* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Orders / Phases of matter
Coherent states wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Path integral formulation wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Renormalization group wikipedia , lookup
Higgs mechanism wikipedia , lookup
Quantum machine learning wikipedia , lookup
Quantum key distribution wikipedia , lookup
Scalar field theory wikipedia , lookup
Quantum teleportation wikipedia , lookup
Quantum state wikipedia , lookup
History of quantum field theory wikipedia , lookup
Hidden variable theory wikipedia , lookup
Quantum chromodynamics wikipedia , lookup
Canonical quantization wikipedia , lookup
Quantum entanglement wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Quantum group wikipedia , lookup
Event symmetry wikipedia , lookup
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