Detecting topological orders from Matrix-Product State based - PowerPoint PPT Presentation

Detecting topological orders from 
 Matrix-Product State based simulations Frank Pollmann Max Planck Institute for the Physics of Complex Systems International Workshop on Tensor Networks and Quantum Many-Body Problems 2016

Detecting topological orders from 
 Matrix-Product State based simulations Overview (1) Matrix-product states and efficient simulations - Review: Entanglement and matrix-product states (MPS) - MPS for infinite systems - Time evolving block decimation (TEBD) (2) Extracting fingerprints of topological order - Symmetry protected topological phases - Characterizing intrinsic topological orders - Symmetry enriched topological phases - (Dynamics of topological excitations)

Matrix-product states ���� ���� ���� ���� ���� ψ j 1 ,j 2 ,j 3 ,j 4 ,j 5 ≈

Entanglement • A generic quantum state has a dimensional Hilbert space d L X | ψ i = ψ j 1 ,j 2 ,...,j L | j 1 i | j 2 i . . . | j L i , j n = 1 . . . d j 1 ,j 2 ,...,j L • Decompose a state into a superposition 
 ... ... of product states ( Schmidt decomposition ) 
 A B X X | ψ i = C i,j | i i A ⌦ | j i B = Λ α | α i A ⌦ | α i B i,j α • Entanglement entropy as a measure for the 
 amount of entanglement 
 S = − P α Λ 2 α log Λ 2 α • Equivalent to with ρ A = Tr B | ψ ih ψ | S = − Tr ρ A log ρ A

Entanglement (a) (b) (a) (b) Area law for ground states of local (gapped) Hamiltonians 
 L R L R in one dimensional systems N S ( L ) = const . N [Srednicki ’93, Hastings ’07] (c) (d) (c) (d) (a) (b) Many body Hilbert space Many body Hilbert space X | ψ i = Λ α | α i A ⌦ | α i B L R α Area law states Area law states N L (c) (d) All ground states live in a tiny corner of the Hilbert space! Many body Hilbert space Area law states

Compression of quantum states N A N B X X X • Example: | ψ � C ij | i � A | j � B = λ γ | φ γ � A | φ γ � B = i =1 j =1 γ • Matrix can represent an image (array of pixel)     0 . 23 0 . 56 0 . 23 0 . 56 · · · · · · . . . . ... ... . . . . C =  =     . . . .    0 . 22 0 . 34 0 . 22 0 . 34 · · · · · · χ = 1200 • Reconstruction of the matrix (image) from a small 
 number of Schmidt states (SVD): ≈

Compression of quantum states χ = 1200 χ = 256 χ = 16 χ = 64 χ = 4 Important features visible already for states! < 16

Compression of quantum states [Mondrian]

Compression of quantum states • Coefficients in the many-body wave function: 
 Rank - L tensor : diagrammatic representation ���� ���� ���������� ψ j 1 ,j 2 ,j 3 ,j 4 ,j 5 = ���� ���� ���� ���� ���� ���� ���� ���� ���������� • Successive Schmidt decompositions: matrix-product states d d A [1] j 1 Λ [1] X X | ψ i = β | j 1 i | β i [2 ,...N ] (a) (b) β L R j 1 =1 β =1 N ���� ���� ���������� (c) (d) Many body Hilbert space Area law states ���� ���� ���� ���� ���� ���� ���� ���� ����������

Matrix-Product States • Matrix-product states: Reduction of variables: ���� ���� ���������� ���� ���� ���������� [M. Fannes et al. 92, Schuch et al ‘08] d L → Ld χ 2 ���� ���� ���� ���� ���� ���� ���� ���� ���������� ���� ���� ���� ���� ���� ���� ���� ���� ���������� A j αβ = ψ j 1 ,j 2 ,j 3 ,j 4 ,j 5 ≈ • Matrix-product operators [Verstraete et al ’04] [1] [2] [3] [4] [5] ���� M M M M M (a) ���� O j 0 1 ,j 0 2 ,j 0 3 ,j 0 4 ,j 0 5 j 1 ,j 2 ,j 3 ,j 4 ,j 5 = � � � � � � � � � � (b) ���� ���� M M M M M ���� ���� � � � � � � � � � � (c) ���� ���� M M M M M ���� ���� ���� ���� * * * * * � � � � � � � � � �

Infinite matrix-product states ∞

Infinite MPS and the canonical form • Infinite and translationally invariant systems: Ld χ 2 → d χ 2 [1] [2] [3] [6] [7] [4] [5] A A A A A A A | ψ i : . . . . . . • MPS is not unique ˜ A i n XA i n X − 1 = = ➡ describes the same state! ˜ A i n

Infinite MPS and the canonical form • Choose a convenient representation in Canonical Form : 
 Bond index corresponds to Schmidt decomposition! [Vidal ‘03] χ X h α | α 0 i = δ αα 0 | ψ i = Λ α | α i L ⌦ | α i R with α =1 A i n • Write tensor as product of αβ (a) : Diagonal matrix with Schmidt values (b) (a) : Tensor relating to Schmidt basis (b) L (a) (b) (c) (d) L ' ' (c) (d) ' � � � ' ' L L ' � � � (c) L (d) L R ' (e) ' L R ' (e) � � � � � L � � L R (e) � �

(a) (b) (a) (b) Infinite MPS and the canonical form L (c) (d) ' • Schmidt states in terms of the MPS: ' L ' � � � (c) (d) L ' ' ' � � � L R • Orthogonality: (e) L R � � L R (e) ' ' ' * * * � �

(a) (a) (b) (b) L L (c) (c) (d) (d) ' ' ' ' ' ' � � � � � � Infinite MPS and the canonical form L L • Conditions for the canonical form: L R L R (e) (e) � � � � • Left and right transfer matrix have dominant eigenvalue one and the corresponding eigenvector is the identity X Λ α Λ α 0 Γ j Γ j � ∗ T L � αα 0 ; ββ 0 = αβ α 0 β 0 j

Infinite MPS and the canonical form • Example: Affleck-Kennedy-Lieb-Tasaki (AKLT) • Ground state the spin-1 Hamiltonian S j +1 + 1 X S j ~ ~ 3( ~ S j ~ S j +1 ) 2 , H = j

Infinite MPS and the canonical form • Example: Affleck-Kennedy-Lieb-Tasaki (AKLT) and χ = 2 d = 3 r r r 4 2 4 3 σ + , Γ 0 = − 3 σ z , Γ 1 = − Γ − 1 = 3 σ − ✓ 1 r ◆ 1 0 = Λ 0 1 2

Infinite MPS and the canonical form • Efficient evaluation of expectation values : Γ Λ 2 Λ 2 h ψ | O n | ψ i = Γ ∗ T R Λ Λ Λ Λ Γ Γ Γ Γ Γ Λ 2 Λ 2 h ψ | O m O n | ψ i = Γ ∗ Λ Γ ∗ Λ Γ ∗ Λ Γ ∗ Λ Γ ∗

(a) (b) L (c) (d) ' ' ' � � � Infinite MPS and the canonical form L • Correlation length of an MPS: L R (e) 1 � � ⇠ = − log | ✏ 2 | , is the second largest eigenvalue of the transfer matrix ✏ 2 • Degeneracy of largest 
 eigenvalue (unity) shows 
 that the MPS is a cat-state 


Efficient numerical simulations i ~ ∂ ∂ t | ψ i = H | ψ i

Efficient numerical simulations • Transverse field Ising model 
 X ( J σ z j σ z j +1 + g σ x H = − j ) Ground state properties • j m = h σ z i Quantum Disordered g/J Ordered Dynamics • σ z ( t ) σ + ... 0 | ψ i ...

Time evolving block decimation • Assume we have a Hamiltonian of the form X h [ j,j +1] H = j Disclaimer: Simple! 
 • Time evolution in real time 
 Not perfect for all uses! | ψ t i = exp( � iHt ) | ψ t =0 i • Time evolution in imaginary time 
 exp( � H τ ) | ψ i i | ψ 0 i = lim || exp( � H τ ) | ψ i i || τ →∞

Time evolving block decimation X h [ j,j +1] • Consider a Hamiltonian H = [Vidal ‘03] j • Decompose the Hamiltonian as H=F+G X F [ j ] ≡ X h [ j,j +1] F ≡ even j even j X X G [ j ] ≡ h [ j,j +1] G ≡ odd j odd j F F F G G [ F [ r ] , F [ r 0 ] ] = 0 ([ G [ r ] , G [ r 0 ] ] = 0) • We observe 
 but [ G, F ] 6 = 0


 Time evolving block decimation • Apply Suzuki-Trotter decomposition of order p 
 
 exp ( − i ( F + G ) δ t ) ≈ f p [exp( − F δ t ) , exp( − G δ t )] f 2 ( x, y ) = x 1 / 2 yx 1 / 2 with , , etc. f 1 ( x, y ) = xy • Two chains of two-site gates Y exp( − iF [ r ] δ t ) = U F even r Y exp( − iG [ r ] δ t ) = U G odd r

Time evolving block decimation • Time Evolving Block Decimation algorithm (TEBD) 1 1 2 3 3 4 5 5 6 0 2 4 7 1 1 2 3 3 4 5 5 6 2 4 7 0 • How do we get the original form back?

Time evolving block decimation • Time Evolving Block Decimation algorithm (TEBD) truncation • Scales with the matrix dimension as χ 3

Time evolving block decimation (i) Γ A Λ A Γ B Λ • Density matrix renormalization group (DMRG) B B Λ Θ ~ (ii) Θ ~ Θ M M E 0 = L R (iii) ~ ~ truncation Θ A X Y Λ SVD ~ ~ ~ (iv) ~ Γ A Λ A Γ B Λ B B B Λ B A Λ X Y Λ Λ -1 -1 ( ) B ( ) B Λ Λ B ~ (v) Γ A Λ • Scales with the matrix dimension as χ 3 ~ L B ~ A * Λ (Γ ) ~ Γ B Λ B ~ R ~ B * Λ B (Γ )