arXiv:0904.2265v1 [nlin.SI] 15 Apr 2009 Factorized finite-size Ising model spin matrix elements from Separation of Variables G von Gehlen† , N Iorgov‡ , S Pakuliak♯♭ and V Shadura‡ † Physikalisches Institut der Universität Bonn, Nussallee 12, D-53115 Bonn, Germany Bogolyubov Institute for Theoretical Physics, Kiev 03680, Ukraine ♯ Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Moscow region, Russia ♭ Institute of Theoretical and Experimental Physics, Moscow 117259, Russia ‡ E-mail: gehlen@th.physik.uni-bonn.de, iorgov@bitp.kiev.ua, pakuliak@theor.jinr.ru, shadura@bitp.kiev.ua Abstract. Using the Sklyanin-Kharchev-Lebedev method of Separation of Variables adapted to the cyclic Baxter–Bazhanov–Stroganov or τ (2) -model, we derive factorized formulae for general finite-size Ising model spin matrix elements, proving a recent conjecture by Bugrij and Lisovyy. 15 April 2009 Submitted to: J. Phys. A: Math. Gen. PACS numbers: 75.10Hk, 75.10Jm, 05.50+q, 02.30Ik 1. Introduction Much work has been done on the 2-dimensional Ising model (IM) during the past over 60 years. Many analytic results for the partition function and correlations have been obtained. These have greatly contributed to establish our present understanding of continuous phase transitions in systems with short range interactions [1, 2, 3, 4, 5, 6]. Recent overviews with many references are given e.g. in [7, 8, 9]. Many rather different mathematical approaches have been used, so that already 30 years ago Baxter and Enting published the “399th” solution for the free energy [10], see also [11]. Spin-spin correlation functions can be written as Pfaffians of Toeplitz determinants. Most work has focussed on the thermodynamic limit and scaling properties since these give contact to field theoretical results and to beautiful Painlevé properties [5, 6, 12]. Only during the last decade more attention has been drawn to correlations and spin matrix elements (form factors [13]) in finite-size Ising systems [14, 15, 16]. Nanophysics experimental arrangements often deal with systems where the finite size matters. Recent theoretical work on the finite-size IM started from Pfaffians and related Clifford approaches. In [17] it has been pointed out that one may write completely factorized closed expressions for spin matrix elements of finite-size Ising systems. One goal of the present paper is to prove the beautiful compact formula conjectured in Eq.(12) of [17], Ising model spin matrix elements from Separation of Variables 2 see (129). For achieving this, we introduce a method which has not yet been applied to the Ising model: Separation of Variables (SoV) for cyclic quantum spin systems. Our approach is the adaption to cyclic models of the method introduced by Sklyanin [19, 20] and further developed by Kharchev and Lebedev [21, 22]. We also make extensive use of the analysis of quantum cyclic systems given in [23]. Little is known about state vectors of the 2-dimensional finite-size IM. Only partial information about these state vectors can be obtained from the work of [3]. Recently Lisovyy [24] found explicit expressions using the Grassmann algebra method. Here we shall present our SoV approach [25, 26, 27] which gives explicit formulas for finitesize state vectors too. However, these come in a basis quite different from the one used in [24]. We shall calculate spin matrix elements by directly sandwiching the spin operator between state vectors. Factorized expressions result if we manage to perform the multiple spin summations over the intermediate states. The prototype of a general N-state cyclic spin model is the Baxter–Bazhanov– Stroganov model (BBS) [28, 29, 30], also known as the τ (2) -model. The standard IM is a very special degenerate case of the BBS model. In order to avoid formulating many precautions necessary when dealing with the very special IM, we shall develop our version of the SoV machinery considering the general BBS model. We chose to do this also because of the great interest in the BBS model due to the fact that its transfer matrix commutes with the integrable Chiral Potts model (CPM) [31, 32] transfer matrix [29, 33]. Obtaining state vectors for the CPM is a great actual challenge [34, 35]. Although the eigenvectors for the transfer matrix of the BBS model with periodic boundary condition are unknown for N > 2, explicit formulas for the eigenvectors of the BBS model with open and fixed boundary conditions have been found [36, 37]. This paper is organized as follows: In section 2 we define the BBS model and its Ising specializations. In section 3 we discuss the Sklyanin SoV method adapted to the BBS model as a cyclic system. We start with the necessary first step, the solution of the associated auxiliary problem. In a second step we obtain the eigenvectors and eigenvalues of the periodic system by Baxter equations. The conditions which ensure that the Baxter equations have non-trivial solutions are formulated as truncated functional equations. Section 4 gives a description of local spin operators in terms of global elements of the monodromy matrix. Starting with Section 5 we restrict ourselves to the case N = 2, for which the BBS model becomes a generalized 5-parameter plaquette Ising model. In section 6 we further specialize to the homogenous case and then to the two-parameter Ising case. Periodic boundary condition eigenvectors are explicitly constructed. Section 7 is devoted to our main result, the proof of the factorized formula for Ising spin matrix elements between arbitrary finite-size states. This is shown to agree with the Bugrij-Lisovyy conjecture. In section 8 we give an analogous formula for the Ising quantum chain in a transverse field. Finally, section 9 presents our Conclusions. Large part of this paper relies on our work in [25, 26, 27]. Sections 4, 2.2 and 6.2 give new material. Ising model spin matrix elements from Separation of Variables 3 2. The BBS τ (2) -model 2.1. The inhomogenous BBS-model for general N We define the BBS-model as a quantum chain model. To each site k of the quantum chain we associate a cyclic L-operator [29, 30] acting in a two-dimensional auxiliary space   1 + λκk vk , λu−1 (a − b v ) k k k k , k = 1, 2, . . . , n. (1) Lk (λ) =  uk (ck − dk vk ), λak ck + vk bk dk /κk λ is the spectral parameter, n the number of sites. There are five parameters κk , ak , bk , ck , dk per site. uk and vk are elements of an ultra local Weyl algebra, obeying uj uk = uk uj , uj vk = ω δj,k vk uj , vj vk = vk vj , ω = e2πi/N , N uN k = vk = 1. At each site k we define a N-dimensional linear space (quantum space) Vk with the basis |γik , γ ∈ ZN , the dual space Vk∗ with the basis k hγ|, γ ∈ ZN , and the natural pairing ′ ∗ k hγ |γik = δγ ′ ,γ . In Vk and Vk the Weyl elements uk and vk act by the formulas: uk |γik = ω γ |γik , vk |γik = |γ + 1ik ; k hγ|uk = k hγ| ω γ , k hγ|vk = k hγ − 1| . (2) The monodromy Tn (λ) and transfer matrix tn (λ) for the n sites chain are defined as ! An (λ) Bn (λ) Tn (λ) = L1 (λ) · · · Ln (λ) = , tn (λ) = tr Tn (λ) = An (λ) + Dn (λ). (3) Cn (λ) Dn (λ) This quantum chain is integrable since the L-operators (1) are intertwined by the twisted 6-vertex R-matrix at root of unity   λ − ων 0 0 0   0 ω(λ − ν) λ(1 − ω) 0   R(λ, ν) =  (4) ,   0 ν(1 − ω) λ−ν 0 0 0 0 λ − ων (1) (2) (2) (1) R(λ, ν) Lk (λ)Lk (ν) = Lk (ν) Lk (λ) R(λ, ν), (1) (5) (2) where Lk (λ) = Lk (λ) ⊗ I, Lk (λ) = I ⊗ Lk (λ). Relation (5) leads to [tn (λ), tn (µ)] = 0 . So tn (λ) is the generating function for the commuting set of non-local and non-hermitian Hamiltonians H0 , . . . , Hn : tn (λ) = H0 + H1 λ + · · · + Hn−1 λn−1 + Hn λn . (6) From (5) it also follows that the upper-right entry Bn (λ) of Tn (λ) is the generating function for another commuting set of operators h1 , . . . , hn : [ Bn (λ), Bn (µ) ] = 0, Bn (λ) = h1 λ + h2 λ2 + · · · + hn λn . (7) Observe that H0 and Hn can be easily written explicitly in terms of the global ZN -charge rotation operator Vn n n n Y Y Y bk dk κk , Vn = v1 v2 · · · vn . (8) ak ck + Vn , Hn = H0 = 1 + Vn κk k=1 k=1 k=1 Ising model spin matrix elements from Separation of Variables 4 Here we shall not explain the great interest in the BBS-model due to a second intertwining relation in the Weyl-space indices found in [29] and the related fact that for particular parameters the Baxter Q-operator of the BBS-model is the transfer matrix of the integrable Chiral Potts model, see [29, 33, 38]. We will also not discuss the generalizations of the BBS model introduced by Baxter in [39], and not explain how (1) arises in cyclic representations of the quantum group Uq (sl2 ), see e.g. [23, 40, 41]. The transfer matrix (3) can be written equivalently as a product over face Boltzmann weights [28, 33]: Q ′ ′ tn (λ) = n+1 k=2 Wτ (γk−1 , γk , γk , γk−1 ) with the face Boltzmann weights P ′ ′ Wτ (γk−1 , γk , γk−1 , γk′ ) = 1mk−1 =0 ω mk−1 (γk −γk−1 ) ′ ′ ′ (γk−1 − γk−1 , mk−1 ) Fk′′ (γk − γk′ , mk−1 ) × (−ωtq )γk −γk −mk−1 Fk−1 (9) where mk ∈ {0, 1} and Fk′ (∆γ, mk ) = Fk′′ (∆γ, mk ) = 0 if ∆γ 6= {0, 1}, and the nonvanishing values are ! ! 1 λ ak 1 λ ck ′ ′′ Fk = , Fk = . (10) κk −bk /ω 1 −dk /κk The vanishing of Fk′ (∆γ, mk ) and Fk′′ (∆γ, mk ) for ∆γ 6= {0, 1} means that the vertically neighboring ZN -spins cannot differ by more than one. The equivalence to the transfer matrix defined by (1) and (3) is seen writing the matrix elements of (1) as ′ ′ hγk′ |Lk (λ)mk−1 ,mk |γk i = ω mk−1 γk −mk γk λγk −γk −mk−1 Fk′′ (γk′ − γk , mk−1 )Fk′ (γk′ − γk , mk ). (11) Lk • • ′ ∈ ZN γk−1 mk−2 • • • γk′ •γ k−1 • Fk′ F ′′k • γk ∈ ZN • • mk−1 ′ Fk−1 γk′ • mk ∈ Z 2 • mk−1 • F ′′k F ′k mk • • γk ∈ ZN Figure 1. Illustration of the two versions: left we see the W of (9) indicated by full lines, whereas (3) the Lk of (1) arise if we look at the lattice formed by the dashed lines in the left figure and the dashed rhombus shown at the right. The lattice is built by ZN -spins on the full lines and Z2 -spins in the centers. 2.2. Homogenous BBS-model for N = 2 The integrability of the BBS-model is valid also if the parameters κk , ak , . . . , dk vary from site to site and the construction of eigenvalues and eigenvectors can be performed Ising model spin matrix elements from Separation of Variables 5 for this general case. However, in order to obtain compact explicit formulas for matrix elements, we shall often put all parameters equal: κk = κ, . . . , dk = d and call this the homogenous model. In [42] it has been shown that for N = 2 the general homogenous BBS-model can be rewritten as a generalized plaquette Ising model with Boltzmann weights   P W (σ1 , σ2 , σ3 , σ4 ) = a0 1 + 1≤i<j≤4 aij σi σj + a4 σ1 σ2 σ3 σ4 , (12) subject to the free-fermion condition a4 = a12 a34 − a13 a24 + a14 a23 . For N = 2 the Weyl elements can be represented by Pauli matrices. Fixing κ = 1 the L-operator becomes ! λ σkz (a − b σkx ) 1 + λ σkx Lk (λ) = σkz (c − d σkx ) λac + σkx b d degenerating at λ = b/a: Lk (b/a) = 1 + b/a σkx σkz (c − dσkx ) !   1 , b σkz . The matrix elements of the corresponding transfer-matrix are n Y
′ δσk ,σk′ (1 + b c σk−1 σk′ ) + δσk ,−σk′ b/a (1 − a d σk−1 σk′ ) , h{σ } | tn(b/a) | {σ}i = k=1 where {σ} = {σ1 , . . . , σn } and {σ ′ } = {σ1′ , . . . , σn′ } are the values of the spin variables ′ of two neighboring rows, σk = (−1)γk , σk′ = (−1)γk ∈ {+1, −1}, and the identifications ′ σn+k = σk , σn+k = σk′ are used. • ′ σk−1 • Kd • • • Ky Kx σk−1 σk′ • σk Figure 2. Transfer-matrix for the triangular Ising lattice. Solid lines show the interaction between spins. The matrix elements of the transfer-matrix of the Ising model on the triangular lattice (see Fig. 2) are ′ h{σ }|t△ |{σ}i = n Y k=1 exp(Kx σk−1 σk + Ky σk σk′ + Kd σk−1 σk′ ) . (13) Ising model spin matrix elements from Separation of Variables 6 The k-th factor of this product, taken at σk = σk′ is exp(Ky ) exp((Kx + Kd ) σk−1 σk′ ) = = exp(Ky ) cosh(Kx + Kd ) (1 + tanh(Kx + Kd ) σk−1 σk′ ) , and at σk = −σk′ is exp(−Ky ) exp((Kd − Kx ) σk−1 σk′ ) = = exp(−Ky ) cosh(Kd − Kx ) (1 + tanh(Kd − Kx ) σk−1 σk′ ) . Now it is easy to compare the transfer-matrices tn (b/a) and t△ : t△ = exp(nKy ) coshn (Kx + Kd ) tn (b/a) , tanh(Kx + Kd ) = b c , exp(−2Ky ) cosh(Kd − Kx ) = b/a , cosh(Kd + Kx ) tanh(Kx − Kd ) = a d . Although we considered tn (λ) at the special value of the spectral parameter λ = b/a, the transfer-matrix eigenstates are independent of this choice of λ. So the eigenstates of the transfer-matrix of the Ising model on the triangular lattice appear as eigenstates of the general homogenous BBS-model for N = 2 (the parameter κ and one of the parameters a, ..., d in the case of homogeneous periodic BBS model can be absorbed by a rescaling of the other parameters and using a diagonal similarity transformation of the L-operators). The formulas for them will be given later. Unfortunately, factorized formulas for the matrix elements of the spin operator in this general case have not been found. There are only two special cases for which such formulas are available: • Row-to-row transfer-matrix for the Ising model on the square lattice: a = c, b=d: Kd = 0, e−2Ky = b/a , tanh Kx = a b . (14) This case will be the main object of our attention. It is the most general case where we have the factorized formula for the spin operator matrix elements found by Bugrij and Lisovyy. • Diagonal-to-diagonal transfer-matrix for the Ising model on the square lattice: a = c, b = −d : Kx = 0, e−2Ky = b/a , tanh Kd = a b . (15) It is known [43] that such transfer-matrices with different parameters Ky = L, Kd = K (and corresponding a, b) constitute a commuting set of matrices having common eigenvectors provided 1 a2 − b2 = (16) 1 − a2 b2 k′ is fixed. Thus in this case the eigenvectors depend on k′ only. Therefore, in order to find the eigenvectors and the corresponding matrix elements of the spin operator it is sufficient to fix a = c = 1/(k′ )1/2 and b = d = 0 and so to obtain a special case of the formulas for the row-to-row transfer-matrix of the Ising model on the square lattice. Note that we get [27] the same matrix elements in the case of the sinh 2K sinh 2L = Ising model spin matrix elements from Separation of Variables 7 quantum Ising chain in a transverse field with strength k′ because the corresponding Hamiltonian commutes with the transfer-matrices having the same k′ . Another remark: with the restriction a = c, b = −d, κ = 1, the transfer-matrices commute among themselves at independent values of two spectral parameters: λ and the parameter which uniformizes (16) (a parameter on the elliptic curve with modulus k′ , see [43]). 3. Separation of Variables for the cyclic BBS-model 3.1. Solving the auxiliary system (7): Eigenvalues and eigenvectors of Bn (λ). We start giving a summary of the SoV method as applied to the general inhomogenous ZN -BBS model [25]. The aim is to find the eigenvalues and eigenstates of the n-site periodic transfer matrix tn (λ) of (3), and the idea [19, 20, 21, 22] is first to construct a basis of the N n -dimensional eigenspace from eigenstates of Bn (λ), see (7). This can be done by a recurrent procedure. Then the eigenstates of tn (λ) are written as linear combinations of the Bn (λ)-eigenstates. The multi-variable coefficients are determined by Baxter T −Q-equations which by SoV separate into a set of single-variable equations. From (7) the eigenvalues of Bn (λ) are polynomials in the spectral variable λ. Factorizing this polynomial, for n ≥ 2 we get Bn (λ) |Ψλi = λλ0 n−1 Y k=1 (λ − λk ) |Ψλi; λ = {λ0 , λ1 , . . . , λn−1 }, (17) where λ1 , λ2 , . . . , λn−1 are the n − 1 zeros of the eigenvalue polynomial and λ0 is a normalizing factor. We can label the eigenvectors by λ. An overview of the space of eigenstates of Bn (λ) is easily obtained using the intertwining relations (5). It follows from (5) that the operators An (λ) and Dn (λ) of the monodromy (3), taken at a zero λ = λk , are cyclic ladder operators with respect to the kth component of λ in |Ψλi. To see this consider e.g. the intertwining relation (λ − ωµ)An (λ)Bn (µ) = ω(λ − µ)Bn (µ)An (λ) + µ(1 − ω)An (µ)Bn (λ) (18) which is a component of (5). Fixing λ = λk , k = 1, . . . , n − 1 , in (18) and acting on Ψλ, the last term in (18) vanishes and we obtain Q Bn (µ) (An (λk ) |Ψλi) = µ λ0 (µ − ω −1λk ) s6=k (µ − λs ) (An (λk ) |Ψλi).(19) This means that An (λk ) |Ψλi = ϕk · |Ψλ0 , ... , ω−1 λk , ... , λn−1 i . (20) Later we shall give an explicit expression for the proportionality factor ϕk . Similarly, from another component of (5) and with another factor ϕ ek we get Dn (λk )|Ψλi = ϕ ek · |Ψω−1 λ0 , ... , ω λk , ..., λn−1 i. (21) Furthermore, acting by (18) on |Ψλi and extracting the coefficient of λn+1 µn we get Vn |Ψλi = |Ψω−1 λ0 , λ1 , ... , λn−1 i . (22) Ising model spin matrix elements from Separation of Variables 8 Assuming generic parameters in Lk such that all proportionality factors are nonvanishing, by repeated application of An (λk ), Dn (λk ) and Vn to any eigenstate |Ψλi we span the whole N n -dimensional space of states. Later, when we give explicit expressions for ϕk and ϕ ek we can check whether these factors can vanish. So, if for a given set of parameters ak , bk , ck , dk , κk , (k = 1, . . . , n) there is an eigenvector with the eigenvalue polynomial determined by the zeros λ, then there are also eigenvectors to all eigenvalue polynomials determined by the zeros {λ0 ω ρn,1 , . . . , λn−1 ω ρn,n−1 } with ρn = (ρn,0 , . . . , ρn,n−1) ∈ (ZN )n . (23) Let us therefore write the zeros as λn,k = −rn,k ω ρn,k , (24) where for n fixed, the n real numbers rn,k are determined by the 5n parameters al , . . . , κl . For fixed parameters the N n in all following calculations we shall label the eigenvectors by the ρn instead of our previous λn,k . For given parameters, the set of the eigenvalues is determined by the rn,0 , . . . , rn,n−1. The eigenvalue equation for Bn (λ) becomes Bn (λ)|Ψρn i = λ rn,0 ω −ρn,0 n−1 Y k=1
λ + rn,k ω −ρn,k |Ψρn i , (25) In order to calculate the rn,k in terms of the parameters, we don’t need the full quantum transfer matrix and the Lk -operators involving the Weyl variables. Rather, by the following averaging procedure [23] Q O(λN ) = h O(λN ) i = s ∈ ZN O(ω sλ). (26) we associate to a spectral parameter dependent quantum operator O(λ) a classical counterpart O(λN ) . We define the classical BSS model by the L-operator Lm (λN ) ! ! N N N N N h L i h L i ) − b λ −ǫλ (a 1 − ǫκ 00 01 m m m Lm (λN ) = = (27) N N N N N N h L10 i h L11 i cN bN m − dm m dm /κm − ǫλ am cm where ǫ = (−1)N . Analogously, we define the classical monodromy Tn by ! N N A (λ ) B (λ ) n n Tn = L1 (λN ) L2 (λN ) · · · Ln (λN ) = Cn (λN ) Dn (λN ) (28) Proposition 1.5 of [23] tells us that the classical polynomials An (λN ), Bn (λN ), Cn (λN ) and Dn (λN ) are the averages of their counterparts in (3): An (λN ) = h An (λ) i, etc. So for n ≥ 2 we have m−1 Y N m N N N Bm (λ ) = (−ǫ) λ rm,0 (λN − ǫ rm,s ). (29) s=1 It is easy to derive [25] a three-term recursion which expresses Bm (λN ) in terms of Bm−1 (λN ) and Bm−2 (λN ). Using the initial values B1 (λN ) = −ǫ λN r1N ; B0 (λN ) = 0 N N and defining r1N = aN 1 − b1 , this gives a (n − 1)th-degree algebraic relation for the rm,s . For the homogenous model (the constants are taken to be site-independent) this can be replaced by just a quadratic equation, see the Appendix of [25]. Ising model spin matrix elements from Separation of Variables 9 3.2. Solving the auxiliary system: Explicit construction of the eigenvectors of Bn (λ). The stepwise construction of the eigenvectors, starting with one-site, then two-site as linear combination of products of two one-site eigenvectors etc. is tedious because we have to go to 4 sites before the general rule emerges. Let us start finding the one-site right eigenvectors |ψρ i1 of B1 (λ) as linear combination of spin states |γi1, γ ∈ ZN , writing X wp (γ − ρ) |γi1 , ρ ∈ ZN . (30) |ψρ i1 = γ ∈ ZN Applying on the left B1 from (1) and on the right (25), we demand X X wp (γ − ρ) |γi1 . wp (γ − ρ) |γi1 = λ r1,0 ω −ρ1,0 λ u1−1 (a1 − b1 v1 ) (31) γ ∈ ZN γ ∈ ZN Applying (2) and shifting the left hand summation for the term with |γ + 1i1 , we get (a1 − r1,0 ω γ−ρ ) wp (γ − ρ) = b1 wp (γ − ρ − 1). (32) This is a difference equation for the function wp (γ) [44]: y wp (γ) = ; wp (γ − 1) 1 − ωγ x γ ∈ ZN , wp (0) = 1 ; (33) where we have put y = b1 /a1 , r1,0 = x a1 and chose the initial value wp (0) = 1. The cyclic property wp (γ) = wp (γ + N) imposes the Fermat condition xN + y N = 1 on the two-component vector p = (x, y ). We indicate p as a subscript on the functions wp (γ). We shall consider the case of “generic parameters”, so in particular we exclude N the case aN k − bk = 0, and the “superintegrable” case ak = ω −1 bk = ck = dk = κk = 1, since in the latter cases degenerations appear. We write the analogous left eigenvector as X 1 1 hγ|, 1 hψρ | = w p (γ − ρ − 1) γ ∈Z N (34) ρ ∈ ZN (35) with the same functions wp (γ), just now p = (r1,0 /a1 , ω −1 b1 /a1 ). The Fermat vector dependent functions wp (γ) play an important role for cyclic models. They are root-ofunity analogues of the q-gamma function. By a similar calculation, the two-site eigenvectors are found to be: X ω −(ρ2,0 +ρ2,1 −ρ1 )(ρ2,0 −ρ2 ) |Ψ ρ2,0 , ρ2,1 i = |ψρ1 i1 ⊗ |ψρ2 i2 . (36) wp 2, 0 (ρ2,0 − ρ1 − 1)wp̃ 2 (ρ2,0 + ρ2,1 − ρ2 − 1) ρ , ρ ∈Z 1 2 N where p2, 0 = (x2, 0 , y2, 0 ), p̃2 = (x̃2 , ỹ2 ) and x2, 0 = a2 c2 r1 , r2, 0 y2, 0 = κ1 r2 , r2, 0 x̃2 = r2 , r2, 0 r2, 1 ỹ2 = b2 d2 r1 . (37) κ2 r2, 0 r2, 1 The condition that p2, 0 and p̃2 are Fermat vectors determines r2,0 and r2,1 . Ising model spin matrix elements from Separation of Variables 10 The explicit formula for both the left- and right eigenvectors of Bn (λ) for general number of sites n has been proved by lengthy induction and is given in [25]. A byproduct of these calculations are the formulas for ϕk and ϕ ek introduced in (20),(21): An (λn,k ) |Ψρn i = ϕk (ρ′n ) |Ψρ+k i, n Dn (λn,k )|Ψρn i = ϕ̃k (ρ′n ) |Ψρn+0,−k i, ϕk (ρ′n ) n−2 Y n,k r̃n−1 −ρ̃n +ρn,0 ω Fn (λn,k /ω) yn−1,s ,(38) = − rn s=1 ϕ̃k (ρ′n ) n−1 ω ρ̃n −ρn,0 −1 Y =− Fm (λn,k ), Qn−2 n,k r̃n−1 s=1 yn−1,s m=1 rn Fn (λ) = ( bn + ωan κn λ) ( λ cn + dn /κn ) . (39) (40) On the left of (38) and (39) the eigenvectors Ψρn of Bn (λ) are labeled by the vector ρn = (ρn,0 , . . . , ρn,n−1) ∈ (ZN )n . (41) ρ±k denotes the vector ρn in which ρn,k is replaced by ρn,k ± 1: n ρ±k n = (ρn,0 , . . . , ρn,k ± 1, . . . , ρn,n−1 ), r̃n = rn,0 rn,1 . . . rn,n−1 and k = 0, 1, . . . , n − 1, Pn−1 ρ̃n = k=0 ρn,k . (42) (43) ρ′n denotes the vector ρn without the component ρn,0 : ρ′n = (ρn,1 , . . . , ρn,n−1 ) ∈ (ZN )n−1 . (44) n,k n,k n,k are components of a Fermat vector pn,k The yn−1,s n−1,s = (xn−1,s , yn−1,s ) defined by xn,k n−1,s = rn,k /rn−1,s , see Section 2.4 of [25]. The Fm (λ) which appears in (38) and (39) is a factor of the quantum determinant: n Y Fm (λ), (45) An (ωλ)Dn (λ) − Cn (ωλ)Bn (λ) = Vn · m=1 0 n From (1) we can read off directly the λ - and λ -coefficients of the polynomial An (λ): An (λ) = 1 + . . . + κ1 κ2 . . . κn V λn . (46) Then using (38), the general action of An (λ) on Bn eigenvectors can be written as an interpolation polynomial  n−1 n−1 Y Y λ i+ (λ − λn,s ) |Ψρ+0 |Ψρn i + λκ1 · · · κn 1− An (λ)|Ψρn i = n λn,s s=1 s=1 ! n−1 Y λ − λn,s X λ + ϕk (ρ′n ) |Ψρ+k i. (47) n λ n,k − λn,s λn,k s6=k k=1 Considerable effort is needed to present the norm of an arbitrary state vector |Ψρn i in factorized form, since multiple sums over the intermediate indices have to be performed. The norms are independent of the phase ρn,0 and their dependence on ρ′n is: Cn Cn . (48) = Q hΨρn |Ψρn i = Q −ρn,m − r ω −ρn,l ) n,l l<m (λn,l − λn,m ) l<m (rn,m ω Ising model spin matrix elements from Separation of Variables 11 The normalizing factor Cn is independent of ρn and can be written recursively [26]. The two lowest values are:  N −1  N −1 x2 N3 N x1 , C2 = C1 . (49) C1 = ω y1 ω y2 ỹ2 y2,0 3.3. Periodic model: Baxter equation and truncated functional equations In the auxiliary problem we looked for eigenfunctions of Bn . Bn does not commute with i. Now we are looking for eigenfunctions of tn which Vn (8), see (22): Vn |Ψρn i = |Ψρ+0 n commutes with Vn . By Fourier transformation in ρn,0 we build a basis diagonal in V, where the Fourier transformed variable ρ ∈ ZN is the total ZN -charge.: P |Ψ̃ρ,ρ′n i = ρn,0 ∈ZN ω −ρ·ρn,0 |Ψρn i, Vn |Ψ̃ρ,ρ′n i = ω ρ |Ψ̃ρ,ρ′n i . (50) We now write the eigenfunctions |Φρ,E i of tn (λ) as linear combination of the |Ψ̃ρ,ρ′n i. The eigenvalues of tn (λ) on these states are again order n polynomials in λ: tn (λ)|Φρ,E i = (E0 + E1 λ + · · · + En−1 λn−1 + En λn )|Φρ,E i (51) Since the values of E0 and En can be read off immediately from (8): Q Q Q E0 = 1 + ω ρ nm=1 bm dm /κm , En = nm=1 am cm + ω ρ nm=1 κm , (52) we combine the remaining coefficients into a vector E = {E1 , . . . , En−1 } , and label the eigenvectors just by the charge ρ and E: X tn (λ) |Φρ,E i = tn (λ|ρ, E) |Φρ,E i, |Φρ,E i = QR (ρ′n | ρ, E) |Ψ̃ρ,ρ′n i . (53) ρ′n Now, in order to achieve SoV of the multi-variable functions QR , we split off from QR (ρ′n | ρ, E) Sklyanin’s separating factor: Qn−1 R Q (ρn,k ) R ′ . (54) Q (ρn | ρ, E) = Qn−1 k=1 k s,s′ =1 wpn,s′ (ρn,s − ρn,s′ ) (s6=s′ ) n,s We shall not give the detailed calculation and just indicate the main mechanism. We express tn (λ) as an interpolation polynomial through the zeros λn,k of Bn (λ): ) (  n−1 n−1 Y Y λ 1− (λ − λn,s ) Ψ̃ρ,ρ′n + (An (λ) + Dn (λ))|Ψ̃ρ,ρ′n i = E0 + λ En λn,s s=1 s=1 ! n−1  X Y λ − λn,s λ  ρ ′ ′ + i . (55) i+ ω ϕ̃k (ρn ) |Ψ̃ρ,ρ′−k ϕk (ρn ) |Ψ̃ρ,ρ′+k n n λn,k − λn,s λn,k k=1 s6=k When we evaluate (55) successively at the n − 1 values λ = λn,k , k = 1, . . . , n − 1, the terms on the right of the first line of (55) do not contribute. Due to the Sklyanin-factor the brackets involving the differences λn,k − λn,s are made to cancel, leading to SoV. This results in the n − 1 single-variable λn,k Baxter equations (k = 1, . . . , n − 1) + R − R tn (λn,k |ρ, E) QR k (ρn,k ) = ∆k (λn,k ) Qk (ρn,k + 1) + ∆k (ωλn,k ) Qk (ρn,k − 1) . (56) Ising model spin matrix elements from Separation of Variables 12 Starting from the left eigenvectors the analogous left Baxter equations are L 1−n + tn (λn,k |ρ, E) QLk (ρn,k ) = ω n−1∆− ∆k (ωλn,k ) QLk (ρn,k − 1) ,(57) k (λn,k ) Qk (ρn,k + 1) + ω where we abbreviated ρ 1−n ∆+ k (λ) = (ω /χk ) (λ/ω) n−1 Y n−1 ∆− Fn (λ/ω) . k (λ) = χk (λ/ω) Fm (λ/ω) , (58) m=1 χk collects several factors (partly arising from ϕk and ϕ ek ) determined by constants κk , ak , . . . , dk alone. Now note that the left hand side of (56) more explicitly reads
Pn−1 Es λsn,k + En λnn,k QR E0 + s=1 k (ρn,k ) = . . . where the E are unknown and have to be determined from the system of homogenous equations (56) together with the n − 1 functions QR k (ρn,k ). In order to have a nontrivial solution, the coefficient determinants have to be degenerate. Fix a k, then from the determinant we may get one relation among E0 , . . . , En . All n − 1 systems for different k should be sufficient to determine all components of E. Fortunately, the condition for non-trivial solutions to (56) can be written as well-known truncated functional equations: Define τ (2) (λ) = t(λ)‡ and construct a fusion hierarchy [45, 33] by setting τ (0) (λ) = 0, τ (1) (λ) = 1, and τ (j+1) (λ) = τ (2) (ω j−1λ) τ (j) (λ) − ω ρ z(ω j−1λ) τ (j−1) (λ), where z(λ) = ω −ρ ∆+ (λ) ∆− (λ) = Qn m=1 j = 2, 3, . . . , N, Fm (λ/ω). (59) (60) Then it can be shown [25] that if τ (N +1) (λ) satisfies the truncation identity τ (N +1) (λ) − ω ρ z(λ) τ (N −1) (ωλ) = An (λN ) + Dn (λN ) (61) with An (λN ) + Dn (λN ) given in (28), then the system (56) has a non-trivial solution for all k. This truncated hierarchy can be used to find the transfer matrix eigenvalues [46, 47]. In our construction we have even more: for every solution of (59),(61) we can construct an eigenvector. 4. Action of uk and vk on eigenstates of Bn (λ) Our main aim is to calculate matrix elements of the local operators uk and vk between eigenstates |Φρ,E i of tn (λ) . Since we know how to get these states from the Bn (λ) eigenstates (53),(54),(56), we first set out to find the action of the local operators on the |Ψρn i. Since we built our auxiliary states successively from one-site to n-site, the formulas will not be symmetric between e.g. uj and uk with j 6= k. For un we can calculate its action directly. Starting from −ρn u−1 |ψρn in , n (an − bn vn )|ψρn in = rn ω ‡ This definition in [29] is the origin of calling the BBS model the τ (2) -model Ising model spin matrix elements from Separation of Variables 13 we get the formula for the action of un on one-site eigenvectors: ω ρn (62) (an |ψρn in − bn |ψρn +1 in ) . un |ψρn in = rn Using then the explicit recursion formula relating |Ψρn i to |Ψρn−1 i one finds [26]: an bn κ1 κ2 · · · κn−1 |Ψρn i − |Ψρ+0 i+ (63) n −ρ̃ n r̃n ω rn,0 ω −ρn,0 n−1 X an bn ϕk (ρ ′n ) Q |Ψρ+k i. + −ρn,0 λ n r (λ n,0 ω n,k (bn + an κn λn,k ) n,k − λn,s ) s6 = k k=1 un |Ψρn i = We shall derive this result in a simpler way expressing the local operators uk and vk in terms of the global entries An and Bn of monodromy matrix, taken at particular values of λ. There is a well-known method elaborated by the Lyon group [48]. However, this method requires the fulfillment of the condition R(0) = P with R the quantum R-matrix intertwining two L-operators in quantum spaces and P the permutation operator. This requirement is fulfilled for the cyclic L-operators only at special values of parameters where the R-matrix is the product of four weights of the Chiral Potts model [29]. Another requirement regards the possibility to obtain such a R-matrix by fusion in the auxiliary space of the initial L-operator. This requirement can not be fulfilled for the cyclic L-operators (1) because the fusion in the auxiliary space [49] gives L-operators with the highest weight evaluation representations of the corresponding quantum affine algebra, but we need cyclic type representation in the auxiliary space. We will use an idea borrowed from a paper of Kuznetsov on SoV for classical systems [50]. What we can do is the following: Consider the inverse of the operator Lk (λ):   ω λ ak ck + vk bk dk /κk −λ u−1 (a − b v ) k k k k   · (detq Lk (λ))−1 , L−1 (64) k (λ) = −ω uk (ck − dk vk ), 1 + ω λ κk vk where detq Lk (λ) = vk Fk (λ) , Fk (λ) = (bk + ωλak κk )(λck + dk /κk ) . ′′ ′ The expression for L−1 k (λ) is singular at zeros λk = −bk /(ωak κk ) and λk = −dk /(ck κk ) of Fk (λ). Of course, Tn−1 (λ) = Tn (λ) L−1 n (λ) . (65) Therefore at the zeros of Fn (λ) the left-hand side is regular in λ and the right-hand side also has to be regular. At λ = λ′n = −bn /(ωan κn ) we get ′ An (λ′n ) u−1 n bn /(ωκn ) + Bn (λn ) = 0 . Hence we have a formula for un : un = λ′n an Bn−1 (λ′n ) An (λ′n ) . (66) From the condition of the regularity of the right-hand side of (65) at λ = λ′′n = −dn /(cn κn ) we get ′′ An (λ′′n )(−λ′′n )u−1 n (an − bn vn ) + Bn (λn )(1 − dn /(ωcn )vn ) = 0 . Ising model spin matrix elements from Separation of Variables 14 Excluding un by means of (66), we obtain the formula for vn : vn = −1/(ωκn ) (An (λ′n ) Bn (λ′′n ) − An (λ′′n ) Bn (λ′n ))−1 × × (An (λ′n ) Bn (λ′′n )/λ′′n − An (λ′′n ) Bn (λ′n )/λ′n ). (67) Using the RTT-relations following from (5), we can permute An and Bn−1 in (66) to get an equivalent formula un = ω λ′n an An (ωλ′n ) Bn−1 (ωλ′n ) . (68) Using (47) and (25) we get (63). We can also get the formulas for un−1 and vn−1 . We express L−1 n (λ) in terms of An (λ) and Bn (λ) using (66) and (67). Now the formula (65) allows to find expressions for An−1 (λ) and Bn−1 (λ) in terms of An (λ) and Bn (λ). Finally we substitute these expressions to (66) and (67) in which the indices n are replaced by n − 1. This gives us expressions for un−1 and vn−1 in terms of An (λ) and Bn (λ). The described procedure can be iterated to express the local operators uk and vk in terms of An (λ) and Bn (λ). For example, the result for un−1 is:   ′ ′ 2 ′ ′ un−1 = ωλn−1an−1 An (ωλn−1)(ω λn−1 an cn + vn bn dn /κn ) − Bn (ωλn−1 )ωun (cn − dn vn )  −1 ′ 2 ′ × −An (ωλ′n−1)ωλ′n−1u−1 (a − b v ) + B (ωλ )(1 + ω λ κ v ) , n n n n n n n n−1 n−1 where ω λ′n−1 = −bn−1 /(an−1 κn−1 ) and the expressions (68) and (67) for un and vn have to be substituted. It gives the action of un−1 on |Ψρn i. We see that the formula gets quite involved. However, u1 can be easily expressed in terms of Dn and Bn :     d1 d1 1 −1 Bn − . (69) Dn − u1 = c1 c1 κ1 c1 κ1 For our purpose of finding matrix elements of spin operator between eigenstates |Φρ,E i of homogeneous tn (λ) we can choose any spin operator uk because they all are related by the action of translation operator having the same eigenstates |Φρ,E i. In what follows we consider matrix elements of the spin operator un because the corresponding formula for the action (63) is the simplest. At the end of this section we would like to mention some similarity of our formulas with the formulas from the paper [51], where the local operators for the quantum Toda chain are expressed in terms of quantum separated variables with the use of a recursive construction of the eigenvectors [22]. 5. The general inhomogenous N = 2 BBS-model In the N = 2 case we have two charge sectors ρ = 0, 1. Following the language of e.g. [14, 17, 24] the sector ρ = 0 will be called the Neveu-Schwarz (NS)-sector, and ρ = 1 the Ramond (R)-sector. We are going to show that the spin matrix elements can be written in a fairly compact, although not yet factorized form (85),(86). The full factorization will be achieved later for the homogenous Ising case. Ising model spin matrix elements from Separation of Variables 15 5.1. Solving the Baxter equations and norm of states Let us fix an eigenvalue polynomial t(λ|ρ, E) of t(λ) corresponding to a right eigenvector |Φρ,E i (since in the following our chain will have the fixed length n we often shall skip the index n. Also sometimes we shall suppress the arguments ρ, E in t). In order to find |Φρ,E i explicitly we have to solve the associated n − 1 systems (k = 1, 2, . . . , n − 1) of (right) Baxter equations:
R + − t(−rn,k ) QR (0) = ∆ (−r ) + ∆ (r ) Qk (1), n,k n,k k k k
R + − t(rn,k ) QR (70) k (1) = ∆k (rn,k ) + ∆k (−rn,k ) Qk (0). Since t(λ|ρ, E) is eigenvalue polynomial, the functional relation (61) ensures the existence of non-trivial solutions to (70) with respect to the unknown variables QR k (0) R and Qk (1) for every k = 1, 2, . . . , n − 1. In the N = 2 case, this means that for every k we have one independent linear equation (in case of degenerate eigenvalues, possibly no equation). In the case of generic parameters, both hand sides of each equation will R be non-zero. So, fixing QR k (0) = 1 we obtain two equivalent expressions for Qk (1): QR k (1) = − ∆+ t(−rn,k ) k (rn,k ) + ∆k (−rn,k ) = . − t(rn,k ) ∆+ k (−rn,k ) + ∆k (rn,k ) (71) Analogously from the left-Baxter equations, fixing QLk (0) = 1 we obtain QLk (1) = − ∆+ (−1)n−1 t(−rn,k ) k (−rn,k ) + ∆k (rn,k ) = . − (−1)n−1 t(rn,k ) ∆+ k (rn,k ) + ∆k (−rn,k ) Since for generic parameters t(rn,k |ρ, E) 6= 0 these explicit formulas give ρn,k (n−1) QLk (ρn,k ) QR t((−1)ρn,k rn,k )/t(rn,k ) . k (ρn,k ) = (−1) To get the periodic state, we have to insert the Skylanin-separation factor (54). Now for N = 2 the functions wp are simple: wp (0) = 1 , wp (1) = 1−x y = , 1+x y (wp (1))2 = 1−x . 1+x (72) n,m n,m In the Sklyanin factor we have to use the Fermat point pn,m n,l = (xn,l , yn,l ) defined by n,m the coordinate xn,m n,l = rn,m /rn,l . Here it can be expressed it in terms of xn,l only and we get Q Qn−1 L R X n−1 (−1)ρn,l +ρn,m (rn,m + rn,l )2 hΦρ,E |Φρ,E i l<m k=1 Qk (ρn,k )Qk (ρn,k ) . (73) = Qn−1 ρn,l r 2 ρn,m r hΨ̃ρ,ρ′n |Ψ̃ρ,ρ′n i n,m ) n,l + (−1) l<m ((−1) ρ′ n We can normalize to a convenient reference state. For the moment, simple formulas arise if for the normalization we chose the auxiliary state |Ψ̃0,0 i where 0 = (0, 0, . . . , 0). From (48) we get Qn−1 (rn,m (−1)ρn,m + rn,l (−1)ρn,l ) hΨ̃ρ,ρ′n |Ψ̃ρ,ρ′n i . (74) = l<m Qn−1 (r + r ) hΨ̃0,0 |Ψ̃0,0 i n,m n,l l<m Ising model spin matrix elements from Separation of Variables 16 Combining all these formulas we get for the left-right overlap of the transfer matrix eigenvectors of the periodic BBS model at N = 2: Qn−1 Qn−1 X (−1)ρn,l t((−1)ρn,l rn,l ) hΦρ,E | Φρ,E i l<m (rn,m + rn,l ) = . (75) Qn−1 Qn−1 l=1 ρ ρn,l r ) n,m r hΨ̃0,0 |Ψ̃0,0 i n,m + (−1) n,l l=1 t(rn,l ) l<m ((−1) ρ′ n This formula is not yet very useful since from (53) it contains the summation over the n − 1 Z2 -variables ρ′n defined in (44). However, in [26] it is shown how to perform this sum explicitly, and the fully factorized result is Qn−1 n Y hΦρ,E | Φρ,E i l<m (rn,m + rn,l ) n−1 ′ = 2 r̃n Qn Qn−1 (µi + µj ) , (76) hΨ̃0,0 |Ψ̃0,0 i l=1 (rn,l + µk ) i<j k=1 where −µi are the zeros of the eigenvalue polynomial of t(λ|ρ, E): t(λ|ρ, E)|Φρ,Ei = Λ n Y i=1 (λ + µi ) |Φρ,E i. (77) We don’t specify the factor Λ, since in the following it will cancel. 5.2. Matrix elements between eigenvectors of the periodic N = 2 BBS model In (63) we obtained the action of un on an eigenvector |Ψρn i of Bn (λ): the result is a linear combination of the original vector plus a sum of vectors which each have one component of ρn shifted. In order to get the matrix elements of un in the periodic model, using (50) we first pass to charge eigenstates hΨ̃ρ,ρ′n |, |Ψ̃ρ,ρ′n i: hΨ̃ρ,ρ′n | = hΨ0,ρ′n | + (−)ρ hΨ1,ρ′n | , |Ψ̃ρ,ρ′n i = |Ψ0,ρ′n i + (−)ρ |Ψ1,ρ′n i. (78) Since ω = −1, un anti-commutes with Vn so that only matrix elements of un between states of different charge ρ can be nonzero. In the following we shall chose the right eigenvector from ρ = 1, then the left eigenvector must have ρ = 0 (the opposite choice gives a different sign in (79)). Using (63), we find hΨ̃0,ρ′n |un |Ψ̃1,ρ′n i hΨ̃0,ρ′n |Ψ̃0,ρ′n i |un |Ψ̃1,ρ′n i hΨ̃0,ρ′ +k n hΨ̃0,ρ′n |Ψ̃0,ρ′n i κ1 κ2 · · · κn−1 bn an ′ (−1)ρ̃n − , r̃n rn,0 = r̃n−1 an bn cn = rn rn,0  (−1)ρn,k dn 1+ κn cn rn,k  (79) ′ Qn−2 n,k yn−1,l (−1)ρ̃n l=1 Q . ρ ρn,s ) n,k + r n,s (−1) s6=k (rn,k (−1) (80) Of physical interest are the matrix elements between periodic eigenstates. To get these we have to form linear combinations determined by the solutions of the Baxter equations: P R ′ Recall (53): |Φρ,E i = ρ′n Q (ρn | ρ, E) |Ψ̃ρ,ρ′n i and the corresponding left equations. Let hΦ0 | be a left eigenvector of the transfer-matrix tn (λ) with ρ = 0 and |Φ1 i be a right eigenvector with ρ = 1 (often suppressing the subscripts E, E′ ): hΦ0,E′ | t(λ|0, E′) = t(0) (λ) hΦ′0,E | , t(λ|1, E) |Φ1,E i = t(1) (λ) |Φ1,E i. (81) Ising model spin matrix elements from Separation of Variables L(0) 17 R(1) Let Qk (ρn,k ) and Qk (ρn,k ) be the solutions of Baxter equation corresponding to these two eigenvectors. After some simplification we get for the matrix elements (keeping the normalization by the auxiliary “reference” state): !   X n−1 X a κ κ · · · κ b h Φ0 |σnz | Φ1 i ′ n 1 2 n−1 n = N (ρ′) R0 (ρ′ ) + Rk (ρ′ ) , (82) (−1)ρ̃ − r̃ r h Ψ̃0,0 | Ψ̃0,0 i 0 ′ k=1 ρ where ′ N (ρ′ ) = (−1)nρ̃ n−1 Y l<m rl + rm , rl (−1)ρl + rm (−1)ρm R0 (ρ′ ) = n−1 Y L(0) Ql R(1) (ρl )Ql (ρl ), (83) l=1 n−1 Y an bn cn L(0) R(1) L(0) R(1) Qk (ρk + 1) Qk (ρk ) Ql (ρl ) Ql (ρl ) × r0 l6=k   n−1 ν χk dn Q k . (84) × 1− κn cn νk s6=k (νk − νs ) Rk (ρ′ ) = − with rk = rn,k , ρk = ρn,k , k = 0, 1, . . . , n − 1, νk = −rk (−1)ρk , r̃ = r0 r1 · · · rn−1 and ρ̃′ = Pn−1 k=1 ρk . The origin of the different terms in (82) is: the sum over ρ′ comes from (53), N (ρ′) is the normalization factor from (74). The terms at R0 (ρ′ ) arise from the first line of (63): the shift in ρn,0 affects the charge sector only. The sum over k and expression for Rk (ρ′ ) come from the second line in (63). Now, the sum over k can be performed. Indeed, as shown in [27], using the Baxter equations, some cancellations take place and (82) can be written as an X h Φ0 | un | Φ1 i N (ρ′ ) R0 (ρ′ ) R(ρ′ ) (85) = 2 r0 ′ n−1 h Ψ̃0,0 | Ψ̃0,0 i ρ ∈Z2 with t(1) (ζn ) t(0) (−ζn ) + Qn−1 , ρl r ) ρl r ) (−ζ + (−1) (ζ + (−1) l l n n l=1 l=1 R(ρ′ ) = Qn−1 ζn = bn . an κn (86) Despite the simple appearance, for the general inhomogenous N = 2 BBS-model, performing the sums over the Z2 variables explicitly seems to be a presently hopeless task. However, for the homogenous Ising model we shall show this to be possible. 6. Homogeneous N = 2 BBS-model 6.1. Spectra and zeros of the Bn - and tn -eigenvalue polynomials We now specialize to N = 2 taking all parameters site-independent (“homogenous”): am = a, bm = b, cm = c, dm = d, κm = κ, rm = r, Lm (λ2 ) = L(λ2 ), ∀ m. Then the classical monodromy is An (λ2 ) Bn (λ2 ) Cn (λ2 ) Dn (λ2 ) !
n = L(λN ) . (87) (88) Ising model spin matrix elements from Separation of Variables 18 Consider trace, determinant and eigenvalues x± of L: τ (λ2 ) = tr L(λ2 ) = 1 + b2 d2 − λ2 (κ 2 + a2 c2 ), κ2 (89) δ(λ2 ) = det L(λ2 ) = (b2 /κ 2 − λ2 a2 ) (d2 − λ2 c2 κ 2 ) = F (λ) F (−λ), (90) √ F (λ) = (b − aκλ)(λc + d/κ). (91) x± = 21 (τ ± τ 2 − 4 δ), From the matrix L(λ2 ) we obtain Bm (λ2 ) = −λ2 (a2 − b2 ) (xn+ − xn− )/(x+ − x− ), (92) so that the zeros of Bm are at x+ /x− = ei m φn,s with φn,s = 2πs/n, s = 1, 2, . . . , n − 1, s 6= 0. (93) Using τ 2 = 4 δ cos2 (φ/2) and (89), (90) we can translate the zeros labeled by φn,s by a quadratic equation in λ2 into zeros λn,s . Now we solve the functional relations (59),(61) for the transfer matrix spectrum. Using (59) for j = 2 and eliminating τ (3) by (61) we get the functional relation t(λ) t(−λ) = (−1)ρ (z(λ) + z(−λ)) + An (λ2 ) + Dn (λ2 ) (94) which we shall use to find t(λ). In terms of (89) and (90) this reads
n n t(λ) t(−λ) = (−1)ρ δ+ + δ− + xn+ + xn− . (95) where δ± = (b ± aκλ) (d ∓ cκλ); δ+ δ− = δ(λ2 ) = x+ x− . Introducing q taking the n values π(2s + 1 − ρ)/n, s = 0, . . . , n − 1 , we can write (94) as Y Y
t(λ) t(−λ) = (−1)n (eiq δ+ − τ (λ2 ) + e−iq δ− ) = (−1)n A(q)λ2 − C(q) + 2i B(q)λ q with q A(q) = a2 c2 − 2κ ac cos q + κ 2 ; B(q) = (ad − bc) sin q ; C(q) = 1 − 2(b d/κ) cos q + b2 d2 /κ 2 . Factorizing the polynomial in λ we get Y t(λ) t(−λ) = (−1)n A(q) (λ − sq ) (λ + s−q ) (96) (97) q 1 p ( D(q) − iB(q)), D(q) = A(q) C(q) − B(q)2 , (98) A(q) p D(q) requires a special convention, see [25]) and after some (fixing the sign of arguments we find the spectrum Q t(λ) = (an cn + (−1)ρ κ n ) q (λ ± sq ), (99) with sq = where the signs are not yet fixed. Comparing the λ-independent term in (51) t(λ) = 1 + (−1)ρ bn dn /κ n + E1 λ + · · · + En−1 λn−1 + λn (an cn + (−1)ρ κ n ). (100) with the corresponding term in (99) shows that the number of minus signs in (99) must be even (odd) for the NS-sector ρ = 0 (R-sector ρ = 1). Ising model spin matrix elements from Separation of Variables 19 It is useful to introduce the following notion: The eigenvalue (99) with all +-signs is called to possess “no quasi-particle” excitations. Each factor labeled by q having a minus sign is said to contribute the “excitation of the q-quasi-momentum”. We shall accordingly label the minus signs by a set of variables σq ∈ Z2 , where for unexcited (excited) levels q we put σq = 0 (σq = 1). So instead of (99), we shall write more precisely Q t(ρ) (λ) = (an cn + (−1)ρ κ n ) q (λ + (−1)σq sq ). (101) The corresponding eigenvectors have been considered for the inhomogenous case in Subsection 5.2. 6.2. Functional relation for the diagonal-to-diagonal Ising model transfer-matrix In this subsection we specialize the results of the previous subsection to the case of the diagonal-to-diagonal transfer-matrix of the Ising model on a square lattice (15). So, we set a = c, b = −d, κ = 1 and λ = b/a. Let us calculate the ingredients of the functional relation (94). We have Fm (λ) = −(b − aλ)2 . Therefore due to (60), z(λ) = (−1)n (b + a λ)2n , z(b/a) = (−1)n (2b)2n , z(−b/a) = 0 and the averaged L-operator (27) at λ2 = b2 /a2 becomes   !   1 − b2 /a2 , −b2 /a2 (a2 − b2 ) 1  = Lk (b2 /a2 ) =  · (1 − b2 /a2 ) · 1, −b2 . a2 a2 − b2 , b2 (b2 − a2 ) Hence An (b2 /a2 ) + Dn (b2 /a2 ) = tr Tn (b2 /a2 ) = (1 − b2 /a2 )n (1 − a2 b2 )n . Substituting these expressions into (94), we get the following functional relation t(b/a) t(−b/a) = (−1)ρ+n (2b)2n + (1 − b2 /a2 )n (1 − a2 b2 )n . We want to compare this with the functional relation equation (7.5.5) in [43]: V (K, L) V (L + iπ/2, −K) C = (2i sinh 2L)n I + (−2i sinh 2K)n R , where C is the operator of translation, R is the operator of spin flip Vn and V (K, L) is the transfer-matrix (13) with Kx = 0, Ky = L, Kd = K, e−2L = b/a, tanh K = a b. Therefore V (K, L) = exp(nL) coshn K tn (b/a). Similar analysis gives V (L + iπ/2, −K)C = in exp(n L) coshn K tn (−b/a). Now taking into account that the eigenvalues of R are (−1)ρ and a2 − b2 4ab , 2 sinh 2L = , 1 − a2 b2 ab we see that both functional relations are identical. 2 sinh 2K = exp(−2L) = (1 − a2 b2 ) b/a , cosh2 K Ising model spin matrix elements from Separation of Variables 20 6.3. Ising model: Spectra and zeros of the Bn (λ)- and tn (λ)-eigenvalue polynomials We now specialize further to the Ising case (14) as already advertised in Subsection 2.2: aj = cj = a, bj = dj = b, κj = 1; ∀j. (102) In the Ising case (102) the 2n eigenvalues of (101) with (98) can be written (2n−1 in each sector ρ = 0, 1): s Y b4 − 2 b2 cos q + 1 , (103) t(ρ) (λ) = (a2n + (−1)ρ ) (λ + (−1)σq sq ), sq = s−q = 4 − 2 a2 cos q + 1 a q where the quasi-momentum q in each sector takes n values: 2π q= m, m integer for ρ = 1 (R); m half-integer for ρ = 0 (NS). (104) n Recall that we found from (100) that in the NS (R) sector, the eigenstates of t(λ) have Q an even (odd) number of excitations: q (−1)σq = (−1)ρ . For q = 0 (this occurs for R-sector only) and q = π we define b2 + 1 b2 − 1 , sπ = 2 . (105) s0 = 2 a −1 a +1 q = π is in the R sector for n even. However, for n odd it is in the NS sector. The different presence of factors (λ ± s0 ) and (λ ± sπ ) in (103) for n even or odd often makes it necessary to consider the cases n-even and n-odd separately. In the following we shall reserve the notation λq for λq = (−1)σq sq and otherwise use sq as defined in (103). The zeros λn,k of the Bn (λ) eigenvalue polynomial are determined by (93),(89),(90): τ (λ2n,k ) = 4 cos2 qn,k F (λn,k ) F (−λn,k ), Since now F (λ) = F (−λ) = b2 − a2 λ2 ; we get rn,k = q qn,k = π k/n, k = 1, . . . , n−1.(106) τ (λ2 ) = 1 + b4 − (1 + a4 ) λ2 , (b4 − 2 b2 cos qn,k + 1)/(a4 − 2 a2 cos qn,k + 1) = sqn,k . (107) (108) Observe that sq and rn,k may coincide. 6.4. Ising model state vectors from Baxter equations In order to obtain the eigenvectors of t(λ), we have to solve Baxter’s equations. For our restricted parameters (102) we have F (λ) = F (−λ) and the left and right Baxter equations (57), (56) become identical. Omitting the superscripts L and R on Qk and recalling λn,k = −(−1)ρn,k rn,k , ρn,k = 0, 1 we obtain:   (−1)ρ F n−1 (λn,k ) n−1 + (−λn,k ) χk F (λn,k ) Qk (ρn,k + 1). (109) tn (λn,k ) Qk (ρn,k ) = (λn,k )n−1 χk From (109) we get the following compatibility condition:  2 (−1)ρ F n−1 (rn,k ) n−1 n−1 t(−rn,k )t(rn,k ) = (−1) + (−rn,k ) χk F (rn,k ) , (rn,k )n−1 χk Ising model spin matrix elements from Separation of Variables 21 if t(λ) is an eigenvalue from the sector ρ. If (−1)k = (−1)ρ+1 then the quasi-momentum q = qn,k belongs to the sector ρ and for rn,k = sqn,k we have t(−rn,k ) t(rn,k ) = 0. This implies a relation not depending on a particular t(λ) and its ρ: 2(n−1) χ2k rn,k = (−1)n+k+1 F n−2(rn,k ) . (110) Although the eigenvalue polynomial t(λ) is known from (103), to solve (109) for the Qk (ρn,k ) can meet a difficulty if tn (λn,k ) vanishes or if, due to (110), the big bracket on the right of (109) vanishes. All this can happen and we have to distinguish four cases (we suppress n and write just rk = rn,k and ρk = ρn,k ): (i) (−1)ρ = (−1)k : This is the easy case, since from (104) and (106) tρ (rk ) 6= 0 and tρ (−rk ) 6= 0, and we may normalize and solve QL,R k (0) = 1 , QL,R k (1) = (−1)n−1 tρ (−rk ) . 2χk rkn−1 F (rk ) The other three cases occur for (−1)ρ = (−1)k−1 : (ii) tρ (rk ) 6= 0, tρ (−rk ) = 0: tρ (λ) contains a factor (λ + rk )2 (both q = ±qk not excited), we may normalize QL,R k (0) = 1, QL,R k (1) = 0 . (iii) tρ (rk ) = 0, tρ (−rk ) 6= 0: tρ (λ) contains a factor (λ − rk )2 (both q = ±qk are excited), we cannot choose QL,R k (0) = 1, but we may normalize QL,R k (0) = 0 , QL,R k (1) = 1 . (iv) tρ (rk ) = tρ (−rk ) = 0: tρ (λ) contains (λ2 − rk2 ) (either q = +qk or q = −qk is excited): A L’Hôpital procedure, using a slight perturbation of (102) as described in [26], is required (to obtain eigenvectors of translation operator), leading to QR k (0) = QLk (0) = 1, QR k (1) = −QLk (1) (−1)n+σqk +1 2i sin qk tρq̌k (−rk ) = n χk rkn−1 A(qk ) L (observe that from the L’Hôpital-limit QR k (1) = − Qk (1)), where tρ (λ) = tρq̌k (λ) (λ + (−1)σqk sqk )(λ − (−1)σqk s−qk ), A(q) = a4 − 2a2 cos q + 1. (111) In the following we shall consider only the three cases which allow the normalization QL,R k (0) = 1. Case (iii) can be treated too, but requires a special treatment, which here we shall not enter. According to which case the corresponding eigenvalue polynomial b (ρ) , D (ρ) : belongs, let us define the sets D̆ (ρ) , D k ∈ D̆ (ρ) if tρ has a factor (λ + rk )2 , i.e. we have case (ii), b (ρ) if tρ has a factor (λ − rk )2 , case (iii), and k∈D k ∈ D (ρ) if tρ has a factor (λ2 − rk2 ), i.e. we have case (iv). By D = |D| we denote the number of elements in D = D (0) ∪ D (1) , similarly for D̆, etc. Ising model spin matrix elements from Separation of Variables 22 7. Calculation of the matrix elements of σnz in the homogeneous Ising model 7.1. Explicit evaluation of the factors N (ρ′) R0 (ρ′ ) R(ρ′ ) in (85) We now start to evaluate (85) with (83) and (86) for the homogenous Ising model where the parameters simplify drastically. Now ζ = b/a, r02 = (a2 − b2 )(a4n − 1)/(a4 − 1). (112) and un is represented by the Pauli σz . We had agreed to consider initial states from the R-sector. Then for matrix elements of σz the final state must be NS. We specify the initial state by the momenta which are excited, i.e. by the σk which are one, analogously the final state. Excluding for the time b (ρ) to be empty. being case (iii), we take D On the right of (83) we have to evaluate the factors N (ρ′ ) R0 (ρ′ ) R(ρ′ ) . Let us Qn−1 (0) (1) Ql (ρl )Ql (ρl ) . start with R0 (ρ′ ) = l=1 (0) (0) For any choice of excitations, always one of the factors Ql (ρl ) or Ql (ρl ) is from case (i) of Subsection (6.4). Since we exclude for the moment case (iii), the other factor (0) (1) then must be from (ii) or (iv). So always Ql (0)Ql (0) = 1. For l ∈ D̆, case (ii), we (0) (1) (0) (1) have Ql (1)Ql (1) = 0 since either Ql (1) = 0 or Ql (1) = 0 depending on the parity of l. So, in (83) the summation reduces to the summation over ρl for l ∈ D only, with fixed ρl = 0 for l ∈ D̆. R0 (ρ′ ) receives non-trivial contributions from Qk (1) of cases (i) and (iv). However, these can be written in a simple way if we use the explicit formulae for t(ρ) (−rk ) . For both values ρl = 0, 1 the result is (−1)ρl rl + ξl Y (−1)ρl rl + rk (0) (1) · , (113) Ql (ρl ) Ql (ρl ) = (−1)(n−1)ρl rl + ξ l rl + rk k∈D̆ where we get different results according to whether s0 or sπ or both (105) are excited:  b2 − eiq    (−1)σ0 2 a − eiq ξl = for (−1)σ0 = ±(−1)σπ ; q̃l = (−1)σql +|D|+l ql . (114) 2 iq b e − 1   σ 0  (−1) a2 − eiq Now, multiplying by N (ρ′ ) , it is easy to see that the products k ∈ D̆ in (113) cancel (recall that ρk = 0 for k ∈ D̆ ) and we get finally ρl Y Y rl + rm ρl (−1) rl + ξl ′ . (115) (−1) N (ρ) · R0 (ρ ) = ρ l rl + ξ l (−1) rl + (−1)ρm rm m∈D,m>l l∈D In the calculation of R(ρ′ ) in (86) we have to insert our explicit expressions for t(0) (−ζn ) and t(1) (ζn ) from (103). Here, as already mentioned after (105), the cases of even n and odd n give different formulas. E.g. the factor (λ − (−1)σπ sπ ) is present only for R n even and NS n odd. So Y Y NS, n odd: t(0) (−ζ) = (a2n + 1)(−ζ + (−1)σπ sπ ) (−ζ + rk )2 (ζ 2 − rl2 ) ,(116) k∈D̆ (0) l∈D (0) Ising model spin matrix elements from Separation of Variables 23 (for even n omit the bracket with sπ ), since in the NS-sector only odd k appear, and these fall into one of the classes (ii) and (iv), class (iii) being momentarily excluded. Analogously: Y Y (ζ + rk )2 (ζ 2 − rl2 ). (117) R, n odd: t(1) (ζ) = (a2n − 1)(ζ + (−1)σ0 s0 ) l∈D (1) k∈D̆ (1) By slight manipulation we can move the ρl -dependent terms such that they appear only in one place each in the numerator and get  Q R(n odd) (ρ′ ) = R · (−1)σπ (a2 + 1) (−ζ + (−1)σπ sπ ) l∈D ((−1)ρl rl + ζ) Q Q −(−1)σ0 (a2 − 1) (ζ + (−1)σ0 s0 ) l∈D ((−1)ρl rl − ζ) · k∈D̆ ((−1)k ζ + rk ) (118) with R = (αβ)−(n−1)/2 an−1 (a4n − 1)/(a4 − 1), α = a2 − b2 , β = 1 − a2 b2 . (119) The first term in the curly bracket comes from the NS-sector final state, the second from the R initial state. The formula for R(n even) is similar. 7.2. Summation, square of the matrix element Combining (115) with (118) the spin matrix element is given by a multiple sum over the components of ρ′ : ! X Y Y h Φ0 |σnz | Φ1 i = Rν+ ((−1)ρl rl + ζ) + Rν− ((−1)ρl rl − ζ) × h Ψ̃0 | Ψ̃0 i n−1 ′ l∈D l∈D ρ ∈ Z2 × ′ Y (−1)ρl rl + ξl (−1)ρl rl + ξ l l∈D Y m∈D,m>l Rν± . rl + rm , ρm l + (−1) rm (−1)ρl r (120) with some ρ -independent factors The superscript ν is there to remind us that we have different expressions for n even and n odd, respectively. Now, in [27] it is shown that this sum can be performed, resulting in a factorized expression. As an example here we quote the summation formula for the multiple summation over ρl with l ∈ D if the dimension of D is odd and ξl defined by the upper formula of (114): Q ρl ρl ρl X l∈D (−1) ( (−1) rl + ξl ) ( (−1) rl + ζ) Q ρm ρl l<m, l,m∈D ( (−1) rl + (−1) rm ) ρl , l∈D = C (b ± a) Q iq̃j j∈D e  Q 2 iq̃l − 1)(D−1)/2 (eiq̃l − a2 )(D−3)/2 l∈D (2 rl /a) (a e Y ∓ ab , (121) (±(eiq̃l +iq̃m − 1)) l,m∈D, l<m 2 /4 C = α−(D−1)(D−3)/4 β −(D−1) . The case of D even is similar, see (52) of [27]. Ising model spin matrix elements from Separation of Variables 24 In the following, we shall be interested in the product of the matrix elements of the spin operator between arbitrary periodic states, which does not depend on normalization of the left and right eigenstates, i.e. we want to calculate hΦ0 |un |Φ1 ihΦ1 |un |Φ0 i . (122) hΦ0 |Φ0 ihΦ1 |Φ1 i Taking the absolute squares, several factors in (121) can be re-written, e.g. |a2 eiq̃l − 1 |2 = |eiq̃l − a2 |2 = A(q̃l ) = a4 − 2a2 cos q̃l + 1, iq̃l +iq̃m |e 2 sin 21 (q̃l + q̃m ) rm − rl2 A(q̃m ) A(q̃l ) − 1| = αβ sin 12 (q̃l − q̃m ) 2 (123) (124) and all factors α, β and A(q̃m ) cancel. So we get for arbitrary n and σ0 = σπ : Y 2 rl h Φ0 | σnz | Φ1 i h Φ1 | σnz | Φ0 i = (λ2π − λ20 )(D−δ)/2 (λ0 + λπ )δ × (λ h Ψ̃0,0 | Ψ̃0,0 i2 0 + rl ) (λπ + rl ) l∈D 1 Y rl + rm sin 2 (q̃l − q̃m ) (125) × · rl − rm sin 12 (q̃l + q̃m ) l<m, l,m∈D where δ = 1. In a similar way we can find the product of matrix elements in the case of σ0 6= σπ . The final result is (125) with δ = 0. Observe that the explicit appearance of excitations of type (ii), i.e. k ∈ D̆ has disappeared from our formula (recall that we b still exclude k ∈ D). 7.3. Normalization of the periodic states, final result in terms of λ0 , λπ , rk and q̃l In order to compare (125) to the results obtained by A. Bugrij and O. Lisovyy [17, 18] we change the normalization and calculate the ratio (122). To do this we have to divide (125) by hΦ0 |Φ0 ihΦ1 |Φ1 i/hΨ̃0,0 |Ψ̃0,0 i2 . (126) However, formula (76) cannot be used directly in our degenerate Ising case (102). As in the case (iv) we have first to go off the Ising point and consider ad − bc = η and apply l’Hopital’s rule for η → 0. For n odd the result is Q Q n Y (λ0 ± rn,k ) hΦ0 |Φ0 i hΦ1 |Φ1 i k−odd (λπ ± rn,k ) |D| · Qk−even × = 2 (2rn,k ) · Q (λπ + rn,k ) (λ0 + rn,k ) hΨ̃0,0 |Ψ̃0,0 i2 k−even k−odd k=1  Q   Q k<l,k,l−odd (rn,k + rn,l )(±rn,k ± rn,l ) k<l,k,l−even (rn,k + rn,l )(±rn,k ± rn,l )   , × Q (±r + r )(r ± r ) n,k n,l n,k n,l k−odd,l−even and similar for n even, see [26]. Including also the hitherto excluded case (iii), our final formula for the matrix element is Y  rl + rm sin 1 (q̃l − q̃m )  hΦ0 |σnz |Φ1 ihΦ1 |σnz |Φ0 i 2 2 (D−δ)/2 δ 2 × = (λπ − λ0 ) · (λ0 + λπ ) 1 hΦ0 |Φ0 ihΦ1 |Φ1 i r − r sin (q̃ + q̃m ) l m 2 l l<m l,m∈D Ising model spin matrix elements from Separation of Variables 25   Q ( −̇r +̇r )( +̇r −̇r ) k l k l k odd, l even Λn  Q  , ·Q × DQ 2 k∈D (+̇2rk ) (+̇rk +̇rl )(−̇rk −̇rl ) (+̇rk +̇rl )(−̇rk −̇rl ) k<l,k,l odd k<l,k,l even (127) where Q (λ +̇r ) (1) (λπ +̇rk ) 0 k k∈D k∈D Q Q ·Q , for odd n, Λn = Q 2 2 2 2 (1) (λ0 +̇rk ) (0) (λπ +̇rk ) (1) (λ0 − rk ) (0) (λπ − rk ) k∈D k∈D k∈D k∈D Q (0) (λ0 +̇rk )(λπ +̇rk ) k∈D Q Q Λn = , for even n. (λ0 + λπ ) k∈D(1) (λ0 +̇rk )(λπ +̇rk ) k∈D(1) (λ20 − rk2 )(λ2π − rk2 ) Q (0) Here we used a superimposed dot: ±̇rm as the short notation for rm if m ∈ D̆, for ±rm b respectively. For composite sets of momentum levels k if m ∈ D and for −rm if m ∈ D, b D(0) = D̆ (0) ∪ D b (0) , D(1) = D̆ (1) ∪ D b (1) . we write D = D̆ ∪ D, 7.4. Final result in terms of momenta Let {q1 , q2 , . . . , qK } and {p1 , p2 , . . ., pL } be the sets of the momenta of the excitations presenting the states |Φ0 i from the NS-sector and |Φ1 i from the R-sector, respectively. After some lengthy but straightforward transformations of (127) we obtain h Φ0 | σnz | Φ1 ih Φ1 | σnz | Φ0 i = J (sπ + s0 ) (s2π − s20 )(K+L−1)/2 × h Φ0 | Φ0 i h Φ1 | Φ1 i Q NS Q R2 QK QL K L R 2 Y Y N PqNS P Mq ,p pl q6=|qk | q,qk p6=|pl | Np,pl k , (128) × · · QK k=1 l=1QL k l QR Q NS 2 N 2 N ′ Mqk ,qk′ ′ Mpl ,pl′ k<k l<l p,q q,p k=1 l=1 k l p q where NS/2 (R/2) is the subset of quasi-momenta from NS (R) taking values in the segment 0 < q < π , NS/2 (R/2) containing qk with odd k (even k): For n odd: sα + sβ sin α+β s2α (s20 − s2π ) 2 Mα,β = , M = · , α,−α sα − sβ sin α−β (s2π − s2α )(s20 − s2α ) 2 Q NS Q R Q NS 2 2 2 2 sα + sβ q p (sq + sp ) q (s0 + sq ) Nα,β = · Q NS . , J = QR Q R2 sα − sβ 2 (s + s ) 2 ′ ′ ) ) (s + s (s + s 0 p ′ ′ q q p p p q,q p,p sp sq , q 6= π, PpR = , p 6= 0, PqNS = (sπ − sq )(s0 + sq ) (sπ + sp )(s0 − sp ) QR 2 1 p (sπ + sp ) P0R = PπNS = J, , J = Q NS sπ + s0 2 (s + s ) π q q for n even: PqNS sq = , (sπ + sq )(s0 + sq ) P0R = −PπNS = 1 , sπ − s0 sp , p 6= 0, π, (sπ − sp )(s0 − sp ) Q NS 2 p (sπ + sq ) J = QR J. 2 q (sπ + sp ) PpR = Ising model spin matrix elements from Separation of Variables 26 7.5. Bugrij–Lisovyy formula for the matrix elements In [18] the following formula for the square of the matrix element of spin operator for the finite-size Ising model was conjectured: | NSh q1 , q2 , . . . , qK | σnz | p1 , p2 , . . . , pL iR |2 = QR  (K−L)2 /2 γ(pl )+γ(p) γ(qk )+γ(q) L K QNS Y Y ty − t−1 p6=pl sinh y q6=qk sinh 2 2 · × = ξ ξT QR QNS γ(qk )+γ(p) γ(pl )+γ(q) tx − tx−1 p sinh q sinh l=1 n k=1 n 2 2 × K Y qk −qk′ 2 2 γ(qk )+γ(qk′ ) k<k ′ sinh 2 sin2 L Y pl −pl′ 2 2 γ(pl )+γ(pl′ ) l<l′ sinh 2 sin2 Y sinh2 γ(qk )+γ(pl ) 2 . 2 qk −pl sin 1≤k≤K 2 (129) 1≤l≤L In this formula the states are labelled by the momenta of the excitations. The factors in front of the right hand side of (129) are defined by !1/4 QNS QR 2 γ(q)+γ(p) sinh p q 2 , ξ = ((sinh 2Kx sinh 2Ky )−2 − 1)1/4 , ξT = QNS γ(q)+γ(q′ ) QR γ(p)+γ(p′ ) sinh ′ q,q′ sinh p,p 2 2 where γ(q) is the energy of the excitation with quasi-momentum q: cosh γ(q) = −1 ty − t−1 (tx + t−1 y x )(ty + ty ) cos q , − −1 2(tx − tx ) tx − t−1 x (130) and tx = tanh Kx , ty = tanh Ky . Formula (129) can be easily derived from (128) if one takes into account the identification of parameters (14). In particular we have tx = a b, ty = (a − b)/(a + b) and the relation a sq + b eγ(q) = (131) a sq − b between the energy γ(q) of the excitation with quasi-momentum q and the corresponding zero sq of the t(λ)-eigenvalue polynomial (103). The following formulas give the correspondence between the different parts of (129) and (128): 1/2  ty − t−1 ty − t−1 sinh2 γ(α)+γ(β) ξ ξT y y 2 Mα,β . = J , = − 1 α−β 2 −1 tx − t−1 sinh 2 (γ(0) + γ(π)) tx − tx sin 2 x Q NS γ(qk )+γ(q) NS 2 P sinh s + s q q6=|qk | Nq,qk q6=qk 0 π k 2 , = Q QR γ(qk )+γ(p) γ(0)+γ(π) 2 N sinh n R sinh p,qk p 2 2 p QNS Q R2 γ(pl )+γ(p) R P sinh s + s p p6=|pl | Np,pl 0 π p6=pl l 2 , = QNS Q NS γ(0)+γ(π) γ(pj )+γ(q) 2 N sinh n q sinh q,pl 2 2 q QR For more details, see [27]. Ising model spin matrix elements from Separation of Variables 27 8. Matrix elements for the diagonal-to-diagonal transfer-matrix and for the quantum Ising chain in a transverse field In this section we derive the matrix elements of the spin operator between eigenvectors of the diagonal-to-diagonal transfer-matrix for the Ising model on a square lattice (see Sect. 2.2). In this case the parameters are given by (15). As has been explained there, if we vary the parameters a and b in such a way to have fixed (a2 − b2 )/(1 − a2 b2 ) = 1/k′ , the eigenvectors (and therefore matrix elements) will not change. So we fix a = c = k′ −1/2 and b = d = 0. Expanding the transfer-matrix (3) with such parameters we get: b = −1 H 2 2λ b H + ··· , tn (λ) = 1 − g n X z (σkz σk+1 + g σkx ) , k=1 b is the Hamiltonian of the periodic quantum Ising chain in a transverse field. where H From (103) we get the spectrum of this Hamiltonian: 1X ± ε(q) (132) E =− 2 q where the energies of the quasi-particle excitations are  q 1/2 2 ε(q) = (1 − 2 k′ cos q + k′ )1/2 = (k′ − 1)2 + 4 k′ sin2 , 2 ε(0) = k′ − 1, q 6= 0, π , ε(π) = k′ + 1 . In (132), the sign +/− in the front of ε(q) corresponds to the absence/presence of the excitation with the momentum q. The NS-sector includes the states with an even number of excitations, the R-sector those with an odd number of excitations. The momentum q runs over the same set as in (103). Since we have a = c and b = d, the formula (128) with sq = k′ /ε(q) for matrix elements for σnz can be applied. After some simplification we get the analogue of (129), now for the quantum Ising chain: | z NS hq1 , q2 , . . . , qK | σm × where and K Y k<k ′ 2 | p1 , p2 , . . . , pL iR | = k q −q 2 sin k 2 k′ ε(qk ) + ε(qk′ ) !2 L Y l<l′   41 ′2 ξ = k −1 , η(q) e = ′ QNS q′ QR p (K−L)2 2 p −p 2 sin l 2 l′ ε(pl ) + ε(pl′ ) ξT = QNS (ε(q) + ε(q′ )) (ε(q) + ε(p)) q,q′ L K Y eη(qk ) Y e−η(pl ) × ξ ξT n ε(qk ) l=1 n ε(pl ) k=1 !2 2 K Y L  Y ε(pl ) + ε(qk ) , pl −qk 2 sin 2 k=1 l=1 QNS QR q p (ε(q) 1 (ε(q) + ε(q′ )) 4 (133) 1 + ε(p)) 2 QR p,p′ 1 (ε(p) + ε(p′ )) 4 . Formally, all these formulas are correct for the paramagnetic phase where k′ > 1, and for the ferromagnetic phase where 0 ≤ k′ < 1. But for the case 0 ≤ k′ < 1 it is natural to Ising model spin matrix elements from Separation of Variables 28 redefine the energy of zero-momentum excitation as ε(0) = 1 − k′ to be positive. From (132), this change of the sign of ε(0) in the ferromagnetic phase leads to a formal change between absence-presence of zero-momentum excitation in the labelling of eigenstates. Therefore the number of the excitations in each sector (NS and R) becomes even. Direct calculation shows that the change of the sign of ε(0) in (133) can be absorbed to obtain formally the same formula (133), but with new ε(0) and even L (the number of the excitations in the R-sector) and new ξ = (1 − k′ 2 )1/4 . Formulas (129) and (133) allow to re-obtain well-known formulas for the Ising model, e.g. the spontaneous magnetization [1, 2]. Indeed, for the quantum Ising chain in the ferromagnetic phase (0 ≤ k′ < 1) and in the thermodynamic limit n → ∞ (when the energies of |vaciNS and |vaciR coincide, giving the degeneration of the ground state), we have ξT → 1 and therefore the spontaneous magnetization z 1/2 = (1 − k′ 2 )1/8 . NS h vac | σm | vac iR = ξ 9. Conclusions We have shown that finite-size state vectors of the Ising (and generalized Ising) model can be obtained using the method of Separation of Variables and solving explicitly Baxter equations. The Ising model is treated as a special N = 2 case of the ZN -BaxterBazhanov-Stroganov τ (2) -model. Finite-size spin matrix elements between arbitrary states are calculated by sandwiching the operators between the explicit form of the state vectors. For the standard Ising case this gives a proof of the fully factorized formula for the form factors (129) conjectured previously by Bugrij and Lisovyy. We also extend this result to obtain a factorized formula for the matrix elements of the finite-size Ising quantum chain in a transverse field. We show how specific local operators can be expressed in terms of global elements of the monodromy matrix. The truncated functional relation guaranteeing non-trivial solutions of the Baxter equation is compared to Baxter’s [43] Ising model functional relation. Acknowledgements The authors are grateful to the organizers of the International Conference in memory of Alexei Zamolodchikov: ”Liouville field theory and Statistical models” for their warm and friendly hospitality. The authors wish to thank Yu.Tykhyy for his collaboration in [26, 27]. G.v.G. and S.P. have been supported by the Heisenberg-Landau exchange program HLP-2008. S.P. has also been supported in part by the RFBR grant 08-01-00392 and the grant for the Support of Scientific Schools NSh-3036.2008.2. The work of N.I. and V.S. has been supported in part by the grant of the France-Ukrainian project ’Dnipro’ No.M/17-2009, the grants of NAS of Ukraine, Special Program of Basic Research and Grant No.10/07-N, the Russian-Ukrainian RFBR-FRSF grant. 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