北京师范大学：拓扑序及其量子相变（PPT课件讲稿）Topological Order and its Quantum Phase Transition

Topological Order and its Quantum Phase transition Su-Peng Kou Beijing normal university Colla borators: X.G. Wen. Jing yu. m. levin

Topological Order and its Quantum Phase Transition Su-Peng Kou Beijing Normal university Collaborators: X. G. Wen, Jing Yu, M. Levin

Outline Introduction to topological orders Quantum phase transition for topological orders Conclusion and open questions

Outline ❖ Introduction to topological orders ❖ Quantum phase transition for topological orders ❖ Conclusion and open questions

Classical orders CDW、FM、AFM、 Crystal NbSe 2 Ferromagnetism Antiferromagnetism Charge density wave

Classical orders： CDW、FM、AFM、Crystal

The phase transition between different phases are always accompanied with symmetry breaking Landau Ferromagnet T< To ↑↑↑↑↑↑ Spin rotation symmetry breaking

The phase transition between different phases are always accompanied with symmetry breaking Spin rotation symmetry breaking!

序参量 液一气密度差,自发磁化强度,超导序参量,… Universal law for classical phase transitions 第一类相变 连续相变(临界现象) 'o The critical parameters between a classical continuous phase transition are determined by the dimension of the system and the degree freedom of the order parameters

❖ The critical parameters between a classical continuous phase transition are determined by the dimension of the system and the degree freedom of the order parameters. Universal law for classical phase transitions

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