Multiparameter statistical models from N 2 × N 2 braid matrices : Explicit eigenvalues of transfer matrices T ( r ) , spin chains , factorizable scatterings for all N

For a class of multiparameter statistical models based on N2 × N2 braid matrices the eigenvalues of the transfer matrix T(r) are obtained explicitly for all (r,N). Our formalism yields them as solutions of sets of linear equations with simple constant coefficients. The role of zero-sum multiplets constituted in terms of roots of unity is pointed out and their origin is traced to circular permutations of the indices in the tensor products of basis states induced by our class of T(r) matrices. The role of free parameters, increasing as N2 with N , is emphasized throughout. Spin chain Hamiltonians are constructed and studied for all N . Inverse Cayley transforms of Yang-Baxter matrices corresponding to our braid matrices are obtained for all N . They provide potentials for factorizable S-matrices. Main results are summarized and perspectives are indicated in the concluding remarks.


Introduction
Statistical models "exact" in the sense of Baxter [1] satisfy "star-triangle" relations leading to transfer matrices commuting for different values of the spectral parameter. Crucial in the study of such models is the spectrum of eigenvalues of these matrices. But even for the extensively studied 6-vertex and 8-vertex models based on 4 × 4 braid matrices (the braid property guaranteeing star-triangle relations), after the first few simple steps, one has to resort to numerical computations. (We cannot adequately discuss here the vast associated literature but refer to texts citing major sources [2,3].) But the basic reason for such situation is that, starting with 2 × 2 blocks T ab of the transfer matrix and constructing 2 r × 2 r blocks via coproduct rules for order r (r = 1, 2, 3, . . .) one faces increasingly complicated non-linear systems of equations to be solved in constructing eigenstates and eigenvalues. Even when a systematic approach is available, such as the Bethe Ansatz for the 6-vertex case, it only means that the relevant non-linear equations can be written down systematically. The task of solving them remains. For the 8-vertex case one explores analytical properties to extract informations. (See Ref. 4, for example.) Also the number of free parameters remains strictly limited for such models, including the multistate generalization of the 6-vertex one [2].
Our class of models exhibits the following properties: 1. A systematic construction for all dimensions. One starts from N 2 × N 2 braid matrices leading to N r × N r blocks of T (r) , the transfer matrix of order r for N = 2, 3, 4, 5, . . .; r = 1, 2, 3, . . . with no upper limit.
2. The number of free parameters increases with N as N 2 . This is a unique feature of our models.
3. For all (N, r) one solves sets of linear equations with systematically obtained simple constant coefficients to construct eigenvalues of T (r) = N a=1 T (r) aa . This is a consequence of our starting point: braid matrices constructed on a basis of "nested sequence" of projectors [5].
The total dimension of the base space on which such equations have to be solved increases as N r . This will be seen to break up into subspaces closed under the action of T (r) , thus reducing the work considerably. Evidently one cannot continue to display the results explicitly, as N r increases. But it is possible to implement a general and particularly efficient approach. This will be illustrated for all N and r = 1, 2, 3, 4, 5. The generalization to r > 5 will be clearly visible. This is the central purpose of our paper. Certain other features will be explored with comments in conclusion.

Braid and transfer matrices
The roots of our class of braid matrices and the transfer matrices they generate are to be found in the nested sequence of projectors [5]. In Ref. 5 it was noted that the 4 × 4 projectors providing the basis of the 6-vertex and 8-vertex models (i.e. the braid matrices leading to those models) can be generalized to higher dimensions -all higher ones (N odd or even). Braid matrices for odd dimensions were exhaustively constructed on such a basis [6] and related statistical models were studied [7]. Then even dimensional ones were presented [8] and the corresponding braid matrices and statistical models were studied [9]. In previous works the eigenvalues and eigenfunctions of the transfer matrices were presented mostly for the lowest values of N, namely the 4 × 4 and 9 × 9 braid matrices. We present below a general approach for all N.
We start by recapitulating the construction of our nested sequence of projectors, leading to the remarkable properties of our solutions: A. Even dimensions: Let N = 2n (n = 1, 2, 3, . . .). Define ((ab) denoting a matrix with unity on row a and column b) where i, j ∈ {1, · · · , n}, ǫ = ±,ī = 2n + 1 − i,j = 2n + 1 − j. Interchanging j ⇋j on the right one obtains P (ǫ) ij . One thus obtains a complete basis of projectors satisfying (with P ij be an arbitrary set of parameters satisfying the crucial constraint The θ-dependence entering exclusively through the exponentials as coefficients as above. Such construction guarantees [7,8,9] the braid property where, in standard notations, R 12 = R ⊗ I and R 23 = I ⊗ R, I denoting the N × N identity matrix. Define the permutation matrix P = a,b (ab) ⊗ (ba). Then satisfies the Yang-Baxter equation. The monodromy matrix of order 1 (r = 1) is given by The N × N blocks are (with a, b ∈ {1, . . . , 2n}) Higher orders (r > 1) are obtained via coproduts For some essential purposes it is worthwhile express (2.8) as (with i, j ∈ {1, . . . , n}) The transfer matrix is the trace on (a, b): The foregoing construction assures the crucial commutativity This implies that the eigenstates of T (r) (θ) are θ-independent (sum of basis vectors with θ-independent, constant relative coefficients).
C. Free parameters: A unique feature of our constructions is the number of free parameters m (ǫ) ij , increasing as N 2 with N. For our choice of normalizations (apart from a possible altered choice of an overall factor, irrelevant for (2.5)) the exact numbers are This is to be contrasted with the multistate generalization of the 6-vertex model where the parametrization remain fixed at the 6-vertex level (Ref. 2 and sources cited here). One of our principal aims is to display the roles of our free parameters concerning the basic features of our models.

Explicit eigenvalues of Transfer Matrices for all dimensions N and orders r
We present below a unified approach for all N and illustrate it in some detail for r = 1, 2, 3, 4, 5. For r > 5 the extension will be evident.
A. Even dimensions: In sec. 5 of Re. 9 it was pointed out that the antidiagonal matrix relates the blocks of T (1) as follows: For N = 2 this led to an iterative construction of eigenstates and eigenvalues of the transfer matrix T (r) for increasing r. This works efficiently only for N = 2 (the 4 × 4 R (θ)-matrix). For our present approach the crucial ingredient shows up immediately in the structure (arising from those of our projectors and the coproducts rules) where each ǫ is independently (±). From (3.3), one obtains immediately, Consider the basis state (1): If any one (even a single one) of the indices b i = a i+1 or a i+1 (cyclic) then the On the other hand (2.9) plays a crucial role. Gathering together all the on (3.5). All essential features for the general case can be read off the first few simples examples: Note that the bars correspond to the indies of the ǫ's. One has thus Consequently only even number of additional bars appear on the right. Starting with even or odd number of bars on the left, lead to "even " and "odd" closed subspaces.
(2): It is important to note that the preceding features are independent of N -for a given r they are valid for all N. For r > N all the indices (b 1 , . . . , b r ) cannot be distinct. But even for r < N some or all of them can coincide. This general validity permits a unified treatment for all N.
(3): Note also the crucial cyclic permutation of the indices under the action of T (r) : . This is a consequence of the trace condition in (2.12) incorporated in (3.3). It will be seen later how this leads to an essential role of the r-th roots of unity e i k·2π r (k = 0, 1, . . . , r − 1) in our construction of eigenstates and also of eigenvalues as factors accompanying exponentials of the type appearing in (3.6-9).
We continue the study of the particular cases r = 1, 2, 3, 4, 5 each one for any N. General features will be better understood afterwards.
• r = 2: We start with (3.6). Separating "even" and "odd" spaces, define (for i = j when (i, j,ī,j) are all distinct) The solutions are, for (c 1 , c 2 , c 3 , Summing over all i, consistently with the general rule, one obtains Tr contributions coming only for i = j. For r = 2, the r-th roots of unity are ±1 and their role above is not visible very distinctly due to the presence of ±1 also from ǫ. The role of the roots of unity will be more evident from r = 3 onwards. The simple exercise above gives all eigenvalues and eigenstates for T (2) for all N(= 2n). For N = 2, r = 2, writing out the 16 × 16 T (2) in full and effectively diagonalizing it we have confirmed the results above.
• r = 3: Now the starting point is (3.7). Define, for the even subset, and hence λ 3 = 1, and also 1 + e i 2π 3 + e i 4π 3 = 0. Corresponding to (i 1 , i 2 , i 3 ) ⇋ (ī 1 ,ī 2 ,ī 3 ), define for the odd-subset, The respectively. Now The eigenfunctions and eigenvalues are (for mutually orthogonal basis states) For V λ one follows exactly similar steps and obtains the same set of eigenvalues (with i ⇋ī in the eigenstates). Combining all the solutions above we obtain complete results for r = 3 and all even N.
• r = 4: The key result (3.8) indicates the following construction (a direct generalization of (3.17)). Define, for the even subset, and similarly, implementing circular permutations as in (3.17), where for non-zero results (1 + ǫ 12 ǫ 23 ǫ 34 ǫ 41 ) = 2. For the 8 possibilities remaining for the ǫ's and the 4 values of λ one obtains the full set of 32 eigenstates and the corresponding eigenvalues. For the complementary "odd" subspace formed by and the sets related through circular permutations of the indices one obtains, entirely in evident analogy of (3.25), One obtains, for distinct indices, the same set of eigenvalues as in (3.27 ii +2m . (3.31) In fact, for each r one obtains for ǫ = 1, e rm (+) ii θ (1, 1) giving All other multiplets have each one zero sum. Concerning the zero sum multiplets, comparing (3.31) and (3.30), we note that: 1. For r = 3 one has only triplets.
Such a feature is worth signalling since it generalizes. When r is a prime number the zero-sum multiplets appear only as r-plets (with r-th roots of unity). How they consistently cover the whole base space (along with the doublets giving the trace as in (3.32)) has been amply discussed in our previous papers [7,9] under the heading "An encounter with a theorem of Fermat". When r has many factors the multiplicity of sub-multiplets corresponding to each one is difficult to formulate giving, say, an explicit general prescription.
• r ≥ 5: Another problem concerning systematic, complete enumeration one encounters already for r = 5. Starting with (3.9) the generalizations of (3.24-27) are quite evident. We will not present them here. But as compared to(3.30) one has now for the 32-dimensional subspace (for i 1 = i 2 = · · · = i 5 ) (That the right hand side is an integer for r a prime number is guaranteed by a theorem of Fermat, as has been discussed before.) A general explicit prescription for the sequence (n 2 , n 4 , . . . , n r−1 ) is beyond the scope of this paper. The situation is as above for prime numbers r. For factorizable r the presence, in addition, of submultiplets has already been pointed out. For r = 4, only such submultiplets was (1, −1). For r = 6, one can have (1, −1), 1, e i 2π 3 , e i2· 2π 3 corresponding to the factors 6 = 2 × 3 respectively. For r = 30 = 2 × 3 × 5 one can have also submultiplets corresponding to 5. And so on.
But apart from the above mentioned limitations concerning multiplicities of zerosum ("roots of unity") multiplets and submultiplets we can claim to have elucidated the spectrum of eigenvalues for all (N, r). For the generic case with distinct indices for any (N, r), one has eigenvalues where (1 + ǫ 12 ǫ 23 · · · ǫ r−1,r ǫ 1,r ) = 2 and λ k = e ik· 2π r (k = 0, 1, 2, . . . , r − 1). Here each m is a free parameter. We solve only sets of linear equations with quite simple constant coefficients to obtain the eigenstates and the eigenvalues.

B. Odd dimensions:
We refer back to (2.14-16). When the special index n (n = n) for N = 2n − 1 is not present in the basis state |b 1 b 2 . . . b r and hence in the subspace it generates via cyclic permutations as in the forgoing examples, the foregoing constructions can be taken over wholesale. When n is present the modifications are not difficult to take into account. The trace is now Tr T (r) = 2 n−1 i=1 e rm (ǫ) ii θ + 1, since the central state |n ⊗|n ⊗. . . |n ⊗ ≡ |nn . . . n contributes with our normalization 1 to the trace. Various aspects have been studied in considerable detail in our previous papers [6,7] on odd N. Here we just refer to them.

Spin Chain Hamiltonians
Spin chains corresponding to our braid matrices have already been studied in our previous papers [7,8]. Here we formulate a unified approach for all dimensions (N = 2n − 1, 2n).
The basic formula (see sources cited in Refs. 7, 8) is where for circular boundary conditions, for k + 1 = r + 1 ≈ 1. For even N (N = 2n), from (2.4),˙ the projectors being given by (2.1) and the remark below (2.1). For N odd (N = 2n − 1; n = 2, 3, . . .) we introduce a modified overall normalization factor to start with. Such a factor is trivial concerning the braid equation, but not for the Hamiltonian (since a derivative is involved) if the factor is θ-dependent. Multiply (2.16) by e mθ and redefine ij , . . . are arbitrary to start with. Now for odd N (with the ranges of (i, j), (ī,j) of (2.16)) (

4.4)
This extends the m = 0 case by including the presence of P nn . In H,˙ R (0) k,k+1 acts on the basis |V (k) ⊗ |V (k+1) . One can denote, using standard ordering of spin components for each k, for N = 2n, 2n − 1 respectively. Without being restricted to spin one can consider more generally any system with N orthogonal states. We will continue however to use the terminology of spin. At each site one can consider a superposition at each level such as il |l k . To keep the notation tractable we will just denote and keep possible significances of |i k in mind. Each index k is acted upon twice by H, namely byṘ k−1,k (0),Ṙ k,k+1 (0) and, for closed chains, the index 1 is also thus involved inṘ 12 (0) andṘ r1 (0). Some simple examples are: and hence in evident notationṡ , (4.8) (ii) N = 4: In the notation of sec. 7 of Ref. 9 wherê (4.10) Hence,  , (4.14) Denote the basis factors for each k as, and .

(4.15)
One obtainṡ . (4.16) For N = 5, 6, . . . one can generate such results systematically. They, considering the action of all the terms of (4.1), furnish the transition matrix elements and expectation values for possible states of the chain and permit a study of correlations.
One can also consider higher order conserved quantities (Sec. 1.5 of Ref. 10) given by . (4.17) For l = 2, (4.1) is generalized by the appearance of factors of the type (apart from non-overlapping derivatives) For l > 2 this generalizes in an evident fashion. A study of spin chains for the "exotic" SØ3 can be found in Ref. 11. It is interesting to compare it with the 4 × 4 (for N = 2) case briefly presented above by exploring the latter case in comparable detail. One may note that for our class of braid matrices, for even N, as compared to (4.2), and for odd N, as compared to (4.4) where λ (θ) = ±e Now one can write down the Lagrangians for scalar and spinor fields. For the spinor case, for example, The simpler scalar case can be written analogously. The scalar Lagrangian has an interaction term of the form φ a φ c V ab,cd (φ b φ d ). For N = 2n, one obtains the nonvanishing elements of V as where a, b ∈ {1, . . . , N} and ((a, b) = (n, n) if N = 2n − 1. When N is odd (N = 2n − 1), one has for the special index n, nn are taken to be zero. For our models, the scattering process can be schematically, presented as (see figure)

Remarks
We have obtained explicit eigenvalues of the transfer matrix T (r) corresponding to our class of N 2 × N 2 braid matrices for all (r, N). Starting with our nested sequence of projectors we obtained a very specific structure of T (r) . Exploiting this structure fully we obtained the eigenvalues and eigenstates. The zero-sum multiplets parametrized by sets of roots of unity arose from the circular permutations of the indices in the tensor products of the basis states under the action of T (r) . The same structure led to systematic constructions of multiparameter Hamiltonians of spin chains related to our class of braid matrices for all N. Finally we constructed the inverse Cayley transformation for the general case (any N) giving potentials compatible with factorizability of S-matrices. The contents of such matrices (generalizing sec. 5 of Ref. 7) will be further studied elswhere.
In the treatment of each aspect we emphasized the role of a remarkable feature of our formalism -the presence of free parameters whose number increase as N 2 with N. A single class of constraints (θ ≥ 0, m ab > m (−) ab , a, b ∈ {1, . . . , N}) assures non-negative Boltzmann weights in the statistical models. But our constructions also furnish directly some important properties of such models. Thus Tr T (r) = 2 n i=1 e rm (+) ii θ (N = 2n) and Tr T (r) = 2 n−1 i=1 e rm (+) ii θ +1 (N = 2n−1) (for the normalization (2.16)). The eigenvalues can be ordered in magnitude by choosing the order of values of the parameters. Thus choosing m (+) 11 > m (+) 22 > · · · the largest eigenvalue of T (r) is for θ > 0, say e rm (+) 11 θ the next largest is e rm (+) 22 θ and so on. We intend to study elsewhere the properties of our models more thoroughly.
In previous papers [8,14] we pointed out that for purely imaginary parameters (im ab real) our braid matrices are all unitary, providing an entire class with free parameters for all N. Here we have not repeated this discussion. But the fact that they generate parametrized entangled states is indeed of interest.
Unitary matrices provide valid transformations of a basis (of corresponding dimension) of quantum states. One may ask the following question: If such a matrix, apart from being unitary, also satisfies the braid equation what consequence might be implied? Link between quantum and topological entanglements have been discussed by several authors [15,16] cited in our previous papers [8,14]. For the braid property to be relevant a triple tensor product (V ⊗ V ⊗ V ) of basis space is essential. We hope to explore elsewhere our multiparameter unitary matrices in such a context.
Note added: Prof. J.H.H. Perk has kindly pointed out that a class of multi-parameter generalization of the 6-vertex model is provided by sl (m|n) ones [17,18]. As Perk-Schultz models they have been studied by many authors and have led to various important applications. (Relevant references can be easily found via arXiv.) In this class the source of parameters are multi-component rapidities. The study of eigenvectors and eigenvalues were pioneered in Ref. 19; the most recent follow-up being Ref. 20, where other references can be found. Such studies may be compared to our systematic explicit constructions for all (r, N).