Boundary theories for spins in lattices J. IGNACIO CIRAC GGI, - PowerPoint PPT Presentation

Boundary theories for spins in lattices J. IGNACIO CIRAC GGI, Florence, May 24, 2012 IC, Poilblanc, Schuch, and Verstraete, Phys. Rev. B 83, 245134 (2011) Poilblanc, Schuch, Perez-Garcia, IC, arXiv:1202.0947 Schuch, Poilblanc, IC, Perez-Garcia, arXiv:1203.4816

TENSOR NETWORKS N spins: States and observables can be written in terms of tensors ∑ Ψ 〉 = 〉 ,..., i i i | | ,..., c i i 1 N i 1 ,..., N i 1 c 1 N i i N ∑ j i = 〉〈 ,..., j j | ,..., ,..., | X X j j i i 1 N 1 1 j ,..., j X 1 i ,..., i 1 N 1 N N 1 N i ,..., i , j i 1 i j N N N Expectation values are tensor contractions: ∑ ∗ 〈 Ψ Ψ 〉 = j ,..., j i ,..., i | | X c X c 1 1 N N j ,..., j i ,..., i 1 1 N N i , j

TENSOR NETWORKS Rewrite tensors in terms of smaller tensors: STATES: i Tensor network states 1 i 1 = or i Tensor product states N i N OBSERVABLES: similarly a b c Why? Efficient description: N d D Guiding principle: entanglement

TENSOR NETWORKS EXAMPLES Multi-scale ENTANGLEMENT Projected ENTANGLED-pair states renormalization ansatz MERA: G.Vidal PEPS: F.Verstraete, I. Cirac i 11 i i 1 N i 1 N i NN

TENSOR NETWORKS PEPS Projected ENTANGLED-pair states Thermal equilibrium N 2 Local interactions Arbitrary dimensions (Hastings) physically relevant Numerical algorithms

TENSOR NETWORKS PEPS Projected ENTANGLED-pair states Thermal equilibrium N Local interactions 2 Arbitrary dimensions (Hastings) physically relevant Many-body physics i A n αβγδ γ β α i δ n

PEPS Projected entangled-pairs: | Ψ〉 Physical spins:

PEPS Projected entangled-pairs: | Ψ〉 Φ〉 ⊗ N | Physical spins: Auxiliary spins:

PEPS Projected entangled-pairs: | Ψ〉 Φ〉 ⊗ N | Physical spins: Auxiliary spins: ∑ = 〉〈 α β γ δ i | , , , | P A i αβγδ P‘s act locally Contain the information about the state Similar to AKLT construction

PEPS Projected entangled-pairs: | Ψ〉 Φ〉 ⊗ N | Physical spins: Auxiliary spins: ∑ = 〉〈 α β γ δ i | , , , | P A i αβγδ γ α β i A n αβγδ i δ n

PEPS Area law: A ρ  ( ) S N ∂ A A

PEPS Area law:

PEPS Area law: A Only the auxiliary particles at the boundary contribute Linear maps P cannot increase entanglement

PEPS Area law: Only the auxiliary particles at the boundary contribute Linear maps P cannot increase entanglement ρ  ( ) S N ∂ A A # degrees of freedom in the bulk scale with the size of boundary

PEPS Bulk-boundary correspondence: Bulk Boundary A U ⊗ → ⊗∂ UU = U U = A A : isommetry † † U H h 1 h 1 H ρ σ ∂ = ρ † U U A A A ∂ = † X x UX U A A A

PEPS Bulk-boundary correspondence: Bulk Boundary ρ σ ∂ = ρ † U U A A A ∂ = † X x UX U A A A Expectation values: 〈 〉 = ρ = ρ = σ = 〈 〉 † † tr( ) tr( ) tr( ) X X UX U U U x x ∂ ∂ ∂ A A A A A A A A Boundary Hamiltonian: ρ = − σ ∂ = − H H ∂ e e A A A A ∂ = † H UH U A A σ = σ Entanglement spectrum:: ( ) ( ) H H ∂ A A The standard ES is exactly the spectrum of the boundary Hamiltonian The boundary Hamltonian has a physical meaning

PEPS Boundary theory: The boundary operators can be determined using PEPS algorithms σ ∂ = − H ∂ e A A The boundary Hamiltonian reflects properties of the original state | Ψ〉 θ ⇒ = Ψ〉 = Ψ〉 i † | | U H U H Symmetries: u e g ∂ ∂ g A g A g Topology: Non-local projector Criticality: Ψ If is the ground state of a GAPPED LOCAL Hamiltonian, then the boundary Hamiltonian is LOCAL

PEPS Examples: AKLT model in 2D - Auxiliary particles s=1/2 H ∂ is the 1D Heisenberg Hamiltonian (2) su - Symmetry: A - Finite correlation length ES corresponds to c=1 CFT Kitaev‘s toric code - Auxiliary particles s=1/2 σ ∂ ( ) is a non-local projector Z Z - Symmetry: 2 A 2 - Finite correlation length - Topological ES is flat RVB on a Kagome lattice - Auxiliary particles s=1 σ ∂ ( ) contains a non-local projector 2 Z (2) su - Symmetry: A - Finite correlation length H ∂ is a 1D t-J model A - Topological

PEPS Z spin liquids 2 Kitaev‘s toric code o-RVB (RK) (square lattice) (Kagome lattice) RVB (Kagome lattice)

PEPS Z spin liquids 2 local unitary Kitaev‘s toric code o-RVB (RK) (square lattice) (Kagome lattice) local invertible RVB (Kagome lattice) They correspond to the same phase RVB is ground state of local (FF) Hamiltonian (4-fold degeneracy)

OUTLINE How to determine the boundary theory for a PEPS Symmetries Finite correlation length Examples

PEPS BOUNDARY THEORY Reduced state: Combine the tensors of B regions A and B A A B | Ψ〉 boundary Polar decomposition: Reduced state: = σ σ V U A B σ ∂ = σ σ σ ρ = Ψ〉〈Ψ σ σ σ † tr (| |) U U A A B A A B A B A

PEPS BOUNDARY THEORY In practice: 1.- Contract the tensor A with ist complex conjugate = σ σ , 2.- Determine A B

PEPS BOUNDARY THEORY Cylinder: Exact calculations N ×∞ A σ = σ Reflection symmetry: A B σ ∂ = σ 2 A A with MPS algorithms ∞×∞

PEPS SYMMETRIES Gauge : Y − 1 = X − 1 X Y i i A B αβγδ n n αβγδ Different tensors give rise to the same state: Under general conditions, the above is the only possibility (Perez-Garcia, Sanz, Gonzalez, Wolf, IC, 2009)

PEPS SYMMETRIES θ Ψ〉 = Ψ〉 i | | U e g Symmetry : g must be related by a local Gauge trafo u ⊗ = N Global symmetry: U g g u g † w = g † v v g g w g where the v‘s and w‘s are (projective) representations of the same group ⊗∂ ⊗∂ σ = σ † A A v v Boundary operator has the same symmetry: ∂ ∂ A g A g

PEPS FINITE CORRELATION LENGTH Cylinder : α  1, α  2, α  RG 3, σ ? A = For pure states (MPS): full classification (Verstraete, IC, Latorre, Rico, Wolf, 2005)

PEPS FINITE CORRELATION LENGTH RG for mixed states MPDO : entangled states Trace-preserving CPM ∑ − σ = ⊕ h + Boundary theory: n n , 1 e ∂ A Local Hamiltonian Degeneracy and topology

PEPS EXAMPLES 2D AKLT in a 2-leg ladder/square lattice: N N v v N h S = spin 2 ∑ = ∆ ∆ ∆ ∆ Deformed AKLT Hamiltonian ( ) ( ) ( ) ( ) H Q Q P Q Q n m n m , m n < > , n m (2) / (1) Symmetry: su u nematic deformation projector onto S=4 subspace ∆ = e − ∆ 2 8 S ( ) Q z Ground state: PEPS with D=2 ∂ = ∑ ∑ all possible terms with range-r interacctions Boundary Hamiltonian: H d A r r

PEPS EXAMPLES 2D AKLT in a 2-leg ladder/square lattice: For AKLT the boundary Hamiltonian is s=1/2 Heisenberg Similar results with other models

PEPS EXAMPLES Kitaev toric code: Degenerate ground state. Gapped. Z Symmetry: 2 Ground state: PEPS with D=2 Boundary state σ ∂ = ⊕ P P A even odd Non-local operator Boundary Hamiltonian: trivial (up to the projector)

PEPS EXAMPLES 2D RVB on a Kagome lattice: 1. single spin ½ at each edge Parent Hamiltonian acting on two stars PEPS with D=3 1 (2) su 2 ⊕ representation 0 Boundary Hamiltonian: t-J model 2. Three spins ½ at each edge: dimers are orthogonal (related to KR model)

PEPS EXAMPLES 2D RVB on a Kagome lattice: 1. single spin ½ at each edge

PEPS EXAMPLES Ψ θ 〉 | ( ) 2D: interpolation RVB-oRVB: Ψ 〉 = 〉 Ψ 〉 = 〉 | (0) | RVB | (1) | oRVB correlation function fidelity RVB and toric code seem to be in the same phase

PEPS EXAMPLES ‚Uncle‘ Hamiltonians Fernandez, Schuch, Wolf, IC, Perez-Garcia, arXiv:1111.5817 An order parameter for gapped phases in 1D Haegeman, Perez-Garcia, IC, Schuch, arXiv:1201.4174 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ = 〈Ψ ⊗ ⊗ ⊗ ⊗ Ψ〉 N N N N N N | ( 1 ) (1 ) | o u u F u u 1 2 3 1 2 1 g g 13 h h Parent Hamiltonian for Laughlin spin state in a lattice poster Anne Nielse, IC, German Sierra, arXiv:1201.3096

SUMMARY and CONCLUSIONS N 2 physically relevant CONCLUSION: Thermal equilibrium and local interaction spins can be efficiently described by PEPS Numerical algorithms New perspective HERE: Area law: bulk-boundary correspondence Boundary reflects properties of the bulk: criticality, topology, etc Finite correlation length implies locality of boundary Hamiltonian Locality + symmetries dictate entanglement spectrum Applicaton: contraction of PEPS is efficient

TENSOR NETWORKS PROJECTED ENTANGLED-PAIR STATES Physical interpretation: ∑ = 〉〈 α β γ δ i | , , , | P A i αβγδ = ∑ Why do they provide efficient descriptions? H h , n m < > , n m M   β − ∏ h − β ⊗ ⊗ Ψ 〉 ϕ 〉 = ϕ 〉  n m , H N N   | lim | | e e M 0 β →∞  

TOPOLOGICAL PHASES Symmetries: w g = † v v g g u g † w g Gauge symmetries: † w w g g = = † ⇔ v v v v g g g g w † w g g