Download Orders / Phases of matter

Orders / Phases of matter
Topological Orders / Phases
Landau Symmetry Breaking
-non-symmetry breaking orders
-Non-local order parameter of macroscopic property – topological probes:
1.) Topological ground state degeneracy n
- only in the thermodynamic limit N→ ∞
- not related to symmetry
- stable against local perturbations
2.) non-Abelian geometric phases
- unitary nxn matrix (where n is the ground state degeneracy)
- can be calculated e.g. from geometric deformation loops
-phases not distinguished by symmetry
- concept introduced by Thouless et al., PRL 49, 405 1982 (is this the first paper on the concept of topological order?)
-Local order parameter
-phases distinguished by (GH,GΦ)
where GH=symmetry group of H
and GΦ=symmetry group of Φ
Quantum phases (only T=0?)
Topology of Fermi-surface (NO!)
-probably does not work
Gapless Systems
Gapped Systems
Short-range
Entanglement
-e.g. via string-net condensation
Long-range
Entanglement
-states with LRE may not
-local unitary transformations
be connected via local unitary transf.
lead to product states
(what kind of product states?)
-all states with short-range entanglement
belong to the same phase
-symmetry protected topological order / phases belong to this class
-examples are
1.) Haldane phase (spin-1-chain where edge states have spin-1/2=fractionalization),
2.) the AKLT phase?, and
3.) topological insulators (like integer quantum Hall states, Haldane's graphene model with magnetic field that has the same periodicity as the lattice,
also top. Kondo insulators?)
See Hasan, Kane, RMP 82, 3045 (2010)
→ bulk is insulating but surface is not
→ due to breaking of time-reversal symmetry (e.g. in integer quantum Hall state) or due to spin-orbit interaction (Kane-Mele model)
→ Topological probes:
A) Chern number = Berry flux through Brillouin zone
B) crossings of chiral edge states through Fermi-level
- even number of crossings: edge states can be removed from the band gap area
- odd number of crossings: edge states cannot be removed completely from band gap area
→ that's the topologically protected chiral edge states
→ bulk-boundary correspondence: number of edge states = #_left mover - #_right mover = difference in topological invariants of the bulk materials that are separated by the surface (like Chern number)
Topological invariants can only change via a quantum phase transition where the gap closes
→ gapless excitations at the boundary between two materials with different topological order
Classical systems
Berenzski-Kosterlitz-Thouless
phase transitions:
- XY model
- liquid crystals
(smectic, nematic, etc.)
- grain boundaries