Eigenstates and eigenvalues of chain Hamiltonians based on multiparameter braid matrices for all dimensions (Q1653867)

Summary: We study chain Hamiltonians derived from a class of multidimensional, multiparameter braid matrices introduced and explored in a series of previous papers. The \(N^{2} \times N^{2}\) braid matrices (for all \(N\)) have \((1/2)N^2\) free parameters for even \(N\) and \((1/2)(N+1)^2-1\) for \(N\) odd. We present systematic explicit constructions for eigenstates and eigenvalues of chain Hamiltonians for \(N=2,3,4\) 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.