Exact solutions for 1D lattice models with topological complicated configuration

In this work a transfer matrix method is developed to study 1D lattice models within the tight binding framework. Employing this method we show, from simple to difficult, the solutions of a semi-infinite wire, a finite open wire, a single closed ring and two coupled rings. We start by studying local properties of solutions in a homogeneous region. It is found that a calculation of the exponential of the transfer matrix is necessary for us to obtain the general form of wave functions in the entire homogenous region. Then by matching wave functions at boundaries or connecting junctions we can get equations that determine the spectrum. In this way we have solved the problem of two coupled rings (a topologically complicated configuration) and it is shown there can exist bound state in this system.