Localization-protected quantum order
More recently, we have also made great strides in understanding topological phases of matter which lie outside Landau's framework: the order in topological phases cannot be characterized by local patterns of symmetry breaking, and is instead encoded in global patterns of quantum entanglement.
Traditionally, phase structure is studied by examining the behavior of ``order parameters" in equilibrium states.
At zero temperature, these are evaluated in the ground state of the system, and different phases correspond to different quantum orders (topological or otherwise).
As an example, the Peierls-Mermin-Wagner theorems prove that a one dimensional system cannot spontaneously break a continuous symmetry at any non-zero temperature.
Recent progress on the phenomenon of many-body localization has revealed classes of generic (typically disordered) many-body systems which never reach local thermal equilibrium, and thus lie outside the framework of equilibrium statistical mechanics.
Remarkably, the answer is affirmative, and out-of-equilibrium systems can also display a rich phase structure.
[1] The recent discovery of time-crystals in periodically driven MBL systems is a notable example of this phenomenon.
Indeed, a thermodynamic ensemble average isn't even appropriate in MBL systems since they never reach thermal equilibrium.
In MBL systems, the suppression of thermal fluctuations means that the properties of highly excited eigenstates are similar, in many respects, to those of ground states of gapped local Hamiltonians.
This enables various forms of ground state order to be promoted to finite energy densities.
We note that in thermalizing MB systems, the notion of eigenstate order is congruent with the usual definition of phases.
On the other hand, MBL systems do not obey the ETH and nearby many-body eigenstates have very different local properties.
This is what enables individual MBL eigenstates to display order even if thermodynamic averages are forbidden from doing so.
Localization enables symmetry breaking orders at finite energy densities, forbidden in equilibrium by the Peierls-Mermin-Wagner Theorems.
term introduces interactions, and the system is mappable to a free fermion model (the Kitaev chain) when
These display long-range order: At any finite temperature, thermal fluctuations lead to a finite density of delocalized domain walls since the entropic gain from creating these domain walls wins over the energy cost in one dimension.
In other words, the domain walls get pinned by the disorder, so that a generic highly excited eigenstate for
[1][2] Note that a spin-spin correlation function evaluated in this state is non-zero for arbitrarily distant spins, but has fluctuating sign depending on whether an even/odd number of domain walls are crossed between two sites.
, the Anderson insulator remains many-body localized and order persists deep in the PM/SG phases.
Strong enough interactions destroy MBL and the system transitions to a thermalizing phase.
While the discussion above pertains to sharp diagnostics of LPQO obtained by evaluating order parameters and correlation functions in individual highly excited many-body eigenstates, such quantities are nearly impossible to measure experimentally.
saturates to a non-zero value even for infinitely late times in the symmetry-broken spin-glass phase, while it decays to zero in the paramagnet.
Indeed, a nice example of this is furnished by recent experiments[15][16] detecting time-crystals in Floquet MBL systems, where the time crystal phase spontaneously breaks both time translation symmetry and spatial Ising symmetry, showing correlated spatiotemporal eigenstate order.
gauge theory in 2D is an example of the former, and the topological order in this phase can be diagnosed by Wilson loop operators.
The topological order is destroyed in equilibrium at any finite temperature due to fluctuating vortices--- however, these can get localized by disorder, enabling glassy localization-protected topological order at finite energy densities.
[12] On the other hand, symmetry protected topological (SPT) phases do have any bulk long-range order, and are distinguished from trivial paramagnets due to the presence of coherent gapless edge modes as long the protecting symmetry is present.
In equilibrium, these edge modes are typically destroyed at finite temperatures as they decohere due to interactions with delocalized bulk excitations.
It has been shown that periodically driven or Floquet systems can also be many-body localized under suitable drive conditions.
However, with MBL, this heating can be evaded and one can again get non-trivial quantum orders in the eigenstates of the Floquet unitary, which is the time-evolution operator for one period.
The most striking example of this is the time-crystal, a phase with long-range spatiotemporal order and spontaneous breaking of time translation symmetry.
