Recurrence relations in one-dimensional Ising models

The exact finite-size partition function for the nonhomogeneous one-dimensional (1D) Ising model is found through an approach using algebra operators. Specifically, in this paper we show that the partition function can be computed through a trace from a linear second-order recurrence relation with nonconstant coefficients in matrix form. A relation between the finite-size partition function and the generalized Lucas polynomials is found for the simple homogeneous model, thus establishing a recursive formula for the partition function. This is an important property and it might indicate the possible existence of recurrence relations in higher-dimensional Ising models. Moreover, assuming quenched disorder for the interactions within the model, the quenched averaged magnetic susceptibility displays a nontrivial behavior due to changes in the ferromagnetic concentration probability.

Figure 1

Temperature dependence of the inverse magnetic susceptibility (solid curves) and its high-temperature assymptotic limit (dashed curves) for several probability values $x$.

Figure 2

Temperature dependence of the magnetic susceptibility (solid curve) from lowest $x=0$ (bottom) to highest $x\approx 0.31$ (top) ferromagnetic probability. Localization of maximum and minimum temperatures (dashed curve) determined by $\eta$ in Eq. (27). Note the appearance of a threshold temperature whose reduced value is slightly below unity. Inset shows minimum and maximum given by $\eta$ as a function of the probability $x$.