A nested sequence of projectors (2): Multiparameter multistate statistical models, Hamiltonians, S-matrices 6 0 B. Abdesselama,1,2 and A. Chakrabartib,3 0 2 n a Laboratoire de Physique Quantique de la Mati`ere et de Mod´elisations Math´ematiques, a Centre Universitaire de Mascara, 29000-Mascara, Alg´erie J 8 and 2 Laboratoire de Physique Th´eorique, Universit´e d’Oran Es-S´enia, 31100-Oran, Alg´erie ] A b Centre de Physique Th´eorique, CNRS UMR 7644 Q Ecole Polytechnique, 91128 Palaiseau Cedex, France. . h t Abstract a m Our starting point is a class of braid matrices, presented in a previous paper, [ constructed on a basis of a nested sequence of projectors. Statistical models associated 3 to such N2 N2 matrices for odd N are studied here. Presence of 1 (N +3)(N 1) v × 2 − 4 freeparameters isthecrucialfeatureof ourmodels,setting themapartfromotherwell- 8 known ones. There are N possible states at each site. The trace of the transfer matrix 5 is shown to depend on 1 (N 1) parameters. For order r, N eigenvalues consitute the 1 2 − 0 trace and the remaining (Nr N) eigenvalues involving the full range of parameters − 6 come in zero-sum multiplets formed by the r-th roots of unity, or lower dimensional 0 multiplets corresponding to factors of the order r when r is not a prime number. The / h modulus of any eigenvalue is of the form eµθ, where µ is a linear combination of the t a free parameters, θ being the spectral parameter. For r a prime number an amusing m relation of the number of multiplets with a theorem of Fermat is pointed out. Chain : v Hamiltonians and potentials corresponding to factorizable S-matrices are constructed Xi starting from our braid matrices. Perspectives are discussed. r a math.QA/0601584 January 2006 1This work was carried out at the Centre de Physique Th´eorique de l’Ecole Polytechnique, Palaiseau. 2Email: [email protected] and [email protected] 3Email: [email protected] 1 1 Introduction The most salient feature of the class of braid matrices presented in ref. [1], setting it apart from other known examples, is the number of free parameters. This class was obtained for N2 N2 braid matrices for odd × N = (2p 1), (p = 1,2,...). (1.1) − Such matrices, depending on a spectral parameter θ and satisfying the braid equation R (θ θ′)R (θ)R (θ′) = R (θ′)R (θ)R (θ θ′) (1.2) 12 23 12 23 12 23 − − have 1 (N +3)(Nb 1) free pabrametebrs when tbhe overabll normbalization is fixed. Thus for 2 − N = 3, 5, 7,..., the respective number of parameters are 6, 16, 30,.... These parameters appear in the coefficients of the N2 projectors (the ”nested sequence” defined in ref. [1]) providing the basis of R(θ). The projectors are defined as follows. Let (ij) be the N N matrix with a single non- × zero element 1 on rowbi and column j. Then N2 projectors are defined, with ǫ = and ± i = N i+1, as − P = (pp) (pp), pp ⊗ 2P = (pp) (ii)+(ii)+ǫ (ii)+(ii) , pi(ǫ) ⊗ 2P = (ii)+(ii)+ǫ (ii)+(ii) (pp), ip(ǫ) (cid:2) (cid:0) (cid:1)(cid:3) ⊗ 2P = (ii) (jj)+(ii) (jj)+ǫ (ii) (jj)+(ii) (jj) , ij(ǫ) (cid:2) (cid:0) (cid:1)(cid:3) ⊗ ⊗ ⊗ ⊗ 2P = (ii) (jj)+(ii) (jj)+ǫ (ii) (jj)+(ii) (jj) , (1.3) ij(ǫ) ⊗ ⊗ (cid:2) ⊗ ⊗ (cid:3) (cid:2) (cid:3) where, from (1.1), 1 i = 1, 2,..., p 1, i = N i+1 = 2p 1, 2p 2,..., p+1, p = (N +1). (1.4) − − − − 2 The projectors satisfy (with α, β standing for triplets (i,j,ǫ)) P P = δ P , P2 = I . (1.5) α β αβ α α N2×N2 α X Their total number is 1+4(p 1)+4(p 1)2 = (2p 1)2 = N2. (1.6) − − − For our class of solutions, normalizing to 1 the coefficient of P , pp R(θ) = Ppp + em(pǫi)θPpi(ǫ) +em(ipǫ)θPip(ǫ) + em(ijǫ)θPij(ǫ) +em(ijǫ)θPij(ǫ) , (1.7) Xi,ǫ (cid:16) (cid:17) Xi,j,ǫ (cid:18) (cid:19) b 2 with the crucial constraint (ǫ) (ǫ) m = m , j = N j +1 = 2p j . (1.8) ij ij − − (cid:0) (cid:1) This sufficient and necessary constraint concerning the coefficient of θ in the exponents, leaves 1 (N +3)(N 1) (1.9) 2 − free parameters. For N = 3 one thus obtains, with 6 free parameters, a 0 0 0 0 0 0 0 a + − 0 b 0 0 0 0 0 b 0 + − (cid:12) (cid:12) (cid:12) 0 0 a 0 0 0 a 0 0 (cid:12) + − (cid:12) (cid:12) (cid:12) 0 0 0 c 0 c 0 0 0 (cid:12) + − (cid:12) (cid:12) R(θ) = (cid:12) 0 0 0 0 1 0 0 0 0 (cid:12) (1.10) (cid:12) (cid:12) (cid:12) 0 0 0 c 0 c 0 0 0 (cid:12) − + (cid:12) (cid:12) b (cid:12) 0 0 a 0 0 0 a 0 0 (cid:12) − + (cid:12) (cid:12) (cid:12) 0 b 0 0 0 0 0 b 0 (cid:12) (cid:12) − + (cid:12) (cid:12) a 0 0 0 0 0 0 0 a (cid:12) (cid:12) − + (cid:12) (cid:12) (cid:12) where (cid:12) (cid:12) (cid:12) (cid:12) 1 1 1 a± = em(1+1)θ em(1−1)θ , b± = em(1+2)θ em(1−2)θ , c± = em(2+1)θ em(2−1)θ . 2 ± 2 ± 2 ± (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (1(cid:17).11) The parameters a are each repeated in (1.10) according to (1.8), since ± (±) (±) m = m , 1 = 3 . (1.12) 11 11 This is the case we will study mostly in the follow(cid:0)ing se(cid:1)ctions. The corresponding results for N > 3 will be indicated briefly. For example, the generalization of the considerations below in this section for N > 3 is entirely straight-forward. To explore the statistical model associated to (1.10) one starts by constructing explicit representations of the monodromy (r) matrices t (θ) of successive orders (r = 1,2,3,...) obtained by taking coproducts of the ij fundamental 3 3 blocks (with the same θ for each factor) × (r) t = t t t . (1.13) ij ij1 ⊗ j1j2 ⊗···⊗ jr−1,j j1,X...,jr−1 For N = 3, (r) (r) (r) t t t 11 12 11 (cid:12) (cid:12) t(r) = (cid:12) t(r) t(r) t(r) (cid:12). (1.14) (cid:12) 21 22 21 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (r) (r) (r) (cid:12) (cid:12) t t t (cid:12) (cid:12) 11 12 11 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 3 (cid:12) (r) If the Rtt equation for the blocks t (appendix C), ij b R(θ θ′) t(r)(θ) t(r)(θ′) = t(r)(θ′) t(r)(θ) R(θ θ′) (1.15) − ⊗ ⊗ − is satisfied for r = 1, then(cid:0)the coproduct co(cid:1)nstr(cid:0)uction (1.13) ens(cid:1)ures that (1.15) is satisfied b b for all higher values r = 2, 3,.... The solution for r = 1 is given by t(1)(θ) t(θ) = PR(θ) = R(θ), (1.16) ≡ where P is the permutation matrix b P = (ij) (ji) (1.17) ⊗ ij X and R(θ) is the Yang-Baxter (YB) matrix. This is a standard result valid generally for solutions of (1.2). (See appendix B of ref. [1] for sources cited.) The transfer matrix, for each order r, is defined to be the trace (with argument θ) T(r) = t(r) +t(r) +t(r). (1.18) 11 22 11 The properties of the model depend crucially on the eigenvalues of T(r). Refs. [2–4] provide ample information citing numerous basic sources. So our basic task will be to construct the eigenstates and eigenvalues of T(r)(θ). Re- markable feature following from (1.10) (and more generally from (1.7)) will be presented in the following sections and appendices. We will also construct chain Hamiltonians and potentials leading to factorizable S- matrices starting from our class of R(θ). Concerning each aspect we will try to display the role of our multiple parameters. For all b (+) (−) m θ m θ (1.19) ij ≥ ij the elements of R(θ)and hence the Boltzmann weights are non-negative, consistent with physical interpretations. For definiteness we consider the sector, say b (+) (−) (+) (−) (+) (−) m > m > m > m m > m , θ 0 (1.20) 11 11 12 12 21 21 ≥ of (1.10). The eigenvalues will be ordered differently for other sectors. They can be consid- ered separately. 2 Transfer matrix, eigenvectors, eigenvalues (N = 3): crucial features We start by signalling some crucial features to be encountered below in the explicit con- structions restricted (in this section) to N = 3. 4 1. The trace of the transfer matrix (1.17) of order r will turn out to be tr T(r)(θ) = 2erm(1+1)θ +1. (2.1) (cid:0) (cid:1) (±) (±) (±) (+) Of the six parameters m ,m ,m of (1.11) only m appears in the trace. 11 12 21 11 A simple explanation of(cid:16)this fact will be(cid:17)given after discussing the generalization for N > 3. 2. The eigenvalue erm(1+1)θ is obtained exactly twice for each r and the value 1 only once. 3. The remaining (3r 3) eigenvalues occur in multiplets of zero sum due to the presence − of roots of unity. Hence they do not contribute to the trace. For r a prime number there will be ”r-plets” (and possibly ”nr-plets”, n being an integer) eµθ 1,e2rπi,e2rπi·2,...,,e2rπi·(r−1) , (2.2) (cid:16) (cid:17) (±) where µ is a linear combination of the parameters m . When r is factorizable lower ij order multiplets can be present corresponding to the factors. Thus for r = 4 one obtains both doublets and quadruplets eµ2θ(1, 1), eµ4θ(1,i, 1, i), (2.3) − − − with appropriate linear combinations µ , µ to be displayed below. 2 4 4. Apart from possible roots of unity phase factors the modulus of each eigenvalue is a simple exponential of the type eµθ of (2.2). For r = 3, for example, one obtains for µ the values (appendix A) (+) (+) (−) 3m , m +2m , 11 11 11 (cid:16) (cid:17) (+) (+) (+) (+) (−) (−) m +m +m , m +m +m , 11 12 21 11 12 21 (cid:16) (−) (+) (−)(cid:17) (cid:16) (−) (−) (+)(cid:17) m +m +m , m +m +m , 11 12 21 11 12 21 (cid:16) (cid:17) (cid:16) (cid:17) (+) (+) (−) (−) m +m , m +m , 12 21 12 21 (cid:16)0. (cid:17) (cid:16) (cid:17) (2.4) Along with roots of unity factors these provides all the 27 eigenvalues as will be shown below. 5. The values of µ depend crucially on the subspaces, to be introduced below, which are invariant under the action of T(r)(θ), the transfer matrix. 5 2.1 Construction of T(r)(θ) for N = 3 The standard construction of the fundamental 3 3 block matrices t (θ) implementing ij × (1.15), (1.16), (1.10), (1.11) leads to (for t(1)(θ) t(θ) with 1 = 3) ≡ a 0 0 0 0 0 0 0 a + − t (θ) = 0 0 0 , t (θ) = c 0 c , t (θ) = 0 0 0 , 11 (cid:12) (cid:12) 12 (cid:12) + − (cid:12) 11 (cid:12) (cid:12) (cid:12) 0 0 a (cid:12) (cid:12) 0 0 0 (cid:12) (cid:12) a 0 0 (cid:12) − + (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 0 b 0 0 0 0 0 b 0 (cid:12) + (cid:12) (cid:12) (cid:12) (cid:12) − (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) t (θ) = 0 0 0 , t (θ) = 0 1 0 , t (θ) = 0 0 0 , (2.5) 21 (cid:12) (cid:12) 22 (cid:12) (cid:12) 21 (cid:12) (cid:12) (cid:12) 0 b 0 (cid:12) (cid:12) 0 0 0 (cid:12) (cid:12) 0 b 0 (cid:12) − + (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 0 0 a 0 0 0 a 0 0 (cid:12) +(cid:12) (cid:12) (cid:12) (cid:12) − (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) t (θ) = 0 0 0 , t (θ) = c 0 c , t (θ) = 0 0 0 , 11 (cid:12) (cid:12) 12 (cid:12) − + (cid:12) 11 (cid:12) (cid:12) (cid:12) a 0 0 (cid:12) (cid:12) 0 0 0 (cid:12) (cid:12) 0 0 a (cid:12) − + (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) where, from (1.11), (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (a+ a−) = em(1±1)θ, (b+ b−) = em(1±2)θ, (c+ c−) = em(2±1)θ. (2.6) ± ± ± One now has to be implement these in (1.13), (1.14) and (1.18) to obtain T(r)(θ). Then one proceeds to construct eigenvalues of T(r)(θ). 2.2 Subspaces invariant under the action of T(r)(θ) Westartbyintroducingconvenient, compactnotations. Thestatevectorsofthefundamental representation (2.5) are denoted as 1 0 0 0 , 1 , 0 1 , 2 , 1 . (2.7) (cid:12) (cid:12) (cid:12)  ≡ (cid:12) 0 + (cid:12) 0 + (cid:12) 1 + (cid:12) (cid:12) (cid:12) (cid:0)(cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11)(cid:1) (cid:12) (cid:12) (cid:12)  (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Tensor products for higher orders are constructed as (cid:12) (cid:12) (cid:12) 1 1 , 1 2 , 1 1 ,... 11 , 12 , 11 ,... (2.8) | i⊗| i | i⊗| i | i⊗ ≡ | i | i (cid:0) (cid:12) (cid:11) (cid:1) (cid:0) (cid:12) (cid:11) (cid:1) and so on in evident continuation. The or(cid:12)der of the labels (1,2,1(cid:12)) will indicate the tensor product structure. Thus, for example, 1 1 2 1 1 11212 (2.9) | i⊗| i⊗| i⊗ ⊗| i ≡ (cid:12) (cid:11) (cid:12) (cid:11) Corresponding to the r-th order coproduc(cid:12)t, T(r)(θ) a(cid:12)cts on a space spanned by 3r states (for N = 3). Let S(r,k), (k = 0,1,...,r) (2.10) 6 denote thesubspaces labeled by k, the multiplicityofthe index 2. Thecoefficients of different power of x in the expansion r (x+2)r = 1 xr +2rxr−1 + +2r−k xk + +2r (2.11) · ··· r k ··· (cid:18) − (cid:19) give the number of states in the respective subspaces. Setting x = 1 one obtains the total number (1+2)r = 3r. (2.12) For example, for r = 3, one obtains the subspaces, S(3,3) : 222 | i S(3,2) : 221 , 221 , 212 , 212 , 122 , 122 , | i | i | i S(3,1) : 211 , 211 , 211 , 211 (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) | i (cid:12) (cid:12) (cid:12) 121 , 121 , 121 , 121 (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) | i (cid:12) (cid:12) (cid:12) 112 , 112 , 112 , 112 (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) | i (cid:12) (cid:12) (cid:12) S(3,0) : 111 , 111 , 111 , 111 (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) | i (cid:12) (cid:12) (cid:12) 111 , 111 , 111 , 111 (2.13) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:12) (cid:12) A striking and most helpful conse(cid:12)quen(cid:11)ce(cid:12) of (cid:11)the(cid:12) str(cid:11)uc(cid:12)ture(cid:11)of the matrices (2.5) and their (cid:12) (cid:12) (cid:12) (cid:12) coproducts is: each subspace S(r,k) is invariant under the action of T(r)(θ). This facilitates considerably the construction of eigenstates. One works on lower dimensional spaces. One possible approachisasfollows: Oneselects any onestatefromthe 2r−k r states of r−k S(r,k) and computes the action of T(r)(θ) on it. One gets on the r.h.s. a linear combination (cid:0) (cid:1) of states belonging to S(r,k). Thus, for example, T(4)(θ) 1111 = a4 +a4 1111 +2a2a2 1111 + 1111 + 1111 + | i + − | i + − a2 +a2 a a 1111 + 1111 + 1111 + 1111 , (2.14) (cid:0) + −(cid:1) + − (cid:0)(cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11)(cid:1) (cid:12) (cid:12) (cid:12) where a± = 21 em(1+1)θ ±em(cid:0)(1−1)θ as (cid:1)noted b(cid:0)e(cid:12)(cid:12)fore.(cid:11) Ne(cid:12)(cid:12)xt on(cid:11)e c(cid:12)(cid:12)ompu(cid:11)tes (cid:12)(cid:12)the a(cid:11)c(cid:1)tion of T(4) successively on (cid:16)the other states(cid:17)appearing on the right. This continues until one obtains the coefficientsforaclosedsubsystem. Thenonesearchesforlinearcombinationssuchthatunder T(4) it is reproduced to within a factor. Thus one systematically obtains all eigenvectors and eigenvalues, for the subspace S(r,k). For our class one has to solve systems of linear equations with fairly simple coefficient. Even the 81 eigenstates and eigenvalues for r = 4 were obtained directly without using a computer program and without any real difficulties. We have thus obtained exhaustive solutions for r = 1, 2, 3, 4. The corresponding 3, 9, 27 and 81 eigenvalues are presented in appendix A. We have also obtained explicitly all the corresponding eigenstates. For brevity they are not presented here. The eigenvalues of the appendix A fully illustrate the crucial properties (1) to (5) signalled at the start of this section. In the following section we indicate a related but somewhat differently formulated approach for various comparisons. 7 3 Linear constraints for eigenvectors for N = 3 and comparison with algebraic Bethe ansatz In section 2 we have noted how, exploiting the invariance of the subspaces S(r,k) defined by (2.10) one can construct step by step all the eigenstates. The comments following (2.14) indicate how the relevant linear equations are obtained. We formulate below the approach in a systematic, explicit fashion. (1) Starting with (2.5) and (2.6) for t (θ) = t (θ) we define the operators ij ij 0 1 0 U = b ( θ)t (θ)+b ( θ)t (θ) = 0 0 0 , + − 21 − − 21   0 0 0 0 0 0   A = t (θ) = 0 1 0 , 22   0 0 0   0 0 0 D = b ( θ)t (θ)+b ( θ)t (θ) = 0 0 0 . (3.1) − − 21 + − 21   0 1 0   Then (suppressing arguments θ of t ) ij t (A,U,D) = (0,a U,a D) 11 + − t (A,U,D) = (0,c A,c A) 12 + − t (A,U,D) = (0,a D,a U) 11 + − t (A,U,D) = (b U +b D,0,0) 21 + − t (A,U,D) = (A,0,0) 22 t (A,U,D) = (b U +b D,0,0) 21 − + t (A,U,D) = (0,a D,a U) 11 − + t (A,U,D) = (0,c A,c A) 12 − + t (A,U,D) = (0,a U,a D). (3.2) 11 − + Also from (2.7) and (3.1) U 2 = 1 , A 2 = 2 , D 2 = 1 . (3.3) | i | i | i | i | i (cid:12) (cid:11) For any r, starting with S(r,r) one obtains the basic eigenstate(cid:12)(trivially since S(r,r) is of dimension 1), T(r)(θ) 22...2 = 1 22...2 . (3.4) | i | i Now one moves up in (r k) stepwise. − 8 S(r,r 1) (dim2r): With 2r coefficients (u ,d ) (i = 1,...,r) one can label the states as i i − (u U +d D) A A 1 1 ⊗ ⊗···⊗ +A (u U +d D) A A  . ⊗ 2 2 ⊗ ⊗···⊗  22...2 . (3.5) .. | i    +A A A (urU +drD)   ⊗ ⊗···⊗ ⊗    The action of T(r)(θ) on these leads to a linear system of equations in (u ,d ) corresponding i i to eigenstates. For S(r,r 1) the solution is particularly simple. Define − A A A (U +ǫD) ⊗ ⊗···⊗ ⊗ +ωA A (U +ǫD) A  ⊗ ⊗···⊗ ⊗  ω,ǫ = +ω2A A (U +ǫD) A A 22...2 , (3.6) | i  .. ⊗ ⊗···⊗ ⊗ ⊗ | i  .     +ωr−1(U +ǫD) A A   ⊗ ⊗ ⊗···⊗    where ǫ = and ω can have r values (as a r-th root of unity) ± ω = 1,ei2rπ,...,ei2rπ·(r−1) . (3.7) (cid:16) (cid:17) One obtains T(r)(θ) ω,ǫ = ωr−1e(m(1ǫ2)+m(2ǫ1))θ ω,ǫ . (3.8) | i | i The 2r eigenvalues are e(m(1ǫ2)+m(2ǫ1))θ 1,ei2rπ,...,ei2rπ(r−1) . (3.9) The next step is k = r 2. (cid:16) (cid:17) − S(r,r 2) (dim2r(r 1)): A set of states spanning this subspace is given by − − (A A (u U +d D) A (u U +d D) A A) 22...2 . (3.10) i i j j ⊗···⊗ ⊗ ⊗ ⊗···⊗ ⊗ ⊗···⊗ | i i6=j X The parameters (u,d) have to be constrained to obtain eigenstates. At each step one obtains sets of linear constraints. The pattern is now evident. At each step one inserts, as in (3.10), (r k) factors of the type (u U +d D) excluding their coincidence. i i − Finally for k = 0, one has S(r,0) of dimension 2r. Here a basis spanning the subspace can be labeled as u(i)U +d(i)D u(i)U +d(i)D u(i)U +d(i)D 22...2 . (3.11) 1 1 ⊗ 2 2 ⊗···⊗ r r | i ! Xi (cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1) Since there are no label 2 left, it can be shown from (3.2) that T(r)(θ), acting on S(r,0) | i simplifies to T(r) t(r) +t(r) t t(r−1) +t t(r−1) +t t(r−1) +t t(r−1). (3.12) ≈ 11 11 ≈ 11 ⊗ 11 11 ⊗ 11 11 ⊗ 11 11 ⊗ 11 9 In successive steps (t(r−1) t t(r−2) and so on) only the indices 1,1 need be retained. → ⊗ They only give non zero contributions. From (3.2) (cid:0) (cid:1) t u(i)U +d(i)D = a u(i)U +a d(i)D X(i)(s), 11 s s + s − s ≡ 11 t11(cid:0)us(i)U +d(si)D(cid:1) = (cid:0)a−d(si)U +a+u(si)D(cid:1) ≡ X1(1i)(s), t11(cid:0)us(i)U +d(si)D(cid:1) = (cid:0)a+d(si)U +a−u(si)D(cid:1) ≡ X1(1i)(s), t11(cid:0)us(i)U +d(si)D(cid:1) = (cid:0)a−u(si)U +a+d(si)D(cid:1) ≡ X1(1i)(s). (3.13) (cid:0) (cid:1) (cid:0) (cid:1) Define, with indices taking values 1,1 only, (cid:0) (cid:1) (i) (i) (i) (i) X (1,2,...,r) = X (1) X (2) X (r). (3.14) ab ab1 ⊗ b1b2 ⊗···⊗ br−1b b1,X...,br−1 The action of T(r)(θ) on the generic state (3.11) finally reduces to r (i) (i) X (1,2,...,r)+X (1,2,...,r) 22...2 . (3.15) 11 11 | i ! Xi=1 (cid:16) (cid:17) It is of particular interest to see what parameterizations in (3.11) corresponds to the two eigenstates (and two only for any r) that contribute to the trace. For r = 2, T(2)(θ) 11 + 11 = e2m(1+1)θ 11 + 11 , | i | i T(2)(θ)(cid:0) 11 +(cid:12)11(cid:11)(cid:1) = e2m(1+1)θ(cid:0) 11 +(cid:12) 11(cid:11)(cid:1) . (3.16) (cid:12) (cid:12) (cid:0)(cid:12) (cid:11) (cid:12) (cid:11)(cid:1) (cid:0)(cid:12) (cid:11) (cid:12) (cid:11)(cid:1) Along with (cid:12) (cid:12) (cid:12) (cid:12) T(2)(θ) 22 = 22 (3.17) | i | i the two states of (3.16) yield tr T(2)(θ) = 2e2m(1+1)θ +1. (3.18) (cid:0) (cid:1) For other six states provide the 3 zero sum doublets of (A.4), (A.5), (A.6). For r = 3. Apart from 8 zero sum triplets of eigenvalues (see appendix A) and the corresponding eigenstates one obtains for V = 111 + 111 + 111 + 111 (3.19) 1 | i V = 111 + 111 + 111 + 111 (3.20) 2 (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:12) (cid:12) T(3)(θ(cid:12)(cid:12))(V(cid:11)1,V2(cid:12)(cid:12)) =(cid:11)e3m(1(cid:12)(cid:12)+1)θ((cid:11)V1,V(cid:12)(cid:12)2). (cid:11) (3.21) 10