On the algebraic Bethe ansatz approach to correlation functions: the Heisenberg spin chain

On the algebraic Bethe ansatz approach to correlation functions: the Heisenberg spin chain V. Terras CNRS & ENS Lyon, France People involved: N. Kitanine, J.M. Maillet, N. Slavnov and more recently: J. S. Caux, K. Kozlowski, G. Niccoli... ENIGMA School 07 Lalonde les Maures

Outline Introduction 1 Introduction The Heisenberg spin-1/ chain Exact computations of correlation functions Correlation functions in the finite chain Elementary blocks in the thermodynamic limit A simple example: the emptiness formation probability 3 Analytical + Numerical methods Analytical resummations for the two-point function

The Heisenberg spin chain The Heisenberg spin-1/ chain Exact computations of correlation functions Model for magnetism in solids (Heisenberg, 198) Crystals with effective one-dimensional magnetic properties Can be tested via inelastic neutron scattering experiments Archetype of quantum integrable models Spectrum resolution via Bethe ansatz (1931) and its developments Links to two-dimensional statistical mechanics (vertex models generalizing Ising) Very rich (non-commutative) algebraic structures Yang-Baxter algebras, R-matrices, Quantum groups They appear in different situations eventually far from magnetism (Gauge and String theories and AdS/CFT correspondence) Link to combinatorics in special point (ice model)

The spin-1/ XXZ Heisenberg chain The Heisenberg spin-1/ chain Exact computations of correlation functions The XXZ spin-1/ Heisenberg chain in a magnetic field is a quantum interacting model defined on a one-dimensional lattice with M sites, with Hamiltonian, H = H (0) hs z, H (0) = S z = 1 M { σ x m σm+1 x + σmσ y y m+1 + (σz mσm+1 z 1) }, m=1 M σm, z [H (0), S z ] = 0. m=1 Quantum space of states : H = M m=1 H m, H m C, dim H = M. are the local spin operators (in the spin- 1 representation) at site m: they act as the corresponding Pauli matrices in the space H m and as the identity operator elsewhere. σ x,y,z m + periodic boundary conditions

The Heisenberg spin-1/ chain Exact computations of correlation functions Correlation functions of Heisenberg chain Free fermion point = 0: Lieb, Shultz, Mattis, Wu, McCoy, Sato, Jimbo, Miwa,... From 1984: Izergin, Korepin,... (first attempts using Bethe ansatz for general ) General : multiple integral representations 199-96 Jimbo and Miwa from q-vertex op. and qkz eq. 1999 Kitanine, Maillet, Terras from Algebraic Bethe Ansatz Several developments since 000: Kitanine, Maillet, Slavnov, Terras; Boos, Korepin, Smirnov; Boos, Jimbo, Miwa, Smirnov, Takeyama; Göhmann, Klümper, Seel; Caux, Hagemans, Maillet...

Correlation functions The Heisenberg spin-1/ chain Exact computations of correlation functions O = tr ( ) H O e H/kT ( tr ) H e H/kT = ψ g O ψ g at T = 0 where ψ g is the state with lowest eigenvalue. Why is it so difficult? (Bethe ansatz already 75 years old...!) Main problems to be solved to achieve this : Compute exact eigenstates and energy levels of the Hamiltonian (Bethe ansatz) Obtain the action of local operators on the eigenstates: main problem since eigenstates are highly non-local! Compute the resulting scalar products with the eigenstates

The methods... Introduction The Heisenberg spin-1/ chain Exact computations of correlation functions q-kz and q-vertex operators : Valid (with some hypothesis) for infinite (and semi-infinite) chains, zero magnetic field and zero temperature Elementary blocks of correlation functions (static) and form factors (massive case) Multiple integrals and recently algebraic solutions of q-kz Bethe ansatz Valid for finite and infinite chains, with magnetic field and temperature, and with impurities or with integrable boundaries (open chain) Determinant representation of form factors (finite chain), multiple integrals for correlation functions (infinite chain), master formula for spin-spin correlation functions. Some results for a continuum model (NLS)

Correlation functions in the finite chain Elementary blocks in the thermodynamic limit A simple example: the emptiness formation probability Algebraic Bethe ansatz and correlation functions Compute ψ g j σαj j ψ g? 1 Diagonalise the Hamiltonian using ABA (Faddeev, Sklyanin, Takhtajan, 1979) key point : Yang-Baxter algebra A(λ), B(λ), C(λ), D(λ) eigenstates: B(λ 1 )... B(λ n ) 0 Act with local operators on eigenstates problem: relation between B (creation) and σj α a priori very complicated! solve the quantum inverse problem (Kitanine, Maillet, V.T., 1999): σ αj j = f αj j (A, B, C, D) = (A, B, C, D) use Yang-Baxter commutation relations 3 Compute the resulting scalar products (Slavnov; Kitanine, Maillet, V.T.) 4 Thermodynamic limit elementary building blocks of correlation functions as multiple integrals (000) 5 Two-point function : further analysis... (since 00)

Correlation functions in the finite chain Elementary blocks in the thermodynamic limit A simple example: the emptiness formation probability Diagonalization of the Hamiltonian via ABA ( ) A(λ) B(λ) σn α monodromy matrix T (λ) = C(λ) D(λ) with T (λ) T a,1...m (λ) = L am (λ ξ N )... L a (λ ξ )L a1 (λ ξ 1 ) ) L an (λ) = ( sinh(λ + ησ z n ) sinh η σ n sinh η σ + n sinh(λ ησ z n) [a] [a] a auxiliary space C n local quantum space at site n Yang-Baxter algebra: generators A, B, C, D commutation relations given by the R-matrix of the model R ab (λ, µ) T a (λ)t b (µ) = T b (µ)t a (λ) R ab (λ, µ) commuting conserved charges: t(λ) = A(λ) + D(λ) [t(λ), t(µ)] = 0 H = sinh η λ log t(λ) λ= η + c for all ξ j = 0 construction of the space of states by action of B (creation) and C (annihilation) on a reference state 0... eigenstates : ψ = k B(λ k) 0 with {λ k } solution of the Bethe equations.

Action of local operators on eigenstates Correlation functions in the finite chain Elementary blocks in the thermodynamic limit A simple example: the emptiness formation probability Solution of the quantum inverse scattering problem (σn α T (λ)) n 1 n σn = t(ξ k ) B(ξ n ) t 1 (ξ k ) n 1 σ n + = t(ξ k ) C(ξ n ) n t 1 (ξ k ) n 1 σn z = t(ξ k ) (A D)(ξ n ) n t 1 (ξ k ) use the Yang-Baxter commutation relations for A, B, C, D to get the action on arbitrary states: 0 C(λ k ) m T εj,ε (λ N+j) = Ω j P ({λ}, {ɛ j, ɛ j }) 0 C(λ b ) j=1 b P P {λ} correlation functions = sums over scalar products

Computation of scalar products Correlation functions in the finite chain Elementary blocks in the thermodynamic limit A simple example: the emptiness formation probability Scalar product 0 C(µ l ) B(λ k ) 0 l=1 }{{} arbitrary state } {{ } eigenstate with U ab = λa τ(µ b, {λ k }), V ab = = det U({µ l}, {λ k }) det V ({µ l }, {λ k }), 1, 1 a, b N, sinh(µ b λ a ) where τ(µ b, {λ k }) is the eigenvalue of the transfer matrix t(µ b ). m-point elementary blocks for the correlation functions in the finite chain: m ψ g E ɛ j,ɛ j j ψ g =... Ω m ({λ}, {ɛ j, ɛ j j=1 }{{} }) det m M m sums with (E ɛ,ɛ ) lk = δ l,ɛ δ k,ɛ

Matrix elements of local operators For example : 0 C(µ j ) σn z B(λ k ) 0 j=1 = 0 j=1 n 1 C(µ j ) t(ξ k ) (A D ) (ξ n ) Correlation functions in the finite chain Elementary blocks in the thermodynamic limit A simple example: the emptiness formation probability n t 1 (ξ k ) B(λ k ) 0 Here the sets {λ k } and {µ j } are both solutions of Bethe equations 0 C(µ j ) σn z B(λ k ) 0 = Φ n 0 j=1 = Φ n ψ j=1 C(µ j ) ( A D ) (ξ n ) B(λ k ) 0 B(λ k ) 0 determinant representations of matrix elements (using the scalar product formula)

Correlation functions in the finite chain Elementary blocks in the thermodynamic limit A simple example: the emptiness formation probability Elementary blocks in the thermodynamic limit Sums become integrals: 1 M N j=1 f (λ j ) f (λ)ρ(λ)dλ M C h {λ j } solution of Bethe eq. for the ground state ρ(λ) density of the ground state solution of a linear integral eq. multiple integral representation for the m-point elementary building blocks of the correlation functions ψ g m j=1 E ɛ j,ɛ j j ψ g = d m λ Ω m ({λ k }, {ɛ j, ɛ j}) det S h({λ k }) C m h where Ω m ({λ k }, {ɛ j, ɛ j }) is purely algebraic and S h({λ k }), C h depend on the regime and on the magnetic field h. Proof of the results and conjectures of Jimbo, Miwa et al. + extension to non-zero magnetic field; more recently, extension to time dependent (KMST) and non zero temperature (Göhmann, Klümper, Seel)

What about this result? Correlation functions in the finite chain Elementary blocks in the thermodynamic limit A simple example: the emptiness formation probability A priori, the problem is solved: expression of all elementary blocks ψ g E ɛ 1,ɛ 1 1... E ɛ m,ɛm m ψ g any correlation function = P (elementary blocks) From a practical point of view, there are two main problems: (1) physical correlation function = HUGE sum of elementary blocks at large distances Example: two-point function m 1 Y ψ g σ1 z σm ψ z g ψ g (E1 11 E1 ) (Ej 11 + E re-summation? = X m terms j= j ) {z } propagator (elementary blocks) (E 11 m E m ) ψ g? m

Correlation functions in the finite chain Elementary blocks in the thermodynamic limit A simple example: the emptiness formation probability () each block has a complicated expression Example: emptiness formation probability for h = 0 in the massless regime ( 1 < = cosh ζ < 1) τ(m) ψ g m = ( 1) m( π ζ 1 σk z ψ g m j=1 dependence on m? ) m(m 1) (1)+() need further analysis! d m λ π m a>b sinh π ζ (λ a λ b ) sinh(λ a λ b iζ) sinh j 1 (λ j iζ/) sinh m j (λ j + iζ/) cosh m π ζ λ j

Correlation functions in the finite chain Elementary blocks in the thermodynamic limit A simple example: the emptiness formation probability A simple example: the emptiness formation probability Integral representation as a single elementary block but previous expression not symmetric symmetrisation of the integrand: τ(m) = lim ξ 1,...ξ m iζ 1 m! m a<b d m λ m a,b=1 1 sinh(λ a λ b iζ) sinh(λ a λ b ) sinh(ξ a ξ b ) Z m({λ}, {ξ}) det m [ρ(λ j, ξ k )] where Z m ({λ}, {ξ}) is the partition function of the 6-vertex model with domain wall boundary conditions and ρ(λ, ξ) = [ iζ sinh π ζ (λ j ξ k )] 1 is the inhomogeneous version of the density for the ground state (massless regime = cos ζ, h = 0). (1) Exact computation for = 1/ () Asymptotic behaviour for m

Correlation functions in the finite chain Elementary blocks in the thermodynamic limit A simple example: the emptiness formation probability Exact computation for = 1/ The determinant structure combined with the periodicity properties at = 1/ enable us to separate and compute the multiple integral : τ inh (m, {ξ j }) = ( 1) m m m In the homogeneous limit: ( 1 τ(m) = m a>b m a,b=1 a b ) m m 1 k=0 sinh 3(ξ b ξ a ) sinh(ξ b ξ a ) 1 sinh(ξ a ξ b ) det m (3k + 1)! (m + k)! = with A m - number of alternating sign matrices ( ) m 1 A m ( ) ξ 3 sinh j ξ k. sinh 3(ξ j ξ k ) first exact result for 0 (and proof of a conjecture of Razumov and Stroganov)

Correlation functions in the finite chain Elementary blocks in the thermodynamic limit A simple example: the emptiness formation probability Asymptotic Results: (saddle-point) massless case ( 1 < = cos ζ 1) log τ(m) lim m m = log π ζ + 1 dω ω R i0 1 massive case ( = cosh ζ > 1) log τ(m) lim m m = ζ 3 sinh ω ωζ (π ζ) cosh sinh πω ωζ sinh cosh ωζ log for = 0 = log [ 3 3 log for = 1 Γ( 3 log for = 1 (XXX chain) n=1 ] 4 ) Γ( 1 ) Γ( 1 4 ) e nζ n [ Γ( 3 4 log ) Γ( 1 ) ] ζ 0 Γ( 1 4 ) ζ + sinh(nζ) cosh(nζ) (XXX) (Ising)

Analytical + Numerical methods Analytical resummations for the two-point function Consider the correlation function of the product of two local operators at zero temperature : g 1 = ψ g θ 1 θ ψ g Two main strategies to evaluate such a function: (i) compute the action of local operators on the ground state θ 1 θ ψ g = ψ and then calculate the resulting scalar product: g 1 = ψ g ψ (ii) insert a sum over a complete set of eigenstates ψ i to obtain a sum over one-point matrix elements (form factor type expansion) : g 1 = i ψ g θ 1 ψ i ψ i θ ψ g

Analytical + Numerical methods Analytical resummations for the two-point function Analytical + Numerical methods for dynamical correlation functions in a field (Biegel, Karbach, Müller; Caux, Hagemans, Maillet) Use (ii) form factor expansion over a complete set of intermediate eigenstates ψ i : S α j (t) S β j (0) = i ψ g S α j (t) ψ i ψ i S β j (0) ψ g for a finite chain of length M even, and a ground state ψ g depending on the magnetic field with a fixed number of reversed spins N, and N M. each form factor = explicit determinant of size N, depending on two sets of parameters solutions of Bethe equations and characterizing the states ψ g and ψ i respectively Numerics are then used to compute the determinants and the (finite) sum (control of the results via sum rules) numerical result for the dynamical spin-spin correlation functions

Introduction Analytical + Numerical methods Analytical resummations for the two-point function, successful comparison to neutron scattering experiments for the structure factor (Fourier transform of the dynamical correlation function) S αβ (q, ω) = Z N 1 X iq(j j 0 ) e dte iωt hsjα (t)sjβ0 (0)i N 0 j,j =1 Left: Bethe ansatz data computed for a chain of 500 sites Right: Experimental data for KCuF3 (D.A. Tennant et al) V. Terras Algebraic Bethe ansatz approach to correlation functions

Analytical + Numerical methods Analytical resummations for the two-point function Analytical resummations for the two-point function m 1 σ1 z σm z = φ m ψ g (A D)(ξ 1 ) (A + D)(ξ i ) (A D)(ξ m ) ψ g i= }{{} propagator (1 m) Use (i): compute resummed action of the propagator from site 1 to m on an arbitrary state: m m ψ t κ (x a ) = ψ n (κ) a=1 n=0 with t κ (x) = (A + κd)(x) twisted transfer matrix partial resummation in the thermodynamic limit: σ1 z σm z = (multiple integrals) (instead of m terms) m+1 terms master formula for the finite chain

Analytical + Numerical methods Analytical resummations for the two-point function Example Generating function Q κ 1,m for σz correlation functions 1 (1 σz 1)(1 σ z m+1) = κ `Q κ 1,m+1 Q κ 1,m Q κ,m+1 + Q κ,m κ=1 with my Q1,m κ 1 + κ = + 1 κ «σn z n=1 my my = (A + κd) (ξ a) (A + D) 1 (ξ b ) a=1 b=1 to compute: m Q1,m κ = φ m ψ g t κ (ξ a ) ψ g with t κ (x) = (A + κd)(x) a=1

Analytical + Numerical methods Analytical resummations for the two-point function Master equation for σ z correlation functions Let the inhomogeneities {ξ} be generic and the set {λ} be an admissible off-diagonal solution of the Bethe equations (cf.tarasov - Varchenko). Then there exists κ 0 > 0 such, that for κ < κ 0 : Q κ 1,m = 1 N! j=1 Γ{ξ} Γ{λ} dz j πi a,b=1 sinh (λ a z b ) ( τκ(λ det j {z}) N a=1 z k ) m a=1 τ κ (ξ a {z}) τ(ξ a {λ}) ( ) det τ(zk {λ}) N λ j ( ). Y κ (z a {z}) det Y(λk {λ}) N λ j Notations: τ κ (µ {λ}) = eigenvalue of the κ-twisted transfer matrix t κ (µ) on the eigenstate ψ κ = k B(λ k) 0, for {λ} solution of the (twisted) Bethe equations : Y κ (λ j {λ}) = 0, j = 1,..., N. (κ = 1 no subscript)

Analytical + Numerical methods Analytical resummations for the two-point function The integration contour is such that the only singularities of the integrand within the contour Γ{ξ} Γ{λ} which contribute to the integral are the points {ξ} and {λ}. ways to evaluate the integrals: compute the residues in the poles inside Γ representation of σ1σ z m+1 z as sum of m multiple integrals (previous resummation obtained with approach (i) ) compute the residues in the poles outside Γ (within strips of width iπ) sum over (admissible) solutions of (twisted) Bethe equations form factor expansion of σ1σ z m+1 z (approach (ii)) link between the two approaches

Time-dependent master equation Analytical + Numerical methods Analytical resummations for the two-point function with Q κ 1,m(t) = 1 N! a,b=1 Γ{± η } Γ{λ} j=1 dz j πi b=1 ( τκ(λ det j {z}) N sinh (λ a z b ) a=1 ( ) ( ) e it E(z b ) E(λ b ) +im p(z b ) p(λ b ) z k ) sinh η E(z) = sinh(z η ) sinh(z + η ( ) sinh(λ η p(λ) = i log ) ) sinh(λ + η ) ( ) det τ(zk {λ}) N λ j ( ) Y κ (z a {z}) det Y(λk {λ}) N λ j

Analytical + Numerical methods Analytical resummations for the two-point function Explicit results at = 1 Generating function at = 1 Partial resummation in the inhomogeneous case multiple integrals can be separated and computed: with Q κ (m) = 3m m m sinh 3(ξ a ξ b ) a>b ˆΦ (n) ({ξ γ+ }, {ξ γ }) = sinh 3 (ξ a ξ b ) a γ + m n=0 κ m n {ξ}={ξγ + } {ξγ } γ + =n det m ˆΦ (n) iπ sinh(ξ b ξ a 3 ) sinh(ξ a ξ b ) sinh (ξ b γ b ξ a + iπ 3 ), Φ(ξ j ξ k ) Φ(ξ j ξ k iπ 3 ) Φ(ξ j ξ k + iπ 3 ) Φ(ξ j ξ k ), Φ(x) = sinh x sinh 3x.

Analytical + Numerical methods Analytical resummations for the two-point function If the lattice distance m is not too large, the representations can be successfully used to compute Q κ (m) explicitely. First results for P m (κ) = m Q κ (m) up to m = 9: P 1 (κ) = 1 + κ, P (κ) = + 1κ + κ, P 3 (κ) = 7 + 49κ + 49κ + 7κ 3, P 4 (κ) = 4 + 10004κ + 45444κ + 10004κ 3 + 4κ 4 P 5 (κ) = 49 + 738174κ + 16038613κ + 16038613κ 3 + 738174κ 4 + 49κ 5, P 6 (κ) = 7436 + 9689380κ + 1144474588κ + 4567793398κ 3 + 1144474588κ 4 + 9689380κ 5 + 7436κ 6.

Analytical + Numerical methods Analytical resummations for the two-point function Two-point functions σ z 1 σz m+1 at = 1 m σ1σ z m+1 z Exact σ1σ z m+1 z Asympt. 1 1-0.5000000000-0.5805187860 7 6 0.1093750000 0.11351569 3 401 1-0.0979003906-0.0993588501 4 184453 0.0439770 0.044068654 5 9514949 31-0.0443379157-0.0444087865 6 175875008939 46 0.04993340 0.049365346 7 3083610739677093 60-0.0666845-0.0640495 8 50018849740515343761 78 0.0166105110 0.016564139 and comparison with the values given by the asymptotic prediction: σ1σ z m+1 z 1 1 = π(π ζ) m + ( 1)m with value of A z conjecture by S. Lukyanov A z m π π ζ +

Some other models Introduction Analytical + Numerical methods Analytical resummations for the two-point function Non periodic boundary conditions Open XXZ chain (with diagonal boundary conditions): M 1 { H = σ x m σm+1 x + σmσ y y m+1 + (σz mσm+1 z 1) } + h σ1 z + h + σm z m=1 no translation invariance revisit solution of the inverse Problem multiple integral formulas for elementary blocks, partial resummation for -point correlation functions Master equation? Continuum field theory Master equation valid for all models with the same R-matrix (depend only on commutation relations of the Yang-Baxter algebra) density-density correlation functions of the quantum non-linear Schrödinger model (or one-dimensional Bose gas): Z L H = xψ (x) xψ(x) + c ψ (x)ψ (x)ψ(x)ψ(x) h ψ (x)ψ(x) dx 0

Some open problems... Analytical + Numerical methods Analytical resummations for the two-point function Asymptotic behavior of correlation functions: challenging the conformal limit from the lattice models Continuum (Field theory) models (NLS, ShG,...) : Approach from the lattice Inverse problem for infinite dimensional representations Link to Q operator and SOV methods Even more sophisticated models : XYZ model Hubbard : needs extended Yang-Baxter and ABA or FBA understanding