Jordan-Wigner Transformation and Topological characterization of quantum phase transitions in the Kitaev model Guang-Ming Zhang (Tsinghua Univ) Xiaoyong.

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Presentation transcript:

Jordan-Wigner Transformation and Topological characterization of quantum phase transitions in the Kitaev model Guang-Ming Zhang (Tsinghua Univ) Xiaoyong Feng (ITP, CAS) T. Xiang (ITP, CAS) Cond-mat/0610626

Outline Brief introduction to the Kitaev model Jordan-Wigner transformation and a novel Majorana fermion representation of spins Topological characterization of quantum phase transitions in the Kitaev model

Ground state can be rigorously solved Kitaev Model Ground state can be rigorously solved A. Kitaev, Ann Phys 321, 2 (2006)

4 Majorana Fermion Representation of Pauli Matrices cj, bjx, bjy, bjz are Majorana fermion operators Physical spin: 2 degrees of freedom per spin Each Majorana fermion has 21/2 degree of freedom 4 Majorana fermions have totally 4 degrees of freedom

4 Majorana Fermion Representation of Kitaev Model Good quantum number x y z

2D Ground State Phase Diagram The ground state is in a zero-flux phase (highly degenerate, ujk = 1), the Hamiltonian can be rigorously diagonalized non-Abelian anyons in this phase can be used as elementary “qubits” to build up fault-tolerant or topological quantum computer

4 Majorana Fermion Representation: constraint Eigen-function in the extended Hilbert space

3 Majorana Fermion Representation of Pauli Matrices Totally 23/2 degrees of freedom, still has a hidden 21/2 redundant degree of freedom

Kitaev Model on a Brick-Wall Lattice Brick-Wall Lattice honeycomb Lattice x y x y x y y x y x y x z z z z x y x y y x y x z z z

Jordan-Wigner Transformation Represent spin operators by spinless fermion operators

Along Each Horizontal Chain x y x y

Two Majorana Fermion Representation Onle ci-type Majorana fermion operators appear!

Two Majorana Fermion Representation ci and di are Majorana fermion operators A conjugate pair of fermion operators is represented by two Majorana fermion operators No redundant degrees of freedom!

Vertical Bond No Phase String

2 Majorana Representation of Kitaev Model good quantum numbers Ground state is in a zero-flux phase Di,j = D0,j

Phase Diagram Critical point Single chain Quasiparticle excitation: x y x y 0 1 J1/J2 Single chain Critical point Quasiparticle excitation: Ground state energy

Phase Diagram J3=1 Critical lines Two-leg ladder = J1 – J2

How to characterize these quantum phase transitions? Multi-Chain System J3=1 Chain number = 2 M Thick Solid Lines: Critical lines How to characterize these quantum phase transitions?

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

QPT: Single Chain x y x y Duality Transformation

Non-local String Order Parameter

Another String Order Parameter

Two-leg ladder J3 = 1 = J1 – J2

Phase I: J1 > J2 + J3 In the dual space: W1 = -1 in the ground state

String Order Parameters

QPT: multi chains Chain number = 2 M

QPT in a multi-chain system 4-chain ladder M = 2

Fourier Transformation

q = 0 ci,0 is still a Majorana fermion operator Hq=0 is exactly same as the Hamiltonian of a two-leg ladder

String Order Parameter

q =  ci, is also a Majorana fermion operator Hq= is also the same as the Hamiltonian of a two-leg ladder, only J2 changes sign

String Order Parameter

Summary Kitaev model = free Majorana fermion model with local Ising field without redundant degrees of freedom Topological quantum phase transitions can be characterized by non-local string order parameters In the dual space, these string order parameters become local The low-energy critical modes are Majorana fermions, not Goldstone bosons