Where does crystal collide?

The classic crystal: a solid whose atoms, ions, or molecules sit in a regularly repeating three-dimensional pattern, a lattice with long-range order, so the whole structure is built by copying one small unit cell over and over through space. Bragg and Bragg's 1913 The reflection of X-rays by crystals showed that the sharp, ordered way a crystal reflects X-rays reveals the exact spacing of its atomic planes, which turned this internal periodic arrangement into something that could be measured directly. This referent orders atoms fixed in space, and it is not a pattern that repeats in time, an abstract graph in algebra, or a flowing material with only partial order.[1]

Unit

One crystal, described by a unit cell copied across a lattice

Recovery means

Not applicable

Measured by

Determined by X-ray, neutron, or electron diffraction, which reveal the spacing and symmetry of the atomic planes

A phase of matter whose pattern repeats in time rather than in space: just as an ordinary crystal breaks the smoothness of space by settling into a repeating spatial arrangement, a time crystal breaks the smoothness of time by settling into motion that repeats on its own, without being driven into it. Wilczek proposed this idea in his 2012 Quantum Time Crystals, arguing that a quantum system could spontaneously break time-translation symmetry, and later work realised related phases in periodically driven systems whose response repeats at a multiple of the driving period. Here the order lives in time, not in the fixed atomic positions of a solid crystal, the combinatorial graph of an algebra, or the orientational order of a fluid.[2]

Unit

One phase whose observable motion repeats at a fixed period on its own

Recovery means

Not applicable

Measured by

Detected as a persistent, spontaneous oscillation at a period the driving does not set, observed in trapped ions and interacting spin systems

In the algebra of quantum groups a crystal, or crystal base, is a discrete combinatorial skeleton that captures a representation once a certain parameter is pushed to a limit: it is a set of vertices joined by coloured arrows, a crystal graph, that records how the representation is built while stripping away its continuous, tangled detail. Kashiwara introduced it in his 1991 On crystal bases of the Q-analogue of universal enveloping algebras, taking the limit as the quantum parameter goes to zero to freeze the structure into this clean graph. The name borrows the picture of a clean, ordered lattice, but the referent is a purely abstract object in pure mathematics, with no atoms, no motion in time, and no physical matter of any kind.[3]

Unit

One crystal graph: vertices joined by coloured raising and lowering arrows

Recovery means

Not applicable

Measured by

Built and studied combinatorially, for instance by tableaux and paths, as the zero limit of the quantum parameter acting on a representation

A state of matter that sits between an ordinary liquid and a solid crystal: its molecules flow past one another like a liquid yet still line up along a common direction, so it has orientational order without the full, fixed positional order of a solid. Friedel's 1922 Les états mésomorphes de la matière named and classified these mesomorphic states, the nematic, smectic, and cholesteric phases, as genuine intermediate forms of matter rather than impure solids or liquids. This partial order in a flowing material differs from the rigid atomic lattice of a solid crystal, a pattern that repeats in time, or an abstract graph in algebra, and it is the order behind everyday liquid-crystal displays.[4]

Unit

One mesophase, classed by how far its molecules are ordered in direction and position

Recovery means

Not applicable

Measured by

Identified by optical texture under polarised light and by X-ray scattering that shows orientational but weak or absent positional order