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

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

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