Temperature correlation functions of quantum spin chains at magnetic and disorder field Hermann Boos Bergische Universität Wuppertal joint works with: F. Göhmann, arxiv:0903.5043 M. Jimbo, T.Miwa, F. Smirnov, Y. Takeyama, arxiv:0801.1176 M. Jimbo, T.Miwa, F. Smirnov, arxiv:0903.0115 Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [ 1/25]
Introduction Main question: does factorization of correlation functions of the XXZ model first discovered for zero temperature work in the temperature case also? Some important issues: Multiple integral representation works by Kyoto school (92 93) and Lyon group (98-2000) Suzuki-Trotter formalism and TBA approach to thermodynamics Suzuki (85), Klümper-Pearce-Batchelor (91-92), Destri-De Vega (93)... Generalization of multiple integrals to the temperature case Göhmann-Klümper-Seel (04) Observation of factorization of multiple integrals and relation to the qkz Takahashi (77), Korepin-HB (01-02), Korepin-Smirnov-HB (03-04) Reduced qkz, exponential form and fermionic basis Jimbo-Miwa-Smirnov-Takeyama-HB (05-09), Jimbo-Miwa-Smirnov (08) Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [intro 2/25]
Introduction The Hamiltonian of the infinite spin- 1 2 XXZ chain H XXZ = 1 ( ) 2 σ x jσ x j+1 + σ y j σy j+1 + σz j σz j+1, = 1 2 (q+q 1 ) q = e η = e πiν Integrable structure is generated by R-matrix of the 6-vertex model 1 0 0 0 0 b(λ) c(λ) 0 R(λ) = 2 2 0 c(λ) b(λ) 0 0 0 0 1 b(λ) = sh(λ) sh(λ+η), c(λ) = sh(η) sh(λ+η), Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [intro 3/25]
Partition function of the 6-vertex model on cylinder ylinder is infinite in spacial direction ut corresponds to insertion of local operator O We call q α 0 σ z j O quasi-local operator with tail α κ magnetic field α disorder parameter Z κ{ q α 0 σ } z j O = Matsubara expectation values: ( Tr S Tr M T S,M q κ σ z+α j 0 σ ) z j O ( Tr S Tr M T S,M q κ σ z+α j 0 ) σ z j T S,M is the transfer matrix with H S = 2 and H M = n 2 j=1 Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [Partition function of the 6-vertex model on cylinder 4/25]
Density Matrix Elements in standard Basis Let us define the elements of density matrix taking O = E ε 1 ε E ε n 1 ε n ε D 1,,ε n N ε 1,,ε n (ξ 1,,ξ n ) := Z κ{ q α 0 } E ε n ε n = κ+α T ε 1 ε 1 (ξ 1,κ) T ε n ε n (ξ n,κ) κ T(ξ 1,κ) T(ξ n,κ) κ+α κ σ z j E ε 1 ε 1 with unique eigenvector κ > of transfer matrix corresponding to the largest eigenvalue T(ξ,κ) and horizontal inhomogeneities ξ j = e ν j. First we take arbitrary vertical inhomogeneities τ j = e β j, j = 1,,N, so that the monodromy matrix T a (ζ,κ) = R a,n (λ β N )...R a,1 (λ β 1 )q κσz a, ζ = e λ Density matrix fulfills reduction property { } tr 1 DN (ξ 1,...,ξ n κ,α)q ασz 1 = ρ(ξ1 )D N (ξ 2,...,ξ n κ,α), ρ(ζ) := T(ζ,κ+α) T(ζ,κ) tr n { DN (ξ 1,...,ξ n κ,α) } = D N (ξ 1,...,ξ n 1 κ,α) Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [Density Matrix Elements in standard Basis 5/25]
Q-matrix, TBA and auxiliary function a Let us introduce Q-matrix with eigenvalues Q(λ,κ) satisfying the Baxter s T Q-relation: T(λ,κ)Q(λ,κ) = q κ d(λ)q(λ+η,κ)+q κ a(λ)q(λ η,κ) with a(λ) := N j=1 sinh(λ β j + η) and d(λ) := N j=1 sinh(λ β j). The right and left eigenvectors κ and κ+α can be constructed by means of the algebraic Bethe ansatz. They are parameterized by two sets {λ} = {λ j } N/2 j=1 and {µ} = {µ j } N/2 j=1 Q(λ,κ) = of Bethe roots N/2 N/2 sinh(λ λ j ), Q(λ,κ+α) = sinh(λ µ j ) j=1 j=1 which are special solutions to the Bethe ansatz equations 2κ d(λ) a(λ,κ) := q a(λ) Q(λ+η,κ) Q(λ η,κ) = 1, a(λ,κ+α) = 1 Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [Q-matrix, TBA and auxiliary functions a 6/25]
Q-matrix, TBA and auxiliary function a One derives the TBA-equation: ln(a(λ,κ)) = N [ ] sh(λ β j ) dµ (N 2κ)η+ ln K(λ µ)ln(1+a(µ,κ)) j=1 sh(λ β j + η) 2πi with K(λ) = cth(λ η) cth(λ+η) and the integration contour λ Im η 2 Re η 2 Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [Q-matrix, TBA and auxiliary functions a 7/25]
Q-matrix, TBA and auxiliary function a One can introduce temperature via alternating inhomogeneities β 2 j 1 = η β N β j =, j = 1,...,N/2 β 2 j = β N with the inverse temperature β = T 1 and then taking the Trotter limit N ln(a(λ,κ)) = 2κη sh(η)e(λ) T with the bare energy : e(λ) = cth(λ) cth(λ + η) dµ K(λ µ)ln(1+a(µ,κ)) 2πi The function ρ is given through the solution of the above equation { [ ]} ρ(ζ) = q α dµ 1+a(µ,κ+α) exp e(µ λ)ln, ζ = e λ 2πi 1+a(µ,κ) Analytical property: a has essential singularity at λ = 0 in contour as well as at λ = ±η outside the contour and ρ only outside the contour at ζ = q ±1. Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [Q-matrix, TBA and auxiliary functions a 8/25]
The multiple integral representation of the density matrix One can adapt the derivation of the multiple integrals developed by Göhmann, Klümper and Seel to the case with α and obtain the following expression D N ε 1...ε n ε 1...ε n (ξ 1,...,ξ n ) = [ p j=1 dm(λ j ) F + l j (λ j )][ n j=p+1 ] dm(λ j ) Fl j (λ j ) [ det j,k=1,...,n G(λ j,ν k ) ] 1 j<k n sh(λ j λ k η)sh(ν k ν j ) dλ where dm(λ) = 2πiρ(ζ)(1+a(λ,κ)), dm(λ) = a(λ,κ)dm(λ), l j 1 F l ± n ε j (λ) = sh(λ ν k ) + j j = 1,..., p sh(λ ν k η), l j = k=1 k=l j +1 j = p+1,...,n ε n j+1 with ε + j the jth plus in the sequence (ε j ) n j=1, ε j the jth minus sign in the sequence (ε j )n j=1 and p the number of plus signs in (ε j) n j=1. Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [The multiple integral representation of the density matrix 9/25]
The multiple integral representation of the density matrix The function G is a solution to the linear integral equation G(λ,ν) = q α cth(λ ν η) ρ(ξ)cth(λ ν)+ where ξ = e ν and the kernel: is deformed version of the original kernel K. The case n = 1 dm(µ)k α (λ µ)g(µ,ν) K α (λ) = q α cth(λ η) q α cth(λ+η) There are only two non-vanishing density matrix elements. From the reduction we have D+ +(ξ) 1 = q α 1 1 D q (ξ) α q α q α 1 ρ(ξ) When we insert it into the integral representation, we get an interesting identity: dm(µ)g(µ,ν) = q α ρ(ν) q α q α and hence lim Reλ ± G(λ,ν) = 0 Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [The multiple integral representation of the density matrix 10/25]
Factorization preliminary remarks Factorization means certain reducibility of multiple integrals representing correlation functions to a combination of one-dimensional integrals with some coefficients of algebraic nature. Example for XXX at T = 0,α = 0,κ = 0: vac S1 z Sz 3 vac = 1 4 4ln2+ 9 4 ζ(3) Takahashi (77) We know that all correlation functions for the XXX and XXZ model at zero temperature factorize in this sense. Korepin, HB (01-02), Korepin, Smirnov, HB (03-04), Jimbo, Miwa, Smirnov, Takeyama, HB (04-08) We know that some particular correlation functions factorize for the XXX and XXZ model at finite temperature and α = 0 Göhmann, Klümper, Suzuki, HB (06-07) The claim is that it generally works for all correlation functions for the XXZ model with temperature, magnetic and disorder field. We continue with the n = 2 case. Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [Factorization preliminary remarks 11/25]
Factorization of the density matrix for n = 2 There are six non-vanishing elements of the density matrix for n = 2: one for p = 0, four for p = 1 and one for p = 2. We take the case p = 1. After substituting w j = e 2µ j and ξ j = e ν j, j = 1,2, the corresponding integrals are all of the form I = 1 ξ 2 2 ξ2 1 dm(µ 1 ) dm(µ 2 )det [ G(µ j,ν k ) ] r(w 1,w 2 ) where r(w 1,w 2 ) = p(w 1,w 2 ) w 1 q 2 w 2, p(w 1,w 2 ) = c 0 w 1 w 2 + c 1 w 1 + c 2 w 2 + c 3 The coefficients c j are different for the four different matrix elements. Inserting dm(µ) = dµ 2πiρ(e µ ) dm(µ) here and taking into account that ρ(eµ ) is analytic and non-zero inside we obtain I(ξ 2 2 ξ 2 1) = dm(µ) det G(µ,ν 1) G(µ,ν 2 ) r(w,ξ 2 1 ) r(w,ξ2 2 ) dm(µ 1 ) dm(µ 2 )det [ G(µ j,ν k ) ] r(w 1,w 2 ), w = e 2µ Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [Factorization of the density matrix for n = 2 12/25]
Factorization of the density matrix for n = 2 The key idea is then to find rational functions f(w) and g(w) such that r(w 1,w 2 ) r(w 2,w 1 ) = f(w 1 ) f(w 2 )+g(w 1 )K α (µ 1 µ 2 ) g(w 2 )K α (µ 2 µ 1 ) and taking residues at w 1 = q 2 w 2 we obtain a difference equation for g g(q 2 w)y 1 g(w)y = p(q2 w,w) 2q 2, y := qα w which can be solved: g(w) = g + w+g 0 + g w with the coefficients g + = c 0y 2(q 2 y 2 ), g = c 3y 2(1 q 2 y 2 ), g 0 = (c 1+q 2 c 2 )y 2(1 y 2 ) ) and hence f(w) = (y y 1 ) (g + w g w. ollecting all terms, we arrive at I = g(ξ2 2 )Ψ(ξ 2,ξ 1 ) g(ξ 2 1 )Ψ(ξ 1,ξ 2 ) ξ 2 2 ξ2 1 where Ψ(ξ 1,ξ 2 ) = + (c 1 q 2 c 2 )(ρ(ξ 1 ) ρ(ξ 2 )) 2(ξ 2 2 ξ2 1 )(y y 1 ) + (y 1 ρ(ξ 1 ))(y ρ(ξ 2 )) f(ξ 2 2 ) (y 1 ρ(ξ 2 ))(y ρ(ξ 1 )) f(ξ 2 1 ) (ξ 2 2 ξ2 1 )(y y 1 ) 2, dm(µ)g(µ,ν 2 ) ( q α cth(µ ν 1 η) ρ(ξ 1 )cth(µ ν 1 ) ) Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [Factorization of the density matrix for n = 2 13/25]
From fermionic operators to fermionic basis The factorized elements of the density matrix look complicated since the standard basis might be not so convenient. We suggested another basis based on fermionic operators. Jimbo, Miwa, Smirnov, Takeyama, HB (08 09) onsider a space W (α) = W α s,s where W α s,s is the space of s= quasi-local operators of spin s with tail α s. The creation operators t (ζ), b (ζ), c (ζ) and annihilation operators b(ζ), c(ζ) act on W (α). They are one-parameter families of operators of the form t (ζ) = (ζ 2 1) p 1 t p, p=1 b (ζ) = ζ α+2 b(ζ) = ζ α (ζ 2 1) p 1 b p, c (ζ) = ζ α 2 (ζ 2 1) p 1 c p, p=1 p=1 (ζ 2 1) p b p, c(ζ) = ζ α (ζ 2 1) p c p p=0 p=0 Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [From fermionic operators to fermionic basis 14/25]
From fermionic operators to fermionic basis The operator t p commutes with all other operators, the operators b p, c p, b p, c p with p > 0 are fermions with canonical anti-commutation relations. Fourier modes have block structure: t p : W α s,s W α s,s b p,c p : W α s+1,s 1 W α s,s, c p,b p : W α s 1,s+1 W α s,s. We call the operator q α 0 σ z j a primary field. The annihilation operators kill the primary field. The creation operators acting on the primary field produce different quasi-local operators. We proved that the operators ( τ m t p 1 t p j b q 1 b q k c r 1 c r k q α 0 ) σ z j, τ = t 1/2, m Z, j,k Z 0, p 1 p j 2, q 1 > > q k 1, r 1 > > r k 1 constitute a basis of W α,0. Length of quasi-local operator is not bigger than p m + q m + r m. learly, this description of W α,0 reminds that of FT. Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [From fermionic operators to fermionic basis 15/25]
From fermionic operators to fermionic basis Important theorem was proved by Jimbo, Miwa and Smirnov (09) Z κ{ t (ζ)(x) } = 2ρ(ζ)Z κ {X}, Z κ{ b (ζ)(x) } = 1 2πi Z κ{ c (ζ)(x) } = 1 2πi Γ Γ ω(ζ,ξ) Z κ{ c(ξ)(x) } dξ 2 ξ 2, ω(ξ,ζ) Z κ{ b(ξ)(x) } dξ 2 where the contour Γ goes around ξ 2 = 1 and ω is some explicit function. These formulae allow one to explicitly calculate Z κ{ t (ζ 0 1) t (ζ 0 p)b (ζ + 1 ) b (ζ + q )c (ζ q ) c (ζ 1 )( q α 0 σ z j)} = p = i=1 2ρ(ζ 0 i )det ω(ζ + i,ζ j ) i, generating function for series in ζ 2 1 j=1, q ξ 2 Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [From fermionic operators to fermionic basis 16/25]
The function ω The above theorem states that any correlation function corresponding to some quasi-local operator O can be reduced to a combination of two transcendental functions ρ and ω. They depend on temperature and magnetic field in contrast to the basis which is pure algebraic. the basis is independent of inhomogeneities in the Matsubara direction one can take a finite Trotter number N and arbitrary inhomogeneities there and find ρ and ω. In this case ω was found to be related to the deformed abelian integrals. It satisfies the so-called normalization condition T(ζ,κ)ω(ζ,ξ)Q Γm (ζ,κ+α)q + (ζ,κ)ϕ(ζ) dζ2, m = 0,,N ζ2 where Γ 0 encircles 0 and Γ m encircle two poles ζ 2 = τ 2 m and ζ 2 = τ 2 m/q 2, Q ± are eigenvalues of Q-matrices corresponding to two linear independent 1 solutions of the TQ-relation and ϕ(ζ) := (ζ 2 /τ 2 m 1)(ζ 2 /τ 2 m/q 2 1). Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [The function ω 17/25] N m=1
The function ω The above formulation of the ω-function is very beautiful but it is not quite clear how to take the Trotter limit N after specifying the inhomogeneities. The further Theorem stays: F. Göhmann, HB (09) The function ω has an alternative representation in terms of the TBA-functions ωrat(ξ 1,ξ 2 ) = 2(ξ 1 /ξ 2 ) α Ψ(ξ 1,ξ 2 ) ψ(ξ)+2 ( ρ(ξ 1 ) ρ(ξ 2 ) ) ψ(ξ) ω(ξ 1,ξ 2 ) = ωrat(ξ 1,ξ 2 )+D ζ D ξ 1 ζ ψ(ζ/ξ) where ψ(ξ) = ξα (ξ 2 +1) 2(ξ 2 1), ξ is the difference operator whose action on a function f is defined by ξ f(ξ) = f(qξ) f(q 1 ξ) and D ζ g(ζ) = g(qζ)+g(q 1 ζ) 2ρ(ζ)g(ζ) Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [The function ω 18/25]
The function ω Sketch of the proof: First we return to the case of finite N and arbitrary inhomogeneities. We use the fact that the function (ζ/ξ) α ω(ζ,ξ) is a rational function of ζ 2 of the form P(ζ 2 )/Q(ζ 2 ) with polynomials P,Q of degree at most N + 2. The zeros of Q are the N zeros of the transfer matrix eigenvalue T(ζ,κ) and two additional zeros at q ±2 ξ 2. Then we prove that the function ω satisfies the normalization condition: ω(ζ,ξ) ζ=τ j + ρ(τ j )ω(ζ,ξ) ζ=q 1 τ j = 0 corresponding to the contours Γ m with m = 1,,N and [ lim ζ 0 (ζ/ξ) α ω(ζ,ξ) = 2q κ q κ + q κ ρ(ξ) q α +(q α q α ) ] dm(µ)g(µ, ν) corresponding to the contour Γ 0. All together we have to provide N + 3 relations. We have N + 1 normalization conditions. Two more relations come from considering the residues at ζ 2 = q ±2 ξ 2. = 0 Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [The function ω 19/25]
The exponential form preliminary remarks The formula by Jimbo, Miwa and Smirnov makes possible the calculation of arbitrary correlation functions because of completeness of the fermionic basis. It also proves the factorization of the correlation functions. Sometimes it is still more convenient to avoid the creation operators and to have an explicit formula for the correlation functions in the standard basis. We know that such a formula exists in the zero temperature case: vac q α 0 σ z j O vac vac q α 0 σ z j vac = tr α{ exp(ω) ( q α 0 σ z jo )} where Ω = dζ 2 1 dζ 2 2 ω(ζ 2πiζ 2 1 2πiζ 2 1 /ζ 2,α)b(ζ 1 )c(ζ 2 ) 2 with some explicit function ω and tr α (X)= tr α 1 tr α 2 tr α 3 (X), tr α (x)=tr ( q 1 2 ασ3 x ) /tr ( q 1 2 ασ3 ), x End( 2 ) Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [The exponential form preliminary remarks 20/25]
The exponential form preliminary remarks onjecture: Similar formula does exist for the temperature case also Z κ{ q α 0 σ } z j O = tr α{ exp(ω) ( q α 0 σ z jo )} where Ω = Ω 1 + Ω 2 Ω 1 has a similar structure as in T = 0 case Ω 1 = dζ 2 1 2πiζ 2 1 dζ 2 2 2πiζ 2 (ω 0 (ζ 1 /ζ 2 α) ωrat(ζ 1,ζ 2 κ,α))b(ζ 1 )c(ζ 2 ) 2 ( ) 1 q α 2 ω 0 (ζ α) = 1+q α ζ ψ(ζ) The second part should be of the form: Ω 2 = dζ 2 2πiζ 2 log(ρ(ζ))t(ζ) where the operator t is yet to be determined. Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [The exponential form preliminary remarks 21/25]
The exponential form preliminary remarks Let us list some of the most important expected properties of the operator t. First, we expect that like t (ζ) the operator t(ζ) is block diagonal, t(ζ) : W α,s W α,s We will deal below mostly with the sector s = 0. Then we expect t(ζ) to have simple poles at ζ = ξ j. Let us define: t j = res ζ=ξ j t(ζ) dζ2, t ζ 2 j = t (ξ j ) In contrast to t j the operator t j is well defined only if it acts on the states X [m,n] with m n < j. Let us denote t [m,n] (ζ) and respectively t j[m,n] the operators defined on the interval [m,n] with m j n. We also expect the R-matrix symmetry s i t [m,n] (ζ) = t [m,n] (ζ)s i for m i < n, s i = K i,i+1 Ř i,i+1 (ξ i /ξ i+1 ) Ř i,i+1 (ξ i /ξ i+1 )(X) = Ř i,i+1 (ξ i /ξ i+1 )XŘ i,i+1 (ξ i /ξ i+1 ) 1 K i, j stands for the transposition of arguments ξ i and ξ j. Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [The exponential form preliminary remarks 22/25]
The exponential form preliminary remarks The further properties are: ommutation relations: [t j,t k ] = [t j,b(ζ 1 )c(ζ 2 )] = 0 Projector property: t 2 j = t j Relations with t : t j t k = t jt k for j k Reduction properties: t j t j = t j, t jt j = 0 t 1[1,n] (q ασz 1 X[2,n] ) = q ασz 1 X[2,n] t j[1,n] (q ασz 1 X[2,n] ) = q ασz 1 t j[2,n] (X [2,n] ) for 1 < j n t j[1,n] (X [1,n 1] ) = t j[1,n 1] (X [1,n 1] ) for 1 j < n t n[1,n] (X [1,n 1] ) = 0. Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [The exponential form preliminary remarks 23/25]
The exponential form preliminary remarks Let us comment on these relations. First of all the commutation relations lead to the factorization of the exponential exp(ω) = exp(ω 1 + Ω 2 ) = exp(ω 1 )exp(ω 2 ). As we know the operator Ω 1 becomes nilpotent when it acts on states of finite length. In contrast, the operator Ω 2 is not nilpotent, but due to the commutation relations and the projector property exp(ω 2 ) ( q α 0 σ z ) n jx [1,n] = where ρ j = ρ(ξ j ). j=1 (1 t j[1,n] + ρ j t j[1,n] ) ( ) α 0 σ z j X [1,n] q The reduction properties look standard except for the first one. It is easy to see that we need all of them in order to have the reduction property of the density matrix. For the moment we know explicit expression of t only for some particular cases. Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [The exponential form preliminary remarks 24/25]
onclusions The main purpose of my talk was to show that the factorization works for the temperature case of the XXZ model at magnetic and disorder field despite of all complications coming from the TBA-approach. I think it could be a hint that factorization of correlation functions might be a fundamental property of integrable models. In case of the XXZ model we could find rather suitable fermionic basis for which the determinant formula is valid. It looks similar to the Slater-determinant of the free fermion model. The whole non-trivial information about the structure of the Matsubara space is hidden in two transcendental functions ρ and ω. Now we know that one can take the scaling limit of these functions and get the results for the one-point correlator of FT. joint paper with M.Jimbo, T.Miwa and F. Smirnov (09) Temperature correlation functions of quantum spin chains at magnetic and disorder field Stony Brook, 20/1/2010 [onclusions 25/25]