%0 Journal Article
%T Eigenstates and Eigenvalues of Chain Hamiltonians Based on Multiparameter Braid Matrices for All Dimensions
%A B. Abdesselam
%A A. Chakrabarti
%J ISRN Mathematical Physics
%D 2013
%R 10.1155/2013/410859
%X We study chain Hamiltonians derived from a class of multidimensional, multiparameter braid matrices introduced and explored in a series of previous papers. The N2 ¡Á N2 braid matrices (for all N) have free parameters for even N and for N odd. We present systematic explicit constructions for eigenstates and eigenvalues of chain Hamiltonians for 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 . 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. 1. Introduction In a series of previous paper [1¨C4], we have formulated and studied a class of braid matrices ( ) with free parameters whose numbers increase as . 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 recapitulate briefly the constructions of the braid matrices satisfying (in standard notations used in [1¨C4]) Our class has a nested sequence of projectors as a basis which are defined as follows. For even ( , ), where , , , , . Interchanging on the right, one obtains with (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 give an braid matrix with free parameters. For odd ( ; ), using our definitions below (2), The braid matrix is now ( , , , , ). 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 sites ( ) is given by the standard where acts on the sites . For circular boundary conditions (or periodic), The Hamiltonians inherit the free parameters (see (6), (9)) of . 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
%U http://www.hindawi.com/journals/isrn.mathematical.physics/2013/410859/