Realization of supersymmetric quantum mechanics in inhomogeneous Ising models

Supersymmetric quantum mechanics is well known to provide, together with the so-called shape invariance condition, an elegant method to solve the eigenvalue problem of some one-dimensional potentials by simple algebraic manipulations. In the present paper, this method is used in statistical physics. We consider the local critical behaviour of inhomogeneous Ising models and determine the complete set of anomalous dimensions from the spectrum of the corresponding transfer matrix in the strip geometry. For smoothly varying perturbations, the eigenvalue problem of the transfer matrix indeed takes the form of a Schr\"odinger equation, and the corresponding potential furthermore exhibits the shape invariance property for some known extended defects. In these cases, the complete spectrum is derived by the methods of supersymmetric quantum mechanics.