Detection of symmetry protected topological phases in 1D Frank - PowerPoint PPT Presentation

Detection of symmetry protected topological phases in 1D Frank Pollmann Max-Planck-Institut für komplexer Systeme, Dresden, Germany FP. and A. M. Turner, arxiv:1204.0704 Florence, May. 29, 2012

Detection of symmetry protected topological phases in 1D Overview • Introduction: Symmetry protected topological phases • Non-local order parameters • Summary Florence, May. 29, 2012

Symmetry protected topological phases • Quantum phases : Two gapped quantum states belong to the same phase if they are adiabatically connected • Phases in condensed matter are usually understood using local order parameters (“symmetry breaking”) - Magnets : spin rotation and TR symmetry broken Magnetization as order parameter • Topological phases not characterized by any symmetry breaking • We introduce non-local order parameter for symmetry protected topological phases in 1D

Symmetry protected topological phases • Example: Spin-1 chain [Haldane ‘83] ... ... j � S j · � j ( S z j ) 2 H = P S j +1 + D P | S z = ± 1 � , | S z = 0 � (time reversal, inversion, symmetry, ...) Z 2 × Z 2 “Haldane Phase” “Large D Phase” | 0 i | 0 i | 0 i | 0 i AKLT no symmetry Phase no symmetry D [Affleck ‘87] broken transition broken • Hidden symmetry breaking [Kennedy-Tasaki ’92] Z 2 × Z 2 • String order parameter [den Nijs ‘89]

Symmetry protected topological phases • Spin-1 chain with less symmetries [Gu et al. ‘09] j ) 2 + B x j � S j · � j ( S z j S x H = J P S j +1 + D P P j   ➡ no symmetry Z 2 × Z 2       ➡ Haldane phase still well defined          Which symmetries are required? How to detect “topological” phases? ➡ Idea: Use entanglement and matrix- product states (capturing non-local properties)

Symmetry protected topological phases Schmidt decomposition (SVD ) C = UDV † • Decompose a state into a | ψ i A B superposition of product states: • Schmidt states: , Schmidt values: • are eigenstates of the reduced density matrix with

Symmetry protected topological phases • Example: Spin-1 Heisenberg chain j ~ S j · ~ H = P S j +1 0 10 N A N B X X | ψ 0 � C ij | i � A | j � B ... ... = A B i =1 j =1 X − 5 λ γ | φ γ � A | φ γ � B 10 = α γ λ 2 γ X ! λ 2 γ = 1 γ − 10 10 0 20 40 60 80 100 γ α • Schmidt values decay rapidly in ground states of gapped, local Hamiltonians ( area law! [Hastings et al. ’07] ): Matrix-Product representation

Symmetry protected topological phases • Matrix product state (MPS) representation X B T A j 1 . . . A j L B | Ψ i = | j 1 , . . . , j L i | {z } j 1 ,...,j L ψ j 1 ,...,jL • Matrices not uniquely defined: Canonical Form is directly related to the Schmidt decomposition: A j = Γ j Λ [Vidal ’02]           1 . . . χ   ψ ...,j 1 ,j 2 ,... = 1 . . . d

Symmetry protected topological phases • Matrices are directly related to the Schmidt decomposition             [ φ α ] j 1 ,j 2 ... = ... ... α A B • Left/right transfer matrices T have largest eigenvalue one with the identity as corresponding eigenstate                 | {z } T Σ ( αα 0 );( ββ 0 )

Symmetry protected topological phases • Transformation of an MPS under symmetry operations [Perez-Garcia ’07]       Σ Σ              , [ U Σ , Λ ] = 0                                                          ...wave function only changes by a phase                     • Time reversal ( ) and inversion ( ) Γ j → Γ T Γ j → Γ ∗ j j • Matrices are projective representations which tell U Σ us about topological phases [FP et al. ’10, Chen et al ’11]

Symmetry protected topological phases Use projective representations to classify phases! • Ground state is invariant under a symmetry group | ψ 0 � G with elements g 1 , g 2 , . . . , g n • Projective representation of the symmetry group U g j g j g k = g l : U g j U g k = e i φ jk U g l

Symmetry protected topological phases Use projective representations to classify phases! • Ground state is invariant under a symmetry group | ψ 0 � G with elements g 1 , g 2 , . . . , g n • Projective representation of the symmetry group U g j g j g k = g l : U g j U g k = e i φ jk U g l • Phase ambiguities classify the phases (Schur classes) ➡ Complete classification of topological phases in 1D [FP , A. Turner, E. Berg, M. Oshikawa ’10, Chen et al ’11]

Symmetry protected topological phases • Which symmetries stabilize topological phases? • Example : Rotation about single axis Z n R n = 1 ⇒ U n R = e i φ 1 ➡ Redefining removes the phase U R = e − i φ /n U R ˜ • Example : Phase for pairs Z 2 × Z 2 R x R y = R y R x ⇒ U R x U R y = e i φ xy U R y U R x ➡ Phases cannot be gauged φ = 0 , π away: topological phases

Symmetry protected topological phases • Example S=1 AKLT state [Affleck ‘87] ➡ | ψ � =             • Matrix-product state representation Γ i = σ i , i = x, y, z • Rotations represented by Pauli matrices and Z 2 × Z 2 thus U R x U R y = − U R y U R x • Inversion symmetry with : U I = σ y U I U ∗ I = − 1 • Time reversal with : U T R U ∗ T R = − 1 U T R = σ y

Symmetry protected topological phases • Framework to classify topological phases in 1D by looking at the “entanglement states” / MPS • “Topological” phase in a S=1 chain protected by - Z 2 × Z 2 - Inversion symmetry - Time reversal symmetry FP , E. Berg, A. M. Turner, and M. Oshikawa, Phys. Rev. B 81 , 064439 (2010) • Symmetry protected topological phases exist only in the presence of certain symmetries (not topologically ordered!)

Symmetry protected topological phases • How can we detect which phase a given state belongs to? • We discuss two ways to detect topological phases : (1)Directly extract the projective representations from a matrix-product state representation (very useful for iTEBD / iDMRG ) [Vidal ’07] [McCulloch ’08] (2)Non-local order parameters for inversion, and time reversal symmetry and a generalized string-order for internal symmetries

Non-local order parameter (1) • Get from the dominant eigenvector U Σ X of the generalized transfermatrix ( ) U Σ = X †                    X Σ jj 0 ˜ T Σ Γ j 0 , αβ Γ ∗ j, α 0 β 0 Λ β Λ β 0 ( αα 0 );( ββ 0 ) =           j,j 0                   • Overlap with transformed Schmidt states                                                   Σ       U Σ = X †           ⇔                                         | {z } | {z } T Σ X = X T 1 = 1

Non-local order parameter (1) • S=1 chain j ) 2 + B P j ~ S j · ~ j ( S z j S x H = P S j +1 + D P • stabilizes Haldane phase if Z 2 × Z 2 B = 0 ⇢ and 0 if symmetry broken O Z 2 × Z 2 = if symmetry not broken . 1 � U x U z U † x U † � χ tr z                  iMPS obtained using the iTEBD / iDRMG method

Non-local order parameter (1) • S=1 chain j ) 2 + B P j ~ S j · ~ j ( S z j S x H = P S j +1 + D P • Inversion symmetry stabilizes Haldane phase if B 6 = 0 ⇢ 0 if symmetry broken and O I = 1 χ tr ( U I U ∗ I ) if symmetry not broken .                  iMPS obtained using the iTEBD / iDRMG method