Kitaev honeycomb lattice model: honeycomb lattice model: Kitaev - PowerPoint PPT Presentation

Kitaev honeycomb lattice model: honeycomb lattice model: Kitaev from A to B and beyond from A to B and beyond Jiri Vala Vala Jiri Department of Mathematical Physics Department of Mathematical Physics National University of Ireland at Maynooth Maynooth National University of Ireland at Postdoc: Postdoc : Graham Kells Graham Kells PhD students: PhD students: Ahmet Bolukbasi Ahmet Bolukbasi Niall Moran Niall Moran Collaborators: Collaborators: Joost Slingerland , NUIM & DIAS Joost Slingerland , NUIM & DIAS Ville Lahtinen Ville Lahtinen , Leeds , Leeds Jianis Pachos , Leeds , Leeds Jianis Pachos

Outline Toric code and Kitaev honeycomb lattice model • introduction • relation between both models • vorticity • loop symmetries • Abelian phase: summary of results Exact solution of the Kitaev honeycomb model • map onto spin-hardcore boson system • Jordan-Wigner fermionization • adding magnetic field • ground state as BCS state with explicit vacuum Further developments • ground states on torus • ground state degeneracy in Abelian and non-Abelian phase

From A … … From A toric code code toric and and Kitaev honeycomb lattice model honeycomb lattice model Kitaev

Toric code Toric code - spin 1/2 particles on the edges of a square lattice (green) H TC =-J eff ( � star Q s + � plaquettes Q p ) A.Y.Kitaev, Fault-tolerant quantum computation by anyons , Ann. Phys. 303, 2 (2003). Unitarily equivalent toric code - spin 1/2 particles on the vertices of a square lattice (blue) - connects naturally with the Kitaev honeycomb model H TC =-J eff � q Q q Q q = � y left(q) � y right(q) � z up(q) � z up(q) Pauli matrices

Toric code code Toric H TC =-J eff � p Q p Q p = � z p � y p+ � y p+ � z Hamiltonian p+ + n x n y n x n y [ H TC , Q p ] = 0 [ Q p , Q q ] = 0 “Symmetries” Eigensvalues the operators Q p have eigenvalues Q p = ±1; for all p we have { Q p } � . |{ Q p }> is characterized completely by the eigenvalues Q p : Q p |{ Q p }> = Q p |{ Q p }> p Ground state is stabilized by Q p for all p Q p |{ Q p }> = |{ Q p }> |{ Q p }> TC = |{ Q p = 1}for all plaquettes> On torus , we have � Q p = 1, and two additional homologically nontrivial symmetries |{ Q p }, l x , l y > TC The energy does not depend on the eigenvalues of the homologically nontrivial symmetries; this implies four-fold ground state degeneracy .

Quasiparticles Quasiparticles m m Toric code quasiparticle excitations, Q p =-1, are e • “magnetic” (living on blue plaquettes) or “electric” (white plaquettes), e • are created in pairs by acting on the ground state with Pauli operators. m m Operator C L,m to move a single “magnetic” excitation in a contractible loop L is the product of all “electric” plaquette operators e enclosed by the loop (and vice versa). e If the initial state |{Q p }> contains an “electric” excitation then moving a magnetic excitation around it returns the initial state with the phase changed by -1 implying that: • “e-m” composite is a fermion • “magnetic” and “electric” particles are relative semions m m m m e e e e • “e-m” fermion behaves as semion when braided with an “e” or “m” particle

Kitaev honeycomb lattice model H 0 = J x � i,j � x i � x j + J y � i,j � y i � y j + J z � i,j � z i � z j x-link y-link z-link = � � J � � i,j � � i � � j = � � J � � i,j K � x -link � -link: ij z -link J z = 1,J x = J y = 0 y -link A Phase diagram: • phase A - can be mapped perturbatively onto B the toric code; A A • phase B - gapless. J y = 1, J x = 1, J y = J z = 0 J x = J z = 0 Adding magnetic field: • parity and time-reversal symmetry are broken • phase B acquires a gap and becomes H = H 0 + H 1 = H 0 + � i � � =x,y,z B � � � ,i non-abelian topological phase of Ising type The leading P and T breaking term in perturbation theory occurs at the third order: A.Y.Kitaev, Ann. Phys. 321, 2 (2006).

Mapping abelian phase onto toric code “dimers” D Effective spins - are formed by ferromagnetic ground states of -J z � j z � k z A.Y.Kitaev, Fault-tolerant quantum computation by anyons , Ann. Phys. 303, 2 (2003).

Mapping abelian phase onto toric code Effective Hamiltonian (no magnetic field) first non-constant term of perturbation theory occurs on the 4 th order toric code defined on the square lattice with effective spins on the vertices Toric code quasiparticles and vortices of the honeycomb lattice model e e e e e e e e e e m m m m m m m m e e e e e e e e e e m m m m m m m m e e e e e e e e e e

Vortex operators in the honeycomb model � x � y 3 W p = � x 1 � y 2 � z 3 � x 4 � y 5 � z 6 = � z 2 4 p y -link 1 5 = K z 1,2 K x 2,3 K y 3,4 K z 4,5 K x 5,6 K y x -link 6 6,1 z -link [H 0 , W p ] = 0 (K � K � k+1,k+2 K � k,k+1 = - K � k,k+1 K � k,k+1 ) 2 = 1 k+1,k+2 H 0 |n> = E n |n> w p = <n|W p |n> = +1 w p = <n|W p |n> = -1 p p

Vortex sectors Each energy eigenstate |n> is characterized by some vortex configuration {w p = <n|W p |n> = ±1} for all plaquettes p also the vortices are always excited in pairs, i.e. even-vortex configurations are relevant on closed surfaces or infinite plane, the Hilbert space splits into vortex sectors, i.e. subspaces of the system with a particular configuration of vortices = 1 � L L w w 1 ,...., m w w ,......., m … … vortex free sector examples from two-vortex sectors full vortex sector

Products of vortex operators � x � y 3 7 � z 2 4 8 p p+1 9 y -link 1 5 x -link 6 10 z -link (we used (K � k,k+1 ) 2 = 1) Products of vortex operators generate closed loops � (1) � (2) � ( �� 1) � ( � ) K i,j K j,k …K p,q K q,i On a torus, this gives the condition � p W p = 1

Loop symmetries on torus For a system of N spins on a torus (i.e. a system with N/2 plaquettes), � p W p = 1 implies that there are N/2-1 independent vortex quantum numbers {w 1 , … , w N/2-1 }. � (1) � (2) � ( �� 1) � ( � ) K i,j K j,k …K p,q K q,i Loops on the torus - all homologically trivial loops are generated by plaquette operators - in addition, two distinct homologically nontrivial loops are needed to generate the full loop symmetry group (the third nontrivial loop is a product of these two). The full loop symmetry of the torus is the abelian group with N/2+1 independent generators of the order 2 (loop 2 =I), i.e. Z 2 N/2+1 . All loop symmetries can be written as C (k,l) = G k F l (W 1 , W 2 , … , W N-1 ) where k is from {0,1,2,3} and G 0 = I , and G 1 , G 2 , G 3 are arbitrarily chosen symmetries from the three nontrivial homology classes, and F l , with l from {1, …, 2 N/2-1 }, run through all monomials in the W p operators.

Results on the Abelian phase 1) The symmetry structure of the system is manifested in the effective Hamiltonian obtained using the Brillouin-Wigner perturbation theory. The longer loops occur at the higher order of the perturbation expansion: trivial � N / 2 2 3 2 � � = H c G ( z , y ) F ( Q , Q ,....... Q ) W p Q p � eff i , j i j 1 2 N / 2 2 = = i 0 j 1 G. Kells, A. T. Bolukbasi, V. Lahtinen, J. K. Slingerland, J. K. Pachos and J. Vala, nontrivial Topological degeneracy and vortex manipulation in the Kitaev honeycomb model , - reflects topology Phys. Rev. Lett. 101 , 240404 (2008). 2) Fermions of the Abelian phase can be moved efficiently using the K strings from the symmetries. A. T. Bolukbasi, et al., in preparation. 3) The symmetry structure of the effective Hamiltonian allows to classify all finite size effect, intrisic to the system of sizes <36 spins: for example N=16 spins. G. Kells, N. Moran and J. Vala, Finite size effects in the Kitaev honeycomb lattice model on torus, J. Stat. Mech. – Th. Exp., (2009) P03006

… to B to B … … … exact solution exact solution of the Kitaev Kitaev honeycomb lattice model honeycomb lattice model of the

Effective spins and hardcore bosons Effective spins and hardcore bosons New perspective: spin-hardcore boson representation Schmidt, Dusuel, and Vidal (2008) Pauli operators: In the A z -phase, J z >> J x , J y , the bosons are energetically suppressed, thus at low energy Vortex and plaquette operators: |{W q }, 0> = |{ Q q }> the low-energy perturbative Hamiltonian equals to toric code This allows to write down an orthonormal basis of the full system in terms of the toric code stabilizers : |{W q }, {q}> where {W q } lists all honeycomb plaquette operators and {q} lists the position vectors of any occupied bosonic modes. On a torus, the homologically nontrivial symmetries must be added |{W q }, l 0 (x) , l 0 (y) , {q}> G. Kells, et al., arXiv:0903.5211 (2009)

Jordan-Wigner Wigner transformation transformation Jordan- Bosonic and effective spin Hamiltonian can be written in terms of fermions and vortices by applying a Jordan-Wigner transformation where on a plane Importantly, presence of a fermion indicates an anti-ferromagnetic configuration of z-link

Magnetic field Magnetic field • breaks parity and time-reversal symmetry • opens a gap in phase B and turns it into non-abelian topological phase of Ising type • H 1 commutes with the plaquette operators, so stabilizer formalism can still be used