Crystals |

**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]; > 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.

@ __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]; news ls(17)mar, Phys(17)dec [experiments]; Sacha & Zakrzewski RPP(17)-a1704 [rev]; > 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].

@ __Special models__: Li et al PRL(12)-a1206 + news pw(12)jul [4D spacetime crystals of trapped ions]; 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].

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