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Classification of Topologically ordered Phases Ching-Yu Huang (⿈黃靜瑜) C.N. Yang Institute for Theoretical Physics State University of New York at Stony Brook, Stony Brook ! "Topological phases of matter, quantum field theory, and quantum information school/ workshop’’ Jan,02, 2015 1 Outline • Introduction: phase transition, topological order… • Symmetry-protected quantum state renormalization Detection of symmetry enriched topological order Summary • • 2 Matter occurs in different phases • • • Most phases of matter can be understood through the lens of spontaneous symmetry-breaking. For example: Crystals: translation and rotation symmetry broken Magnets: spin rotation and time-reversal symmetry broken Local order parameters distinguish different phases [Landau] ordered [Gu, Levin,& Wen’ 08] 3 disordered Topological order: Beyond the Landau Paradigm • • • • Fractional quantum Hall effect No symmetry breaking No local order parameters Characterized by - Ground state degeneracy [Tsui, Stormer, & - Fractional statistics of quasiparticles (anyons) - Topological entanglement entropy - Long-range entanglement (LRE) [Chen, Gu, & Wen’ 10] 4 Gossard ’82] Local unitary transformation and quantum state • For gapped system, a quantum phase transition can happen only when energy gap closes • Two gapped ground states, |Ψ1> and |Ψ2> , belong to the same phase - if they can connected by an adiabatic evolution that does not close energy gap - if they are related by a local unitary transformation. |Ψ1> = U |Ψ2> [Chen, Gu & Wen ’10] 5 Quantum states classification • Local unitary (LU) transformation Quantum information view – Entanglement • Short-range entanglement(SRE) Long-range entanglement (LRE) • Symmetric local unitary transformation (I) (II) LRE topological order (III) with symmetry (I) (a) LRE SRE SET order (I) (III) (II) breaking symmetry (I) SRE (III) (II) (II) (III) SPT order (I) (II) (III) breaking symmetry 6 (b) [Chen, Gu & Wen ’10] Matrix/Tensor product states • • Question: how to represent efficiently quantum states of a quantum lattice system Solution: divide the coefficient of the quantum state in a “Tensor Network” that only depends on a small number of parameters | Ψ〉 = ! • d d ∑ ...∑ c i1 =1 in =1 i1 ...in | i1...in 〉 The numerical implementation for finding the quantum states of 1D/2D spin systems are based on the matrix/tensor product states. 7 PART II : Symmetry-protected quantum state renormalization work with Xie Chen and Feng-Li Lin [Huang, Chen, & Lin PRB. 88,205124 (2013) ] 8 Symmetry-protected topological order phases • • • • Symmetry-protected topological order phases (SPT) Example: Topological insulator, Haldane phase in spin-1 chains SPT ground states have short-range entanglement which is protected by symmetry The local unitary transformations preserve some particular symmetry 9 Motivation • • Tensor Product State and quantum state RG provide a useful numerical tool to identify or even classify the topological order How to identify the symmetry-protected topological phases with this ideas? trivial phase Topologically ordered phase [ Chen, Gu & Wen ’10] 10 Quantum state renormalization group The usual renormalization group : effective Hamiltonian The wave function • • ! • Quantum state RG is to remove SRE and coarse grain • Symmetry-protected Quantum state renormalization group 11 1-Dimensional AKLT phase [I. Affleck, T. Kennedy, E. H. Lieb, & H. Tasaki . 1987] • The 1-dimensional AKLT state • Merging neighboring sites : the valence bond solid The fixed-point state of the QSRG [Verstraete et al. ’05] • Preserving symmetry transformations trivial (product) state 12 non-trivial state Symmetry-protected Quantum state renormalization group (SP-QSRG) • Schematic procedure ! ! ! ! [Huang, Chen, & Lin’13] ! • The key point is to make sure the physical leg be in the same projective class (half integer and integer) • 1D AKLT and 1D dimer state flow to the same fixed point 13 2-Dimensional AKLT phase • • Spin 3/2 particle on the honeycomb lattice The Hamiltonian of the system is ! ! • AKLT point 14 2-Dimensional (SP)- QSRG • Remove SRE - the dimer on the middle bond gets removed ! ! ! • coarse grain ! • Ensure the physical indices j1 and j2 are in the same class of projective representation [Huang, Chen, & Lin’13] 15 2-Dimensional AKLT-like model • Phase diagram ! ! ! ! 2 ! ! 3 H = ∑ Si ⋅ S j + J 2 ∑ ( Si ⋅ S j ) + J 3 ∑ ( Si ⋅ S j ) <ij > <ij > <ij > ! • • TPS method Tensor renormalization group 1.4 1.2 1.5 1.0 1.0 0.8 0.5 0.6 0.4 0.2 0.0 -1.0 16 AKLT phase 0.0 J2=0.5 -0.1 0.1 0.2 J3 MZ MZs λ1−λ2 -0.5 0.0 AKLT phase 0.0 J3 0.5 1.0 2-Dimensional AKLT-like model • Phase diagram , fix J2=0.5 • • (SP)-QSRG results √ ! 17 Summary • • • We try to promote the QSRG as a useful numerical tool in finding and classifying the SPT phase via MPS/TPS. A key observation is that preserving symmetry in the QSRG allows us to study fix points of symmetry protected topological phases We have consider the 1- and 2-dimensional AKLT phases as the examples. 18 outlook • • • Generalize QSRG procedure to SET phases Generalize this QSRG procedure to quantum dimer models Quantum Dimer Model on the Triangular Lattice has Z2 fixed point | i = (1 g)| icol + g| irvb 19 PART III : Detection of symmetry enriched topological order work with Xie Chen and Frank Pollmann [Huang, Chen, & Pollmann. PRB, 90,045142 (2014)} 20 Topological order: Topological degeneracy • • Topological degeneracy depends on the topology of space g Ground state degeneracy (N ): ! ! • g=0 g=1 g=2 For example: υ=1/3 FQH - Degeneracy on torus= 3 Z2 liquid - Degeneracy on torus= 4 [Wen & Niu ‘90 ] [Laughlin ‘83] Degeneracy sector: winding number ( Even and odd) 21 Topological order: anyons • • Topological order characterized by its quasiparticle excitations “anyons” (with nontrivial braiding statistics) Particle Exchange iθ e Boson/ Fermion ψ (x1 ,x 2 ) = ± ψ (x 2 ,x1 ) Abelian anyons ψ (x ,x ) = e ψ (x ,x ) Non-Abelian anyons ψ a (x1 ,x 2 ) = M ab ψ b (x 2 ,x1 ) iθ 1 2 2 22 1 Topological entanglement entropy • Topological entanglement entropy (TEE) with smooth boundary, circumference L S = cL - γ ! ! B A ! • ϒ >0, ϒ=log D (D: total quantum dimension ) KP S = S + S + S + S − S topo A B C D AB • Z2 gauge theory: ϒ=Log 2 − S BC − SCD − S AD + S ABCD • General partition [Kitaev &Preskill ’06; Levin & Wen ’06; Li & Haldane ’08] Minimally entangled states • A bipartition with “non-contractible” boundary, where the TEE depends on the ground state ! A S = cL - γ ' B Ψ = ∑ α λα φα ,a A φα ,a B ! • • There is a special basis of ground state for an “Nontrivial bipartition”- Minimally Entangled States (MES) The minimally entangled state can be identified with the quasiparticles of the topological phase [Zhang et al. ’12; Grover et al. ’12; Cincio & Vidal ’12; Wen ’90] 24 Motivation Topologically ordered systems have a richer structure when symmetries are present - Symmetry enriched topological order (SET) phase • Trivial phases Topological order B • Topological order A Topological order A2 Topological order A3 Topological order A1 Toric code : Z2 liquid How to detect the SET phase? Order parameter, some quantities ? 25 RVB (resonating valence bond state) liquid: Z2 liquid + SU(2) symmetry Fractionalization Symmetry fractionalization • Laughlin’s liquid - When an electron is added to a FQH state • RVB state on the kagome lattice - a pair of excitations carrying a spin ½ each Electron charge= e Quasiparticles (charge = e/3 for υ=1/3 ) Quasiparticles carry “fractional “symmetry representation ! 26 Symmetry enriched topological order • Hamiltonian and ground state have the same symmetry ! • U G H =Gψ For abelian group, fractionalization of symmetry operators - Quasiparticles carry projective representations ! • Classify symmetry fractionzation with Second cohomology group [Chen,Gu & Wen ’10; Pollmann & Turner ‘12 ] 27 Ug g ∈ Hψ + g a a∗ Z2 topologically ordered phase • • • Z2 topologically ordered phase has four quasiparticle types Four minimally entangled state Tensor product state representation of four anyons can be obtained from the complete set of ground states ! • • Consider “spin 1/2 representations on the Z2 gauge charges” How to detect symmetry enriched topological ordered phases? - projective representation and non-local order parameter 28 Spin-1 Bosons on the hexagonal lattice • • • Z2 loop state of S = 1 AKLT chains on the honeycomb (AKLT string state) An excited state with two defects which carry a spin 1/2 each. It can be represented by χ=3 tensor product state ω = 01 − 10 + 22 (S=1/2) ⊕ ( S = 0) d = 4 (ϕ , s = 1, 0, −1) 29 Spin-1 Bosons on the hexagonal lattice • The non-local order parameter Projective representations of quasiparticles ! a (g, X L , X R ) = O ! ! ! • • ! • Projective representations from the minimally entangled states 30 lim n→∞ $ n −1 % R Ξ X (1) ' ∏ g (k ) ( X (n) Ξ a ) k =2 * a L Selection rule forces the nonlocal order to vanish if edge spins are fractionalized RVB phase on kagome lattice Z2 topological order A singlet can fractionalize into two spinons which carry a spin ½ Topological entanglement entropy • • • Projective representation • ! • Non-local parameter • In the same SET phase as the spin-1 model above 31 Conclusion and outlook • The symmetries can enrich the structure of topological phases. • Propose methods to detect the symmetry enriched topological order phases by measuring projective representations and the non-local parameters ! • Generalize this method to non-abelian group • If we have a Hamiltonian (with topological order), numerically, how to classify all possible state? • The applications : Quantum computation? 1D AKLT state: [Chen, Duan, Ji, & Zeng’ 10 , Else, Schwarz, Bartlett, & Doherty’ 12] 2D AKLT state [Wei, Affleck, & Raussendorf ’11 ; Wei, Affleck, & Raussendorf ’12] 32 Thank You 33