2 Landau's theory cannot describe all the continuous phase transition such as the transitions between quantum ordered phase Orders Symmetry brea king orders Nonsymmetry breaking orders ° Particle" condensation Symmetry goup Nambu-Gokstone mode Quantum system Classical system Quantum orders Gapped Topobgical orders Fermi liquids String-net condensation Topobgical field theory Fermi surface topology COnformal algebra, ??>) Projective sym metry group Gapless Gauge bosons/Fermions. Exotic superconductor, Lifshiz phase transition Spin liquid, FQHE Spin liquid
Landau's theory cannot describe all the continuous phase transition such as the transitions between quantum ordered phase Spin liquid Lifshiz phase transition Exotic superconductor, Spin liquid, FQHE
I\ Introduction to topological orders The 2D Topological order is a quantum state with the following key properties 4 All excitations are gapped g Topological degeneracy 今 Exotic statistics . Stable against all kinds of perturbations X.G. Wen prB. 65 ,o No global symmetry 165113(2002)
I、 Introduction to topological orders The 2D Topological order is a quantum state with the following key properties : ❖ All excitations are gapped ❖ Topological degeneracy ❖ Exotic statistics ❖ Stable against all kinds of perturbations ❖ No global symmetry X. G. Wen PRB, 65, 165113 (2002)
Examples for topological order o Fractional Quantum hall states Chiral ptip and d+id superconductors Topological orders for spin liquids: chiral spin liquid, Z2 spin liquid or others
Examples for topological order: ❖ Fractional Quantum Hall states ❖ Chiral p+ip and d+id superconductors ❖ Topological orders for spin liquids : chiral spin liquid, Z2 spin liquid or others
Three types of topological orders in 2D abelian topological orders without time reversal symmetry anyon Non-abelian topological orders without time reversal symmetry non-Abelian anyon o Z2 topological orders with time reversal symmetry Z2 vortex and z2 charge
Three types of topological orders in 2D ❖ Abelian topological orders without time reversal symmetry : anyon ❖ Non-Abelian topological orders without time reversal symmetry : non-Abelian anyon ❖ Z2 topological orders with time reversal symmetry : Z2 vortex and Z2 charge
1. FQH states: the first example for the abelian topological orders 2.5 1.5 c 0.5 Magnetic field m) Daniel c. tsui horst l stormer
1. FQH states: the first example for the Abelian topological orders