Correlation functions of the XXZ spin-1/2 chain: recent progress

Correlation functions of the XXZspin-1/2 chain: recent progressJ. M. MailletCNRS & ENS LyonCollaborators :N. Kitanine (Tokyo), N. A. Slavnov (Moscow), V. Terras (Rutgers)– Typeset by FoilTEX – Florence 2003

J. M. Maillet Correlation functionsQuantum integrable modelsInterests• Exacts results not accessible by usual techniques• Direct applications : condensed matter, solid state physics,...• Mathematics : quantum groups, knot theory,...What can we compute?Hamiltonian spectrum, scattering matrix , partition function, critical exponents,....(Bethe, Onsager, Yang, Baxter, McCoy, Faddeev, Zamolodchikov,...)Correlation functions?〈 O 〉 =tr H(O e −H/kT )tr H(e−H/kT )– Typeset by FoilTEX – Florence 2003 1

J. M. Maillet Correlation functionsCorrelation functionsWhy is it so difficult? (Bethe ansatz already 70 years old...!)For example at T=0 only the ground state |ψ g 〉 of H contributes :Three main problems to be solved :〈ψ g | O |ψ g 〉• Compute exact eigenstates and energy levels of the Hamiltonian (Bethe ansatz)• Obtain the action of local operators on the eigenstates : main problem since eigenstatesare highly non-local!• Compute the resulting scalar products with the eigenstatesBeyond Ising type models (there, use the free fermion algebra)...?(Lieb, Shultz, Mattis, Wu, McCoy,...)– Typeset by FoilTEX – Florence 2003 2

J. M. Maillet Correlation functionsA very brief history...Three main lines of approach have been developped :• Bootstrap methods for the form factors + sums over all eigenstates (Karowski, Weisz,Smirnov,....)• Infinite volume + non-abelian quantum symmetries (Jimbo, Miwa,....)• Finite volume + algebraic Bethe ansatz (Izergin, Korepin,...)−→ More recently a new approach :Algebraic Bethe ansatz in finite volume + solution of the quantum inverse scatteringproblem (Kitanine, Maillet, Terras)−→ At zero temperature, and in the thermodynamic limit, it leads to multiple integralrepresentations of the correlation functions for the XXZ spin-1/2 chain– Typeset by FoilTEX – Florence 2003 3

J. M. Maillet Correlation functionsThe spin-1/2 XXZ Heisenberg chainH = ∑ M(m=1 σxm σx m+1 + σy m σy m+1 + ∆ (σz m σz m+1 − 1)) − h ∑ M2 m=1 σz m• Hamiltonian eigenstates( A(λ)Algebraic Bethe ansatz : σ α m −→ T (λ) = C(λ)B(λ)D(λ))T (λ) ≡ T a,1...N (λ) = L aN (λ − ξ N ) . . . L a1 (λ − ξ 1 )L an (λ) being 2 × 2 matrices with entries function of σ x,y,znoperators in site n.Yang-Baxter algebra : R 12 (λ 1 , λ 2 )T 1 (λ 1 )T 2 (λ 2 ) = T 2 (λ 2 )T 1 (λ 1 )R 12 (λ 1 , λ 2 )Commuting conserved charges : t(λ) = A(λ) + D(λ), [t(λ), t(µ)] = 0– Typeset by FoilTEX – Florence 2003 4

J. M. Maillet Correlation functionsHamiltonian : H = 2 sinh η ∂∂λ log t(λ)∣ ∣λ=η2+ c for all ξ j = 0.Eigenstates of t(µ) : |ψ〉 = ∏ k B(λ k)|0〉 with {λ k } solution of the Bethe equations.• Action of local operators on eigenstatesResolution of the quantum inverse scattering problem : σ α m←−T (λ)σ − j=σ + j=∏j−1k=1j−1t(ξ k ) · B(ξ j ) ·∏t(ξ k ) · C(ξ j ) ·k=1j∏t −1 (ξ k ),k=1j∏t −1 (ξ k ),k=1j−1σ z j = ∏t(ξ k ) · (A− D ) (ξ j ) ·k=1j∏t −1 (ξ k ),k=1– Typeset by FoilTEX – Florence 2003 5

J. M. Maillet Correlation functions+ Yang-Baxter algebra for A, B, C, D to get the action on arbitrary states, for example〈0|N∏C(λ k ) A(λ N+1 ) =k=1N+1∑a ′ =1Λ a ′ 〈0|N+1∏C(λ k )k=1k≠a ′• Scalar products :〈0|N∏C(µ j )j=1N∏k=1B(λ k ) |0〉 = detU({µ j}, {λ k })detV ({µ j }, {λ k })for {λ k } a solution of Bethe equations and {µ j } an arbitrary set of parameters, :U ab = ∂ λ aτ(µ b , {λ k }), V ab =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 )– Typeset by FoilTEX – Florence 2003 6

J. M. Maillet Correlation functionsMatrix elements of local operatorsFor example :〈0|N∏N∏C(µ j ) σ z nB(λ k ) |0〉 =j=1k=1= 〈0|N∏C(µ j )n−1∏t(ξ k ) · (A− D ) (ξ j ) ·n∏t −1 (ξ k )N∏B(λ k ) |0〉j=1k=1k=1k=1Here the sets {λ k } and {µ j } are both solutions of Bethe equations −→〈0|N∏N∏C(µ j ) σ z nB(λ k ) |0〉 = Φ n 〈0|j=1k=1N∏C(µ j ) ( A − D ) N∏(ξ j ) B(λ k ) |0〉j=1k=1Hence it leads to determinant representations of these matrix elements (using thescalar product formula)– Typeset by FoilTEX – Florence 2003 7

J. M. Maillet Correlation functionsCorrelation functions : elementary blocksj=1F m ({ɛ j , ɛ ′ j }) = 〈ψ g | m ∏E ɛ′ j ,ɛ jj|ψ g 〉〈ψ g |ψ g 〉E ɛ′ ,ɛlk= δ l,ɛ ′δ k,ɛSolution of the quantum inverse scattering problem + Yang-Baxter algebra of operatorsT (λ) −→ Multiple integral formula for the correlation functionsmF m ({ɛ j , ɛ ′ j }) = ( ∏∫k=1C h kdλ k ) Ω m ({λ k }, {ɛ j , ɛ ′ j }) S h({λ k })where Ω m ({λ k }, {ɛ j , ɛ ′ j }) is purely algebraic and S h({λ k }), C h kregime and the magnetic field h.are depending on the−→ Proof of the results and conjectures of Jimbo, Miwa et al. and extension to the nonzero magnetic field h (a case where the quantum affine symmetry is broken)– Typeset by FoilTEX – Florence 2003 8

J. M. Maillet Correlation functionsPhysical correlation functionsExample :〈ψ g |σ + 1 σ− m+1 |ψ g〉 ≡ 〈ψ g |E 121m∏j=2(E 11j+ E 22j) E21 m+1 |ψ g〉−→ sum of 2 m−1 elementary blocks: 〈ψ g |C( η 2 ) (A + D)m−1 ( η 2 ) B(η 2 )|ψ g〉Compact and symmetrized formula for the multiple action of A + D operators−→ Two point correlation functions :m−1〈σ α 1 σβ m+1 〉 = ∑n=0∮Cz∫ ∫d n+1 z d n λC λCµd 2 µ [f({λ, z})] m Γ αβn({λ, µ, z}) S h({λ, z})−→ Gives directly known free fermion results setting ∆ = 0−→ Asymptotics of two point functions for general ∆?– Typeset by FoilTEX – Florence 2003 9

J. M. Maillet Correlation functionsEmptiness formation probabilityfor ∆ = cos ζ, 0 < ζ < π :τ(m) =limξ 1 ,...ξm→− iζ 21m!τ(m) = 〈ψ g |∫ ∞−∞m∏k=11 − σ z k|ψ g 〉2()Z m ({λ}, {ξ})idet m∏m2ζ sinhsinh(ξ a − ξ b )π ζ (λ j − ξ k )abFor ζ = π/3 : τ(m) = ( )1 m2 ∏ m−12 k=0( √3 ) 3m2(3k+1)!(m+k)! → c 2m − 36, 5m → ∞– Typeset by FoilTEX – Florence 2003 10

J. M. Maillet Correlation functionsIdea of the proof : Observe first that for ζ = π/3,−mZ m ({λ}, {ξ}) = (−1)m2 2m∏ sinh 3(ξ b − ξ a )2 m2 +m sinh(ξa>b b − ξ a ) sinh(ξ a − ξ b )(() det1m×det m (sinh(λ j − ξ k ) sinh(λ j − ξ k − iζ) det m)1sinh(λ j −ξ k + iπ 3 )1sinh 3(λ j −ξ k )) .Usingsinh(3x) = 4 sinh(x) sinh(x + iπ/3) sinh(x − iπ/3).τ(m, {ξ j }) =×∫ ∞−∞( 3i4π) m(−1) m2 −m22 m2 m!m∏a>bsinh 3(ξ b − ξ a )sinh(ξ b − ξ a )m∏a,b=1a≠bsinh −1 (ξ a − ξ b )() ()d m 11λ det msinh(λ j − ξ k + iπ 3 ) det msinh(λ j − ξ k ) sinh(λ j − ξ k − iπ 3 )– Typeset by FoilTEX – Florence 2003 11

J. M. Maillet Correlation functionsτ(m, {ε j }) = (−1) m2 −m2 3 m 2 −m2 m ∏⎛⎜× det m ⎝∫ ∞−∞−mτ(m, {ε j }) = (−1)m2 2 ∏ m2 m2a>bsinh 3(ε b − ε a )sinh(ε b − ε a )m∏a,b=1a≠b1sinh(ε a − ε b )⎞dλ⎟4π cosh(λ − ε j ) sinh(λ − ε k − iπ 6 ) sinh(λ − ε k + iπ 6 ) ⎠a>bsinh 3(ε b − ε a )sinh(ε b − ε a )m∏a,b=1a≠bTake the homogeneous limit ε j → 0 (ξ k = ε k − iπ/6) :τ(m) = (−1) m2 −m2 3 m2 +m2 2−m2m−1⎛1sinh(ε a − ε )·det mb[ ]∏∂(n!) −2 j+k−2sinh x 2det m∂x j+k−2 sinh 3x 2n=0Can be computed using Kuperberg determinant identity⎝ 3 sinh ε j −ε k2sinh 3(ε j −ε k )2x=0⎞⎠– Typeset by FoilTEX – Florence 2003 12

J. M. Maillet Correlation functionsFor arbitrary ζ :limm→∞log τ(m)m 2 = log π ζ + 1 2where cosh η = cos ζ = ∆, 0 < ζ < π.∫R−i0dωωsinh ω ωζ2(π − ζ) cosh22,sinh πω ωζ2sinh2cosh ωζlimm→∞limm→∞For the XXX chain (∆ = 1, ζ = 0):log τ(m)= − 1 log 2, ∆ = 0,m 2 2log τ(m)= 3 m 2 2 log 3 − 3 log 2, ∆ = 1 2 ,limm→∞log τ(m)m 2= log(Γ(34 )Γ(1 2 ))Γ( 1 4 )≈ log(0.5991),– Typeset by FoilTEX – Florence 2003 13

J. M. Maillet Correlation functionsIdea of the proof :τ(m) =×m∏a>b( i2ζ sin ζ) m ( πζ)m 2 −m2sinh π ζ (λ a − λ b )∫Dd m λ · F ({λ}, m)sinh(λ a − λ b − iζ) sinh(λ a − λ b + iζ)(m∏ sinh(λa − iζ 2 ) sinh(λ a + iζ 2 )) m,a=1cosh π ζ λ awithF ({λ}, m) =limξ 1 ,...ξm→− iζ 2m∏a>b()1−i sin ζdet msinh(λsinh(ξ a − ξ b )j − ξ k ) sinh(λ j − ξ k − iζ)Integration domain D ≡ −∞ < λ 1 < λ 2 < · · · < λ m < ∞. Then at the saddle pointdistribution of λ’s can be described by a density function ρ(λ ′ ):ρ(λ ′ j ) = limm→∞1m(λ ′ j+1 − λ′ j )– Typeset by FoilTEX – Florence 2003 14

J. M. Maillet Correlation functionsThus for large m, one can replace sums over the set {λ ′ } by integrals. Now estimate thebehavior of F ({λ ′ }, m) for large m using :det m(−i sin ζsinh(λ ′ j − ξ k) sinh(λ ′ j − ξ k − iζ)(−→ (−2πi) m det m δ jk − K(λ′ j − λ′ k ) ) (2πimρ(λ ′ k ) det m)i2ζ sinh π ζ (λ′ j − ξ k)),with Lieb kernelHence :K(λ) =−i sin 2ζsinh(λ − iζ) sinh(λ + iζ)τ(m) −→( πζ) m2e m2 S 0 +o(m 2) , m → ∞,– Typeset by FoilTEX – Florence 2003 15

J. M. Maillet Correlation functionswithS 0 =+ 1 2∫ ∞−∞∫ ∞−∞dλρ(λ) logdµdλρ(λ)ρ(µ) logand the integral equation for the density ρ(λ)−∞(sinh(λ − iζ/2) sinh(λ + iζ/2)(cosh 2 πζ λ )sinh 2 πζ(λ − µ)sinh(λ − µ − iζ) sinh(λ − µ + iζ)2π πλtanh − coth(λ − iζ/2) − coth(λ + iζ/2)ζ ζ∫ ∞( 2π= V.P.ζ coth π )ζ (λ − µ) − coth(λ − µ − iζ) − coth(λ − µ + iζ) ρ(µ) dµ)Solution :ρ(λ) =πλcosh2ζζ √ 2 cosh πλζ,– Typeset by FoilTEX – Florence 2003 16

J. M. Maillet Correlation functionsConclusions and PerspectivesNew method to obtain correlation functions of quantum integrable models• Generic tools for a large class of models• Explicit results for the Heisenberg spin chainsOpen new perspectives• Asymptotic behavior of correlation functions (under study)• Dynamical correlation functions and depending on temperature (under study)• Applications to many different models (models with impurities, with boundaries, fieldtheories,...)– Typeset by FoilTEX – Florence 2003 17