  We discuss certain quadratic models of spinless fermions on a 1D lattice, and
their corresponding spin chains. These were studied by Keating and Mezzadri in
the context of their relation to the Haar measures of the classical compact
groups. We show how these models correspond to translation invariant models on
an infinite or semi-infinite chain, which in the simplest case reduce to the
familiar XX model. We give physical context to mathematical results for the
entanglement entropy, and calculate the spin-spin correlation functions using
the Fisher-Hartwig conjecture. These calculations rigorously demonstrate
universality in classes of these models. We show that these are in agreement
with field theoretic and renormalization group arguments that we provide.
