Griffiths singularities in the randomly dilute one-dimensional Ising model

Exact solutions for the thermodynamic functions of the randomly dilute $s=\frac{1}{2}$ nearest-neighbor Ising chain in a magnetic field are examined. Both site and bond impurities are treated. Behavior is nonanalytic at $T=h=0\mathrm{}$. The divergences of the pure-chain thermodynamics are replaced at nonzero dilution by essential singularities of the Griffiths type at which all functions are finite and infinitely differentiable. The simplicity of the solution allows the origin and form of the Griffiths singularities to be traced in detail.