The Algebra of Chebyshev Polynomials and the Transfer-Matrix Approach for the One-Dimensional Ising Model with a Defect

We investigate a random field of mutually dependent random variables ("spins"), indexed by a finite one-dimensional lattice, called in physical sciences the one-dimensional Ising model, in which the random variables can take only ±1 values (see the text for a precise definition). One of the couplings, termed a "bond," that describes the mutual influence of two adjacent random variables is altered—it does not equal the others, thereby introducing a single "defect" bond. This defect bond represents a localised perturbation within an otherwise uniform system. Utilising the recurrence relations of Chebyshev polynomials and the bijective  map between the number of spins and the polynomial index, we present a new method for calculations and  systematically explore, using it, the system’s properties across different chain lengths and boundary conditions. As an application, we derive analytical expressions for the dependence of the average values of the random variables on their position within the chain, which we refer to as the "local magnetisation profile". From the findings related to the system with a defect bond, we present a novel result for this profile under free (Dirichlet) boundary conditions and re-derive the corresponding result for antiperiodic boundary conditions.