Valence bond states: link models Quantum Information and Condensed - PowerPoint PPT Presentation

Wednesday, September 16th Valence bond states: link models Quantum Information and Condensed Matter Physics Enrique Rico Ortega Saturday, September 19, 2009 1

Collaborators and References H. J. Briegel R. Hübener S I T Ä R T E V U I N L M U · · O S C D I N E S. Montangero A N R D U O C · · D O O D C N E B. Pirvu N. Moran J. Vala Annals of Physics 323 (2008) 2115- 2131. arXiv:0710.2349 Annals of Physics 324 (2009) 1875- 1896. arXiv:0811.1049 Saturday, September 19, 2009 2

What is this talk about? We look for a 2D spin system in a square lattice with a ground state such that: i. Real singlet state of SU(2) (non-chiral). ii. Homogeneous, translationally and rotationally invariant. iii.With a local spin- 1 representation. iv.Unique ground state of a nearest neighbor Heisenberg-like hamiltonian. Saturday, September 19, 2009 3

Contents Motivation. Quantum spin liquids. (Why are VBS interesting?). Example of spin liquids in 1 D (AKLT model) Entanglement of spins on a square lattice. 2D multipartite valence bond states. Ground state properties and correlations. Field theory: bosonization Numerical methods: D.M.R.G., C.O.R.E, exact diagonalization Antiferromagnetic Mott-Hubbard insulator. Neutron scattering Saturday, September 19, 2009 4

Quantum spin liquids What is the responsible mechanism that causes certain materials to exhibit high-temperature superconductivity? Spin liquids ground states believed to be related to high-temperature superconductivity [P . Anderson, Science, 235: 1196- 1198, 1987] Saturday, September 19, 2009 5

Quantum spin liquids � ˆ m ˆ n � → const � = 0 S α S α Neel state and anti-ferromagnetic spin wave What are the possible ground states of 2D Heisenberg-like models when magnetic long-range order has been destroyed? A spin liquid is a quantum state without magnetic long-range order. A spin liquid is a state without any spontaneous broken symmetry. Saturday, September 19, 2009 6

Quantum spin liquids Example of 1 D spin liquid. - AKLT model [I. Affleck, T. Kennedy, E.H. Lieb, H. Tasaki. Phys. Rev. Lett. 59, 799 (1987)] � | ↑� | α � ǫ αβ | β � = | ↑↓� − | ↓↑� | α � = | ↓� √  2 | + 1 � α = β = ↑  Ψ αβ = | α �| β � + | β �| α �  √ | 0 � α � = β = √ 2  2 | − 1 � α = β = ↓  Saturday, September 19, 2009 7

Quantum spin liquids 1 � � � αβ | x � + σ y σ x αβ | y � + σ z A s √ (Ψ · ǫ ) αβ = αβ | z � = αβ | s � 3 s | x � ∝ | ↑↑� + | ↓↓� = | + 1 � + | − 1 � | y � ∝ | ↑↑� − | ↓↓� = | + 1 � − |− 1 � | z � ∝ | ↑↓� + | ↓↑� = | 0 � σ s j − 1 σ s j σ s j +1 σ s i − 2 σ s i − 1 σ s i σ s i +1 Saturday, September 19, 2009 8

Quantum spin liquids Antiferromagnetic spin- 1 chain m � = δ αβ ( − 1) m + n e − | m − n | / ξ � ˆ n ˆ S α S β No long-range order: � ˆ n � = � ˆ n � = � ˆ Singlet state: n � = 0 S x S y S z Translationally invariant Saturday, September 19, 2009 9

Quantum spin liquids Some properties: i) Composition rules.- s 1 2 ⊗ 1 2 = 0 ⊕ 1 β α 0 ⊗ 1 2 = 1 1 2 ⊗ 1 = 1 ⇒ SU (2) 2 2 2 ii) Boundary conditions and degeneracy.- Periodic boundary conditions = Unique state Open boundary conditions = 4-fold degeneracy Saturday, September 19, 2009 10

Quantum spin liquids iii) Two-point correlation function.- Exponential decay. Correlation length smaller than the lattice spacing iv) Non-local order parameter.- String order parameter and entanglement length den Nijs, Rommelse (1989) Cirac, Martin-Delgado, Popp, Verstraete (2005) Saturday, September 19, 2009 11

Entanglement of spins on a square lattice Experiment shows an anti-ferromagnetic ground state substantially different from “Neel order + minor QM corrections” [N.B. Christensen et al., PNAS, 104: 15264- 15269, 2007] 2D spin- 1/2 system (cuprates) “ substantial deviation” occurred at length scale about “ distance between two sites” deviation believed to be entanglement related Saturday, September 19, 2009 12

2D multipartite valence bond state Requirements: i) Real singlet state of SU(2) (non-chiral). ii) Homogeneous, translationally and rotationally invariant. iii) With a local spin- 1 representation. iv) Ground state of a nearest neighbor Hamiltonian. Minimum spin representation: 3/2 Saturday, September 19, 2009 13

2D multipartite valence bond state 2 We place the physical Hilbert space 1 4 at every link of the lattice 3 � | ↑� Local spin- 1 | α � = representation | ↓� √  2 | + 1 � α = β = ↑  Ψ αβ = | α �| β � + | β �| α �  √ | 0 � α � = β = √ 2  2 | − 1 � α = β = ↓  Saturday, September 19, 2009 14

2D multipartite valence bond state We need a multipartite bond state at every vertex Saturday, September 19, 2009 15

2D multipartite valence bond state Real singlet state of SU(2) (non-chiral). Homogeneous, translationally and rotationally invariant. | α � ǫ αβ | β � = | ↑↓� − | ↓↑� β α β α β α ± = θ γ θ γ θ γ Saturday, September 19, 2009 16

2D multipartite valence bond state i. The physical Hilbert space is placed at the links of the lattice. ii. The Hamiltonian is made out of nearest neighbor Heisenberg- like interactions. iii.It is homogeneous, translationally and rotationally invariant. iv.The ground state is a real singlet state of SU(2) (non-chiral). Saturday, September 19, 2009 17

Ground state properties and correlations ψ α 1 β 1 ψ α 3 β 3 Γ β 1 β 2 Locally.- β 3 β 4 ψ β 2 α 2 ψ β 4 α 4 � � R ij � VBS | VBS � = lk = Z 2 D configuration lattice β 1 β 2 Uncorrelated chains − = β 3 β 4 Critical theory + Saturday, September 19, 2009 18

Ground state properties and correlations Some generalization. - Ψ = a ( s ) σ 0 | 0 � + a ( t ) ( σ x | x � + σ y | y � + σ z | z � ) √ √ 1 − Λ 1 + 3 Λ a ( t ) = i a ( s ) = 2 2 β α β α β α ± = θ γ θ γ θ γ Saturday, September 19, 2009 19

Ground state properties and correlations Any expectation value is obtained via a mapping of the 2D quantum state to a 2D classical statistical model and from there to a 1d quantum mechanical problem using a transfer matrix defined from the 2D quantum state. Saturday, September 19, 2009 20

Ground state properties and correlations First analysis: Continuum limit. - [A.M. Tsvelik. Phys. Rev. B42, 10499 (1990)] a µ ] + H m s [ˇ a 0 ] H = � µ = { x,y,z } H m t [ˇ H = iv eff � � dx (ˇ a L ∂ x ˇ a L − ˇ a R ∂ x ˇ a R ) + im dx (ˇ a L ˇ a R ) 2 The ladder problem is equivalent to four Ising models. The only relevant operator is a mass term. Saturday, September 19, 2009 21

Ground state properties and correlations Relevance of the parameters: Inverse of the gap in the ladder = Correlation length in the 2D VBS. D.M.R.G. results with a sample of 100 points Saturday, September 19, 2009 22

Ground state properties and correlations Two points correlation function: Exponential or algebraic decay? | � rj | ri − � r j ) = � 0 | � S i · � g ( � r i − � S j | 0 � ≃ e − ξ ∆ ≃ 1 ξ ≃ N − θ θ ≃ 0 . 99(4) 2 0 -2 LOG � 2, � � -4 -6 2 4 6 8 scale Numerical results obtained from CORE calculations Data, fitted curve and 95% confidence interval Saturday, September 19, 2009 23

Ground state properties and correlations Two points correlation function: Exponential or algebraic decay? Results from D.M.R.G., C.O.R.E and exact diagonalization of the first energy gap as a function of the length and scale. All plots show a clear linear dependence of the gap with the inverse of the length of the ladder. Saturday, September 19, 2009 24

Ground state properties and correlations Two points correlation function: Relevance of the parameters The plots does not show a linear dependence of the gap with the perturbation Saturday, September 19, 2009 25

Antiferromagnetic Mott-Hubbard insulator. Copper oxide Universal structure.- Ionic configuration.- d 9 Cu 2+ O 2 − p 6 Bi 2 Sr 2 Ca n Cu n +1 O 2 n +6 Saturday, September 19, 2009 26

Antiferromagnetic Mott-Hubbard insulator. Super-exchange mechanism: Anderson 1950 Hybridisation of ionic orbital by covalent mixing E p = � σ p |H| σ p � E d = � σ d |H| σ d � Orbital energies: λ ≃ � σ p |H| σ d � Covalent mixing amplitude: E p − E d Saturday, September 19, 2009 27

Antiferromagnetic Mott-Hubbard insulator. Triplet (parallel) configuration | ↓ p � + λ | ↓ d L � + λ | ↓ d R � √ | ↓ p � covalent-mixing − − − − − − − − − − − → 1 + 2 λ 2 | ↑ p � | ↑ p � . Pauli principle − − − − − − − − − − → 1 E ↑↑ ≃ 1 + 2 λ 2 [( �↓ p | + λ �↓ d L | + λ �↓ d R | ) �↑ p |H| ↑ p � ( | ↓ p � + λ | ↓ d L � + λ | ↓ d R � )] 2 λ 2 = 2 E p + 1 + 2 λ 2 ( E p − E d ) . Saturday, September 19, 2009 28