Only last week I realised that an important and profound property of emergent quantum matter is the emergence of new length scales. These can be mesoscopic - intermediate between microscopic and macroscopic length scales. Say, very roughly between 100 nanometers and 100 microns.
Previously, I highlighted the emergence of new low energy scales in quantum many-body physics. The energy and length scales are often related.
In or near broken symmetry phases, some emergent length scales can be related to the rigidity of the order parameter such as the spin stiffness.
An example is the superconducting coherence length, xi which determines the minimum thickness of a thin film required to sustain superconductivity and the size of vortices in a type II superconductor. It is roughly given by xi ~ hbar v_F/Delta ~ a E_F/Delta
where v_F is the Fermi velocity, Delta is the energy gap, a is a lattice constant, and E_F is the Fermi energy. Since in a weak-coupling BCS superconductor Delta is much less than E_F, the length xi can be orders of magnitude larger than a lattice constant, and so is mesoscopic.
In BCS theory, the coherence length can be interpreted roughly as the "size" of a Cooper pair.
In neutral superfluids, such as liquid 4He or bosonic cold atomic gases, the corresponding length scale is sometimes known as the healing length, and defines the size of quantised vortices. In 4He this length scale is microscopic, being of the order of Angstroms, but in cold atoms it can be mesoscopic.
A second independent emergent length scale associated with superconductivity is the London penetration depth, that determines the scale on which magnetic fields penetrate the superconductor, or are expelled, i.e. the Meissner effect.
Just in case on thinks the emergence of new length scales is trivial in the sense that it is really the same as the emergence of new energy scales, consider the case of the Kondo effect. Clearly there are many experimental and theoretical signatures of the Kondo temperature, T_K.
One can easily construct a "Kondo length", L_K ~ hbar v_F / k_B T_K, and can identify this with the size of the "Kondo screening cloud."
Yet this has never been observed experimentally.
Ian Affleck has a nice review article discussing the relevant issues.
Partly as an aside I include below a nice graphic about length scales, taken from a UK Royal Society report on nanotechnology, from 2004.