In General > s.a. condensed matter / statistical mechanical systems; X Rays [crystallography].
$ Def: Any material whose diffraction pattern is essentially discrete (International Union of Crystallography, 1992)!
* Crystallization process: Landau-Ginzburg model.
* Symmetries: An n-fold symmetry axis can be consistent with translation invariance only for n = 1, 2, 3, 4, 6 (in 4D).
@ General references: Glusker & Trueblood 10; Sidebottom 12 [r PT(13)may].
@ Dynamics: Born & Huang 54 [dynamics of lattices]; Gerstle 15 [peridynamics, computational]; > s.a. Defects; Elasticity.
@ And quantum theory: Grushevskaya & Gurskii qp/06 [many-particle effects on electron states]; Bottesi & Zemba JSM(08)-a0801 [effective theories for electrons]; Gajda et al EPL(16)-a1511 [Pauli crystals, from the exclusion principle]; > s.a. modified coherent states.
@ Symmetries, types: Baake JPA(97) [color symmetries]; Michel PRP(01); Nardone JMP(06); Bradley & Cracknell 10 [mathematical theory]; Allen a1006 [hexagonal, four-index notation]; Hitzer a1306-proc [history]; Honari & Mohades a1601 [symmetries and acoustic spectrum]; Alexandradinata et al PRX(16) [extended classification using generalized symmetries that include quasimomentum translations].
> Specific examples / types: see quasicrystals; water and ice; Wigner Crystal.

Related Topics > s.a. optical technology [photonic crystals]; matter [crystallization, mathematical models]; particles [propagation].
* Liquid crystals: They consist of rod-shaped molecules with the ability to polarize light; An applied voltage lines up the rods and produces a nematic order (directional rather than positional) that shuts off or turns on transmitted light; 1999, seen to be able to produce sound; > s.a. condensed matter [soft matter].
* Quantum crystals: Crystals are made up of atoms (like He) or molecules (like H) that are so light that nuclear quantum effects become important, and new, nonclassical theories are needed to describe them; The lightness of the atoms or molecules gives them, according to quantum mechanics, a large zero-point motion, which results in delocalization of bonds between them; > s.a. Plasticity.
@ In higher dimensions: Parisi JSP(08) [compact regular lattices].
@ Liquid crystals: in Landau & Lifshitz v7; de Gennes & Prost 95; Ondris-Crawford et al AJP(95)sep [RL]; Dunmur & Sluckin 11 [history]; Blinov 11 [structure and properties]; Ravnik Phy(13) [and Hopf fibrations]; news pw(14)jan [living liquid crystals]; Mila Phy(17) [magnetic analog for spins in a copper oxide]; > s.a. Defects; topological defects.
@ Other topics: Michel & Mozrzymas in(78), in Nash & Sen 83, ch8 [Morse theory and symmetry breaking]; Nussbaum AJP(00)oct [Bravais lattice]; Rabson et al FP(03) [and cohomology]; Libbrecht pw(08)jan [snowflakes]; news wisc(13)apr [vaterite is composed of two interspersed crystal structures]; Taioli et al a1511 [non-Euclidean crystallographic group]; news pt(19)jun [forms of a crystalline solid can melt into distinct liquids]; > s.a. Ewald Construction [reciprocal lattice]; Foam [polycrystals]; metamaterials [granular crystals].

Time Crystals
* Idea: A previously unknown state of matter proposed by Wilczek in 2012, whose structure would repeat periodically but in time rather than in space, as an example of spontaneous breaking of time-translation symmetry; The trick to finding an example is to find a system in its ground state which is nevertheless in motion, which means it would be an example of perpetual motion machine (without breaking fundamental laws because it would not be possible to extract energy from them); The closest that modern technology has come to a time crystal is a current-carrying superconductor, but the actual systems set up so far to not qualify as true time crystals.
@ Reviews: Sacha & Zakrzewski RPP(17)-a1704; Khemani et al a1910.
@ General references: Wilczek PRL(12)-a1202, comment Bruno PRL(13)-a1210 [quantum time crystals] + Zakrzewski Phy(12) + Coleman Nat(13)jan; news SA(13)feb, sf(13)apr; Nozières EPL(13)-a1306, Bruno PRL(13)-a1306 + news PhysOrg(13)aug [impossibility of spontaneous rotation]; Watanabe & Oshikawa PRL(15)-a1410 + news PhysOrg(15)jul [no-go result]; Castillo et al a1410 [quantum fluctuations]; Strocchi & Heissenberg a1605 [existence of quantum time crystals]; Yao et al PRL(17)-a1608; Khemani et al PRB(17)-a1612 [new definition and applications]; Richerme Phy(16) [on an approach to non-equilibrium/driven tieme crystals]; Yao & Nayak PT(18)aug [in periodically driven systems]; > s.a. classical systems [classical time crystals].
@ Floquet systems: Else et al PRL(16)-a1603 + news PhysOrg(16)sep [spontaneously broken time-translation symmetry]; Zhang et al Nat(17)mar-a1609 [observation].
@ Experiments, examples: Li et al PRL(12)-a1206 + news pw(12)jul [4D spacetime crystals of trapped ions]; news ls(17)mar, Phys(17)dec; news sn(18)may; Giergiel et al PRA(18)-a1805 [experimental conditions]; Smits et al PRL(18)-a1807 [spacetime crystal]; Kozin & Kyriienko PRL(19) + news sn(19)nov [self-sustaining time crystal, proposal]; Syrvid et al a2010 [not in a chiral soliton model].
@ Special models: news ns(12)jul, csm(12)sep [computer based on a time crystal could outlive the universe]; Borzdov a1410 [electromagnetic]; Smolyaninov EJTP-a1501 [metamaterial model]; Easson & Manton PRD(19)-a1802 [cosmological, stable]; news cosmos(18)jun [and quantum computing].

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