Efficient description of many body systems with prjected entangled pair states

Transcription

1 Efficient description of many body systems with prjected entangled pair states J. IGNACIO CIRAC Workshop on Quantum Information Science and many-body systems, National Cheng Kung University, Tainan, Taiwan, December 19, 2009

2 QUANTUM MANY-BODY SYSTEMS EFFICIENT DESCRIPTIONS: LATTICES h Spins, Fermions Hamiltonian: H = hnm, < nm, > Questions: Ground state: Thermal state: Dynamics: H Ψ = E Ψ ρ e H / T iht Ψ ( t) = e Ψ (0) Physical properties: σ, σσ,... n n m

3 QUANTUM MANY-BODY SYSTEMS EFFICIENT DESCRIPTIONS: LATTICES h Spins, Fermions Problem: 2 N c Hilbert space c c N Exp. number of parameters. Exp. number of operations.

4 QUANTUM MANY-BODY SYSTEMS EFFICIENT DESCRIPTIONS: LATTICES h 2 N Hilbert space but H H( p,..., p ) = 1 N Ψ 0 H =Ψ0 p1 pn ( ) (,..., ) depends on few parameters. We are only interested in some special states

5 QUANTUM MANY-BODY SYSTEMS EFFICIENT DESCRIPTIONS: LATTICES h 2 N Hilbert space physically relevant but H H( p,..., p ) = 1 N Ψ 0 H =Ψ0 p1 pn ( ) (,..., ) depends on few parameters. We are only interested in some special states

6 EFFICIENT DESCRIPTIONS: h 2 N Hilbert space physically relevant Example: spin 1/ σ = σ = i σ = i σ = 0 1 General 2-body interactions: N 3 H = λ σ σ Ψ nm, = 1 αβ, = 0 ( λ ) α, β 0 nm, nm, n m α, β α β Short-range, homogeneous 3 n m n m H = λ σ σ n m < 1 αβ, = 0 α, β α β 2 there are 16N coefficients there are 16R d coefficients

7 QUANTUM MANY-BODY SYSTEMS EFFICIENT DESCRIPTIONS: LATTICES Methods: 2 N Exact Monte Carlo. Perturbation expansions Density Functional Theory Dynamical Mean Field Theory physically relevant Ψ (,..., ) p1 p D Variational - Mean Field, Hartree-Fock, BCS, Laughlin, - Tensor network states

8 QUANTUM MANY-BODY SYSTEMS EFFICIENT DESCRIPTIONS: LATTICES We are interested in a small corner of Hilbert space physically relevant May be one can find an efficient description of those states: Few parameters (eg, scaling polynomically with N) Expectation values can be efficientily calculated. 2 N Ψ (,..., ) p1 p D Applications: Variational family: Numerical algorithms. Describe general properties. But what is such a description?

9 QUANTUM INFORMATION EFFICIENT DESCRIPTIONS: LATTICES TENSOR NETWORKS i 1 i 2 2 N i 3 physically relevant Ψ (,..., ) p1 p D Novel descriptions: Efficient Intuition: entanglement No sign problem: frustrated spin and fermionic systems. Complementary to other methods (cf, Monte Carlo) Extension of DMRG, NRG and other renormalization methods

10 QUANTUM INFORMATION EFFICIENT DESCRIPTIONS: LATTICES PROJECTED ENTANGLED-PAIR STATES (Verstraete, IC 04) Properties: A A A A A A A A A A A A Build to fulfill the area law A A A A Naturally arise in problems with short-range interactions. They contain the relevant states for short-range interactions Infinite homogeneous systems: - Example: AKLT (in any dimension) - In a 2D square lattice: Vertex-Type Matrix Product Ansatz (Sierra and Martin-Delgado, 1998) (different than Interaction-Round the Face used by Nishino and col) (TNS = TPS) For 1D systems, they coincide with MPS (Zittart et al) - Infinite homogeneous systems FCS (Fannes, Nachtergaele and Werner 91).

11 QUANTUM INFORMATION EFFICIENT DESCRIPTIONS: LATTICES PROJECTED ENTANGLED-PAIR STATES (Verstraete, IC 04) A A A A A A A A A A A A A A A A See: Cirac and Verstraete (J. Phys. A, 2009) Verstraete, Murg and Cirac (Adv. Phys, 2008)

12 OUTLINE EFFICIENT DESCRIPTIONS: PROJECTED ENTANGLED-PAIR STATES. AREA LAWS THERMAL EQUILIBRIUM: SYMMETRIES MPS and CONFORMAL FIELD THEORY MODELS OTHER METHODS: STRING-BOND AND PLAQUTTE STATES

13 1. EFFICIENT DESCRIPTIONS: PROJECTED ENTANGLED-PAIR STATES

14 PEPS DEFINITION Verstraete and IC 04 Let us consider our particles as composite objects Atom Spin 1 particle Low energy 87 Rb The three levels are entangled states of many degrees of freedom We obtain them by projecting out the state: s = P Ψ P : H H H H C n el pos el pos Ψ el spin Ψ nuc pos Ψ nuc spin el spin nuc pos nuc spin 3

15 PEPS DEFINITION Verstraete and IC 04 Let us consider our particles as composite objects Spin ½ atoms is composed of electrons, protons, etc Auxiliary particles have different dimensions: D We only see the macroscopic objects: We project onto a 2-dimensional subspace. The auxiliary particles are in a very simple state: maximally entangled. The state is determined by the way we project onto the spin 1/2 space. The state is completely characterized by N projectors.

16 PEPS EXAMPLE Example: GHZ state Ψ = 1,1,1 + 2, 2, 2 Auxiliary construction: 1,1 + 2,2 1,1 + 2,2 P = 1 1, ,2 1,1 1,1 + 1,1 2,2 + 2,2 1,1 + 2,2 2,2 1,1,1 2,2,2 We can specify the state by just giving P = 1 1, ,2

17 PEPS EXAMPLE Verstraete and IC 04 Example: 2D lattice Φ

18 PEPS EXAMPLE Verstraete and IC 04 Example: 2D lattice P i, j Φ D D D D P : C C C C C n 2

19 PEPS EXAMPLE Verstraete and IC 04 Example: 2D lattice P i, j Φ D D D D P : C C C C C n 2 All the information is contained in the P s 4 There are 2D N parameters

20 PEPS EXAMPLE Verstraete and IC 04 Example: 2D lattice written in a basis: 1 i i1,..., in Ψ = c i,..., i N P = A i i αβγδ αβγδ A A A A A A A A A A A A A A A A One can define them in any lattice in any dimension One can define them for Fermions too (Kraus, Schuch, Verstraete, IC, 09)

21 2. AREA LAWS: PROJECTED ENTANGLED-PAIR STATES

22 AREA LAWS EFFICIENT DESCRIPTIONS 2 N general properties: Guiding principle: Area law (Sredniky 93) - There is a spatial structure. - Hamiltonian has short-range interactions, etc - Low temperatures A B Area law: S( ρ ) < O( L A d 1 Conjecture: All physically relevant states at T=0 fulfill this law (up to log corrections). ) ρ = tr Ψ Ψ A B

23 AREA LAWS CORRELATIONS Idea: at long distances, particles are uncorrelated. ξ A B

24 AREA LAWS CORRELATIONS Idea: at long distances, particles are uncorrelated. ξ A B SA E( A, B) ξ N In some way, the existence of a correlation length should give rise to the area law.

25 AREA LAWS MUTUAL INFORMATION (Wolf, Hastings, Verstraete, Cirac 08) Finite T: quantum mutual information A B I( A: B): = S( ρ ) + S( ρ ) S( ρ ) A B AB It measures correlations, but is stronger : I( A: B) ρ ρ ρ 1 2 AB A B 2 1 I( A: B) The mutual information does not overlook correlations (cf data-hidding states) it coincides with the entanglement entropy for T=0. X Y X Y A B A B X A YB 2 It fulfills area laws for Gibbs states. Area law is a consequence of the decay of correlations.

26 AREA LAWS THERMAL EQUILIBRIUM (Wolf, Hastings, Verstraete, Cirac 08) A B i) Hamiltonian with short-range interactions: ii) Finite temperature. ρ T 1 Z H / kbt = e H = hi, j i, j h op I( A: B) N kt B where N = # particles at the border A-B

27 AREA LAWS THERMAL EQUILIBRIUM PROOF: Main tool: FREE ENERGY: F( ρ): = H ρ k B TS( ρ) Free energy is minimized by the Gibbs state: min F( ρ) F( ρ ) where = T ρ T 1 Z H / kbt = e A B F( ρ ) F( ρ ρ ) AB A B H ρ ρ H A B ρab S( ρa ρb) S( ρab) kt Hab ρ ρ Hab ρ I( A: B) kt B h op N kt B B A B AB H = HA + HB + Hab A B

28 AREA LAWS CORRELATIONS (Wolf, Hastings, Verstraete, Cirac 08) A B L For fixed B, I ( B): = I( A: B) decreases with L. L We denote by, the length L such that for all B (sufficiently large) ξ I ( B) I ( B) 1 L= ξ 2 L= 0 (i.e, ξ is the correlation length) I I ( B ) ( B ) L= 0 2 L= 0 1 ξ L ξ I ( ) 4ξ 0 B N If is finite then L=

29 AREA LAWS CORRELATIONS PROOF: The mutual information decreases with L. a A B I( A: B) I( Aa: B) L But cannot decrease too much: I( Aa: B) I( A: B) + 2Sa We choosel = ξ I ( B) I ( B) + 2 LN I ( B) + 2LN 1 L= 0 L= ξ 2 L= 0 I ( ) 4ξ 0 B N L=

30 AREA LAWS T=0 h op I( A: B) N kt B a A B 1D systems: L 1 α Sα ( ρl ) log 2[tr( ρ )] 1 α Renyi entropies: c+ c 1 Critical systems: CFT: Sα ( ρl) = 1+ log2 12 α Non-critical: S ( ρ ) < C α L α L (Vidal, Rico, Kitaev, Latorre) (Calbrese, Cardy, Korepin,..) Higher dimensions: Gaussian theories (Eisert et al) Spin/Fermionic systems (Wolf, Perez-Garcia et al, )

31 AREA LAWS PEPS PEPS is the natural family fulfilling the area law: The mixed state version, also fulfills the area law. Φ They provide the natural way of expressing the area law

32 3. THERMAL EQUILIBRIUM: PROJECTED ENTANGLED-PAIR STATES

33 THERMAL EQUILIBRIUM PEPS Any action between two systems can be described in terms of ancillas: (Kraus, Dur, Lewenstein, and IC 01, Verstraete and IC 04) O P P Φ 1 2 P1 P 2 ab PN O12 O23 O N 1, N Thermal states can be written in terms of a purification: e βh = tr( Ψ Ψ ) Ψ = 1 Φ H e β βh / 2 N with [ e ]

34 THERMAL EQUILIBRIUM PEPS Consider local Hamiltonians: H = h n, n+ 1 Let us assume that the h s commute: n βh βh h h 12 β β 23 N 1, N e = e e... e e βh e βh 12 e βh 23 βh N 1, N e P1 P 2 PN βh βh / M h / M h M If the h s do not commute: [ ] M N 1, N / 12 β β e There are M bonds between neighbors, but they are weakly entangled It can be approximated with a single bond of dimension D. This argument has been made rigorous by Hasting (Hastings 06). It also works for Fermions (Kraus, Schuch, Verstraete, and IC 09). = e e e

35 4. SYMMETRIES: PROJECTED ENTANGLED-PAIR STATES

36 SYMMETRIES TRANSLATIONS Translational: T Ψ = Ψ P i, j Φ We choose all the P equal: D D D D P : C C C C C 2 A single operator P describes the whole system. All physical properties, including symmeries, are encapsultated in P.

37 SYMMETRIES LOCAL Local symmetries: N u g Ψ = e iθ g Ψ where g G For the PEPS: u g PV 4 g = P D D D D P : C C C C C e iφ g where V g is a D-dim representation. 2 Applications: Lieb-Schultz-Mattis theorem: G = su(2) For a 1D systems and semi-integer spin, V g must be reducible. If reducible, then for any H for which Ψ is the ground state, it must be degenerate. It also applies to higher dimensions. (Sanz, Perez-Garcia, Wolf, IC 08) Extensions to other grups: u(1), Z d, etc String order, topological order, etc. (Perez-Garcia, Sanz, Wolf, Verstraete, IC 08)

38 SYMMETRIES LOCAL 2 N Since PEPS represent the states that appear in Nature, we should develop the theory of those states. Whatever we can show to hold for PEPS, it should be true in general (?).

39 5. CONFORMAL FIELD THEORIES: MATRIX PRODUCT STATES (In collaboration with German Sierra, Madrid)

40 MATRIX PRODUCT STATES 1 DIMENSION 1D systems: P1 Pn P n D D : C C C 2 In a basis: P n = D 1 [ An ] x, y= 1s= 1 s x y s x, y, Matrix Product States: (Fannes, Narchtengaele, Werner 08) s1 sn Ψ = tr( A1... A ) s1,..., s s... s 1 N =± 1 N N For finite D, all correlation functions decay exponentially. In order to describe exactly a critical system, we have to take We have taken a QFT for the auxiliary particles. D

41 MATRIX PRODUCT STATES CFT 1D systems: P1 P n Pn D 1 s D 2 C Pn = [ An ] s x, y x, y x, y= 1s= 1 D : C C State: Ψ = s... s 1 N cs 1.. s ( α, z1,..., z ) 1... N N s =± 1 Variational parameters s N V s ( α, z ) = : e n i α sφ ( z n vertex operator φ(z) chiral free boson field z: complex number ) : c s.. (, 1,..., ) (, 1)... (, ) 1 s α z zn = Vs α z V 1 s α z N N N vac The auxiliary particles correspond to a critical theory (CFT c=1). The spin theory, for some values of α, corresponds to a critical theory (c=1) We can use the technology of CFT.

42 MATRIX PRODUCT STATES CRITICALITY Translationally invariant: z equidistant in the unit circle. Correlation functions Area law (Renyi entropy) α ( 0, 1 ] critical with c=1 2 Non-critical otherwise.

43 MATRIX PRODUCT STATES VARIATIONAL CALCULATIONS Anisotropic Heisenberg Model: z equidistant in the unit circle. N H = S x x i S i+1 + S y y i S i+1 1=1 +ΔS z z i S i+1 FM Critical (c=1) Gapped (Neel) Remarkable overlap with exact solution. For Δ=-1 and Δ=0 it is exact.

44 MATRIX PRODUCT STATES VARIATIONAL CALCULATIONS Dimerized Heisenberg Model: z free, α=1/2 H = N 1=1 r r r J 1 S i S i+1 + J 2 S i S r i+2 (J 1 =1) Critical (c=1) Dim Dimerized (spiral phase) Remarkable overlap with exact solution. For J2=J1/2 it is exact (Majumdar-Gosh)

45 MATRIX PRODUCT STATES EXACTLY SOLVABLE MODELS Su(2) invariance: z free, α=1/2 The vertex operators are primary fields of the SU(2) WZW model with spin ½ conformal weight h=1/4 Level k=1 With fusion rule: φ 1/2 φ 1/2 =φ 0 The coefficients c.. ( α s = 1/ 2, 1,..., ) = ( 1)... ( ) 1 s z zn Vs z V 1 s z N N N vac form a conformal block satisfying the Knizhnik-Zamolodchikov Equation: r r N k + 2 Si S j c( z1, K, zn ) = c( z1, K, zn ) 2 z z z i j i Using this equation it is easy to show that: zn zm H = n m ( zn zm) w i n, m ( c n j H Ψ = E Ψ c m r r ) Sn S For the uniform case, we obtain the Haldane-Shastry Hamiltonian. m

46 MATRIX PRODUCT STATES EXACTLY SOLVABLE MODELS Su(2) invariance: z free, α=1/2 Taking random z s: (Monte Carlo, N=1000) Scales like a critical theory (compare with Moore and Refael 05)

47 6. OTHER METHODS: STRING BOND AND PLAQUETTES

48 OTHER METHODS SPINS: MERA (Vidal 07) PEPS + MONTECARLO (Schuch, Wolf, Verstraete, IC, 08) TNS + MONTECARLO (Sandvik and Vidal, 08) EPS + MONTECARLO (Mezzacapo, Bonisegni, Schuch, IC, 09) imps + MONTECARLO (Sierra, IC, 09) FERMIONS: fpeps (Kraus, Schuch, Verstraete, IC, 09, Corboz, Orus, Bauer, Vidal, 09) fmera (Corboz, Evently, Vidal, 09, Eisert et al 09)

49 OTHER METHODS STRING-BOND STATES (Schuch, Wolf, Verstraete, IC, 2008) We first restrict ourselves to certain kind of PEPS: k k A αβγδ = A A i αβγδ = fαδ gβγ or A i αβγδ = fαβ gγδ They can be efficiently contracted using MC This family can be easily extended. N 2 d 2 D 3 Ψ = C{ n ij } C{ n n n kl }... 11,..., LL n... n 11 LL tr ( A n A n '...)

50 OTHER METHODS STRING-BOND STATES 2D results (10x10 lattices): (Schuch, Verstraete, Wolf, and IC, 2008) (Sfondrini, Cerrillo, Schuch, IC, submitted) Ising model XX frustrated model J1 J 2 model E J 2 / J 1

51 OTHER METHODS STRING-BOND STATES 3D results: Ising model (8x8x8) (Sfondrini, Cerrillo, Schuch, IC, submitted) XX frustrated model (6x6x6) magnetization magnetization transv. field transv. field

52 ENTANGLED PLAQUETTE STATES SPIN MODELS (Mezzacapo, Schuch, Bonisegni, IC, 2009) We cover the spins with small overlapping plaquettes Ψ = C{ n ij } C{ n n n kl }... 11,..., LL n... n 11 LL

53 ENTANGLED PLAQUETTE STATES SQUARE LATTICE Heisenberg model: (Mezzacapo, Schuch, Bonisegni, IC, 2009) 10x10 lattice E= (MC) E= (EPS) M=0.324 (MC) M=0.307 (EPS) E E Other models: 2x2 2x3 3x3 3x44x4 Ising model E XX model 10x10 lattice E J1 J 2 model

54 ENTANGLED PLAQUETTE STATES TRIANGULAR LATTICE (preliminary) AF XX model: (Mezzacapo, et al, in preparation) ΔE=0.5% (N=36) E Heisenberg E ΔE=1% (N=36)

55 OUTLOOK i1, i2, μ A Ψ = c i, i,... Ψ ( A μ ) Quantum Mechanics of PEPS: Symmetries Topological order Excitations Gaps Criticality Efficient descripitons: Proofs New families Different problems New computational methods: 2 N

56 THANKS Frank Verstraete (Vienna) Mari Carmen Banuls Christina Kraus Fabio Mezzacapo Valentin Murg Mikel Sanz J.J. Garcia-Ripoll (Madrid) B. Paredes (Munich) D. Perez-Garcia (Madrid) D. Porras (Madrid) N. Schuch (CALTECH) M. Wolf (Copenhagen) R. Fazio s group (Pisa) M. Hastings (LANL) J.I. Latorre (Barcelona) M.A. Martin Delgado (Madrid) G. Ortiz (Indiana) G. Sierra (Madrid) G. Vidal s group (Queensland)

57

58 LATTICE PROBLEMS PEPS Verstraete and IC 04 Let us consider our particles as composite objects Spin ½ neutron is composed of quarks. Auxiliary particles have different dimensions: D We only see the macroscopic objects: We project onto a 2-dimensional subspace. The auxiliary particles are in a very simple state: maximally entangled. The state is determined by the way we project onto the 2d space. The state is completely characterized by N projectors.

59 PEPS DEFINITION Example: 2-dimensional system Verstraete and IC 04 Φ P i, j Φ P = A i i αβγδ αβγδ All the information is contained in the P s There are parameters 4 2D N

60 QUANTUM MANY-BODY SYSTEMS EFFICIENT DESCRIPTIONS: LATTICES h 2 N Hilbert space physically relevant Example: spin 1/ σ = σ = 1 0 i σ = i σ = 0 1 General 2-body interactions: N 3 H = λ σ σ Ψ nm, = 1 αβ, = 0 ( λ ) α, β 0 nm, nm, n m α, β α β Short-range, homogeneous 3 n m n m H = λ σ σ n m < 1 αβ, = 0 α, β α β 2 there are 16N coefficients there are 16R d coefficients

61 OTHER METHODS: STRING-BOND STATES

62 OTHER METHODS STRING-BOND STATES (Schuch, Wolf, Verstraete, IC, 2008) We first restrict ourselves to certain kind of PEPS: P: P = 0 Ψ Ψ1 with 0,1 ϕ0,1 ϕ0,1 Ψ = They can be efficiently contracted using MC This family can be easily extended. N 2 d 2 D 3 Ψ = C{ n ij } C{ n n n kl }... 11,..., LL n... n 11 LL tr ( A n A n '...)

63 OTHER METHODS STRING-BOND STATES 2D results (10x10 lattices): (Schuch, Verstraete, Wolf, and IC, 2008) (Sfondrini, Cerrillo, Schuch, IC, submitted) Ising model XX frustrated model J1 J 2 model E J 2 / J 1

64 OTHER METHODS STRING-BOND STATES 3D results: Ising model (8x8x8) (Sfondrini, Cerrillo, Schuch, IC, submitted) XX frustrated model (6x6x6) magnetization magnetization transv. field transv. field

65 OTHER METHODS: ENTANGLED PLAQUETTE STATES

66 ENTANGLED PLAQUETTE STATES SPIN MODELS (Mezzacapo, Schuch, Bonisegni, IC, 2009) We cover the spins with small overlapping plaquettes Ψ = C{ n ij } C{ n n n kl }... 11,..., LL n... n 11 LL

67 ENTANGLED PLAQUETTE STATES SQUARE LATTICE Heisenberg model: (Mezzacapo, Schuch, Bonisegni, IC, 2009) 10x10 lattice E= (MC) E= (EPS) M=0.324 (MC) M=0.307 (EPS) E E Other models: 2x2 2x3 3x3 3x44x4 Ising model E XX model 10x10 lattice E J1 J 2 model

68 ENTANGLED PLAQUETTE STATES TRIANGULAR LATTICE (preliminary) AF XX model: (Mezzacapo, et al, in preparation) ΔE=0.5% (N=36) E Heisenberg E ΔE=1% (N=36)

69 MATRIX PRODUCT STATES 1 DIMENSION 1D systems: P1 Pn P n D D : C C C 2 In a basis: P n = D 1 [ An ] x, y= 1s= 1 s x s x, y, y Matrix Product States: (Fannes, Narchtengaele, Werner 08) s1 sn Ψ = tr( A1... AN ) s1,..., s s... s 1 N =± 1 N For finite D, all correlation functions decay exponentially. In order to describe exactly a critical system, we have to take We have taken a QFT for the auxiliary particles. D

70 MATRIX PRODUCT STATES CFT 1D systems: P1 P n Pn D 1 s D 2 C Pn = [ An ] s x, y x, y x, y= 1s= 1 D : C C State: Ψ = s... s 1 N cs 1.. s ( α, z1,..., z ) 1... N N s =± 1 Variational parameters s N V s ( α, z ) = : e n i α sφ ( z n vertex operator φ(z) chiral free boson field z: complex number ) : c s.. s α, z1,..., zn ) = Vs ( α, z1)... Vs ( α, zn ) ( 1 N 1 N The auxiliary particles correspond to a critical theory (CFT c=1). The spin theory, for some values of α, corresponds to a critical theory (c=1) We can use the technology of CFT. vac

71 MATRIX PRODUCT STATES CRITICALITY Translationally invariant: z equidistant in the unit circle. Correlation functions Area law (Renyi entropy) α ( 0, 1 ] critical with c=1 2 Non-critical otherwise.

72 MATRIX PRODUCT STATES VARIATIONAL CALCULATIONS Anisotropic Heisenberg Model: z equidistant in the unit circle. N H = S x x i S i+1 + S y y i S i+1 1=1 +ΔS z z i S i+1 FM Critical (c=1) Gapped (Neel) Remarkable overlap with exact solution. For Δ=-1 and Δ=0 it is exact.

73 MATRIX PRODUCT STATES VARIATIONAL CALCULATIONS Dimerized Heisenberg Model: z free, α=1/2 H = N 1=1 r r r J 1 S i S i+1 + J 2 S i S r i+2 (J 1 =1) Critical (c=1) Dim Dimerized (spiral phase) Remarkable overlap with exact solution. For J2=J1/2 it is exact (Majumdar-Gosh)

74 MATRIX PRODUCT STATES EXACTLY SOLVABLE MODELS Su(2) invariance: z free, α=1/2 The vertex operators are primary fields of the SU(2) WZW model with spin ½ conformal weight h=1/4 Level k=1 With fusion rule: φ 1/2 φ 1/2 =φ 0 The coefficients c α s.. s = 1/ 2, z1,..., zn ) = Vs ( z1)... Vs ( zn ) ( 1 N 1 N form a conformal block satisfying the Knizhnik-Zamolodchikov Equation: r r N k + 2 Si S j c( z1, K, zn ) = c( z1, K, zn ) 2 z z z i j i Using this equation it is easy to show that: zn zm H = n m ( zn zm) w i n, m ( c n j H Ψ = E Ψ c m r r ) Sn S For the uniform case, we obtain the Haldane-Shastry Hamiltonian. m vac

75 MATRIX PRODUCT STATES EXACTLY SOLVABLE MODELS Su(2) invariance: z free, α=1/2 Taking random z s: Scales like a critical theory (compare with Moore and Refael 05)

76 7. Numerical methods