Master equation for correlation functions in algebra symmetry \mathfrak{gl}(2|1) related models
aa r X i v : . [ h e p - t h ] F e b SciPost Physics Submission
Master equation for correlation functionsin algebra symmetry gl (2 | related models A. Hutsalyuk a , A. Liashyk b,c a Department of Theoretical Physics,E¨otv¨os Lor´and University Budapest MTA-ELTE “Momentum” Integrable QuantumDynamics Research Group,E¨otv¨os Lor´and University Budapest b National Research University Higher School of Economics, Faculty of Mathematics,Moscow, Russia c Skolkovo Institute of Science and Technology, Moscow, Russia
Abstract
We consider integrable models solved by the nested algebraic Bethe ansatz and asso-ciated with gl (2 |
1) or gl (3) algebra symmetry. The analogue of sum formulae, previouslyformulated for scalar products, is established for the form factors and correlation functions.These formulae are direct generalisation of the some earlier results derived for models with gl (2) symmetric R -matrix. It is also shown that in the case of algebra symmetry gl (2 | Correlation functions and form factors of integrable systems were the object of interest for along time [1–4]. Among the different approaches one of the most successful is algebraic Betheansatz (ABA) developed in [5–7]. Using the ABA correlation functions of integrable systemswere extensively studied [8–16]. Asymptotic behaviour, temperature and time dependence ofcorrelation functions were established in multiple systems.Recently the major interest is attracted by models described by the nested Bethe ansatz (see [17–19]). These models are related to multicomponent systems or systems with theadditional internal degrees of freedom (such as spin, colour charge, etc.) [20–23]. However upto now these systems were much less studied and there are relatively few results on correlationfunctions in case of models described by the nested ABA (see [24], [25]).The problem is the complexity of the method. Thus, typically in order to build the n -point correlation function using the ABA the knowledge of at least the scalar product oftwo eigenvectors is required. Usually such scalar products are given by extremely bulkyexpressions containing multiple summations (so-called Reshetikhin formula [26, 27]) and it isquite complicated to find a compact form for them. In the case of one-component models (orthe same, algebra symmetry gl (2) related models) the problem was solved in [28]. In the caseof multicomponent models (or the same, algebra symmetry gl ( N ) related models) the problemremained unsolved for a long time. Recent advances in this direction [29, 30] finally allowus to build an integral representation for correlation functions in algebra symmetry gl (2 | [email protected], [email protected] ciPost Physics Submission Note that in most of the paper we do not focus on a particular model, since algebraicBethe ansatz approach allows one to describe simultaneously all models associated with aparticular algebra symmetry. See section 1.2 and the comment at the end.The paper is organised as follows. In Section 1 we introduce notation and give a shortdescription of the ABA. In section 2 we prove the sum formula for generation function ofcorrelators, i.e. the analogue of the Reshetikhin formula for correlation functions. In section3 integral representation for correlation functions in algebra symmetry gl (2 |
1) related modelsare established. In section 4 we briefly describe how the result of the previous section canbe derived using the form factor series. In Conclusion the brief outlook of the perspectives isgiven.
Through the paper the following functions are used f ( x, y ) = x − y + cx − y , g ( x, y ) = cx − y , h ( x, y ) = f ( x, y ) g ( x, y ) = x − y + cc ,t ( x, y ) = g ( x, y ) h ( x, y ) = c ( x − y )( x − y + c ) . (1.1)We use the shorthand notation for sets ¯ u = { u , . . . , u k } . Sets are marked with Arabic orRoman numerals or Greek letters, for example ¯ u , ¯ v II , ¯ v α , etc. Individual elements of thesets are labelled by Latin letters, for example u j , v k . We use the following notation forcomplements of the sets ¯ u j = ¯ u \ u j . For arbitrary functions G ( s ), F ( s, t ) and arbitrary sets¯ x, ¯ y the following notation are applied for the products G (¯ y ) = y Y j =1 G ( y j ) , F ( s, ¯ y ) = y Y j =1 F ( s, y j ) , F (¯ x, ¯ y ) = y Y j =1 x Y k =1 F ( x k , y j ) , etc. (1.2)For a = f, g, h the short-hand notation for skew-symmetric products are used∆ a (¯ x ) = Y i>j a ( x i , x j ) , ∆ ′ a (¯ x ) = Y i 1, 2 that are C in thecase of algebra symmetry gl (3) and C | in the case of algebra symmetry gl (2 | 1) related models.Here C | denotes Z -graded vector space with a grading [1] = [2] = 0, [3] = 1 (square brackets2 ciPost Physics Submission denote the parity). Matrices acting in C | in case of gl (2 | 1) algebra symmetry are also gradedwith a grading given by [ e ij ] = [ i ] + [ j ] where we define elementary units ( e ij ) ab = δ ai δ bj . The R -matrix satisfies Yang-Baxter equation (YBE) R ( v, u ) R ( v ) R ( u ) = R ( u ) R ( v ) R ( v, u ) , (1.5)that holds in a tensor product of three spaces (cid:0) C (cid:1) ⊗ (or (cid:0) C | (cid:1) ⊗ in a case of the gl (2 | R ( u, v ) = I + c P u − v , P = X i,j =1 ( − [ j ] e ij ⊗ e ji , I = X i,j =1 e ii ⊗ e jj . (1.6)(For algebra symmetry gl (3) all [ j ] = 0). We denote the Bethe ansatz vacuum by | i andassume normalisation h | i = 1. Vacuum eigenvalues of the diagonal entries of the monodromymatrix T ii are denoted by λ i and their ratios by r i T ii ( t ) | i = λ i ( t ) | i , i = 1 , . . . , , (1.7) r ( t ) = λ ( t ) /λ ( t ) , r ( t ) = λ ( t ) /λ ( t ) . (1.8)We expand agreement (1.2) for the commuting operators, for instance T (¯ u ), T (¯ v ), etc.Note, however, that in case of algebra symmetry gl (2 | 1) related models operators T , T with different arguments do not commute with themselves, thus T ( u ) T ( v ) = T ( v ) T ( u ).Instead, in this case we introduce symmetrised products T j (¯ v ) = ∆ h (¯ v ) − T j ( v ) . . . T j ( v n ) , T j (¯ v ) = ∆ ′ h (¯ v ) − T j ( v ) . . . T j ( v n ) , j = 1 , . (1.9)ABA implies the existence of the special objects called Bethe vectors . In a case of al-gebra symmetry gl (2) related models Bethe vectors are monomials on the matrix element T ( u ) acting on the vacuum and depend on a set of spectral parameters ¯ u (also called Betheparameters) | ¯ u i = T ( u a ) . . . T ( u ) | i . (1.10)For models related to the higher rank algebra symmetries we need to apply the so-called nested Bethe ansatz procedure [17–19]. The Bethe vectors now are given by the specialpolynomials on monodromy matrix entries acting on vacuum, and depend on two sets ofspectral parameters { ¯ u, ¯ v } . We refer to these sets as the Bethe parameters of the first andthe second level of the nesting. The explicit form of the Bethe vectors is given by [31] | ¯ u ; ¯ v i = X λ (¯ u ) λ (¯ v II ) g (¯ v I , ¯ u I ) f (¯ u I , ¯ u II ) g (¯ v II , ¯ v I ) h (¯ u I , ¯ u I ) T (¯ u I ) T (¯ u II ) T (¯ v II ) | i , (1.11)in the case of gl (2 | 1) and [32] | ¯ u ; ¯ v i = X λ (¯ v II ) λ (¯ u ) K n (¯ v I | ¯ u I ) f (¯ v II , ¯ v I ) f (¯ u I , ¯ u II ) T (¯ u I ) T (¯ u II ) T (¯ v II ) | i , (1.12) Here and further subscripts of T ij denote the matrix indices in the auxiliary space, not numbers of spaces. Note that these are the only quantities that depend on a particular model in all the paper. Pay attention that our normalisation differs from used in [27, 30] by the additional factor f (¯ v, ¯ u ) in thenumerator. ciPost Physics Submission in the case of gl (3) algebra symmetry related models, where Izergin-Korepin determinant isdefined as K n (¯ x | ¯ y ) = ∆ ′ g (¯ x )∆ g (¯ y ) h (¯ x, ¯ y ) det n [ t ( x j , y k )] , (1.13)and in both cases sum is taken over partitions ¯ u → { ¯ u I , ¯ u II } , ¯ v → { ¯ v I , ¯ v II } . Dual (left)Bethe vectors can be obtained by mapping ψ : | i → h | , ψ : T ij → ( − [ i ][ j ]+[ i ] T ji and ψ ( AB ) = ( − [ A ][ B ] ψ ( B ) ψ ( A ).We denote the cardinalities of sets as u = a and v = b . In the case sets { ¯ u, ¯ v } satisfy the system of Bethe ansatz equations (BAE) Bethe vectors become eigenvectors of theHamiltonian of the model. We call such Bethe vectors on-shell , otherwise they are called off-shell or generic. Using the shorthand notation Bethe equations can be written as r ( u j ) = 1 κ f ( u j , ¯ u j ) f (¯ u j , u j ) f (¯ v, u j ) , j = 1 , . . . , a,r ( v j ) = 1 κ (cid:18) f (¯ v j , v j ) f ( v j , ¯ v j ) (cid:19) s f ( v j , ¯ u ) , j = 1 , . . . , b, (1.14)where κ = { κ , κ } are twists (see [8, 9]), s = 1 for algebra symmetry gl (3) and s = 0for algebra symmetry gl (2 | 1) related models. In case when κ = 1 and/or κ = 1 BAEtraditionally are explicitly called twisted BAE and on-shell Bethe vectors that correspond tosolutions of such BAE are called twisted on-shell-Bethe vectors .We define (twisted) transfer matrix as a (graded) trace of the monodromy matrix. t κ ( w ) = X i =1 κ i ( − [ i ] T ii ( w ) . (1.15)It posses the commutativity property for arbitrary parameters u , v [ t κ ( u ) , t κ ( v )] = 0 . (1.16)(Twisted) on-shell Bethe vectors are also eigenvectors of the (twisted) transfer matrix t κ ( w ) | ¯ u ; ¯ v i = τ κ ( w | ¯ u ; ¯ v ) | ¯ u ; ¯ v i . (1.17)Further we use a concept of two-site model [8, 9] (also called partial model ). Considerthe system of length L with two subsystems, correspondingly [0 , x ] and [ x, L ]. We denotequantities belonging to the i subsystem by the superscript ( k ) and call them partial. Thus λ ( k ) i ( t ), i = 1 , . . . , k = 1 , k and the vacuum vector is given by | i = | i (2) ⊗ | i (1) . It can be shown that Bethe vectors of the total model can be expressedvia the partial Bethe vectors in a following way | ¯ u ; ¯ v i = X r (2)1 (¯ u I ) r (1)3 (¯ v II ) f (¯ u II , ¯ u I ) f (¯ v II , ¯ v I ) f (¯ v I , ¯ u II ) | ¯ u II ; ¯ v II i (2) ⊗ | ¯ u I ; ¯ v I i (1) . (1.19)The sum is taken over partitions ¯ u → { ¯ u I , ¯ u II } , ¯ v → { ¯ v I , ¯ v II } . The dual Bethe vector has absolutely similar property h ¯ u ; ¯ v | = X r (1)1 (¯ u II ) r (2)3 (¯ v I ) f (¯ u I , ¯ u II ) f (¯ v I , ¯ v II ) f (¯ v II , ¯ u I ) h ¯ u II ; ¯ v II | (2) ⊗ h ¯ u I ; ¯ v I | (1) . (1.18) ciPost Physics Submission Following [8] we use the concept of generalised model in which sets { r ( u k ) } , k = 1 , . . . , a , { r ( v j ) } , j = 1 , . . . , b are treated as sets of free parameters, without any reference to theparticular model.As we already mention, we do not concentrate on a particular model, but rather on algebrasymmetry of the R -matrix and our results in this way does not depend on particular model.In order to specify results for a particular model it is enough to specify λ i defined in (1.7),that can be done at the very end. Thus, solving the problem for R -matrix with algebrasymmetry gl (2 | 1) we can describe both 1D Fermi gas [20, 23] and lattice hopping model[33, 34]. Exception is section 4 that concentrates on the particular vase of t-J model. Introduce operators Q ( k ) i , k = 1 , i = 1 , i inthe k subsystem (and Q (1) i + Q (2) i = Q i ). Operatorsexp( αQ ) = exp (cid:16) − α Q (1)1 + α Q (1)2 (cid:17) , exp( αQ (1) ) = exp (cid:16) − α Q (1)1 + α Q (2)2 (cid:17) (1.20)are generators for two-point correlation functions of the densities of particles.Thus, for the Gaudin-Yang model (spin-1/2 Fermi gas) with a Hamiltonian H = Z L dx X α,β n ∂ψ † α ∂ψ α + 2 cψ † α ψ † β ψ β ψ α o , α, β = ↑ , ↓ , (1.21)and canonical commutation relation { ψ † α ( x ) , ψ ( y ) } = δ ( x − y ) δ αβ we have for the particlesdensities ( q is the total density and q is the density of particles with the projection of spindown) Q (1) i = Z x dz q i ( z ) , Q (2) i = Z Lx dz q i ( z ) , (1.22) h q i ( x ) q i (0) i = − ∂ ∂x ∂ ∂α i D exp (cid:16) αQ (1) (cid:17)E(cid:12)(cid:12)(cid:12)(cid:12) α =0 , (1.23)and for i = j h q i ( x ) q j (0) i = ∂ ∂x ∂ ∂α ∂α D exp (cid:16) − α Q (1)1 + α Q (2)2 (cid:17)E(cid:12)(cid:12)(cid:12)(cid:12) α =0 . (1.24)Notation α = 0 here and further means that α = α = 0. Since by definition Q (2)2 = Q − Q (1)2 and Q | ¯ u B ; ¯ v B i = b | ¯ u B ; ¯ v B i , ( b is just a total number of particles on [0 , L ]) we can writeexp( Q (1) ) = exp( αQ ) | α →− α exp( α Q ) . (1.25)Thus we will concern only about one of generation functions, since the other one can be foundvia (1.25).In the same way correlators in other models can be expressed via the derivatives of h exp( αQ (1) ) i , h exp( αQ ) i (for example correlators of electrons densities in supersymmetric t-Jmodel, densities in the Fermi-Bose mixtures). For the lattice models the derivatives w.r.t. x would be naturally replaced by the finite differences and the integrals over the subsystems bythe sums. 5 ciPost Physics Submission We define scalar product of two Bethe vectors S a,b as S a,b (¯ u C ; ¯ v C | ¯ u B ; ¯ v B ) = h ¯ u C ; ¯ v C | ¯ u B ; ¯ v B i . (2.1)It can be shown that scalar product of the off-shell Bethe vectors can be presented via highestcoefficients (Reshetikhin formula) Z k,n [26] as S a,b (¯ u C ; ¯ v C | ¯ u B ; ¯ v B ) = X r (¯ u B I ) r (¯ u C II ) r (¯ v B I ) r (¯ v C II ) f (¯ u C I , ¯ u C II ) f (¯ u B II , ¯ u B I ) f (¯ v C II , ¯ v C I ) f (¯ v B I , ¯ v B II ) × f (¯ v C I , ¯ u C I ) f (¯ v B II , ¯ u B II ) Z a − k,n (¯ u C II ; ¯ u B II | ¯ v C I ; ¯ v B I ) Z k,b − n (¯ u B I ; ¯ u C I | ¯ v B II ; ¯ v C II ) . (2.2)Here u = a , v = b . The sum is taken over partitions ¯ u B → { ¯ u B I , ¯ u B II } , ¯ v B → { ¯ v B I , ¯ v B II } andthe same for { ¯ u C , ¯ v C } . Coefficients Z m,n do not depend on a particular model but only onthe algebra symmetry of R -matrix.Our goal in this section is to show that for the matrix elements of operator exp( αQ )formula with a structure similar to (2.2) can be established.Using the property (1.19) we immediately arrive at the following representation for h exp( αQ ) ih ¯ u C ; ¯ v C | exp( αQ ) | ¯ u B ; ¯ v B i = X κ − a κ b r (1)1 (¯ u C II ) r (2)1 (¯ u B I ) r (2)3 (¯ v C I ) r (1)3 (¯ v B II ) f (¯ v C II , ¯ u C I ) f (¯ v B I , ¯ u B II ) × S (¯ u C I ; ¯ v C I | ¯ u B I ; ¯ v B I ) S (¯ u C II ; ¯ v C II | ¯ u B II ; ¯ v B II ) f (¯ u C I , ¯ u C II ) f (¯ u B II , ¯ u B I ) f (¯ v C I , ¯ v C II ) f (¯ v B II , ¯ v B I ) , (2.3)where r ( ℓ ) i , ℓ = 1 , , i = 1 , r i of ℓ -subsystem and S ℓ , ℓ = 1 , κ i = e α i , i = 1 , αQ ) only contributes producing the additional factors κ − a , κ b , where a , b are thenumbers of particles of the first and the second type in the first subsystem, i.e. eigenvaluesof Q (1)1 and Q (1)2 .Substituting (2.2) into (2.3) we get h ¯ u C ; ¯ v C | exp( αQ ) | ¯ u B ; ¯ v B i = X κ − a κ b r (1)1 (¯ u C II ) r (2)1 (¯ u B I ) r (2)3 (¯ v C I ) r (1)3 (¯ v B II ) × f (¯ v C II , ¯ u C I ) f (¯ v B I , ¯ u B II ) f (¯ u C I , ¯ u C II ) f (¯ u B II , ¯ u B I ) f (¯ v C I , ¯ v C II ) f (¯ v B II , ¯ v B I ) × r (1)1 (¯ u B ) r (1)1 (¯ u C ) r (1)3 (¯ v B ) r (1)3 (¯ v C ) f (¯ u C , ¯ u C ) f (¯ u B , ¯ u B ) f (¯ v C , ¯ v C ) f (¯ v B , ¯ v B ) × f (¯ v C , ¯ u C ) f (¯ v B , ¯ u B ) Z a − k ,n (¯ u C ; ¯ u B | ¯ v C ; ¯ v B ) Z k ,b − n (¯ u B ; ¯ u C | ¯ v B , ¯ v C ) × r (2)1 (¯ u B ) r (2)1 (¯ u C ) r (2)3 (¯ v B ) r (2)3 (¯ v C ) f (¯ u C , ¯ u C ) f (¯ u B , ¯ u B ) f (¯ v C , ¯ v C ) f (¯ v B , ¯ v B ) × f (¯ v C , ¯ u C ) f (¯ v B , ¯ u B ) Z a − k ,n (¯ u C ; ¯ u B | ¯ v C ; ¯ v B ) Z k ,b − n (¯ u B ; ¯ u C | ¯ v B ; ¯ v C ) , (2.4)here the sets are divided as ¯ u B I → { ¯ u B , ¯ u B } , ¯ u B II → { ¯ u B , ¯ u B } , ¯ v B I → { ¯ v B , ¯ v B } , ¯ v B II → { ¯ v B , ¯ v B } and the same for { ¯ u C I , ¯ u C II , ¯ v C I , ¯ v C II } .Now we want to regroup the factors under the sum over partitions in order to find the newhighest coefficients (HC) for the expectation value (2.3) explicitly and derive the Reshetikhin-like formula structure. We gather the first Z a,b in (2.4) with the last and the second Z a,b withthe third and collect all factors that depend on the same variables as corresponding Z a,b . We do not give here the explicit representation for Z a,b since we do not need them, but it can be found in[35] for algebra symmetry gl (3) related models and in [27] for algebra symmetry gl (2 | 1) related models. Foralgebra symmetry gl (2) related models they coincide with Izergin-Korepin determinant (1.13). ciPost Physics Submission Let us make some simplification of the long factors separately. f (¯ v C , ¯ v C ) f (¯ v C , ¯ v C ) f (¯ v C , ¯ v C ) f (¯ v C , ¯ v C ) f (¯ v C , ¯ v C ) f (¯ v C , ¯ v C )= f (¯ v C , ¯ v C ) f (¯ v C , ¯ v C ) f (¯ v C , ¯ v C ) f (¯ v C , ¯ v C ) . (2.5)Here we denote f (¯ y i,j , ¯ x ) = f (¯ y i , ¯ x ) f (¯ y j , ¯ x ). In the similar way we obtain f (¯ u C , ¯ u C ) f (¯ u C , ¯ u C ) f (¯ u C , ¯ u C ) f (¯ u C , ¯ u C ) f (¯ u C , ¯ u C ) f (¯ u C , ¯ u C )= f (¯ u C , ¯ u C ) f (¯ u C , ¯ u C ) f (¯ u C , ¯ u C ) f (¯ u C , ¯ u C ) , (2.6) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C )= f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) . (2.7)Using equations (2.5)–(2.7) and absolutely similar simplification for sets { ¯ v B , ¯ u B } we arriveat h ¯ u C ; ¯ v C | exp( αQ ) | ¯ u B ; ¯ v B i = X r (1)1 (¯ u C ) r (1)1 (¯ u C ) r (2)1 (¯ u B ) r (2)1 (¯ u B ) r (2)3 (¯ v C ) r (2)3 (¯ v C ) κ − a κ b × r (1)3 (¯ v B ) r (1)3 (¯ v B ) r (1)1 (¯ u B ) r (1)1 (¯ u C ) r (1)3 (¯ v B ) r (1)3 (¯ v C ) r (2)1 (¯ u B ) r (2)1 (¯ u C ) r (2)3 (¯ v B ) r (2)3 (¯ v C ) × [ Z a − k ,n (¯ u C ; ¯ u B | ¯ v C ; ¯ v B ) Z k ,b − n (¯ u B ; ¯ u C | ¯ v B ; ¯ v C )] × [ Z k ,b − n (¯ u B ; ¯ u C | ¯ v B ; ¯ v C ) Z a − k ,n (¯ u C ; ¯ u B | ¯ v C ; ¯ v B )] × f (¯ u C , ¯ u C ) f (¯ u C , ¯ u C ) f (¯ u C , ¯ u C ) f (¯ u C , ¯ u C ) f (¯ u B , ¯ u B ) f (¯ u B , ¯ u B ) f (¯ u B , ¯ u B ) f (¯ u B , ¯ u B ) × f (¯ v C , ¯ v C ) f (¯ v C , ¯ v C ) f (¯ v C , ¯ v C ) f (¯ v C , ¯ v C ) f (¯ v B , ¯ v B ) f (¯ v B , ¯ v B ) f (¯ v B , ¯ v B ) f (¯ v B , ¯ v B ) × f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B ) . (2.8)Regrouping now the factors in (2.8), we can express h ¯ u C ; ¯ v C | exp ( αQ ) | ¯ u B ; ¯ v B i in terms of twonew highest coefficients (factors in the first and the second brackets correspondingly) h ¯ u C ; ¯ v C | exp( αQ ) | ¯ u B ; ¯ v B i = X r (1)1 (¯ u C ) r (1)3 (¯ v B ) r (2)1 (¯ u B ) r (2)3 (¯ v C ) κ k − a − k κ n − n + b × nX r (¯ u C ) r (¯ u B ) r (¯ v C ) r (¯ v B ) Z k ,b − n (¯ u B ; ¯ u C | ¯ v B ; ¯ v C ) Z a − k ,n (¯ u C ; ¯ u B | ¯ v C ; ¯ v B ) × κ a − k κ b − n f (¯ u C , ¯ u C ) f (¯ u C , ¯ u C ) f (¯ u C , ¯ u C ) f (¯ u B , ¯ u B ) f (¯ u B , ¯ u B ) f (¯ u B , ¯ u B ) × f (¯ v C , ¯ v C ) f (¯ v C , ¯ v C ) f (¯ v C , ¯ v C ) f (¯ v B , ¯ v B ) f (¯ v B , ¯ v B ) f (¯ v B , ¯ v B ) × f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B ) o × nX κ k − a κ n − b Z a − k ,n (¯ u C ; ¯ u B | ¯ v C ; ¯ v B ) Z k ,b − n (¯ u B ; ¯ u C | ¯ v B ; ¯ v C ) × f (¯ u C , ¯ u C ) f (¯ u B , ¯ u B ) f (¯ v C , ¯ v C ) f (¯ v B , ¯ v B ) f (¯ v C , ¯ u C ) f (¯ v B , ¯ u B ) o . (2.9)The common summation is taken over partitions ¯ u C → { ¯ u C , ¯ u C } , ¯ v C → { ¯ v C , ¯ v C } and thesame for { ¯ u B , ¯ v B } . In the first brackets summation is taken over partitions ¯ u C → { ¯ u C , ¯ u C } ,¯ v C → { ¯ v C , ¯ v C } and the same for { ¯ u B , ¯ v B } . In the second brackets summation is taken overpartitions ¯ u C → { ¯ u C , ¯ u C } , ¯ v C → { ¯ v C , ¯ v C } and the same for { ¯ u B , ¯ v B } .7 ciPost Physics Submission Let us make further simplifications now. After applying Bethe equations (1.14) for r (¯ u B )and r (¯ v B ) the long factors in the first HC (the first brackets in (2.9)) can be simplified. X f (¯ u C , ¯ u C ) f (¯ u C , ¯ u C ) f (¯ u C , ¯ u C ) f (¯ u B , ¯ u B ) f (¯ u B , ¯ u B ) f (¯ u B , ¯ u B ) f (¯ u B , ¯ u B ) × r (¯ u C ) r (¯ v C ) f (¯ v C , ¯ v C ) f (¯ v C , ¯ v C ) f (¯ v C , ¯ v C ) f (¯ v B , ¯ v B ) f (¯ v B , ¯ v B ) f (¯ v B , ¯ v B ) f (¯ v B , ¯ v B ) × f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B )= f (¯ u B , ¯ u B ) f (¯ u C , ¯ u C ) f (¯ v C , ¯ v C ) f (¯ v B , ¯ v B ) f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B ) × X f (¯ u C , ¯ u C ) f (¯ u C , ¯ u C ) f (¯ u C , ¯ u C ) f (¯ u B , ¯ u B ) f (¯ v C , ¯ v C ) f (¯ v C , ¯ v C ) f (¯ v C , ¯ v C ) f (¯ v B , ¯ v B ) × r (¯ u C ) r (¯ v C ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) f (¯ v B , ¯ u B ) . (2.10)The sum is taken over partitions ¯ u → { ¯ u , ¯ u } , ¯ v → { ¯ v , ¯ v } . The factors with f (¯ v B α , ¯ u B β )here were simplified in the following way: f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B )= f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B ) . (2.11)Now we absorb part of factors in the coefficients r , r r (¯ u C ) −→ ˆ r (¯ u C ) = r (¯ u C ) f (¯ v C , ¯ u C ) f (¯ u C , ¯ u C ) f (¯ u C , ¯ u C ) ,r (¯ v C ) −→ ˆ r (¯ v C ) = r (¯ v C ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ v C ) f (¯ v C , ¯ v C ) . (2.12)Finally, (2.10) can be written as f (¯ u B , ¯ u B ) f (¯ u C , ¯ u C ) f (¯ v C , ¯ v C ) f (¯ v B , ¯ v B ) f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) × f (¯ v B , ¯ u B ) X κ a − k κ b − n ˆ r (¯ u C )ˆ r (¯ v C ) f (¯ u C , ¯ u C ) f (¯ u B , ¯ u B ) f (¯ v C , ¯ v C ) f (¯ v B , ¯ v B ) × f (¯ v B , ¯ u B ) f (¯ v C , ¯ u C ) Z k ,b − n (¯ u B ; ¯ u C | ¯ v B ; ¯ v C ) Z a − k ,n (¯ u C ; ¯ u B | ¯ v C ; ¯ v B ) . (2.13)The last two lines of (2.13) coincide with the scalar product (2.2) where rapidities { ¯ u B , ¯ v B } satisfy (untwisted) Bethe equations, { ¯ u C , ¯ v C } are arbitrary and { r ( u ) , r ( v ) } are modi-fied according to (2.12) and posses twists: r → κ ˆ r , r → κ ˆ r . Partitions are ¯ u →{ ¯ u , ¯ u } , ¯ v → { ¯ v , ¯ v } . We denote this scalar product with the additional normalisation( f (¯ v C , ¯ u C ) f (¯ v B , ¯ u B )) − as ˆΘ αa − k + k ,b − n + n (¯ u C , ¯ u B | ¯ v B , ¯ v C ).The second highest coefficient (second brackets in (2.9)) coincides with a scalar productof the on-shell Bethe vector with parameters { ¯ u B , ¯ v B } and the twisted-on-shell Bethe vectorwith parameters { ¯ u C , ¯ v C } normalised by factor ( f (¯ v B , ¯ u B ) f (¯ v C , ¯ u C )) − . We denote thisnormalised scalar product as Θ αa − k + k ,b − n + n (¯ u C , ¯ u B | ¯ v B , ¯ v C ).Let us rename for brevity the sets from (2.9) as ¯ u → ¯ u , ¯ u → ¯ u , ¯ v → ¯ v and¯ v → ¯ v , and cardinalities of sets are also relabeled as a − k + k → a , a + k − k → a , b − n + n → b and b − n + n → b . Then the matrix element of the operator exp ( αQ )between off-shell and on-shell Bethe vectors can be presented in form of Reshetikhin sum8 ciPost Physics Submission formula where each set is divided into two subsets and the new highest coefficients Θ and ˆΘare given by the scalar products of Bethe vectors h ¯ u C ; ¯ v C | exp( αQ ) | ¯ u B ; ¯ v B i = X κ a κ b r (1)1 (¯ u C ) r (1)3 (¯ v B ) r (2)1 (¯ u B ) r (2)3 (¯ v C ) × f (¯ u B , ¯ u B ) f (¯ u C , ¯ u C ) f (¯ v C , ¯ v C ) f (¯ v B , ¯ v B ) f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) × f (¯ v B , ¯ u B ) f (¯ v C , ¯ u C ) ˆΘ αa ,b (¯ u C , ¯ u B | ¯ v B , ¯ v C ) Θ αa ,b (¯ u C , ¯ u B | ¯ v B , ¯ v C ) . (2.14)Here ¯ u B → { ¯ u B , ¯ u B } , ¯ v B → { ¯ v B , ¯ v B } and the same for sets { ¯ u C , ¯ v C } . This formula is a direct gl (3) analogue of one derived in [36] for the algebra symmetry gl (2) related models (see formulaC.7 there) . It is easy to check that the similar formula holds for the algebra symmetry gl (2 | 1) related models. The only difference is replacement of factors f (¯ v B , ¯ v B ) f (¯ v C , ¯ v C ) by g (¯ v B , ¯ v B ) g (¯ v C , ¯ v C ) and disappearance of the factor f (¯ v C , ¯ v C ) /f (¯ v C , ¯ v C ) in the second line of(2.12). h ¯ u C ; ¯ v C | exp( αQ ) | ¯ u B ; ¯ v B i = X κ a κ b r (1)1 (¯ u C ) r (1)3 (¯ v B ) r (2)1 (¯ u B ) r (2)3 (¯ v C ) × f (¯ u B , ¯ u B ) f (¯ u C , ¯ u C ) g (¯ v C , ¯ v C ) g (¯ v B , ¯ v B ) f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) × f (¯ v B , ¯ u B ) f (¯ v C , ¯ u C ) ˆΘ αa ,b (¯ u C , ¯ u B | ¯ v B , ¯ v C ) Θ αa ,b (¯ u C , ¯ u B | ¯ v B , ¯ v C ) . (2.15) Representations (2.14)-(2.15) allow further simplification in case if explicit compact formulaefor Θ a,b and ˆΘ a,b are known. The compact formulae (determinant representations) for the theΘ αm,n , ˆΘ αm,n are known for algebra symmetry gl (2 | 1) related models. For Θ αm,n it was derivedin [30] while the scalar product of the off-shell and on-shell Bethe vectors that coincides withˆΘ αm,n was derived in [29]. Further in this paper we restrict ourselves by the algebra symmetry gl (2 | 1) case. Our goal now is a derivation from (2.15) integral representation suitable for theasymptotic analysis.Explicit determinant representations of Θ αa,b is given byΘ αa,b (¯ z, ¯ u B | ¯ v B , ¯ y ) = g (¯ u B , ¯ y )∆ g (¯ u B )∆ g (¯ y )∆ ′ g (¯ z )∆ ′ g (¯ y ) h (¯ u B , ¯ u B ) h (¯ y, ¯ u B ) det a,b N , (3.1)where diagonal parts of block matrix N (¯ u B , ¯ v B | ¯ z, ¯ y ) are defined as N = t ( z j , u B k ) f (¯ v B , u B k ) h (¯ z, u B k ) f (¯ y, u B k ) h (¯ u B , u B k ) + 1 κ t ( u B k , z j ) h ( u B k , ¯ z ) h ( u B k , ¯ u B ) , j = 1 , . . . , a, k = 1 , . . . , a, N = δ jk g ( y k , ¯ v B ) g ( y k , ¯ y k ) (cid:18) − κ f ( y k , ¯ z ) f ( y k , ¯ u B ) (cid:19) , j = 1 , . . . , b, k = 1 , . . . , b, (3.2)and antidiagonal as N = 1 κ t ( y k , z j ) h ( u B k , ¯ z ) h ( u B k , ¯ u B ) , j = 1 , . . . , a, k = 1 , . . . , b, N = g ( u B k , ¯ v B ) g ( u B k , ¯ y ) (cid:18) g ( u B k , y j ) + κ / κ h ( y j , u B k ) (cid:19) , j = 1 , . . . , b, k = 1 , . . . , a. (3.3) Particular case of this formula was also derived in algebra symmetry gl (2) related models in [9], but theset ¯ u C , was also on-shell there. ciPost Physics Submission We can introduce two functions Y ( i ) α , i = 1 , Y (1) α ( z j | ¯ z, ¯ y ) = 1 − κ r ( z j ) f (¯ y, z j ) f (¯ z j , z j ) f ( z j , ¯ z j ) , Y (2) α ( y j | ¯ z, ¯ y ) = 1 − κ r ( y j ) f ( y j , ¯ z ) . (3.4)It is easy to check thatRes z j = u Bj det a,b N (¯ u B , ¯ v B | ¯ z, ¯ y ) Y (1) α ( z j | ¯ z, ¯ y ) = − κ h ( u B j , ¯ z j ) h ( u B j , ¯ u B j ) det a − ,b N (cid:0) ¯ u B j , ¯ v B | ¯ z j , ¯ y (cid:1) , Res y j = v Bj det a,b N (¯ u B , ¯ v B | ¯ z, ¯ y ) Y (2) α ( y j | ¯ z, ¯ y ) = − κ g ( v B j , ¯ v B j ) g ( v B j , ¯ y j ) f ( v B j , ¯ z ) f ( v B j , ¯ u B ) det a,b − N (cid:0) ¯ u B , ¯ v B j | ¯ z, ¯ y j (cid:1) . (3.5)The second scalar product is given byˆΘ αa,b (¯ u C , ¯ z | ¯ y, ¯ v C ) = g (¯ v C , ¯ z ) h (¯ v C , ¯ z ) h (¯ z, ¯ z )∆ ′ g (¯ u C )∆ g (¯ z )∆ ′ g (¯ v C )∆ g (¯ v C ) det a,b M , (3.6)where diagonal blocks of matrix M (¯ z, ¯ y | ¯ u C , ¯ v C ) are defined as M = φ ( u C j ) g ( z k , u C j ) − h ( z k , u C j ) ! h ( z k , ¯ u C ) h ( z k , ¯ z ) + g ( u C j , z k ) − φ ( u C j ) h ( u C j , z k ) ! f (¯ y, z k ) h (¯ u C , z k ) f (¯ v C , z k ) h (¯ z, z k ) ,j = 1 , . . . , a, k = 1 , . . . , a, M = δ jk − κ r ( v C j ) f ( v C j , ¯ z ) ! g ( v C j , ¯ y ) g ( v C j , ¯ v C j ) , j = 1 , . . . , b, k = 1 , . . . , b, (3.7)and antidiagonal blocks are M = − φ ( u C j ) g ( v C k , u C j ) + 1 h ( v C k , u C j ) ! h ( v C k , ¯ u C ) h ( v C k , ¯ z ) , j = 1 , . . . , a, k = 1 , . . . , b, M = g ( v C j , z k ) − κ r ( v C j ) f ( v C j , ¯ u C ) h ( v C j , z k ) ! g (¯ y, z k ) g (¯ v C , z k ) , j = 1 , . . . , b, k = 1 , . . . , a, (3.8)with φ ( u j ) = κ r ( u j ) f (¯ v, u j ) f (¯ u j , u j ) f ( u j , ¯ u j ) . (3.9)It is easy to check thatRes z j = u Cj det a,b M (¯ z, ¯ y | ¯ u C , ¯ v C ) Y (1) α ( z j | ¯ z, ¯ y ) = − f (¯ y, u C j ) f (¯ v C , u C j ) h (¯ u C , u C j ) h (¯ z, u C j ) det a − ,b M (cid:0) ¯ z j , ¯ y | ¯ u C j , ¯ v C (cid:1) mod , Res y j = v Cj det a,b M (¯ z, ¯ y | ¯ u C , ¯ v C ) Y (2) α ( y j | ¯ z, ¯ y ) = − g ( v C j , ¯ y ) g ( v C j , ¯ v C ) det a,b − M (cid:0) ¯ z, ¯ y j | ¯ u C , ¯ v C j (cid:1) mod , (3.10)10 ciPost Physics Submission where in the r.h.s. the modification (2.12) of r (¯ u C ), r (¯ v C ) is taken.Substituting (3.1), (3.6) to (2.15) we arrive at h ¯ u C ; ¯ v C | exp( αQ ) | ¯ u B ; ¯ v B i = f (¯ v B , ¯ u B ) X κ a κ b r (1)1 (¯ u C ) r (2)3 (¯ v C ) r (1)1 (¯ u B ) r (2)3 (¯ v B ) × f (¯ u B , ¯ u B ) f (¯ u C , ¯ u C ) g (¯ v C , ¯ v C ) g (¯ v B , ¯ v B ) f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) × f (¯ v B , ¯ u B ) f (¯ v C , ¯ u C ) (cid:2) f (¯ v C , ¯ u B )∆ g (¯ v C )∆ ′ g (¯ v C )∆ g (¯ u B )∆ ′ g (¯ u C ) (cid:3) [ h (¯ u B , ¯ u B ) h (¯ u B , ¯ u B )] × (cid:2) f (¯ v C , ¯ u B )∆ ′ g (¯ u C )∆ g (¯ u B )∆ g (¯ v C )∆ ′ g (¯ v C ) (cid:3) det a ,b N (¯ u B , ¯ v B | ¯ u C , ¯ v C ) det a ,b M (¯ u B , ¯ v B | ¯ u C , ¯ v C ) . (3.11)Here ¯ u B → { ¯ u B , ¯ u B } , ¯ v B → { ¯ v B , ¯ v B } and the same for sets { ¯ u C , ¯ v C } .Now we are in position to formulate one of the main results of the paper. Lemma 3.1. The sum over partitions (3.11) can be expressed via the multiple contour integral h ¯ u C ; ¯ v C | exp( αQ ) | ¯ u B ; ¯ v B i = 1 a ! b ! I ¯ u C ∪ ¯ u B a Y j =1 dz j πi I ¯ v C ∪ ¯ v B b Y j =1 dy j πi × r (1)1 (¯ z ) r (2)3 (¯ y ) r (1)1 (¯ u B ) r (2)3 (¯ v B ) det a,b N (¯ z, ¯ u B | ¯ v B , ¯ y ) det a,b M (¯ u C , ¯ z | ¯ y, ¯ v C ) Q aj =1 Y (1) α ( z j | ¯ z, ¯ y ) Q bj =1 Y (2) α ( y j | ¯ z, ¯ y ) S. (3.12) Here contours encircle points ¯ u C and ¯ u B for ¯ z and do not include any other singularities of theintegrand and around ¯ v C , ¯ v B for ¯ y . Factor S = S (¯ z, ¯ y | ¯ u B , ¯ u C ; ¯ v B , ¯ v C ) is defined as S = S S S with S = h (¯ u B , ¯ u B ) , S = ∆ g (¯ v C )∆ g (¯ u C )∆ g (¯ u B ) ,S (¯ z ; ¯ y ) = f (¯ v C , ¯ u C ) f (¯ v B , ¯ u B ) f (¯ y, ¯ u B ) f (¯ v C , ¯ z ) f (¯ y, ¯ z ) f (¯ y, ¯ z ) . (3.13) Proof. In order to prove the lemma it is enough to calculate integrals explicitly. We shouldtake into account that we can not compute residues at z j = u k and z i = u k for j = k (andthe same for y variables). Taking residues according to (3.5), (3.10) we arrive at four sumsover partitions ¯ u C → { ¯ u C , ¯ u C } , ¯ u B → { ¯ u B , ¯ u B } , ¯ v C → { ¯ v C , ¯ v C } , ¯ v B → { ¯ v B , ¯ v B } b ! X ¯ u C →{ ¯ u C , ¯ u C } X ¯ u B →{ ¯ u B , ¯ u B } I d ¯ y (2 πi ) b det a ,b N (¯ u B ; ¯ v B | ¯ z ; ¯ y ) det a ,b M (¯ z ; ¯ y | ¯ u C ; ¯ v C ) b Q j =1 Y (2) κ ( y j | ¯ z, ¯ y ) × S κ a r (1)1 (¯ z ) r (2)3 (¯ y ) r (1)1 (¯ u B ) r (2)3 (¯ v B ) h (¯ u B , ¯ z ) h (¯ u B , ¯ u B ) f (¯ y, ¯ u C ) h (¯ u C , ¯ u C ) f (¯ v C , ¯ u C ) h (¯ z, ¯ u C ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ¯ z = { ¯ u B , ¯ u C } (3.14)= X ¯ u C →{ ¯ u C , ¯ u C } ¯ u B →{ ¯ u B , ¯ u B } X ¯ v C →{ ¯ v C , ¯ v C } ¯ v B →{ ¯ v B , ¯ v B } det a ,b N (¯ u B ; ¯ v B | ¯ z ; ¯ y ) det a ,b M (¯ z ; ¯ y | ¯ u C ; ¯ v C ) S κ a κ b × r (1)1 (¯ z ) r (2)3 (¯ y ) r (1)1 (¯ u B ) r (2)3 (¯ v B ) h (¯ u B , ¯ z ) h (¯ u B , ¯ u B ) f (¯ y, ¯ u C ) h (¯ u C , ¯ u C ) f (¯ v C , ¯ u C ) h (¯ z, ¯ u C ) g (¯ v B , ¯ v B ) f (¯ v B , ¯ z ) g (¯ v B , ¯ y ) f (¯ v B , ¯ u B ) g (¯ v B , ¯ y ) g (¯ v C , ¯ v C ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ¯ z = { ¯ u B , ¯ u C } , ¯ y = { ¯ v C , ¯ v B } . (3.15)11 ciPost Physics Submission After the substitution ¯ z = { ¯ u B , ¯ u C } , ¯ y = { ¯ v C , ¯ v B } from the explicit form of determinants(3.2), (3.7) it is clear, that they will be reduced to the scalar products of the on-shell andtwisted-on-shell Bethe vectors with sets { ¯ u , ¯ v B } , { ¯ u C , ¯ v C } and the scalar product of the on-shell and off-shell Bethe vectors with sets { ¯ u C , ¯ v C } , { ¯ u B , ¯ v B } correspondingly and modifiedˆ r , ˆ r , as it should be in (3.11). It remains to check all other factors in (3.11). h (¯ u B , ¯ z ) h (¯ u B , ¯ u B ) h (¯ u C , ¯ u C ) h (¯ z, ¯ u C ) h (¯ u B , ¯ u B ) (cid:12)(cid:12)(cid:12)(cid:12) ¯ z →{ ¯ u B , ¯ u C } = h (¯ u B , ¯ u C ) h (¯ u B , ¯ u B ) h (¯ u C , ¯ u C ) h (¯ u B , ¯ u C ) h (¯ u B , ¯ u B ) h (¯ u B , ¯ u B ) h (¯ u B , ¯ u B ) h (¯ u B , ¯ u B )= h (¯ u B , ¯ u B ) h (¯ u B , ¯ u B ) h (¯ u B , ¯ u B ) h (¯ u C , ¯ u C ) . (3.16)Simplify part of factors in (3.11) that contains g (¯ x, ¯ y ) and ∆ g (¯ x ) (here ¯ x, ¯ y = ¯ u B , ¯ u C or ¯ v C , ¯ v B )∆ g (¯ v C )∆ ′ g (¯ v C )∆ g (¯ u B )∆ g (¯ u C ) g (¯ v C , ¯ v C ) g (¯ v B , ¯ v B )∆ g (¯ u C )∆ g (¯ u B ) × ∆ g (¯ v C )∆ ′ g (¯ v C ) g (¯ u C , ¯ u C ) g (¯ u B , ¯ u B ) = ∆ g (¯ v C )∆ g (¯ u C )∆ g (¯ u B ) g (¯ v B , ¯ v B ) g (¯ v C , ¯ v C ) . (3.17)The last expression coincides with factors g coming form residue computation and S in (3.15) S g (¯ v B , ¯ v B ) g (¯ v B , ¯ v C ) g (¯ v C , ¯ v B ) g (¯ v C , ¯ v C ) . (3.18)Finally, simplify part of factors in (3.11) that contains f (¯ v C , ¯ u B ) or f (¯ v B , ¯ u C ) f (¯ v C , ¯ u B ) f (¯ v C , ¯ u B ) f (¯ v B , ¯ u B ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) f (¯ v C , ¯ u C ) f (¯ v B , ¯ u B ) f (¯ v B , ¯ u B )= f (¯ v C , ¯ u C ) f (¯ v B , ¯ u B ) f (¯ v C , ¯ u B ) f (¯ v C , ¯ u B ) f (¯ v C , ¯ u C ) f (¯ v B , ¯ u B ) . (3.19)The last expression coincides with factors f coming from computation of residues and S in(3.15) f (¯ v C , ¯ u C ) f (¯ v B , ¯ u B ) f (¯ v B , ¯ z ) f (¯ y, ¯ u C ) f (¯ v B , ¯ u B ) f (¯ v C , ¯ u C ) f (¯ y, ¯ u B ) f (¯ v C , ¯ z ) f (¯ y, ¯ z ) f (¯ y, ¯ z ) (cid:12)(cid:12)(cid:12)(cid:12) ¯ y →{ ¯ v C , ¯ v B } , ¯ z →{ ¯ u B , ¯ u C } = f (¯ v C , ¯ u C ) f (¯ v B , ¯ u B ) f (¯ v C , ¯ u B ) f (¯ v C , ¯ u B ) f (¯ v C , ¯ u C ) f (¯ v B , ¯ u B ) . (3.20)Thus we convinced ourselves that the direct computation of integrals in (3.12) leads to (3.11). (cid:3) In (3.12) { ¯ u C , ¯ v C } are still free parameters. They may be the solution of Bethe equationsincluding the solution { ¯ u B , ¯ v B } . This allows us compute form factors and zero-temperaturecorrelation function. This formula is a direct analogue of one derived in [36] for the algebrasymmetry gl (2) related models . This result was derived there even for the case of the trigonometric R-matrix. ciPost Physics Submission Similarly to the algebra symmetry gl (2) situation in case of fundamental model where thesolution of quantum inverse problem is known exp( αQ ) could be expressed explicitly in termsof the (twisted) transfer matrices. Then the series (3.12) could be derived in a different way.The method is absolutely similar to the algebra symmetry gl (2) related models, thus we giveonly a very brief description here. More details are given in [13].Solution of the quantum inverse problem is formulated in the following way [37, 38]. Letus denote the elementary units ( e ij ) rs = δ ir δ js , i, j, r, s = 1 , . . . , k as( e ij ) ( k ) , then ( e ij ) ( k ) = t ( c/ k − T ij ( c/ t − k ( c/ , (4.1)where t ( w ) is a transfer matrix (see 1.2).For the fundamental model generation function for the first subsystem consisting of onlyone site m = 1 is given byexp( αQ ) = exp (cid:16) α ( e ) (1) + α ( e ) (2) (cid:17) = κ ( e ) (1) + ( e ) (1) − κ ( e ) (1) . (4.2)In the case of subsystem consisting of sites 1 , . . . , m generation function combining (4.1) and(4.2) we arrive at exp( αQ ) = t m κ ( c/ t − m ( c/ , (4.3)where t κ is a twisted transfer matrix defined in (1.17) and its eigenvalue on eigenvector | ¯ u, ¯ v i is explicitly given by τ ( w | ¯ u, ¯ v ) τ ( w | ¯ u, ¯ v ) = κ λ ( w ) f (¯ u, w ) + λ ( w ) f ( w, ¯ u ) f (¯ v, w ) − κ λ ( w ) f (¯ v, w ) . (4.4)For the fundamental model λ ( w ) = ( w + c/ L and λ ( w ) = λ ( w ) = ( w − c/ L . Momentumis given by p ( w ) = log (cid:18) w + c/ w − c/ (cid:19) . (4.5)Assuming now the completeness of the basis of twisted Bethe eigenstates we can insertbetween the transfer matrices 1 = P ¯ µ, ¯ λ | ¯ µ ; ¯ λ ih ¯ µ ; ¯ λ |h ¯ µ ; ¯ λ | ¯ µ ; ¯ λ i − and obtain the followingexpansion for matrix elements of the generation function h ¯ u C ; ¯ v C | exp ( αQ ) | ¯ u B ; ¯ v B i = X ¯ µ, ¯ λ τ m κ ( c/ | ¯ µ ; ¯ λ ) τ m ( c/ | ¯ u B ; ¯ v B ) h ¯ u C ; ¯ v C | ¯ µ ; ¯ λ ih ¯ µ ; ¯ λ | ¯ u B ; ¯ v B ih ¯ µ ; ¯ λ | ¯ µ ; ¯ λ i , (4.6)where h ¯ u ; ¯ v | ¯ µ ; ¯ λ i denotes scalar products and we use (1.17). Summation is taken over alladmissible (physical) solutions of twisted Bethe equations: Y (1) κ (cid:0) µ i | ¯ µ ; ¯ λ (cid:1) = 0, i = 1 , . . . , a This is the supersymmetric t-J model describing 1D hopping electron gas, see [33, 34] for good description. The proof of completeness requires a separate discussion, we are not aware whether such a proof was evergiven rigorously for algebra symmetry gl (2 | 1) related models. There exist also unphysical solutions of Bethe equations, where both terms in (4.8) are equal to zero. Itwas shown in the case of algebra symmetry gl (2) related model [13] that contributions of such solutions to thesum (4.6) are zero. We omit here this proof since it is identical to the previous one. ciPost Physics Submission and Y (2) κ (cid:0) λ j | ¯ µ ; ¯ λ (cid:1) = 0, j = 1 , . . . , b . The norm of (on-shell) Bethe vector h ¯ µ ; ¯ λ | ¯ µ ; ¯ λ i wascomputed in [30]. We present it here in the following form h ¯ µ ; ¯ λ | ¯ µ ; ¯ λ i = ˜ S det ∂ ˜ Y ( r ) κ (cid:0) w rk | ¯ µ, ¯ λ (cid:1) ∂w rj ! , ˜ S = ∆ g (¯ λ )∆ ′ g (¯ λ ) f (¯ λ, ¯ µ )∆ g (¯ µ )∆ ′ g (¯ µ ) h (¯ µ, ¯ µ ) κ λ (¯ λ ) λ (¯ µ ) h (¯ µ, ¯ µ ) f (¯ λ, ¯ µ ) . (4.7)Here, r, s = 1 , w j = z j , j = 1 , . . . , a , w j = y j , j = 1 , . . . , b and˜ Y (1) κ ( µ j ) = κ λ ( µ j ) h (¯ µ j , µ j ) + λ ( µ j ) h ( µ j , ¯ µ j ) f (¯ λ, µ j ) , ˜ Y (2) κ ( λ j ) = κ λ ( λ j ) + λ ( λ j ) f ( λ j , ¯ µ ) . (4.8)For an arbitrary function F (¯ µ, ¯ λ ) sum can be rewritten as a contour integral around thesolutions of Bethe equations using the trick X ¯ µ, ¯ λ F (¯ µ, ¯ λ ) = 1 a ! b ! I ¯ µ d ¯ z I ¯ λ d ¯ y det ∂ ˜ Y ( r ) κ ( w rj | ¯ z, ¯ y ) ∂w sk ! F (¯ z, ¯ y ) a Q j =1 ˜ Y (1) κ ( z j | ¯ z ; ¯ y ) b Q j =1 ˜ Y (2) κ ( y j | ¯ z, ¯ y ) . (4.9)Integrals on { ¯ z, ¯ y } are taken around all admissible solutions { ¯ µ, ¯ λ } of (twisted) Bethe equa-tions. Factorials a ! b ! appear in order to avoid multiple counting of the Bethe states that differonly by permutations of the spectral parameters inside the set. Thus we can present (4.6) as h ¯ u C ; ¯ v C | exp( αQ ) | ¯ u B ; ¯ v B i = 1 a ! b ! I ¯ µ d ¯ z (2 π ) a I ¯ λ d ¯ y (2 πi ) b τ m κ ( c/ | ¯ z ; ¯ y ) τ m ( c/ | ¯ u B ; ¯ v B ) h ¯ u C ; ¯ v C | ¯ z ; ¯ y ih ¯ z ; ¯ y | ¯ u B ; ¯ v B i ˜ S a Q j =1 ˜ Y (1) κ ( z j | ¯ z ; ¯ y ) b Q j =1 ˜ Y (2) κ ( y j | ¯ z, ¯ y ) . (4.10)Now we can substitute in (4.10) (3.1) for h ¯ z ; ¯ y | ¯ u B ; ¯ v B i , (3.6) for h ¯ u C ; ¯ v C | ¯ z ; ¯ y i and ˜ S defined in(4.7). Also we extract from ˜ Y (1) κ ( z j | ¯ z, ¯ y ) factor λ ( z j ) h ( z j , ¯ z j ) f (¯ y, z j ) and from ˜ Y (2) κ ( y j | ¯ z, ¯ y )factor λ ( y j ) f ( y j , ¯ z ), hereby ˜ Y (1) κ −→ Y (1) κ and ˜ Y (2) κ −→ Y (2) κ . After collecting factors andelementary simplification of prefactors we arrive at h ¯ u C ; ¯ v C | exp( αQ ) | ¯ u B ; ¯ v B i = 1 a ! b ! I ¯ µ a Y j =1 dz j πi I ¯ λ b Y j =1 dy j πi × e im ( p (¯ z ) − p ( ¯ u B )) det a,b N (¯ z, ¯ u B | ¯ v B , ¯ y ) det a,b M (¯ u C , ¯ z | ¯ y, ¯ v C ) Q aj =1 Y (1) κ ( z j | ¯ z, ¯ y ) Q bj =1 Y (2) κ ( y j | ¯ z, ¯ y ) S, (4.11)where S is defined in (3.13), N and M in (3.2), (3.3) and (3.7), (3.8). We also used here that τ ( c/ | ¯ z ; ¯ y ) is nothing but exp( ip (¯ u )) and τ κ ( c/ | ¯ z ; ¯ y ) is κ exp( ip (¯ z )).Note, that at z → ∞ , y → ∞ integrand of (4.11) vanishes. Thus instead of evaluationof integrals by the residues inside the integration contours we can evaluate integrals by theresidue outside the integration contours. One can check that the only poles of the integrandare the first order poles at z j = u C k or z j = u B k , j, k = 1 , . . . , a and y j = v C k or y j = v B k , j, k = 1 , . . . , b . Then switching to integration over these poles and taking into account thatin the fundamental model r ( i )2 = 1, r (1)1 (¯ v ) = e imp (¯ u ) we immediately arrive at (3.12).14 ciPost Physics Submission Conclusion The first result of the paper are new representations for the matrix elements of the generationfunctions of correlators (2.14) and (2.15). These are direct analogues of the Reshetikhinformula (2.2) derived in [26] for the scalar products. We restricted ourselves here only to casesof R -matrices with algebra symmetry gl (3) or gl (2 | R -matrices with the arbitrary gl ( m | n ) algebra symmetry. This is easyto proof using the same simple steps as it was done in section 2 starting from the analoguesof (1.19), (1.18) and (2.2) that are known for arbitrary gl ( m | n ) algebra symmetry (see [39]).We leave this exercise to the interested reader.The second result is integral representation (3.12). In the algebra symmetry gl (2) relatedmodels such integral representation allowed to establish the asymptotic behaviour of corre-lation functions [16, 40]. We have a hope that the new representation will be suitable forthe computation of asymptotic in the algebra symmetry gl (2 | 1) related model. We also havea hope that the similar representation can be established for the case of arbitrary gl ( m | n )algebra symmetry. At least it is clear that integral representation can be written for formfactors of generator Q in case of gl (3) algebra symmetry.Let us also mention also that in the algebra symmetry gl (2) related model integral rep-resentation similar to (3.12) was written also for dynamical case [14]. Without given toomuch details but rather relaying on complete analogy with algebra symmetry gl (2) case wewould argue that dynamical correlation function can be computed using analogue of (3.12)in algebra symmetry gl (2 | 1) related models too. The result is quite similar to the static caseand the only modification consist in presence of the additional factor e itE (¯ z ;¯ y ) (of course thisis nothing but eigenvalue of time propagation operator) under the integral and modificationof of integration contours by adding poles of this factor { e , o } inside the contours h ¯ u C ; ¯ v C | exp( αQ ) | ¯ u B ; ¯ v B i = 1 a ! b ! I ¯ u C ∪ ¯ u B ∪ e a Y j =1 dz j πi I ¯ v C ∪ ¯ v B ∪ o b Y j =1 dy j πi e − it ( E (¯ z ;¯ y ) − E (¯ u ;¯ v )) × κ a κ b r (1)1 (¯ z ) r (2)3 (¯ y ) r (1)1 (¯ u B ) r (2)3 (¯ v B ) det a,b N (¯ z, ¯ u B | ¯ v B , ¯ y ) det a,b N (¯ u C , ¯ z | ¯ y, ¯ v C ) Q aj =1 Y (1) α ( z j | ¯ z, ¯ y ) Q bj =1 Y (2) α ( y j | ¯ z, ¯ y ) S. (4.12)All notations here are the same as in (3.12). Acknowledgments Authors are grateful to F. G¨ohmann, N. Slavnov and K. Kozlowski for fruitful discussions. References [1] J. Honerkamp, An exploration of the correlation functions for finite temperaturein the non-linear Schr¨odinger equation model , Nucl. Phys B , 301 (1981),doi:10.1016/0550-3213(81)90561-7. Being precise, this result was proven there only for the fundamental model but we expect that it is truefor arbitrary models related to the algebra symmetry gl (2). ciPost Physics Submission [2] M. Jimbo, K. Miki, T. Miwa, A. Nakayashiki, Correlation functions of the XXZ modelfor ∆ < − 1, Phys. Lett. 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