Chaikin, P. M. & Lubensky, T. C. Principles of Condensed Matter Physics (Cambridge Univ. Press, 2000).
Yeh, P. & Gu, C. Optics of Liquid Crystal Displays (Wiley, 2010).
Wilczek, F. Quantum time crystals. Phys. Rev. Lett. 109, 160401 (2012).
Shapere, A. & Wilczek, F. Classical time crystals. Phys. Rev. Lett. 109, 160402 (2012).
Wilczek, F. Superfluidity and space-time translation symmetry breaking. Phys. Rev. Lett. 111, 250402 (2013).
Bruno, P. Impossibility of spontaneously rotating time crystals: a no-go theorem. Phys. Rev. Lett. 111, 070402 (2013).
Watanabe, H. & Oshikawa, M. Absence of quantum time crystals. Phys. Rev. Lett. 114, 251603 (2015).
Sacha, K. & Zakrzewski, J. Time crystals: a review. Rep. Prog. Phys. 81, 016401 (2017).
Yao, N. Y. & Nayak, C. Time crystals in periodically driven systems. Phys. Today 71, 40–47 (2018).
Khemani, V., Moessner, R. & Sondhi, S. L. A brief history of time crystals. Preprint at https://doi.org/10.48550/arXiv.1910.10745 (2019).
Sacha, K. Time Crystals (Springer International Publishing, 2020).
Guo, L. Phase Space Crystals: Condensed Matter in Dynamical Systems (IOP Publishing, 2021).
Zaletel, M. P. et al. Colloquium: quantum and classical discrete time crystals. Rev. Mod. Phys. 95, 031001 (2023).
Sacha, K. Modeling spontaneous breaking of time-translation symmetry. Phys. Rev. A 91, 033617 (2015).
Khemani, V., Lazarides, A., Moessner, R. & Sondhi, S. L. Phase structure of driven quantum systems. Phys. Rev. Lett. 116, 250401 (2016).
Else, D. V., Bauer, B. & Nayak, C. Floquet time crystals. Phys. Rev. Lett. 117, 090402 (2016).
Yao, N. Y., Potter, A. C., Potirniche, I.-D. & Vishwanath, A. Discrete time crystals: rigidity, criticality, and realizations. Phys. Rev. Lett. 118, 030401 (2017).
Zhang, J. et al. Observation of a discrete time crystal. Nature 543, 217–220 (2017).
Choi, S. et al. Observation of discrete time-crystalline order in a disordered dipolar many-body system. Nature 543, 221–225 (2017).
Smits, J., Liao, L., Stoof, H. T. C. & van der Straten, P. Observation of a space-time crystal in a superfluid quantum gas. Phys. Rev. Lett. 121, 185301 (2018).
Liao, L., Smits, J., van der Straten, P. & Stoof, H. T. C. Dynamics of a space-time crystal in an atomic Bose-Einstein condensate. Phys. Rev. A 99, 013625 (2019).
Smits, J., Stoof, H. T. C. & van der Straten, P. Spontaneous breaking of a discrete time-translation symmetry. Phys. Rev. A 104, 023318 (2021).
Randall, J. et al. Many-body–localized discrete time crystal with a programmable spin-based quantum simulator. Science 374, 1474–1478 (2021).
Keßler, H. et al. Observation of a dissipative time crystal. Phys. Rev. Lett. 127, 043602 (2021).
Taheri, H., Matsko, A. B., Maleki, L. & Sacha, K. All-optical dissipative discrete time crystals. Nat. Commun. 13, 848 (2022).
Taheri, H., Matsko, A. B., Herr, T. & Sacha, K. Dissipative discrete time crystals in a pump-modulated Kerr microcavity. Commun. Phys. 5, 159 (2022).
Mi, X. et al. Time-crystalline eigenstate order on a quantum processor. Nature 601, 531–536 (2022).
Frey, P. & Rachel, S. Realization of a discrete time crystal on 57 qubits of a quantum computer. Sci. Adv. 8, eabm7652 (2022).
Kongkhambut, P. et al. Observation of a continuous time crystal. Science 377, 670–673 (2022).
Liu, T., Ou, J.-Y., MacDonald, K. F. & Zheludev, N. I. Photonic metamaterial analogue of a continuous time crystal. Nat. Phys. 19, 986–991 (2023).
Chen, Y.-H. & Zhang, X. Realization of an inherent time crystal in a dissipative many-body system. Nat. Commun. 14, 6161 (2023).
Wu, X. et al. Dissipative time crystal in a strongly interacting Rydberg gas. Nat. Phys. 20, 1389–1394 (2024).
Greilich, A. et al. Robust continuous time crystal in an electron–nuclear spin system. Nat. Phys. 20, 631–636 (2024).
Carraro-Haddad, I. et al. Solid-state continuous time crystal in a polariton condensate with a built-in mechanical clock. Science 384, 995–1000 (2024).
Yi, Y., Farrow, M. J., Korblova, E., Walba, D. M. & Furtak, T. E. High-sensitivity aminoazobenzene chemisorbed monolayers for photoalignment of liquid crystals. Langmuir 25, 997–1003 (2009).
Landau, L. D. & Lifshitz, E. M. Statistical Physics (Elsevier, 2013).
de Gennes, P. G. & Prost, J. The Physics of Liquid Crystals (Clarendon Press, 1993).
Reichhardt, C., Reichhardt, C. J. O. & Milošević, M. V. Statics and dynamics of skyrmions interacting with disorder and nanostructures. Rev. Mod. Phys. 94, 035005 (2022).
Smalyukh, I. I. Review: knots and other new topological effects in liquid crystals and colloids. Rep. Prog. Phys. 83, 106601 (2020).
Zhao, H., Tai, J.-S. B., Wu, J.-S. & Smalyukh, I. I. Liquid crystal defect structures with Möbius strip topology. Nat. Phys. 19, 451–459 (2023).
Mundoor, H., Senyuk, B. & Smalyukh, I. I. Triclinic nematic colloidal crystals from competing elastic and electrostatic interactions. Science 352, 69–73 (2016).
Xu, S. & Wu, C. Space-time crystal and space-time group. Phys. Rev. Lett. 120, 096401 (2018).
Gallego-Gómez, F., del Monte, F. & Meerholz, K. Optical gain by a simple photoisomerization process. Nat. Mater. 7, 490–497 (2008).
Brener, E. A. & Marchenko, V. I. Nonlinear theory of dislocations in smectic crystals: an exact solution. Phys. Rev. E 59, R4752–R4753 (1999).
Sohn, H. R. O. & Smalyukh, I. I. Electrically powered motions of toron crystallites in chiral liquid crystals. Proc. Natl Acad. Sci. USA 117, 6437–6445 (2020).
Prigogine, I. Time, structure, and fluctuations. Science 201, 777–785 (1978).
Zhang, R. et al. Spatiotemporal control of liquid crystal structure and dynamics through activity patterning. Nat. Mater. 20, 875–882 (2021).
Zhao, H., Malomed, B. A. & Smalyukh, I. I. Topological solitonic macromolecules. Nat. Commun. 14, 4581 (2023).
Lee, Y.-H. et al. Recent progress in Pancharatnam–Berry phase optical elements and the applications for virtual/augmented realities. Opt. Data Process. Storage 3, 79–88 (2017).
Cohen, E. et al. Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond. Nat. Rev. Phys. 1, 437–449 (2019).
Lyubarov, M. et al. Amplified emission and lasing in photonic time crystals. Science 377, 425–428 (2022).
Chang, T. et al. Cellulose nanocrystal chiral photonic micro-flakes for multilevel anti-counterfeiting and identification. Chem. Eng. J. 446, 136630 (2022).
Huang, W. & Mow, W. H. PiCode: 2D barcode with embedded picture and ViCode: 3D barcode with embedded video. In Proc. 19th Annual International Conference on Mobile Computing & Networking 139–142 (ACM, 2013).
Chang, S. et al. Electrical tuning of branched flow of light. Nat. Commun. 15, 197 (2024).
Poy, G. et al. Interaction and co-assembly of optical and topological solitons. Nat. Photon. 16, 454–461 (2022).
Martinez, A., Mireles, H. C. & Smalyukh, I. I. Large-area optoelastic manipulation of colloidal particles in liquid crystals using photoresponsive molecular surface monolayers. Proc. Natl Acad. Sci. USA 108, 20891–20896 (2011).
Lee, T., Trivedi, R. P. & Smalyukh, I. I. Multimodal nonlinear optical polarizing microscopy of long-range molecular order in liquid crystals. Opt. Lett. 35, 3447–3449 (2010).