Kitaev Model Michele Burrello SISSA Firenze, September 3, 2008 - PowerPoint PPT Presentation

Kitaev Model Michele Burrello SISSA Firenze, September 3, 2008

Contents Kitaev model 1 Majorana Operators Exact Solution Spectrum and phase diagram Gapped Abelian Phase 2 Magnetic Field: Non–Abelian Phase 3 Spectral Gap Edge Modes Non-Abelian Anyons Michele Burrello Kitaev Model

Anyons and topological quantum computation ‘Quantum phenomena do not occur in a Hilbert space. They occur in a laboratory’ Asher Peres Local errors, thermic noise and decoherence are considered the main obstacles in the realization of a quantum computer Topological properties of physical systems seem to be one of the best answer to overcome those problems Qubits encoded in topological states can be insensitive to local perturbations Michele Burrello Kitaev Model

Anyons and topological quantum computation Different applications of topological states have been proposed to encode qubits: Abelian anyons on a torus imply a ground state degeneracy and so the possibility to store quantum information ( toric code ) Non–abelian anyons can be used to implement a universal quantum computer (Kitaev, Freedman,...) Example of non–abelian fusion rule: σ × σ = I + ε Michele Burrello Kitaev Model

Kitaev Model The aim of this talk is to study an example of anyonic system realized through a particular honeycomb spin lattice. The study of the Kitaev model will allow us to understand the main features of non-abelian anyons and we’ll analyze the interplay between a simple anyonic theory defined by fusion and braiding rules and the conformal field theory of the Ising model ( M 3 ). Main features of anyonic systems: Energy gaps which allow the existence of local excitations (exponential decay of correlators) Topological Quantum Numbers which make such excitations stable (anyons as topological defects: for example vortices) Topological Order Michele Burrello Kitaev Model

The Model Alexei Kitaev, Anyons in an exactly solved model and beyond , arXiv: cond-mat/0506438v3 � � � σ y j σ y σ x j σ x σ z j σ z H = − J x k − J y k − J z k x − links y − links z − links � H = − J jk K jk j n . n . k Michele Burrello Kitaev Model

Plaquettes: Integrals of Motion 1 σ y 4 σ y W p = σ x 2 σ z 3 σ x 5 σ z 6 = K 12 K 23 K 34 K 45 K 56 K 61 Commutation rules: [ K ij , W p ] = 0 ∀ i, j, p ⇒ [ H, W p ] = 0 , [ W q , W p ] = 0 ∀ q, p To find the eigenstates of the Hamiltonian it is convenient to divide the total Hilbert space in sectors - eigenspaces of W p : H = w 1 ,...,w m H w 1 ,...,w m ⊕ For every n vertices there are m = n/ 2 plaquettes. There are 2 n/ 2 sectors of dimension 2 n/ 2 . Michele Burrello Kitaev Model

Majorana operators To describe the spins one can use annihilation and creation fermionic � � a ↑ , a † ↑ , a ↓ , a † operators . It is also possible to define their self adjoint ↓ linear combinations: � � c 2 k − 1 = a k + a † a k − a † c 2 k = − i k k The Majorana operators c j define a Clifford algebra: { c i , c j } = 2 δ ij Using these operators we are doubling the fermionic Fock space: � � {|↑� , |↓�} − → | 00 � ↑↓ , | 11 � ↑↓ , | 01 � ↑↓ , | 10 � ↑↓ We need a projector onto the physical space. Michele Burrello Kitaev Model

From Majorana to spin operators For each vertex on the lattice we define: � � � � b x = a ↑ + a † b y = − i b z = a ↓ + a † a ↑ − a † a ↓ − a † ↑ , , ↓ , c = − i ↑ ↓ We can write: σ x = ib x c, σ y = ib y c, σ z = ib z c, D = − iσ x σ y σ z = b x b y b z c D is the gauge operator : [ D, σ α ] = 0 ∀ α Over the physical space D = 1 and the projector over the physical space is: � 1 + D j � � P phys = 2 j Michele Burrello Kitaev Model

Kitaev model Using the Majorana operators we can rewrite: � � K jk = σ α j σ α ib α ( ib α u jk ≡ ib α j b α k = j c j k c k ) = − iu jk c j c k with k And the Hamiltonian reads: � 2 J α jk u jk H = i � if j and k are connected A jk c j c k with A jk ≡ 0 otherwise 4 j,k u jk = − u kj ⇒ A jk = A kj Michele Burrello Kitaev Model

u ij operators u ij are hermitian operators such that: u ij commute with each other u ij commute with H and have eigenvalues u ij = ± 1 We can study the Hamiltonian in an eigenspace of all the operators u jk u ij is not gauge invariant: we need to project onto the physical subspace. D j changes the signs of the three operators u jl linked with j . Michele Burrello Kitaev Model

Gauge invariant operators Wilson loop over each plaquette: � w p = u jk ( j,k ) ∈ p Where j is in the even sublattice and k on the odd one. Path operator : � n � � W ( j 0 , ..., j n ) = K j n j n − 1 ...K j 1 j 0 = − iu j s j s − 1 c n c 0 s =1 u jk can be considered a Z 2 gauge field and w p is the magnetic flux through a plaquette. If w p = − 1 we have a vortex and a Majorana fermion moving around p acquires a − 1 phase. Michele Burrello Kitaev Model

Quadratic Hamiltonian H ( A ) = i � A jk c j c k 4 j,k where A is a real skew-symmetric 2 m × 2 m matrix. Through a transformation Q ∈ O (2 m ) we obtain: m H = i � ε k b ′ k b ′′ k 2 k =1 with: ( b ′ 1 , b ′′ 1 , ..., b ′ m , b ′′ m ) = ( c 1 , c 2 , ..., c 2 m − 1 , c 2 m ) Q and:   0 ε 1 − ε 1 0     ...   Q T A = Q     0 ε m   − ε m 0 Michele Burrello Kitaev Model

Quadratic Hamiltonian H can be diagonalized using creation and annihilation operators: m m � � H = i k a k − 1 � � a † ε k b ′ k b ′′ k = ε k 2 2 k =1 k =1 with: � 1 � � b ′ � � � a † = 1 − i b ′′ a i 1 2 It is possible to define a spectral projector P onto the negative eigenvectors of A which identifies the ground state: � � P = 1 − i I I � ˜ ˜ Q T Q P kj c j | ψ GS � = 0 ∀ k i I 2 I j a † a = cPc � ψ GS | c j c k | ψ GS � = P kj Michele Burrello Kitaev Model

Spectrum In the physical space the energy minimum is reached in the vortex free configuration ( w p = 1 ∀ p ). We can consider the coupling between unit cells: � � H ( q ) = i 0 if ( q ) 2 A ( q ) = − if ∗ ( q ) 0 � � J x e iqn 1 + J y e iqn 2 + J z f ( q ) = Spectrum: ε ( q ) = ± | f ( q ) | ε ( q ) vanishes for some q iff the triangle inequalities hold: | J x | ≤ | J y | + | J z | | J y | ≤ | J z | + | J x | | J z | ≤ | J x | + | J y | Michele Burrello Kitaev Model

Phase diagram Phase B is gapless: there are two values ± q 0 such that ε ( ± q 0 ) = 0 B acquires a gap in presence of an external magnetic field Phases A are gapped and are related by rotational symmetry Michele Burrello Kitaev Model

Gapped Phases In a gapped phase A correlations decay exponentially. There are no long range interactions. Local and distant particles can interact topologically. ( Braiding Rules ) We need to identify the right (stable and local) particles ( Superselection Sectors ) We will apply a perturbation theory study to reduce the Kitaev model to the Toric model Michele Burrello Kitaev Model

Phase A z : Perturbation Theory Let us suppose J z ≫ J x , J y and J z > 0 . � � � σ y j σ y σ z j σ z σ x j σ x H 0 = − J z k , V = − J x k − J y k z − links x − links y − links The strong z − links in the original model (a) become effective spins (b) and can be associated with the links of a new lattice (c). Michele Burrello Kitaev Model

Phase A z , J z ≫ J x , J y : Perturbative results The first 3 orders in the perturbative expansion give just a shift in the spectrum. The fourth order is: eff = − J 2 x J 2 � H (4) y W eff p 16 J 3 z p where: W p = σ x 1 σ y σ z 3 σ x 4 σ y σ z W eff = σ y l σ z u σ y r σ z − → 2 5 6 p d � �� � � �� � σ y σ y r l Michele Burrello Kitaev Model

Phase A z : Toric Code Hamiltonian A. Kitaev, arXiv: quant-ph/9707021 Through unitary transformations the previous effective Hamiltonian can be mapped onto the toric code Hamiltonian :    � � H eff = − J eff A s + B p  vertices plaquettes with: � � σ x σ z A s = j , B p = j j ∈ star ( s ) j ∈ boundary ( p ) and: [ A s , B p ] = [ B p , B q ] = [ A s , A r ] = 0 and the translational invariance is broken. Michele Burrello Kitaev Model

Excitations Ground State: A s | ψ � = + | ψ � B p | ψ � = + | ψ � Excitations: Electric charge e : A s | e s � = − | e s � Magnetic vortex m : B p | m p � = − | m p � Superselection sectors: I (vacuum), e , m , ε = e × m Fusion Rules: e × e = m × m = ε × ε = I e × m = ε ; e × ε = m ; m × ε = e Michele Burrello Kitaev Model

Braiding Rules To create a pair of e , or move an e through a path t we must apply: � S z ( t ) = σ z j j ∈ t To create a pair of m , or move an m through a path t ′ we must apply: � S x ( t ′ ) = σ x j j ∈ t ′ e and m are bosons; Moving an e around an m yields − 1 ; ε are fermions. Michele Burrello Kitaev Model

Gapped Phases We can translate these results into the original model. e and m particles correspond to vortices that live in different rows: e × m = ε e × ε = m m × ε = e e × e = I m × m = I ε × ε = I The Majorana fermions in the original model belong to the superselection sector ε although they are not directly composed of e and m (different energies between c and ε ). Michele Burrello Kitaev Model