Transition in Kitaev model Razieh Mohseninia November, 2013 1
Phase Transition Landau symmetry-breaking theory(1930-1980) different phases different symmetry Liquid solid Paramagnetic ferromagnetic 2
Topological order No symmetry breaking. No local order parameter. There is a gap. Long range entanglement. Degeneracy depends on the topology of space. Degeneracy cannot be lifted by any local perturbations. 3
Local Unitary Transformation and Long Range Entanglement = Topological Classes 4
Toric code L ={ H: =, =! "} # = %& ' & 5
Coding Torus =1 =1 +, --./01,230.!0"0!0 - -3.40 2 2,230 +,3.- 2 Degeneracy = 789 7 89 :8 = 4 ;; = 00, ;= = 01 =; = 10, == = 11 6
Ground States ;; >1? @ 0 7A B CD E F G,= C F G,7 D ;;,.,H 0,1 7
Kitaev model,why? 8
What is I J Kitaev model? L L # = % ( + ) ' ( + ) Torus =1 =1 Degeneracy: J89 J 89:8 =! 7 9
Ground States ;; M>1?? 7 JO=?? @ 0 7A B CD E F G,= C F G,7 D ;;.,H 0,,!$1 10
Classical Potts model(1952) # QCRS = = 7 (1+T CT D C,D ) T C =1, 1 = U B, V C,D # WXX = &U B, V C,D JO= = 1! & & ( C D ) Z C,D Z[; T C =1,\,\ 7,\ ],,\ JO= 11
An open problem(1852-1976) Chromatic polynomial: ^(_, +) Chromatic number :` _ =min (+;^(_,+ >0) ^f 4 >0 for any planar graph 12
Potts model and 4-color problem # WXX = %&U B, V C,D I(h )=&0 h B,V k l B,l V {m B } =^(_,+) 13
Quantum Potts model JO= # W = &&((I C I D L ) Z +(I C L I D ) Z ) C,D Z[; o,! --0= 0 A 1 A 2 A! 1 A degeneracy=d 14
Our problem 15
Our problem # =# rcxst +q # WXX # = % ( + L L ) ' ( + ) q JO= ((I C I L D ) Z +(I L C I D ) Z ) C,D Z[; q 0 Degeneracy! 7! Order topological order Phase transition Ferromagnetic order 16
Potts model in magnetic field # v L u = %&w C +w C JO= q! &&((I CI D L ) Z +(I C L I D ) Z ) ' C C,D Z[; 17
Mean Field min # =z S. Λ # Λ z S. Λ Λ = y 9 8 JO= y= C[; C. z ={Λ # u Λ= J 2 &( C C}= + C}= C ) 4q& C ~ C C 18
Wilson loop ƒ =Mw C Mw D L =M t C D t 19
; >1? @? 7A y ; 0 7A ƒ 1, '.-0 0, ^-- 20
Calculation of Wilson loop in q q ƒ = Ψ ƒ Ψ Ψ Ψ O~G8 Š 0 J 8 21
Calculation of Wilson loop in q q ƒ = Ψ ƒ Ψ Ψ Ψ 1 2 0O Œ =}Ž :8 7Ž : 22
Continuous Unitary Transformations(CUTs) = # L # #, # s #> @ # ; # s # CUTs 23
CUTs # = # 0 L ()!() = ()!!#()! =[,# ] 24
Wegner s generator = # J,# h CD = C # D š C ()= C # C & œ,c œ h œc 7 = 2&(š C š œ ) 7 h C,œ 7 C,œ!#()! =[[# J,# ],# ] 25
Pertubative Continuous Unitary Transformations # ž =Ÿ+ž 1) The unperturbed Hamiltonian Ÿ must have an equidistant spectrum bounded from below.by R the corresponding subspaces are denoted. Ÿ = 2) The perturbing Hamiltonian links subspaces C and D only if. H is bounded from above. There is a number >0such R[}A R[OAF R that can be written as = F R increments(or decrements, if <0) the number of energy quanta by. Ÿ,F R = F R where # ž =Ÿ+ž R[}A R[OAF R 26
Pertubative Continuous Unitary Transformations # ž, =Ÿ+ž () 1) The unperturbed Hamiltonian Ÿ must have an equidistant spectrum bounded from below.by R the corresponding subspaces are denoted. Ÿ = 2) The perturbing Hamiltonian links subspaces C and D only if. H is bounded from above. There is a number >0such R[}A R[OAF R that can be written as = F R increments(or decrements, if <0) the number of energy quanta by. Ÿ,F R = F R where # ž, =Ÿ+ž R[}A R[OAF R () 27
Kitaev-Potts for d=3 # v u = ] L w 7 C C +w C ( ) ž (I C I D L +I C L I D ) C,D 2 E 1 E z = = 2+3 0 E z ; = 2 Ÿ C = 0 0 0 0 1 0 0 0 1 = O( B} B )}7Q ] Ÿ = Ÿ C C 3 2 &w C+w C L C = 9 2 Ÿ 28
F R operators I C I D L 00= 12 +2 21= 00 2 10= 22 +1 11= 20 1 12= 21 &(I C I D L +I C L I D ) C,D =F O7 +F O= +F ; +F }= +F }7 29
PCUT # = 9 2 Ÿ+{F O7 +F O= +F ; +F }= +F }7 } = Ÿ,# = 2F O7 F O= ()+F }= ()+2F }7 () Modified Generator: = Ÿ,# = F O7 F O= ()+F }= ()+F }7 () modified generator preserves the initial block band structure. 30
Flow equation!#()! =,# =[ F O7 F O= ()+F }= ()+F }7 (),H ] F ; =2[F }7 (),F O7 ]+2[F }= (), F O= ] F }7 = 9F }7 +[F }7 (),F ; ] F O7 = 9F O7 [F O7 (),F ; ] F }= = «7 F }= +2[F }7 (),F O= ]+[F }= (), F ; ] F O= = «7 F O= +2[F }= (),F O7 ]+[F ; (), F O= ] 31
Solving flow equation Example: F R = &F R œ () œ[= œo= F œ }7 ()= 9F œ }7 ()+&[F D }7 (),F œod ; () ] D[= F }7 = ()= 9F }7 = () F }7 = =F }7 0 O«# s = «7 Ÿ+F ;+ = «([F }7,F O7 ]+[F }=,F O= ])+ 7 = 8([F }7,[F ;,F O7 ]]+[[F }7,F ; ],F O7 ])- = = ([F }=,[F }=,F O7 ]]+[[F }7,F O= ],F O= ])+ 7 = ([F }=,[F ;,F O= ]]+[[F }=,F ; ],F O= ]) 32
# s =# ; +# = +# 7 + 33
34
Small coupling results 35
References 1. Topological order: from long-range entangled quantum matter to a unified origin of light and electrons, Xiao-Gang Wen. 2. Kitaev, A. Yu. Fault-tolerant quantum computation by anyons. Annals Phys.,303, 2003. 3. James R. Wootton, Jiannis K. Pachos. Universal Quantum Computation with Abelian Anyon Models. ENTCS, 270, 2011. 4. Christian Knetter, Kai P. Schmidt, Goetz S. Uhrig. The Structure of Operators in Effective Particle-Conserving Models. J. Phys. A: Math. Gen, 36, 2003. 5. Christian Knetter, Goetz S. Uhrig. Perturbation Theory by Flow Equations: Dimerized and Frustrated S=1/2 Chain. Eur. Phys. J. B, 13, 2000. 6. S. Dusuel, M. Kamfor, K. P. Schmidt R. Thomale J. Vidal. Bound states in two-dimensional spin systems near the Ising limit: A quantum finite lattice study. Phys. Rev. B 81, 064412 (2010) 36