of the spatial\nordering?\nAuthors:\nWe thank the Referee for this helpful question, in LC systems a strictly 1D spatially periodic lattice\ncannot exist due to thermal fluctuations (Ref. 41). The spatial location where each stripe region\nemerges has a small randomness, leading to a quasi-long-range order in spatial coordinate, which is\nthe characteristic of the well-known smectic phase in LCs. However, a more interesting thing we\nobserved is that when we calculate the temporal correlation of the pattern, the correlation function\nalso exhibits a power-law decay behavior, which indicates a quasi-long-range order (smectic-like) in\ntime (Fig. 9) and (as we see from informal discussions) is already attracting strong interest of\ntheorists as a phenomenon. In the revised manuscript, we have improved the corresponding\ndescriptions to provide a clearer explanation of this phenomenon.\nReferee 4:\n2) The authors introduce two different free energy models to capture the observed phenomena. From\nmy understanding of discussion in the Methods section, only the Landau-de Gennes approach is able\nto capture the dynamical observations. Can the authors expand on the differences between the two\napproaches? Besides a more complex order parameter (built upon the same director as in the Frank-\nOseen approach), what physical content is present in the second free energy that enables the better\nmodeling of the system? At the same time, aspect is captured by the Frank-Oseen approach that is\nnot captured by the Landau-de Gennes one?\nAuthors:\nA major difference between these two approaches is that the Frank–Oseen free energy functional is\nvector based, while the Landau–de Gennes free energy is tensor based. The latter not only includes\nthe orientation information of the director field but also captures the degree of orientational alignment\nrelative to the reference direction, described by the scalar order parameter. For example, the Landau–\nde Gennes free energy model can accurately simulate singular disclination structures (with low scalar\norder parameter), whereas the Frank–Oseen free energy cannot (Refs. 35 and 39). The interactions\nand transformations of the disclinations and topological domain wall solitons under external drive\nplay an important role for the period-doubling phenomenon, which we will describe in detail below\nin our responses to Question #6.\nHowever, the Landau–de Gennes approach involves more parameters than the Frank–Oseen\napproach, making it computationally time-consuming for simulating large-scale structures while\nusing all elastic constants and material parameters known from experiments. Therefore, we use the\nFrank–Oseen free energy to obtain quasi-static configurations, which can be directly compared in\ndetail to experimental POM and 3PEF-PM images, effectively depicting what is the initial state for\nthe dynamic evolution of the discrete time crystal. We thank the referee for this insightful question.\nIn the revised manuscript, we have emphasized the differences between these two free energy-based\nmethods and elaborated on why using both approaches provides a more comprehensive\nunderstanding of our time crystals.\nReferee 4:\n10\n3) The observations correspond to a simultaneous breaking of the spatial and time translation\nsymmetry. Are the observed spatial symmetry breaking patterns known in previous literature? Are\nthey a requirement for the observation of the temporal ordering?\nI ask because there are classes of time crystals that arise from the presence of different symmetry\nbroken states, with the drive serving to toggle the system between the different stable configurations\nand thus induce a period doubling behavior. Is this an example of one such time crystal or are the\nspatial patterns only stable in the presence of the period doubling behavior? If not, are these spatial\npatterns metastable?\nAuthors:\nWe thank the Referee for these helpful questions. We note that similar spatial-temporal patterns have\nnot been observed, to the best of our knowledge. Our system is a common model system, with a\npositive dielectric constant LC and uniform homeotropic boundary condition, which we then\nadditionally dope with ions. When an electric field is applied, the LC molecules tend to align parallel\nto the field following the dielectric coupling. Interestingly, we observed not only the spontaneous\nbreaking of spatial symmetry, leading to the formation of lattice patterns under Floquet drives, but\nalso rich periodic dynamics in time, including time-symmetry unbroken phase, period-doubling\nphenomena, fractional, and disordered response behavior. We find that in both simulations and\nexperiments under a constant external field, the 1D spatially periodic structures can be observed at\nlow voltages (Fig. 3) but become unstable at higher voltages. Additionally, none of the 2D spatial\npatterns remain stable under a constant field. So we believe these spatio-temporal structures only\nemerge as the novel space-time crystal behavior due to many-body interactions and inter-\ntransformations of topological defects.\nReferee 4:\n4) The authors discuss the presence of different \"types\" of DSTCs. Is there a precise distinction\nbetween the different types of DSTCs? While the visual patterns are slightly different, they appear to\nhave the same symmetry properties.\nAuthors:\nThe spatial spacings of type 1 and type 2 DSTCs are 8.2 μm and 11.6 μm, respectively, in a sample\nwith a cell thickness of 10 μm. For type 3, the spatial spacing is 18.4 μm in a sample with a cell\nthickness of 15 μm. We observe that type 1 and type 2 DSTCs can co-exist in different regions of the\nsame sample, while type 3 DSTCs are only observed when the cell thickness exceeds 15 μm. We\nthank the referee for this question and have clarified these distinctions in the revised manuscript.\nReferee 4:\n1) Focusing on Fig, 1h, it is actually unclear that the system is exhibiting a period-doubling\nresponse--in particular, all the snapshots appear to have the same pattern. Why is that a DSTC?\nAuthors:\nThe pattern in the same reference area is surrounded by four rectangular stripes (two in blue and two\nin yellow), exhibiting an open-close or close-open like cycle. We thank the Referee for this helpful\nquestion. In the revised manuscript, we have re-plotted the figures for the type 2 DSTC to more\n11\nclearly demonstrate the period-doubling phenomenon.\nReferee 4:\n5) What happens for larger TE in Fig2a? And smaller temperature for 2b?\nAuthors:\nThe authors thank the Referee for this question. In the experiment, the dynamic phase becomes disordered\nwhen T is increased. However, we did not explore the phase diagram at temperatures lower than\nE\nroom temperature (~23°C), as the liquid crystal 5CB we used may undergo a phase transition from\nthe nematic phase to the solid molecular crystal phase below room temperature.\nReferee 4:\n6) Looking through the videos, it seems like the patterns described in the paper only emerge for very\nbrief moments, with the system mostly looking homogeneous.\n6.1) Do these patterns occur at the max/min of the voltage or near the zero voltage point?\nI think that emphasizing this point in the main text is important.\nAuthors:\nWe thank the Referee for this helpful question. In the revised manuscript, we add a new figure (Fig.\n2f) to illustrate when these patterns emerge. Indeed, these patterns appear when the voltage is close to\nzero. For example, under an external drive with T = 0.5 s (U ∈ [−50 V,+50 V]), a clear pattern\nE\nemerges for around 0.04s when U ~ 0V. This is consistent with the results of our modeling (twist\nstructures of birefringent materials can allow the light to pass through under crossed polarizers).\nReferee 4:\n6.2) How does the system preserve a \"memory\" of the previous pattern even when looking\nhomogeneous?\nAuthors:\nWe thank the Referee for this question, which is also related to the mechanism of period-doubling\nphenomenon in our systems now described above in responses and in the paper in more details.\nUtilizing LdG free energy functional based simulations, we reveal that there are inter-\ntransformations, generations and annihilations of the particle-like topological structures of director\nfield within one temporal driving period with all the details. Compared to the structures of the\ndirector field one temporal driving period before, the twist angle of the disclinations and topological\nsolitons are the same, with half of the spatial period shift, thus every two driving periods the system\nadopts the same configuration, resulting in the period-doubling phenomenon. The memory effect\nstems from the fact that defects and director deformations never fully disappear, just inter-transform.\nEven when the optical images appear almost homogeneous under a high electric field (the director is\nperpendicular to the polarizer), the experiments and simulations reveal differences from the uniform\nbackground, which directly reveals these memory-related transformations.\nelaborated on the origin of the period-doubling phenomenon in our systems, which preserve the\n“memory” by retaining the topological structures.\nReferee 4:\n7) What sets the orientation of the checkerboard pattern in the 2+1D DSTC? Naively, the system has\na rotation symmetry around the z-axis, yet the patterns appear to form along the picture axis. Is this\nsome feature of the boundary conditions of the experiments that are not made explicit.\nAuthors: The boundary conditions on both substrates are uniform and homeotropic. We did not pre-\npattern or pre-coat any structures on the surfaces; the checkerboard patterns emerged spontaneously\nwith spontaneously selected orientations, as shown in Fig. 6a and 6b, which is then retained over time.\nRegarding the lateral boundary effects, the sample size is approximately 10 mm × 10 mm, while the\nspace-time crystal area is larger than 500 μm × 500 μm and is always at least 1 mm away from the\nlateral cell edges. The space-time crystal phase patterns are typically surrounded by disordered regions\n(Fig. 6c). When the voltage and external drive frequency are within the time crystal phase (Fig. 5), we\nobserve that the space-time crystal pattern area grows, as the disordered regions are relatively unstable\ncompared to the space-time crystal patterns. We thank the Referee for this question and have explicitly\ndetailed the boundary conditions in the revised manuscript.\nReferee 4:\n8) I was unable to parse the statement \"While defect regions tend to disappear in the space\ncoordinates, they still occasionally emerge in higher-dimensional space-time coordinates.\". Can the\nauthors clarify the meaning of higher-dimensional in this context?\nAuthors: We thank the Referee for this question, which has helped us clarify the description of\ndefects in the manuscript. Here we refer to the defects in the space-time crystal lattice. In spatial\ncoordinates, for example, a lattice defect in the xy-plane may appear at t = t and disappear at t = t .\n0 1\nIf we consider only the two-dimensional xy-plane, the defect finally vanishes. However, when we\ninclude the time coordinate in an xy-t frame (higher-dimensional space-time coordinates compared to\nthe xy frame), the temporal axis (though intrinsically different from spatial axes) behaves similarly to\nthe z-axis. In this extended frame, the defect does not simply \"appear\" or \"disappear\"; instead, it\nemerges at specific positions along the t-coordinate. Since the related discussion is not central to the\nDSTC behaviours and may cause unnecessary confusion, whereas characterizing it further is outside\nthe scope of this work, we have removed this sentence in the revised manuscript for clarity.\nReferee 4:\n9) \"Φ is the signals [sic] captured by the quasi-long-range the camera.\" In this sentence, what\nconstitutes the signal?\nNaively, there are multiple colors being displayed, so how is that information mapped onto a scalar quantity? Why is such\nquantity not used when displaying the period doubling behavior in Fig 1k, and instead the position of the middle region is\nprefered?\nand blue components, as our videos are captured in RGB mode over 644 external drive periods. We\nconvert the RGB images to grayscale using ImageJ, and the scalar quantity is obtained from the\ngrayscale video. We have added an image of the time correlation functions for the red, green, and\nblue signals separately (Supplementary Fig. S4). We find that both the red and blue signals exhibit\nthe same power-law decay behavior, while the green signal is significantly weaker, as evident from\nFig. 1d, e, g-i, etc. The time correlation function calculation provides information about temporal\ncorrelations when averaged over a large dataset.\nIn Fig. 1m (Fig. 1k in the previous version), they not only show the positions of the particle-like units\nover time but also show their motion direction and displacement. While this information could also\nbe revealed by calculating a space-time correlation function (represented as a 3D image with two\nspatial axes and one temporal axis), in our opinion, such an approach for presenting results is less\nclear and straightforward compared to providing the trajectories shown in Fig. 1m. Additionally, it\nwould require an even larger dataset to achieve reliable results (recording a high resolution video\nover 600 drives demands 1 GB of hard drive space).\nReferee 4:\n10) What is the meaning of the quasi-long-range order in this context? While the system displays a U(1) symmetry (for a\nrotation of the order parameter around the z axis), the behavior in the time direction has a discrete symmetry, so what causes\nthe power-law decay in time? Is it inherited from the spatial decay of correlations?\nAuthors:\nWe thank the Referee for this insightful question. The quasi-long-range order in time is indeed\nrelated to the spatial symmetry. For example, in the case of the 1+1D space-time crystal, stripe-like\npatterns emerge during each drive and recur every two drives. However, the positions of these stripes\nare not always exactly the same, exhibiting slight randomness due to the nature of the 1D periodic\nlattice, because of thermal fluctuations (Ref. 38). When we calculate the local temporal correlation\nfunction, the randomness in the spatial coordinates translates into the temporal domain. Spatially, we\nknow that the correlation function can exhibit a power-law decay if fluctuations are small.\nInterestingly, we find that the temporal correlation function also follows a power-law decay, which is\ninherited from the spatial behavior. Additionally, we note that significant temporal perturbations can\ndestroy this order (Fig. 6e). However, in the time crystal phase with small temporal perturbations\n(Fig. 6d) or without perturbations (Fig. 9), the temporal order does not decay rapidly. We also note\nthat our work calls for the development of theories that could help deepening the understanding of\nsuch time crystalline order, though they are outside the scope of this present study.\nReferee 4:\n11) What is the FT response of the hexagonal order? What is the ratio between the drive frequency\nand the response frequency?\nIt would be very surprising if there there is a non-frational ratio, yet I couldn't find additional data\non this observation.\nAuthors:\nWe thank the Referee for this helpful suggestion. In the revised paper, we have added a new figure to\nexhibit the potential fractional temporal response of the quasi-hexagonal lattice. From the FFT\nanalysis, we find that the frequency f/f ≈ 0.3 (T/T ≈ 3.33). We also calculated the temporal\nE E\n14\ncorrelation function and found that the internal temporal periodicity is around 40T /12 =10T /3\nE E\n(3.33T ) or 33T /10(3.3T ), where the temporal periodicity of the external drive is T . Interestingly,\nE E E E\nalthough our system is very different, a theoretical study on a clean (non-disordered) spin-1/2 system\nand an experimental study on Rydberg atoms also reported fractional responses around 10/3 (e.g., f/f\nE\n≈0.3 in Ref. [43] and f/f =100/29 in Ref. [44]). We thank the Referee again for this question, which\nE\nhas helped us improve our understanding of the fractional response in the quasi-hexagonal lattice.\nThis system could potentially have the fractional space-time crystal properties of other different\nfractional responses, which is awaiting for future studies.\nReferee 4:\n13) How is the \"relative noise\" quantified in the context of the comment in the discussion? The fact\nthat the system is coupled to a bath at some temperature suggests that there is indeed a large source\nof noise that is much bigger than the prethermal regime of an isolated quantum system? Or is this a\ncomment with regards to the coherence time of modern quantum platforms?\nAuthors:\nThe relative noise we mention here refers to the thermal fluctuations within the system. In Fig.4b, we\ncalculate the interaction energy between neighboring quasiparticles; we find it ~10k T of a 10%\nB\ncompression and stretching. As a result, intrinsic thermal fluctuations do not significantly affect the\nstructures unless considered over a large spatial scale or a long temporal scale (for example, quasi-\nlong-range order in space and time). We thank the Referee for this helpful question and have\nprovided a clearer description of this point in the revised manuscript.\nReferee 4:\n14) I'm confused how to compatibilize two claims of the paper: (i) that the DSTC can have day long lifetime (Extended Data\nFig 7), and (ii) that you observe power-law decay of the spatial temporal correlations (Extended Data Fig 8). These two\nobservations seem to be at odd since one seems compatible with very stable long-lived two point correlation function, where\nthe later suggests that there is no sharp way of defining a long lifetime for the behavior being explored. Can the authors\nclarify what they are claiming in this context?\nAuthors:\nIn Fig. 8 (Extended Data Fig. 7 in the previous version), the stripe-like features emerge during each\ndrive and recur every two drives. After several hours, we still observe the time crystal behaviours in\nthe same camera-captured region. However, if we calculate the time correlation over time, we find\nthe power-law decay correlation function (Fig. 9). We thank the Referee for this question and make\nsure to have clarified these points in the revised manuscript.\nReferee 4:\n15) As a non-expert in the physics of liquid crystal, I was confused by the description of the\nmeasurement scheme. Why do different colors get shifted differently by the same pattern of LC\ndirectors and how is that related to the average director along the beam path?\nAuthors:\nThe sample is illuminated by a wide spectrum of visible light from a lamp. The light first passes\nthrough a polarizer, becoming linearly polarized. As it passes through the sample, the light becomes\nelliptically polarized due to the LC's large birefringence. This birefringence induces a phase\n15\naccumulation that depends on both the wavelength of the light and the angle between the LC director\norientation and the polarization of the incident light. Finally, the light passes through a second\npolarizer (analyzer), which is oriented perpendicular to the first polarizer. After passing through the\nanalyzer, the intensity of light at different wavelengths varies, resulting in a superposition of colors\nthat produces the observed color patterns. Such polarized interference analyses are widely used to\nstudy liquid crystalline materials. We thank the Referee for this question. In the revised manuscript,\nwe have expanded and detailed the description of the measurement scheme and moved it to the\nMethods part.\nReferee 4:\n# Comments:\n1) I found the lack of discussion of the periodic drive makes the discussion of the DSTC behavior\nharder to parse. For example, the authors only note the shape of the profile in the figure and never\nthe main text, and do not address how the shape of the profile was chosen and how much it impacts\nthe observations.\nWhile the authors have a mean-field model that is able to capture the observed dynamics, it remains\nunclear what feature of the system/drive is able to generate the time crystalline behavior. As such it is\nhard to understand the origin and nature of the observed phenomena.\nAuthors: In the revised manuscript, we have provided a detailed explanation of the mechanism\nbehind the period-doubling phenomenon, as outlined earlier and in response to Question #2&6. This\nexplanation is supported by both experimental and simulation results. In addition, we have\nnumerically calculated the energy landscape of many-body interactions between neighboring\ntopological solitons (Fig. 4a,b), commonly described as quasiparticles or particle-like objects. We\nhave added additional insets and schematics, making sure that our presentation is clear. We thank the\nReferee again for these insightful questions and comments, which have greatly enhanced our\npresentation of the origin of time crystal behavior in our system.\nReferee 4:\n2) For 1j, the markers might benefit from having two colors that differ more -- i.e. green and orange.\nThis can be combined with different markers for the two parity of periods.\nAuthors: We thank this helpful suggestion, which we account for.\nReferee 4:\n3) Because the processing for the \"center\" of the blue region is somewhat obscure, I might suggest\nthe authors append their data with the value/color of the image as a function of time -- e.g. near the\nblack points of 1g, it is clear that the data fluctuates between blue and purple with a sub-harmonic\nresponse.\nAuthors: We thank this helpful suggestion, in the revised paper we have added the related results in\nFig. 1k. We thank Referee 4 again for the thoughtful report and suggestions and comments that helped\nus to further improve presentation of our findings.\n-----------------------------------------------------------------------------------\nwere raised by the original manuscript. We also feel that they improve the paper significantly. We\nthank the Referees for the thoughtful valuable comments that directed these changes.\nSincerely,\nProf. Ivan I. Smalyukh, Department of Physics, 390 UCB\nUniversity of Colorado at Boulder\nPhone: 303-492-7277 (office); Email: Ivan.Smalyukh@colorado.EDU\nhttp://www.colorado.edu/physics/SmalyukhLab/index.html\nEmergent discrete space-time crystal of Majorana-like quasiparticles in chiral\nliquid crystals\nH.Zhao, R. Zhang and I. I. Smalyukh\nSummary of Changes and Responses to Requests/Remarks\nChanges to Account for the Referees’ Suggestions.\nReport of Referee 1:\nI have carefully reviewed the authors’ responses to the referee reports, including their detailed replies to my\nown comments and concerns. I appreciate the thoroughness with which the authors have addressed the issues\nraised, particularly regarding the terminology around time crystals, the distinction from classical nonlinear\nphenomena, and the broader context and implications of their findings.\nI agree with the authors’ clarifications and the modifications made to the manuscript. The revisions strengthen\nthe presentation and improve the accessibility of the work. I also find the discussions and suggestions from the\nother referees to be constructive and well-integrated into the authors’ revised manuscript.\nIn light of the authors’ comprehensive and satisfactory responses, as well as the work's overall strong\nscientific merit, I recommend that the manuscript be accepted for publication.\nAuthors:\nWe thank the Reviewer #1 for the recommendation and the positive reports.\nReport of Referee 2:\nThe authors responded to all comments raised and thoughtfully explained the missing parts. I can only raise a\npoint regarding the very detailed explanation on the doubling period. I am not an expert in \"inter-\ntransformations, generations, and annihilations of coexisting topological solitons and singular disclinations\"\nwhich seems the founding cause of such phenomena but for a non-expert is really not easy to follow. Could\nyou mitigate the entire added paragraph which I saw transferred 1to1 from the reviewer response to the paper\nin order to be more open to a non-expert audience?\nAuthors:\nWe thank the Reviewer #2 for the remarks and the positive reports. In the revised manuscript, we\nhave modified the paragraphs accordingly (page 6,7) to make the explanation more accessible for\nnon-expert readers.\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part\nof the Nature Communications initiative to facilitate training in peer review and to provide\nappropriate recognition for Early Career Researchers who co-review manuscripts.\nReport of Referee 4:\nIn the reviewed version of the manuscript, the authors have extensively revised their observations,\nappending two new results: the description of the period doubling dynamics in terms of the creation\nand annihilation of pairs of topological \"Majorana-like\" defects, and the observation of a fractional\ndiscrete time crystal whereby the sub-harmonic response is not a precise fraction of the original\nfrequency.\nWhile I thank the authors for their efforts in improving the manuscript, I found that they were not\nable to address some of the questions I raised, and the new results beg a few new points that I think\nthe authors should address before the manuscript should be published.\nIn general, I have found that the paper is exploring an interesting new regime and demonstrates a\nnovel platform for studying novel out of equilibrium phenomena. However, I have found that the\nrelevance of the work is diminished because the paper fails to engage with the broader set conceptual\nand categorization questions that come from the observation of time crystalline phenomena, specially\nin open classical systems. Without such discussion, some of the claims of the paper feel\nunsubstantiated and unclear. I think that to appeal to the broad audience of Nature Comms, such\ndiscussions are relevant, otherwise I agree with Referee 1 that the manuscript is more suitable to a\nmore specialized manuscript.\nIn an attempt to help the authors, I have divided my comments into main concerns and minor\nconcerns, as per the scope my comment.\nAuthors: We appreciate the reviewer for this very thoughtful and detailed report saying that “the\npaper is exploring an interesting new regime and demonstrates a novel platform for studying novel\nout of equilibrium phenomena”. We also thank the reviewer for the helpful suggestions, which we\nhave carefully addressed in the revised manuscript and detailed in our responses below.\nReviewer #4:\nMain Concerns:\n1) Categorization of the time crystalline phenomena and nature of the ordered phase\nAs also alluded by Referee 1, the nature of period doubling (and other time crystalline phenomena) in\nclassical systems is subtle. That is because the role of the external bath, as well as the type and\ninteractions between the different degrees of freedom leads to fundamental different expectations\nabout the nature and the stability of the behavior. For example, in few particle systems, near-\nintegrability can enable the stabilization of long-lived period doubling behavior via the KAM\ntheorem; by contrast, when coupling to a bath that can extract entropy without providing\n2\nfluctuations, the system can be toggled between different stable sectors in a trivial manner. In both\nthese scenarios, the life-time of the DTC behavior is infinitely long-lived owing to the details of the\nsystem.\nThe present work does not fall into these conditions: it is a strongly open system (the authors must\ninclude a meaningful damping rate on their simulations) at a very non-zero temperature, and it treats\nthe dynamics of an extensive number of interacting particles.\nAs a result, the expectation is much more subtle: the interplay of noise and interactions is expected to\ndestroy the time crystalline order, unless the particular model has some novel mechanism for\nstabilizing the expected proliferation of defects and their destruction of the time crystalline phase. In\nthis case, the lifetime of the time crystalline phenomena is finite, albeit very long.\n(These considerations are extensively explored in arXiv:2305.08904, a paper that the authors\nreference but do not engage too strongly with).\nI understand that the current manuscript is not meant to revisit the entire understanding of the time\ncrystalline phenomena, however, I find that the lack of any discussion of these considerations (in\nparticular the lifetime and what determines it), specially where they authors understand their\nobservations to lie, severely limits ones capability to parse their observations and to contextualize\ntheir results. As a result, the authors appear to emphasize the discovery of a completely distinct time\ncrystalline phenomena, but without having strong enough evidence to make such a point.\nFor the clarity of this work and its contextualization with the broader time crystalline community, I\nthink it is important for the authors to directly engage with the question of stability and heating\nwithin the referee report and the manuscript.\nAuthors: We thank the reviewer for these helpful questions and remarks. In the revised manuscript,\nwe have added a discussion on the lifetime of the time crystal in relation to thermal noise. Our\nnumerical simulations show that noise strongly affects the lifetime, as expected, which is consistent\nwith our experimental observations. To numerically quantify this effect, we introduce a white-noise\nstochastic perturbation f (t) that directly couples to the director, with the auto-correlation\nnoise\n<f (t)f (t')>=2η T δ(t-t'), where η is the friction coefficient and T is the effective temperature\nnoise noise 0 eff 0 eff\nof the thermal perturbation. In the simulations, as we increase the effective temperature, we observe\nthat the order of the system decreases, with more defects emerging in the LC bulk (Supplementary\nFig. S9a). We then measure the lifetime τ (correlation time), calculated from the Fourier transform\n0\nspectra (as detailed in Ref. [43]), and find a dramatic decrease in lifetime as the noise increases\n(Supplementary Fig. S9b). Interestingly, what we find is that τ 0 ~𝑒𝛥/𝑘 b 𝑇 eff, where Δ is the quasi-\nenergy activation barrier as defined in Ref. [9] (arXiv:2305.08904), indicating a (thermally)\n“activated” time crystal. Such a dramatic transition is consistent with our experimental results.\nIndeed, in experiments, it is difficult to precisely control the noise level (for example, temporal\nperturbation); as we vary the perturbation, the system behavior changes dramatically from a time\ncrystal state (which can typically persist for an extremely long time) to a disordered state (which\nshows a very short or nearly zero lifetime, Fig. 6d,e). In addition, inspired by this idea, we revisit the\nanalysis of temporal correlation function (Fig. 9) of the 1+1D DSTC and find that G(t) could be fitted\nby 𝑒−𝑡/τ0 (Chapter V.A in Ref [9]). We thank the reviewer again for these helpful remarks, which\nhelp us to categorize our findings.\nReviewer #4:\n1.1) Throughout the paper, the authors utilize the center location of the blue domain to build the two-\npoint correlation function that they use to state the observation of quasi-long-range order. I am still\nconfused about this choice. In the book that they reference, power-law correlations are given in terms\nof the order parameter of the smetic transition (Chapter 6 if I am not mistaken), and they see power-\nlaw in spatial correlations with a power that is controlled by the parameters of the system (namely\ntemperature).\nThe authors claim to see the power-law behavior in the two-time (albeit the evidence is not very\nstrong given the small dynamical range of their fit), using a correlation function over the position\ncenter. What is the relationship between the spatial and the time fluctuations? What is the expected\nform of the power-law exponent in this case? And what does that inform us about the stability of the\ntime crystalline phenomena? And what prevents the authors from considering data for thousands of\ntime steps since they observe the robustness over hours and each period is second scale?\nAuthors:\nWe thank the reviewer for these insightful questions. First, to clearly describe how we calculate the\ntime correlation function, in the revised paper, we have provided a more detailed explanation in the\nMethod section. For the correlation functions in Fig. 9 and Fig. 10e, we did not calculate them\nutilizing the center locations of the blue domain. Instead, the time correlation function G(t) is directly\ncalculated from the video signal Ф (τ) (without any center location tracking). Here, Φ (𝜏) denotes\ni,j 𝑖,𝑗\nthe signal of the pixel at column i, row j at time τ, and G(t)~∑ ∑ Φ (𝜏)Φ (𝜏+𝑡). We also\n𝜏 𝑖,𝑗 𝑖,𝑗 𝑖,𝑗\ncalculate the spatial correlation function G(r)~ ∑ ∑ Φ (𝜏)Φ (𝜏) and spatial-temporal\n𝜏 𝑖,𝑗 𝑖,𝑗+𝑟 𝑖,𝑗\ncorrelation function G(r,t)~ ∑ ∑ Φ (𝜏)Φ (𝜏+𝑡) (Supplementary Fig. S4,8), where j and r are\n𝜏 𝑖,𝑗 𝑖,𝑗+𝑟 𝑖,𝑗\nalong the direction of spatial spacing, and i is perpendicular to the spatial spacing for 1+1D DSTC. In\nsmectic liquid crystal systems with 1D quasi-long-range positional order, the power-law exponent is\nrelated to the inter-molecular interactions and orientational elasticity coefficients and temperature\n(Ref. [43]), and the emergence of defects and lattice imperfections can strongly affect it. The\nemergent system we study shows similar spatial correlations (in fact, this is expected as chiral\nnematic’s elasticity is often mapped to that of smectic LCs as part of coarse-graining - Ref. [43,44]).\nWe note that G(r) can be fitted by a power-law decay in the scale of ~100 μm (Supplementary Fig.\nS4. For the stability, we find that for larger-scale DSTC containing impurity regions, the G(t) and\nG(r,t) decay faster than in well-ordered space-time crystal without impurities (Supplementary Fig.\nS6-8), which informs that the stability is determined by the noise (as described in the previous\ncomment) and lattice impurities. Regarding the question of time steps, we have added related results\nand minor revisions related to methodology in the revised manuscript (Supplementary Fig. S6-8).\nReviewer #4:\n1.2) In the rebuttal, the authors emphasize that the file size for the data is very large, but it seems like\nthey have access to the videos. Is this a statement that the data analysis is very challenging or a\nstatement that they were unable to store videos for all the experiments they conducted and so they\nhad to only store this subset of the information?\nRegardless, a discussion about how the middle of the blue regions is obtained should be more\nprominent as it constitutes a non-trivial piece of data analysis that is important to parse the\nobservations, as well as the correlation function discussed later.\nAuthors:\nrecording of high resolution (Supplementary Fig. S6-8), the experiment demands ~2 GB of hard drive\nspace for 1000 drives, and it is difficult to record an even longer video for the software while\nprecluding various drifts (even though the experiment is done on an optical table). Since this is not\nthe focus of the current study, additional dedicated experiments may be needed in future to test\ntheoretical expectations/models. In the Method section, we explicitly demonstrate how we track the\ncenter position for Fig. 1j,l,m and Fig. 8b using a freeware ImageJ (National Institutes of Health), and\nhow we calculate the correlation function directly using the video signal (without tracking) for Figs.\n9, 10e, and Supplementary Fig. S4, 5, 7,8. We thank the reviewer again for helping us improve the\nclarity of description of these methods.\nReviewer #4:\n2) Role of the \"Majorana-like quasiparticles\"\nOne of the main changes done to the manuscript is the inclusion of a discussion of the period\ndoubling behavior in terms of the dynamics of topological defects within the liquid crystal. These\ndefects, which take half-integer value assuming that the orientation of the liquid crystal is within the\n2d plane, are actually their own anti-particles, owing to the rotation freedom of the liquid crystal\nmolecules in 3d space. The authors can then describe the observation of the time crystalline behavior\nby the creation and annihilation of such pairs of defects along the vertical direction, shifted by L/2\nevery period of the drive.\n2.1) I am a bit confused about the meaning of Fig 4 and the dynamics being depicted. In figs g-i, it\nseems like the dynamics being depicted is the transmutation of the +1/2 into -1/2 defects and vice\nversa, rather than their annihilation and creation. Indeed, from the discussion it appears that the\nnotion of +-1/2 defect is only valid in the large |U| limit where the liquid crystal molecules'\norientation is restricted to the XZ plane. In what way is that language (of +-1/2 defects) meaningful\nin this context?\nAuthors:\nWe thank the reviewer for these remarks. In the 2D cross-sectional plane the defect's (disclination’s)\nwinding number can be related to the so-called twist angle β: β = 0 corresponds to a +1/2 defect and β\n= π corresponds to a -1/2 defect; the definition of these winding numbers in 2D is provided in text. In\n3D samples, these winding numbers are not invariants characterizing distinct topology but just\ngeometric quantifiers of structures. A general defect can transform between the +1/2 and a -1/2 states\nvia tuning β (Ref. [38]). In the revised manuscript, we add the results of β in Fig. 4j: most of the time\nthey are +1/2 and -1/2 geometric embodiments of topological defects since the voltage is high\nenough, and the inter-transformation of the half integer defects exhibit analogues to the inter-\ntransformation of Majorana quasiparticles, which are described in detail in the following comments.\nReviewer #4:\n2.2) It appears that the dynamics are divided into two moments; a very fast one 0 < t < 0.02Te,\naround which the topological defects bounce a lot between the top and the bottom of the system after\nthe quench, and then an entirely different setting where their motion follows smoothly with the\nelectric field intensity, albeit shift by L/2. What guarantees that the oscillations observed in the first\nfew moments die down, and that the defects merge? What determines the position of the formation of\nthe defects as the electric field is ramped up?\nAuthors: We thank the reviewers for these questions and comments. Generally speaking, the\ndifferent dynamics of the defects are related to the temporal gradient (slope in the U vs. time plot)\nof the external voltage. During the dynamic process, except when the disclinations (defect)\nannihilate and re-generate, the +1/2 disclinations are close to the bottom surfaces when U < 0. If the\nvoltage is smoothly changing across U = 0, the disclinations transformation follows the scenario from\nFig. 4g to 4i. However, as the voltage suddenly changes from U to -U at t = T , 2T , the director\nmax max E E…\nfield cannot suddenly deform correspondingly, inducing disclinations annihilation rather than smoothly\ntransformation at the same x positions. During the annihilation process, the director field keeps the\n“memory” of the periodic deformations, the neighboring L/2 positions are more favorable (strong\nlocalized deformations and low scalar order parameter S region) for the re-generation of new pairs of\ndisclinations compared to other positions (spontaneously in simulation and experiments).\nReviewer #4:\nOne reading of the observed dynamics is that the system has two stable configurations,\ncorresponding to two different lattices of domain walls, and the drive is such that the system is moved\nbetween the two and the stability is guaranteed by the stability of the lattice of defects. Is this picture\naccurate? If so what is added by the discussion of the microscopic dynamics of the \"Majorana-like\"\ndefects?\nAuthors: We thank the reviewer for providing a new perspective on the mechanism. The interesting\nfeature of our system is that the building blocks are topological quasiparticles that exhibit intriguing\ninter-transformations within each time period. There may be more than two dynamically stable\nconfigurations in our system. For example, the free energies of the soliton array are the same for a 1D\nlattice as its orientation rotates by any angle along z (energy degeneracy of the lattice orientation due\nto the homeotropic boundary condition). Nevertheless, we did not observe a transformation from the\noriginal orientation to other angles. The local microscopic transformation is very important for clearly\ndemonstrating the dynamics of our systems: not only because the half-integer disclinations are\nanalogous to the Majorana-like quasiparticles and its antiparticles, but also because the smooth\ntransformations as spinors from +1/2 disclinations to -1/2 disclinations follows the Majorana equation\n(Ref. [27]). This concept suggests that the time crystalline order observed in our liquid crystal\nsystems is topologically interesting and, we believe, distinct from other literature.\nReviewer #4:\n2.3) Given these questions (and the previous point about the characterization of the time crystalline\nbehavior), I find that the revised title of the manuscript is not entirely clear. Can the authors clarify\nwhat they mean by \"discrete space-time crystal of Majorana-like quasiparticles\"?\nAuthors:\nWe thank the reviewers for this question. In our systems, the Majorana-like quasiparticles play\npivotal roles: in the spatial coordinate, these Majorana-like quasiparticles and their antiparticles self-\nassemble to the lattice; in the temporal coordinate, the smooth transformation of the Majorana-like\nquasiparticles follow the Majorana equation; these spatial-temporal elements form the space-time\ncrystal.\nReviewer #4:\n2.4) Finally, I am a bit confused about the (apparent) inconsistency between Fig 3 and Fig 4. In Fig\n3, the steady state obtained by the mean field method is such that the defect structure is alternating\n(+1/2, -1/2, +1/2, ...), whereas in Fig 4, the system now exhibits a fixed pattern of topological defects\n6\n(+1/2, +1/2, +1/2). What is the difference between the two settings? Given that the authors claimed\nthat the mean field accurately captures the dynamics, I am confused by the observations.\nAuthors:\nThe structures in Fig.3 are the director field configurations without applying voltage, they are the\ninitial configurations that are used to obtain the dynamic process shown in Fig. 4. In addition, these\nalternating +π- and –π-rotation Néel domain walls (topological solitons) in Fig. 3 locate at the x-y\nplane, which are not the +1/2 and -1/2 disclinations (singular defects) and domain walls (Néel or\nBloch domain wall depended on the external voltage) embedded within the x-z plane. We have\nslightly revised the text description and assured that this is clear to readers and thank the reviewer for\nthis question, we revised corresponding parts to clearly show their differences and topological nature.\nReviewer #4:\n3) Observation of the Fractional DTC physics\nThe other novel result is the observation of the fractional discrete time crystal. Looking through Fig\n10, it seems like the system returns to its initial configuration after 3 periods, and thus should exhibit\na 1/3 sub-harmonic response. The shift in the correlation function appears to occur due to the\npresence of a non-negligeable number of defects in the hexagonal lattice that is being prepared.\nThe assertion that the system is at a fraction p/q suggests that the system performs p orbits over q\nperiods -- this is not supported by the observations, since the response is clearly affected by the\npresence of defects.\nUnless the authors have additional evidence that they have not included in the current version of the\nmanuscript, I am unconvinced of the claim that they have a \"fractional discrete time crystal\".\nIndeed, returning to the point of the characterization of the behavior using the center of the blue\nregion, I find that the authors' analysis further complicates one's ability to parse the significant of\nthis response.\nAuthors:\nWe thank the Reviewer #4 for these comments. Although the POM images with Δt=3T look similar,\nE\nthey are not identical. In the revised manuscript, we added a new supplementary figure\n(Supplementary Fig. S10) with Δt=9T : if the system has a rigorous 1/3 sub-harmonic response, they\nE\nshould not show such a difference after 9 periods (no defect emerging during Δt), which indicates\nthat the system may not have a period-tripling phenomenon. In addition, as mentioned in previous\nquestions, the Fourier transform spectrum and time correlation function are obtained directly from the\nsignals of video pixels, both of them do not exhibit a clear integer response.\nWe agree that due to the presence of a non-negligeable number of defects, it is difficult to\nunambiguously conclude whether they are a fractional discrete time crystal. In the revised\nmanuscript, we tune down the statements and emphasize that the quasi-hexagonal case potentially\ncould be a candidate of fractional discrete time crystals, which awaits further studies.\nReviewer #4:\nMinor Comments:\n1) In the rebuttal, the authors claim. \"After several hours, we still observe the time crystal behaviours\nin the same camera-captured region.\" Can they make precise what they mean? Locally (in time) the\n7\nsystem is still toggling between two configurations? Or there is some form of locking with the initial\noscillation in the system? What does this mean about the stability of the time crystalline phenomena?\nAuthors:\nWe thank the Reviewer #4 for these questions. In the experiments, we apply the external drive for\nhours to one day (continuously switching on), and monitor the camera-captured region, the system\ncan exhibit the period-doubling phenomenon locally in time as shown in Fig. 8a and Supplementary\nVideo S9. However, it is difficult to claim that these period-doubling observed after several hours is\nlocked to the very first period-doubling, since sometimes small impurity/defect regions emerge (Fig.\n7a). Nevertheless, the impurity/defect regions disappear after tens of drives, and the system recurs to\nperiod-doubling behaviors. This indicates that our time crystal is very stable against small spatial-\ntemporal\nfluctuations.\nReviewer #4:\n2) In the rebuttal, the only time the authors engage with the concern about the analysis of the data is\nin the addition of the oscillation of the amount of blue within an arbitrarily chosen square of the\nimage. It is not explained why this is the correct metric (the blue channel or the chosen region), nor\nhow it relates to the metric being analyzed. I find that the authors engaged with the letter of my\nprevious comment but didn't engage with the point that I was raising, namely that the data analysis\npipeline is complex so it is hard to parse the importance of the correlation function\nAuthors: We thank Reviewer #4 for these questions and previous suggestions. In Fig. 1k, as\nsuggested by reviewer #4 in the previous report, the blue signal shows a clear period-doubling\nphenomenon, where the vertical axis represents the average blue signal from the region marked in\nFig. 1j. This selected region cannot be too large or too small, if the size is too large (for example, the\nfull image of Fig. 1j), it’s hard to verify the period doubling only from the blue signals; if the marked\nregion size is too small, the signal fluctuation will be larger. Therefore, to clearly show the period-\ndoubling behavior and how the signals auto-correlates in time, we calculate the time correlation\nfunction of each pixel from the full images over time (Fig. 9), and then sum the results from all\npixels, as explicitly described in the Method section and in the previous comments. We thank the\nreviewer again for helping us improve the clarity of the analysis, and inspiring us to exhibit the\nperiod-doubling phenomenon in different ways.\nReviewer #4:\n3) In the rebuttal, the authors note the experimental settings when asked about the \"types\" of DSTCs.\nI am confused about this response since the authors continue to emphasize that they observe different\n\"types\" of DSTCs, but cannot highlight what is the sharp feature that distinguishes them. It seems like\nall exhibit some square pattern of correlations that shifts between different periods of the drive.\nWhile the shape of the pattern is different, the symmetry properties seem to be the same, they all live\nin a square lattice. Is there something that I am missing? How do the authors define the different\ntypes?\nAre these patterns captured by the mean-field numerics?\nAuthors: We thank Reviewer #4 for these questions and comments, we agree that all of these\npatterns are square lattice and have same symmetry properties, the differences of patterns indicate\nthat they have different director field configurations. To clearly indicate the difference and avoid the\nmisunderstanding, in the revised manuscript we labelled them as configurations 1-3.\nRegarding for the numerical simulations, to reconstruct the space-time configuration of 1+1D DSTC,\nthe simulation box size of the director field is 300×150, and the time step is 5000 within one temporal\n8\nperiod (the simulations contain over 100 temporal periods). To capture the 2+1D DSTC dynamics\nusing the tensor-based simulation, the box size should be larger than 300×300×150, which is beyond\nour simulation ability. Although the quasi-static structures (Fig. 3k-o) from vector-based simulation\nshows similarity of 2+1D DSTC patterns, we cannot reconstruct a whole spatial-temporal dynamic\nfrom it.\nReviewer #4:\n4) The authors now emphasize the defect perspective on the dynamics of the 1+1D DSTC. Does\nsomething apply to the 2+1D case? If not why? What about the hexagonal case?\nAuthors:\nWe thank the reviewers for these insightful questions. For the 1+1D DSTC, we find that the dynamic\nis related to the Majorana-like quasiparticles and domain wall solitons (1D topological solitons),\nwhich are labelled as first homotopy group 𝜋 (𝕊2/ℤ )= ℤ (the transformation follows the 1+1D Majorana\n1 2 2\nequation). For 2+1D, the basic units also consist of pairs of 𝜋 (𝕊2/ℤ )= ℤ topological defects and skyrmion\n2 2\ntubes (2D topological solitons), which are labelled as second homotopy group. In liquid crystal systems,\nsuch an assembly is called toron [31,57], which exhibits similar optical patterns compared to Fig. 1.\nHowever, these phenomena are rather complex and more disciplinary, with the power of numerical\nmodelling still insufficient to capture details that could be presented in an accessible form. We therefore\nleave this aspect to future more specialized articles.\n-----------------------------------------------------------------------------------\nWe believe that the above-described revisions address all remarks and account for all suggestions that\nwere raised by the original manuscript. We also feel that they improve the paper significantly. We\nthank the Referees for the thoughtful valuable comments that directed these changes.\nSincerely,\nProf. Ivan I. Smalyukh, Department of Physics, 390 UCB\nUniversity of Colorado at Boulder\nPhone: 303-492-7277 (office); Email: Ivan.Smalyukh@colorado.EDU\nhttp://www.colorado.edu/physics/SmalyukhLab/index.html\n9"}, {"round": 2, "reviewer_comments": {"reviewer_2": "I reviewed the manuscript on “Emergent discrete space-time crystal of Majorana-like quasiparticles in chiral liquid crystals”\nin its entirety, in light of all the changes made due to all review feedback in the last round. The result is a well rounded\noverview of the period-doubling phenomena found in experiment. While I am not an expert on time crystals, and therefore\nhesitant to apply this terminology, the computational modeling based on Frank-Oseen and Landau-de Gennes theory\nappears to capture the overall dynamics well.\nIn particular, my request for a description suitable to a more general audience is addressed satisfactory. I have no further\nremarks that would improve the manuscript within a reasonable time frame.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature\nCommunications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career\nResearchers who co-review manuscripts.", "reviewer_4": "The author's have significantly improved their paper, adding important discussions and clarifying some important aspects of\ntheir work. At this point, I understand their experiment and observations and I find them novel and interesting to the broader\naudience. However, I still find some points confusing or ill-explained in the manuscript. Pending addressing these points, I\nsupport publication in Nature Communications.\nClarification of the definition of the Majorana-like quasiparticles\n1) I remain confused about the definition of the Majorana quasi-particle and its relationship to the the director orientation\nprofile. The authors emphasize that the homotopy class of the director in 3D is $\\pi^1(S_2/Z_2) = Z_2$, which leads either\ndefect or no-defect. In this language the origin of the Majorana is somewhat clear, since two defects annihilate one another --\nthis is also the discussion within the PNAS cited by the paper.\n2) Despite being equivalent, when the applied field is large, the director appears to be restricted to the zx plane and the\nwinding number becomes well-defined to the point that one can assign +-1/2 charge to each \"Majorana-like\" particle. The\nauthors emphasize how the DTC dynamics can be understood in terms of the annihilation and creation of pairs of such pairs of\nparticles (which is the more general language are the Majorana quasiparticles)\n3) This begs three questions:\n1) This discussion all occurs in the translationally invariant case, where these defects are not point-like but rather extended\nobjects along the y direction. In this case the director orientation is ill-defined, not on a single point, but along a line,\nchanging the dimensionality of the defect. In this case, why does the same characterization of the defect still persist?\n2) The authors also emphasize (in Page 6) how the nature of the domain wall is also captured by the same homotopy class.\nThese, however, are extended two dimensional objects. How does the homotopy class of the solitons affects our\nunderstanding of their dynamics? Does it simply state that any two solitons can annihilate one another? At the same time, is\nthere some important connection between the presence of the aforementioned defect lines and the solitons (more on this\nbelow)? And what is the relationship between the solitons and the Neel/Bloch domains walls emphasized in Fig4?\n3) The numerical evidence (and ability to pear into the dynamics of the director) seems to be limited to the 1+1D case. In this\ncase, I understand how the line defects (labeled by +-1/2) paint a simple description of the dynamics. However, the authors\nextend their claims of the Majorana description to the 2+1D DTC (see abstract quote below), where the authors have no\npicture or description in terms of the director topological defect.\nIn the report, they have also stated that \"these phenomena are rather complex and more disciplinary, with the power of\nnumerical modelling still insufficient to capture details that could be presented in an accessible form\".\nTo this end, I think that either additional evidence / descriptions are added to the paper, or the claim of a time crystal driven\nby the dynamics of Majorana-like particles should be restricted to the analysis of the 1+1D case (upon a clarification of the\nabove point)\n> \"Here we describe the observation of classical analogues of both 1+1-dimensional and 2+1-dimensional discrete space-\ntime crystals in a liquid crystal system driven by a Floquet electrical signal. These classical time crystals comprise\nparticle-like18 structural features and exist over a wide range of temperatures and electrical driving conditions. The\nphenomenon-enabling period-doubling effect comes from their topological Majorana-like quasiparticle features\"\n---------\nConfusing point about the initial state:\nIn my previous reply I alluded to some confusion with regards to the relationship between the profile in Fig 3 and 4 which I\nnow better understand how the two correspond to the initial state vs a description of the dynamical process, respectively.\nFrom reading the text, I still find these two plots and their relationship hard to parse (magnified by the lack of description of\nFig3 in the main text, something that is more extreme for Fig 2 which I would recommend moving to the Methods or SM).\nHowever, even with this clarification, the role of the initial state remains confusing. Let me first start by laying out my\nunderstanding/confusion.\n1) In Fig 3 the authors emphasize the presence of soliton domain whose winding along the XY plane is alternative between\n+ and - \\pi and there are no defect lines of the form presented in Fig 4 (the \\beta = +-1/2 defects). Indeed, in Fig.4 it seems\nlike the directors predominantly point along the z axis even when the applied field is small (around t = 0.5TE Fig 4h).\n2) This suggests that the first period is very important because it takes a symmetry breaking state (lattice of solitons) and\nmaps it into a completely different symmetry braking state (a lattice of Néel domain walls) which are embedded in a different\nplane of the liquid crystal (note that in the response, the authors still called both domain walls, Néel domains, although they\nemphasize they are distinct because of the plane of the orientation).\n3) The different defects and their transformations are made extra confusing when looking at Fig1d where the 1+1D response\nlooks the same as the initial state presented in Fig3a, however, the authors are emphasizing that the director ordering is\nactually very different (Fig3g vs Fig4h).\n4) Moreover, it seems to be the case that the same initial state (1D symmetry breaking) induces the dynamics of the 2+1D\nstate (\"When we set the 1+1D DSTC spatial structures as initial configurations and apply the time-dependent Floquet\nelectrical field8 (sawtooth wave, maximum amplitude Umax = 0.2V and temporal periodicity TE = 0.5s), the 2+1D DSTC\n(configuration 2) spatial configurations spontaneously emerge (Fig. 3m-o) in each drive\"). This point is stated in the methods,\nbut further emphasizes that there is a lot of complexity in the way that the initial state induces the time crystalline behavior.\nThis distinction between the initial state and the dynamical state is very important, but it is not\naddressed. Much like a careful discussion of the nature of the defects (and a clearer justification of their Majorana nature), it\nwarrants more space in the text to guide the reader through the differences between the different defects considered (the two\ndomain walls and the Majorana defects), what their relationship is and how to understand the transformation between the\ninitial state and the dynamical steady state.\n---------\nMinor points:\n- In S7, the authors emphasize having data averaged over 1500 drives (I assume this means 1500 periods since the caption\nemphasized different driving durations), yet G(t) only goes to 100. This axis should be extended to help emphasize the\nnature of the decay.\n- Ln 9. Page 1: \"period of the driven system\" should read \"period of the system's dynamics\""}, "author_response": "Nature Communications manuscript NCOMMS-24-59238C\nEmergent discrete space-time crystal of Majorana-like quasiparticles in chiral\nliquid crystals\nH. Zhao, R. Zhang and I. I. Smalyukh\nSummary of Changes and Responses to Requests/Remarks\nReport of Referee 2:\nI reviewed the manuscript on “Emergent discrete space-time crystal of Majorana-like quasiparticles\nin chiral liquid crystals” in its entirety, in light of all the changes made due to all review feedback in\nthe last round. The result is a well rounded overview of the period-doubling phenomena found in\nexperiment. While I am not an expert on time crystals, and therefore hesitant to apply this\nterminology, the computational modeling based on Frank-Oseen and Landau-de Gennes theory\nappears to capture the overall dynamics well.\nIn particular, my request for a description suitable to a more general audience is addressed\nsatisfactory. I have no further remarks that would improve the manuscript within a reasonable time\nframe.\nAuthors:\nWe thank the Reviewer #2 for the recommendation and the positive reports.\nReport of Referee 3:\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part\nof the Nature Communications initiative to facilitate training in peer review and to provide\nappropriate recognition for Early Career Researchers who co-review manuscripts.\nReport of Referee 4:\nThe author's have significantly improved their paper, adding important discussions and clarifying\nsome important aspects of their work. At this point, I understand their experiment and observations\nand I find them novel and interesting to the broader audience. However, I still find some points\nconfusing or ill-explained in the manuscript. Pending addressing these points, I support publication\nin Nature Communications.\nAuthors: We appreciate the reviewer for this positive report saying that “The author's have\nsignificantly improved their paper, adding important discussions and clarifying some important\naspects of their work” and our experiment and observations are “novel and interesting to the broader\naudience”. We also thank the reviewer for the helpful suggestions, which we have carefully\naddressed in the revised manuscript.\nReviewer #4:\nClarification of the definition of the Majorana-like quasiparticles\n1\n1) I remain confused about the definition of the Majorana quasi-particle and its relationship to the\nthe director orientation profile. The authors emphasize that the homotopy class of the director in 3D\nis $\\pi^1(S_2/Z_2) = Z_2$, which leads either defect or no-defect. In this language the origin of the\nMajorana is somewhat clear, since two defects annihilate one another -- this is also the discussion\nwithin the PNAS cited by the paper.\n2) Despite being equivalent, when the applied field is large, the director appears to be restricted to\nthe zx plane and the winding number becomes well-defined to the point that one can assign +-1/2\ncharge to each \"Majorana-like\" particle. The authors emphasize how the DTC dynamics can be\nunderstood in terms of the annihilation and creation of pairs of such pairs of particles (which is the\nmore general language are the Majorana quasiparticles)\n3) This begs three questions:\n1) This discussion all occurs in the translationally invariant case, where these defects are not point-\nlike but rather extended objects along the y direction. In this case the director orientation is ill-\ndefined, not on a single point, but along a line, changing the dimensionality of the defect. In this case,\nwhy does the same characterization of the defect still persist?\nAuthors: We thank the reviewer for these questions and helpful remarks. The homotopy group\n𝜋 (𝕊2/ℤ )= ℤ (Ref. [44]) characterizes both the singular point defects in 2D, where director is\n1 2 2\nallowed to adopt 3D orientations according to the nonpolar director's order parameter space 𝕊2/ℤ , as\n2\nwell as the translationally invariant line defects in 3D samples, in which case the cross-section has\nthis configuration. The latter are found experimentally in our system, as well as are modelled\nnumerically. Here 𝜋 refers to the measuring circle (one-sphere), which is the same for\n1\ntranslationally invariant line defects in 3D and point defects in 2D. Our system is slightly different\nfrom that in the PNAS article but the same principles apply and we vividly see the dynamics of\ntransformations. In the revised manuscript, we slightly expand the discussion and further clarify this\nmatter.\nReviewer #4:\n2) The authors also emphasize (in Page 6) how the nature of the domain wall is also captured by the\nsame homotopy class. These, however, are extended two dimensional objects. How does the\nhomotopy class of the solitons affects our understanding of their dynamics? Does it simply state that\nany two solitons can annihilate one another? At the same time, is there some important connection\nbetween the presence of the aforementioned defect lines and the solitons (more on this below)? And\nwhat is the relationship between the solitons and the Neel/Bloch domains walls emphasized in Fig4?\nAuthors: We thank the reviewer for these questions, which help us in making our writing more\naccessible to the readership. Because the one-dimensional space ℝ1 can be compactified on 𝕊1 (one-\nsphere mathematical notation, which is a circle, corresponding to 𝜋 ) for the case of uniform\n1\nembedding background (one-point compactification), the 1D solitons can be classified by the same\nfirst homotopy group 𝜋 (𝕊2/ℤ )= ℤ . It is natural that the 𝜋 (𝕊2/ℤ )= ℤ solitonic walls are\n1 2 2 1 2 2\nterminated by the topologically analogous line defects 𝜋 (𝕊2/ℤ )= ℤ described within the same\n1 2 2\nhomotopy theory classification. In our observations, the topological solitons and singular lines inter-\ntransform between geometrically different but topologically equivalent states synchronously. This is\nbecause different geometric states of the soliton (Néel or Bloch type) and the states of singular defect\nlines with different cross-sectional structures (which, in a purely 2D system that allow for only 2D\norientations of director as at some values of applied field, would correspond to defects of opposite-\nsign winding number, but are topologically analogous and deformable to one another when 3D\nrotations of director are allowed, e.g. at other values of instantaneous voltage of our driving) are all\n2\ndescribed by 𝜋 (𝕊2/ℤ )= ℤ . The solitons have geometric embodiments of Néel and Bloch types and\n1 2 2\nalso the Majorana quasiparticle nature, similar to the singular defect lines. In the revised manuscript,\nwe slightly expand the discussion and further clarify this matter.\nReviewer #4:\n3) The numerical evidence (and ability to pear into the dynamics of the director) seems to be limited\nto the 1+1D case. In this case, I understand how the line defects (labeled by +-1/2) paint a simple\ndescription of the dynamics. However, the authors extend their claims of the Majorana description to\nthe 2+1D DTC (see abstract quote below), where the authors have no picture or description in terms\nof the director topological defect.\nIn the report, they have also stated that \"these phenomena are rather complex and more disciplinary,\nwith the power of numerical modelling still insufficient to capture details that could be presented in\nan accessible form\".\nTo this end, I think that either additional evidence / descriptions are added to the paper, or the claim\nof a time crystal driven by the dynamics of Majorana-like particles should be restricted to the\nanalysis of the 1+1D case (upon a clarification of the above point)\n> \"Here we describe the observation of classical analogues of both 1+1-dimensional and 2+1-\ndimensional discrete space-time crystals in a liquid crystal system driven by a Floquet electrical\nsignal. These classical time crystals comprise particle-like18 structural features and exist over a\nwide range of temperatures and electrical driving conditions. The phenomenon-enabling period-\ndoubling effect comes from their topological Majorana-like quasiparticle features\"\nAuthors: We thank the reviewer for these helpful questions and remarks. In the revised abstract and\nmain text, we further clarified that the 1+1D space-time crystals are related to the reported dynamic\ntransformations of the Majorana-like quasiparticle features. In the revised abstract we updated the\nstatement to read: \"The phenomenon-enabling period-doubling effect in 1+1-dimensional discrete\nspace-time crystals comes from their topological Majorana-like quasiparticle features.\"\nIn the manuscript's conclusions and future perspectives part we now also mention that: \"The future\nstudies should extend the detailed analyses of the reported period-doubling effect in 1+1D space-\ntime crystals to high-dimensional cases, including 2+1D that we already observed experimentally.\"\nWe appreciate referee’s remarks that guided these improvements of presentation of our findings.\nReviewer #4:\nConfusing point about the initial state:\nIn my previous reply I alluded to some confusion with regards to the relationship between the profile\nin Fig 3 and 4 which I now better understand how the two correspond to the initial state vs a\ndescription of the dynamical process, respectively. From reading the text, I still find these two plots\nand their relationship hard to parse (magnified by the lack of description of Fig3 in the main text,\nsomething that is more extreme for Fig 2 which I would recommend moving to the Methods or SM).\nAuthors: We thank the reviewer for these helpful remarks. We have added additional details in the\ndescription of figures, including a more detailed discussion of Fig. 2 while fully accounting for this\nfeedback.\nReviewer #4:\nby laying out my understanding/confusion.\n1) In Fig 3 the authors emphasize the presence of soliton domain whose winding along the XY plane\nis alternative between + and - \\pi and there are no defect lines of the form presented in Fig 4 (the\n\\beta = +-1/2 defects). Indeed, in Fig.4 it seems like the directors predominantly point along the z\naxis even when the applied field is small (around t = 0.5TE Fig 4h).\n2) This suggests that the first period is very important because it takes a symmetry breaking state\n(lattice of solitons) and maps it into a completely different symmetry braking state (a lattice of\nNéel domain walls) which are embedded in a different plane of the liquid crystal (note that in the\nresponse, the authors still called both domain walls, Néel domains, although they emphasize they are\ndistinct because of the plane of the orientation).\nAuthors: We thank the reviewer for these helpful remarks. We provide additional details in the\ndescription of the transformation between the initial state and the time crystalline structures that are\nthe focus of our study. The sample has perpendicular boundary conditions on confining substrates,\ndefining a topologically neutral background. The director structures are topologically neutral of the\ninitial state as it consists of alternating + π and –π domain-wall solitons in the x-y plane. In the 1+1D\nDSTC phase, pairs of disclination lines can be annihilated (Fig. 4l), which also indicates a\ntopologically neutral state, dictating that the topological objects' invariants self-compensate. Both in\nsimulations and experiments, we observe that the initial state transforms into the 1+1D DSTC state\nwithin tens of driving periods. Indeed, the symmetry breaking becomes different during the time\ncrystal formation. We have now mentioned this detail and appreciate this referee's remark.\nReviewer #4:\n3) The different defects and their transformations are made extra confusing when looking at Fig1d\nwhere the 1+1D response looks the same as the initial state presented in Fig3a, however, the authors\nare emphasizing that the director ordering is actually very different (Fig3g vs Fig4h).\n4) Moreover, it seems to be the case that the same initial state (1D symmetry breaking) induces the\ndynamics of the 2+1D state (\"When we set the 1+1D DSTC spatial structures as initial\nconfigurations and apply the time-dependent Floquet electrical field8 (sawtooth wave, maximum\namplitude Umax = 0.2V and temporal periodicity TE = 0.5s), the 2+1D DSTC (configuration 2)\nspatial configurations spontaneously emerge (Fig. 3m-o) in each drive\"). This point is stated in the\nmethods, but further emphasizes that there is a lot of complexity in the way that the initial state\ninduces the time crystalline behavior.\nThis distinction between the initial state and the dynamical state is very important, but it is not\naddressed. Much like a careful discussion of the nature of the defects (and a clearer justification of\ntheir Majorana nature), it warrants more space in the text to guide the reader through the differences\nbetween the different defects considered (the two domain walls and the Majorana defects), what their\nrelationship is and how to understand the transformation between the\ninitial state and the dynamical steady state.\nAuthors: We thank the reviewer for these helpful remarks. Taking into consideration these questions\nand comments, we have explicitly clarified the details of initial states and dynamic steady states,\ntopological defects and topological solitons (together with the Majorana nature) in the revised\nmanuscript. We have also moved Fig. 3a–d to the Supplementary Materials following the reviewer’s\nsuggestion, which we appreciate.\nReviewer #4:\nMinor points:\n- In S7, the authors emphasize having data averaged over 1500 drives (I assume this means 1500\nperiods since the caption emphasized different driving durations), yet G(t) only goes to 100. This axis\nshould be extended to help emphasize the\nnature of the decay.\nAuthors: We thank the reviewer for this helpful remark. We have extended the time axis in the\nrevised Supplementary Fig. S8 to 150 drives - the results are consistent with the previous ones. We\nalso mention that extending to larger number of drives will require enhancing system controls to\npreclude issues like thermally or vibrations-induced drift, etc.\nReviewer #4:\n- Ln 9. Page 1: \"period of the driven system\" should read \"period of the system's dynamics\"\nAuthors: We thank the reviewer for this helpful remark, which we accounted for.\n-----------------------------------------------------------------------------------\nWe believe that the above-described revisions address all remarks and account for all suggestions that\nwere raised by the original manuscript. We also feel that they improve the paper significantly. We\nthank the Referees for the thoughtful valuable comments that directed these changes.\nSincerely,\nProf. Ivan I. Smalyukh, Department of Physics, 390 UCB\nUniversity of Colorado at Boulder\nPhone: 303-492-7277 (office); Email: Ivan.Smalyukh@colorado.EDU\nhttp://www.colorado.edu/physics/SmalyukhLab/index.html\n5"}, {"round": 3, "reviewer_comments": {"reviewer_4": "The authors have carefully engaged with the comments I have made in previous reports and have expanded and clarified their observations and conclusions.\nAt present, I do not have more comments that I think are crucial for the understanding of the paper, and, as such, I am happy supporting the publication of the manuscript in Nature Communications."}, "author_response": "Nature Communications manuscript NCOMMS-24-59238C\nEmergent discrete space-time crystal of Majorana-like quasiparticles in chiral\nliquid crystals\nH. Zhao, R. Zhang and I. I. Smalyukh\nResponses to Referee Remarks\nReport of Referee 4:\nThe authors have carefully engaged with the comments I have made in previous reports and have\nexpanded and clarified their observations and conclusions.\nAt present, I do not have more comments that I think are crucial for the understanding of the paper,\nand, as such, I am happy supporting the publication of the manuscript in Nature Communications.\nAuthors:\nWe thank the Reviewer #4 for the recommendation and the positive reports.\nSincerely,\nProf. Ivan I. Smalyukh, Department of Physics, 390 UCB\nUniversity of Colorado at Boulder\nPhone: 303-492-7277 (office); Email: Ivan.Smalyukh@colorado.EDU\nhttp://www.colorado.edu/physics/SmalyukhLab/index.html"}]

Decision letter