Eigenstates and Eigenvalues of Chain Hamiltonians Based on Multiparameter Braid Matrices for All Dimensions

We study chain Hamiltonians derived from a class of multidimensional, multiparameter braid matrices introduced and explored in a series of previous papers. The N 2× N 2 braid matrices (for all N) have (1/2)N2 free parameters for even N and (1/2)(N + 1)2 − 1 forN odd.We present systematic explicit constructions for eigenstates and eigenvalues of chain Hamiltonians forN = 2, 3, 4 and all chain lengths r. We derive explicitly the constraints imposed on these states by periodic (circular) boundary conditions. Our results thus cover both open and closed chains. We then indicate how our formalism can be extended for all (N, r). The dependence of the eigenvalues on the free parameters is displayed explicitly, showing how the energy levels and their differences vary in a particular simple way with these parameters. Some perspectives are discussed in conclusion.


Introduction
In a series of previous paper [1][2][3][4], we have formulated and studied a class of  2 ×  2 braid matrices ( ≥ 2) with free parameters whose numbers increase as  2 .Chain Hamiltonians corresponding to these matrices were also presented.Here, we undertake systematic study of chain Hamiltonians derived from these braid matrices presenting iterative and explicit constructions of eigenstates and eigenvalues for all dimensions  and for all orders (chain lengths) .
(We use the notation () for a matrix with only one nonzero element, unity, on row  and column .)On such an orthonormal complete basis of projectors, one defines (with domains defined below (2)) ()   ( ()   +  ()  ) .
The crucial constraints on the free parameters The braid matrix is now (,  ∈ {1, . . .,  − 1}, ,  ∈ {2 − 1, . . .,  + 1},  = ±).The number of free parameters is now (An overall factor     and redefinitions of the 's,  ()   +   →  ()   , etc. convert our previous convention for odd  to the present one, which is more convenient for what follows.) Defining the Hamiltonian acting on a chain of +1 sites ( ≥ 1) is given by the standard where Ṙ,+1 (0) acts on the sites (,  + 1).For circular boundary conditions (or periodic), The Hamiltonians inherit the free parameters (see (6), ( 9)) of R.This is the most striking features of our construction.The eigenvalues will be seen to depend linearly on these parameters (given by simple sums of the 's).So, by varying them, one can vary the differences between the energy levels.We will first present, in the following sections, explicit results for the simplest cases.Then, an iterative formalism will be implemented to generalize them.Also, we will start with open chains and subsequently impose the constraints (12) for periodicity.
The base spaces (for different 's) will be spanned by the tensor products of the fundamental states vectors Using the notations of our previous papers, and so on.We now increase  stepwise as follows.

Cyclic Boundary Conditions (CBC
The condition and the sum imposed above leave only two eigenstates with the common eigenvalue It will be shown below that for all  the extreme eigenvalues ( (+) 11 ,  (−) 11 ) are eliminated under (CBC) and the dimension of the base space is divided by a factor 4 (for  = 2).

𝑁 = 4
We continue to study even dimensions.For  = 2, we had to deal with one pair of indices (|1⟩, |1⟩ of Section 2).

CBC.
Circular boundary conditions can be imposed now.As compared to the case  = 2, along with strong analogies, there are crucial differences.One evident difference is the larger domains of the indices.The constraints  +1 =  1 leave, for  = 2, the possibilities Now, one can have For  = 2, one can have But there is a somewhat more subtle difference.The fundamental constraint (5) and the symmetries of the projectors (see (3)) lead to for our free parameters.But we do not indent to impose, diminishing number of free parameters, which are an essential feature of our formalism, additional restrictions as  ()  =  ()   ( ̸ = ), and so on.Let us explore the consequences for (CBC) for  > 2.
ISRN Mathematical Physics But for  = (, ), these are not eigenstates of  (2) if  12 ̸ =  21 .For  = (, ) only, one obtains eigenvalues They are eigenstates only under constraints (independent of each other).The first and the second terms of (72) or (3.30) contribute to differing eigenvalues by which vanishes for (73) (along with (66)).Hence the results.For  = 2, there is one constraint less.The restrictions on the eigenvalues and on the dimensions of the base spaces can now be obtained in a straightforward fashion.Along with, as in Section 2,  1   = −1, there are now (71) and (73).
We have preferred to study the case  = 4 (rather than  = 3) directly after  = 2, since all even ( = 2) cases can be treated coherently together while all odd ones ( = 2−1,  = 2, 3, . ..) have features in common which are absent for  = 2.The prototype is provided by  = 3.We study this case in some detail in the following section.Then, we will only try to understand generalizations necessary for arbitrary dimensions  = (2, 3, 4, 5, 6, . ..).

𝑁 = 3
The definition below ( 2 While, for all other indices,  ̸ = .This is the central fact inducing special properties for odd  (for even  no  = ).

𝑟 = 2.
We continue to elucidate the role of (84) by treating this case in detail.Here, For states involving only (1, 1), one proceeds exactly as for  = 2 (see ( 24)-(32)).One thus recovers formally the eight eigenstates not involving |2⟩.But now, one also has the following possibilities in the iterative structure of the eigenstates: The total action of  (2) is evident form the left and the right factorizations displayed.The eigenvalues are, respectively, (91) The corresponding states, eigenvalues, and multiplicities are easily written down factorizing out more and more 's.For example, for (, , , ) = (2, 2, , ), one has with eigenvalue 11 . ( One ends with The situation is analogous for all , with ever increasing possibilities and subcases as  increases.We do not intend to track down such proliferations systematically.Let us, however, briefly indicate some essential features.Now, . One, correspondingly, starts with base states When each (, ) is (1, 1), |2⟩ being absent, one proceeds as in Section 2. The multiplicity of |2⟩ can now be (when present) (0, 1, 2, . . .,  + 1) .
The number of possibilities (generalizing results as (92)) for the subsets is, respectively, One ends with 4.2.CBC.The Index 2 (2) again plays a special role concerning CBC for  = 3.For states involving only (1, 1), one proceeds exactly as for  = 2 (Section 2).But |2⟩ now leads to additional possibilities as shown below.

𝑟 = 1. Apart from
one now has also 4.2.2. = 2. Apart from periodic states (corresponding to (39)) one now has also In (43), the sum on the left eliminates the (NP) part of (42).For states of the class (as compared to (42)) a sum like (43) is no longer necessary.One has a periodic state with if ( 2 , . . .,   ) are each (1, 1).If one or more of them are (2), the corresponding 's factor out as (1 + ) giving (for  = +1) a set of periodic states spanning a space of corresponding lesser dimension.Since a sum like (43) is not here necessarily, a problem like that analyzed for  = 4 (see (67)-(74)) does not arise.

(𝑁, 𝑟)
We are now in a position to consider the situation for  and for any chain length .In fact, after our detailed study of the cases  = 2, 3, 4 in the previous sections for all  values ( = 2, 3, . ..), no really new technic is necessary for  > 4. As the domains of the indices labeling the states spanning the base space increase as the dimensions of the base space increase as  +1 displaying explicitly all eigenvalues, and each one with its specific multiplicity becomes prohibitive, not due to subtle new features, but due to the sheer length of enumeration necessary.We do not intend to meet this aspect head on.But we claim that the essential problems have been solved in the preceding sections.let us consider examples.For  = 6,  = (1, 1) , (2, 2) , (3,3) .
This crucial result is to be implemented in Here,  1 is our previous .Now, we set For  = 2, for example, one obtains for a -chain (as a generalization of ( 11)) Implementing the basic result (110), one can collect together the coefficients in a particular simple form.
Let us start with a simple example, which, however, clearly indicates how to perform generalization.For ( = 2,  = 2) with  = (, ) and so on, as compared to ( 23 This involves a systematic use of (111) and a stepwise generalization of (114).We now briefly study the higher order Hamiltonians that modify the transition matrix elements.We again concentrate on a particular simple example.With in (110)   = 0( > 2) and  = 2 in (115), ( 116 with Note that, for  = 2, the coefficient of |⟩ is no longer zero (as for  = 1) for general values of parameters.
Decomposing | 1  2 ⋅ ⋅ ⋅    +1 ⟩ as in (117) to give factors (|  ⟩ +   |  ⟩) ( = 1, 2, . . ., ) which appear in eigenfunctions (115), implementing the action of   (from (116)) and then again collecting together coefficients of states generalizing the right hand members of (120), one obtains transition matrix elements for all (, ).We will not try to present such results explicitly.The foregoing ones clearly indicate the successive steps.However, some general features are worth noting.When  increases, the action of (112) on basis states induced of simultaneous flips between different closely or well separately sites.The corresponding transition matrix elements can be obtained systematically for the full action of (112) if so desired.Short and long range correlations can thus be extracted and explicitly formulated.

Discussions
The central feature of our class of braid matrices and associated Hamiltonians is the number of free parameters (of the order of  2 ) coexisting with simple symmetries permitting systematic, explicit construction of eigenstates and eigenvalues for all dimensions and chain lengths (, ).
There exists a rich class of multidimensional, multiparameter Yang-Baxter (and hence braid) matrices [7,8], where progress has been made in the construction of eigenstates of transfer matrices [9].(See the discussion added in [5].)For this class, however, the parameters enter via multiple rapidities (in contrast to our single ), making the situation basically different.For our case the symmetries leading to complete, explicit solutions restrict the properties of the models.But the solutions show how one can pass from one sector to another by varying the relative magnitudes of the parameters.
The simplest case,  = 2 becomes trivial for  (+) 11 =  (−) 11 .For  (+)  11 >  (−) 11 , the highest and the lowest eigenvalues (for chain length ) are, respectively,  (+)  11 and  (−) 11 .The situation is reversed for  (+) 11 <  (−) 11 .The relative spacing of the levels depend on  (+)  11 −  (−) 11 .It is also clearly seen how the highest and lowest levels  (±)  11 are excluded for closed chains (as noted below (41)).For  > 2, the number of parameters, increasing as  2 , leads to elaborating possibilities for classifying accessible sectors.For  = 4, one already has 8 parameters and the general expression for the eigenvalues for chain length  (from (61)) is ( +1 ), where each  can be (1, 1, 2, 2) and each  can be (+, −) independently.This and its direct generalization for all  display the possibilities concerning the number of sectors and crossovers according to 's chosen.We cannot propose physical significances corresponding to such multisector patterns.One may contrast it, however, with the domains of the single parameter of the 6-vertex model and the associated (antiferromagnetic, critical, and ferromagnetic) regimes [10,11].We have presented a thorough study of eigenstates and eigenvalues.But other directions remain to be explored.We intend to study elsewhere correlations in our models.It would be interesting to compare the situation with those encountered for famous familiar cases [12].
It would be interesting to study higher order conserved magnitudes obtained through higher order -derivatives of the braid matrix [6].We intend to generalize in another direction, the nested sequence of projectors (2) that can be generalized by including parameters.A 4 × 4 example can be found in the paper where nested sequences were introduced [13].It appeared in the context of U  ( ŝ 2 ) (Section 5, [13]).We aim to present a systematic generalization of higher dimensions and corresponding Baxterized braid matrices.
Finally, we briefly point out that the Hamiltonians studied here can be carried over without change for a class of unitary braid matrices presented in the context of entanglement [14].It was pointed out [3,14] that for all 's imaginary ( ()   → i ()   , with 's real on the right) R() of (4) becomes unitary.A further overall factor −i gives the same Ṙ(0) as for real 's.Hence, the result, in [14], another class of unitary R(), was constructed.Namely,