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. Spin rotation symmetry breaking!
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. Exotic superconductor, Lifshiz phase transition
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, (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
Shortly after the discovery of the FQHE, Bob Laughlin solved the problem by writing down essentially the exact many-body quantum mechanical wave function. The important property of this wave function is that there are 3 zeros, which can also be thought of as 3 vortices, attached to each electron. If one imagines now adding 1 extra vortex (or zero) at some point, there is a missing 1/3 of an electron to go with this vortex. This defect is called a quasihole and carries -1/3 of the electron charge. By removed one vortex, one obtains a quasiparticle carrying +1/3 the charge of the electron. So, in this semiconductor system, the electron breaks up into 3 particles carrying fractional charge.

12. flux
Anyon and fractional statistics

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
As a final note on the quantum Hall effect, I want to mention that
on open surfaces, the topological order shows up in edge states or new particles which move on the edges. These edge modes are gapless with a spectrum resembling that of a relativistic Dirac particle.
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 Mutual semion statistics
Z2 charge
flux
Z2 vortex
Mutual semion statistics

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/

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

24. The Abelian gapped phases Ax, Ay, Az are all Z2 topological orders
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, (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, (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. String net condensation for the topological order
The string operators：
For the ground state, the closed-strings are condensed

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
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 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
一般的量子相变有对称破缺
(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. Duality between the 2D Wen-plaquette model and the 1D transverse Ising model
The Hamiltonian for 2D transverse Wen-plaquette 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

51. The correlation function at the QPT
The long-distance limit of the static correlator for
the 2D transverse Wen-plaquette model can be described
by the dual correlator of the 1D transverse Ising model.
For a diagonal case

52. 1D properties for the QPT

53. Conclusion
The ground state for the Wen-plaquette model is the Z2 topological order with string-net condensation, topological degeneracy, mutual semion statistics
A duality between the 2D transverse Wen-plaquette model and the 1D transverse Ising model
A new type of QPT between a topological ordered phase and a non-topological ordered phase :
No conventional Landau-type local order parameters, but non-local order parameters ( open-string-closed-string duality)

54. Further Issues Which properties are universal?
How to realize new type of topological orders?
What’s the effective theory of the quantum phase transition between a topological order and a non-topological order?
Can the quantum entanglement describe the Quantum phase transition?

55. Thanks very much for Your Attention

56. Quantum Dimer Models Square lattice (Rokhsar-Kivelson, ’88)
Assume dimer configurations are orthogonal
RK ’88
Leung et al, ‘96

57. QDM on triangular lattice
Moessner and Sondhi, ’01
RVB spin liquid with gapped spectrum
Topological degeneracy

58. Rokhsar-Kivelson model on a triangular lattice
The ground state of “Rokhsar-Kivelson” Type RVB spin liquid is a Z2 topological order : all excitations are gapped; four-fold degeneracy on a torus
D.S. Rokhsar and S. Kivelson, Phys. Rev. Lett. 61, 2376 (1988)
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991);
X. G. Wen, Phys. Rev. B 44, 2664 (1991).

59. 2. The quantum phase transition for the Kitaev toric-code model
A．Y．Kitaev，Annals Phys. 303, 2 (2003)

60. A second order quantum phase transition between a spin-polarized phase and a topological order
S. Trebst, P. Werner, M. Troyer, K. Shtengel, C. Nayak,
Phys. Rev. Lett. 98, (2007).

61. The condensation of `magnetic' excitations, and the confinement of
`electric' charges of the phase transition out of the topological phase

62. 3. The quantum phase transition for Kitaev Model
A. Kitaev, Ann Phys 321, 2 (2006)
The Abelian gapped phases Ax, Ay, Az are all Z2 topological orders

63. Phase Diagram for single-chain limit for Kitaev model
J1/J2
Single chain
Critical point
x y x y
Quasiparticle excitation:
Xiao-Yong Feng, Guang-Ming Zhang,
Tao Xiang, Phys. Rev. Lett. 98, (2007).

64. Phase Diagram for two-leg ladder limit for Kitaev model
J3 = 1
= J1 – J2

65. multi chains limit
Chain number = 2 M

66. Non-local String Order Parameter