Phase transition in the generalized Fibonacci quantum Ising models

We prove that quantum Ising models, built on the scheme of generalized Fibonacci chains, undergo a magnetic phase transition whenever the roots of the characteristic equation of the associated substitution rules satisfy the Pisot-Vijayaraghavan (PV) property that only one root in absolute value is greater than one. The proof can be generalized in a straightforward manner to arbitrary two-letter substitution rules. We also conjecture that these models behave as random systems if the mentioned roots fail to satisfy the PV property.