Topological Order and its Quantum Phase Transition

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

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

3. Classical orders： CDW、FM、AFM、Crystal

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

5. Universal law for classical phase transitions • 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.

6. Landau's theory cannot describe all the continuous phase transition such as the transitions between quantum ordered phase Exotic superconductor, Spin liquid, FQHE Lifshiz phase transition Spin liquid

7. 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).

8. 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

9. 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

10. 1. FQH states: the first example for the Abelian topological orders

11. Laughlin wave-function and anyon

12. Anyon and fractional statistics flux

13. Topological degeneracy for the Abelian topological orders • Topological degeneracy on a torus (periodic boundary condition) is always q

14. Topological degeneracy for FQH states • The topological degeneracy is determined by the genus of the Riemann surface, • g is the genus

15. Edge states The electrons are confined in a 2D space Chiral Luttinger liquid  massless, relativitistic

16. 2. Chiral P-wave superconductor an example of non-Abelian topological order • Strong superconductivity( v=5/2 FQH state)：Chemical potential µ<0 Non-Abelian Topologial order • Weak superconductivity ： Chemical potential µ>0 • Z2 topological order

17. Skyrmion in momentum space

18. Topological quantum compatation is proposed by Kitaev. Application of nonAbelian topological order.

19. 3. Z2 topological orders and spin liquid Z2 charge Mutual semion statistics flux Z2 vortex

20. Topological degeneracy for Z2 topological orders: Red line denotes “half flux tube”

21. The topological degeneracy for the Z2 topological orders is determined by the genus of the Riemann surface, • g is the genus

22. Quantum exotic states for the spin models G. Misguich, C. Lhuillier, cond-mat/0310405.

23. The Kitaev toric-code model A．Y．Kitaev，Annals Phys. 303, 2 (2003)

24. The Kitaev Model A. Kitaev, Ann Phys 321, 2 (2006) The Abelian gapped phases Ax, Ay, Az are all Z2 topological orders

25. The Wen-plaquette model X. G. Wen, PRL. 90, 016803 (2003).

26. The energy gap • For g>0, the ground state is The ground state energy is E0=Ng The elementary excitation is The energy gap for it becomes

27. The statistics for the elementary excitations • There are two kinds of Bosonic excitations: • Z2 vortex • Z2 charge • Each kind of excitations moves on each sub-plaquette: • Why?

28. There are two constraints (the even-by-even lattice): One for the even plaquettes, the other for the odd plaquettes • The hopping from even plaquette to odd violates the constraints : You cannot change a Z2 vortex into a Z2 charge

29. The mutual semion statistics between the Z2 Vortex and Z2 charge • When an excitation (Z2 vortex) in even-plaquette move around an excitation (Z2 charge) in odd-plaquette, the operator is • it is -1 with an excitation on it • This is the character for mutual mutual semion statistics X. G. Wen, PRD68, 024501 (2003).

30. Topological degeneracy on a torus (even-by-even lattice) : • On an even-by-even lattice, there are totally states • Under the constaints, the number of states are only • For the ground state , it must be four-fold degeneracy.

31. The string operators： For the ground state, the closed-strings are condensed String net condensation for the topological order

32. Closed strings • Open strings

33. Conpare the two kinds of topological orders • Abelian type：FQF states Effective theory：Chern-Simons theory Fractional charge： Topological degeneracy： • Z2 type : “RVB” spin liquids Effective theory：mutual Chern-Simons theory Fractional charge： Topological degeneracy： S. P. Kou, M. Levin, X. G. Wen, preprint.

34. The topological order and new physics 电子分数化： 拓扑元激发及其 诱导量子数 ？Nonperturbative physics 弦网凝聚： Loop理论 Topological order Wave functions Gauge theories 拓扑场论及其规范场的 零模动力学 投影的平均场波函数及其 动量空间拓扑

35. II. Quantum phase transition for topological orders • Quantum phase transition • Quantum phase transition for Wen-plaquette model • Quantum phase transition for Kitaev toric-code model • Quantum phase transition for Kitaev model

36. 1. Quantum phase transition • The transition between different ground states • Caused by quantum fluctuations (a) (b)

37. Quantum phase transition for the transverse Ising model • The Hamitonian： • J is the energy scale， • g is the dimensionless parameter

38. Classifications of continuous phase transitions • Conventional: Landau-type • Symmetry breaking • Local order parameters • Topological: • Both phases are gapped • No symmetry breaking • No local order parameters

39. Universal natures for the quantum phase transition of the topological orders • Gap closes at the QPT • Topological degeneracy is removed • The massless fermion modes at the QPT • String-condensation and non-local order parameters

40. Quantum phase transition for topological orders

41. 2. The quantum phase transition for Wen-plaquette model Jing. Y, S. P. Kou, X. G. Wen, priprent.

42. The Hamiltonian for 2D transverse Wen-plaquette model : Duality between the 2D Wen-plaquette model and the 1D transverse Ising model （1） The Hamiltonian for the 1D transverse Ising model （2） The two terms for the two models have the same commutation relations

43. The transverse Wen-plaquette model on a square lattice is dual to the 1D transverse Ising chain “a” denotes the chain index，h is the energy scale，gI=g/h is a dimensionless parameter

44. For a N×N lattice, the transverse Wen-plaqeutte model is dual to N decoupled Ising chains

45. Jordan-Wigner transformation • By the Jordan-Wigner transformation of the spin operators to spinless fermions, the effective Hamitonian becomes

46. The energy spectrum is The energy gap is ： The scaling law near the critical point is

47. The global phase diagram

48. The the expectation value for and

49. The numerical results:

50. Non-local order parameters • Open-string order parameter • Closed-string order parameter open-string-closed-string duality