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

Classical orders： CDW、FM、AFM、Crystal

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

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.

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

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: 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 Z2 topological orders with time reversal symmetry : Z2 vortex and Z2 charge

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

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.

flux Anyon and fractional statistics

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

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

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

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

Skyrmion in momentum space

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

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

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

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

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

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

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

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

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

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?

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

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).

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.

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

Closed strings Open strings

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.

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

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

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)

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

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

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

Quantum phase transition for topological orders

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

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

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

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

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

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

The global phase diagram

The the expectation value for and

The numerical results:

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

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

1D properties for the QPT

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)

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?

Thanks very much for Your Attention

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

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

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).

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

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, 070602 (2007).

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

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

Phase Diagram for single-chain limit for Kitaev model 0 1 J1/J2 Single chain Critical point x y x y Quasiparticle excitation: Xiao-Yong Feng, Guang-Ming Zhang, Tao Xiang, Phys. Rev. Lett. 98, 087204 (2007).

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

multi chains limit Chain number = 2 M

Non-local String Order Parameter