  We present an exact solution for a class of one-dimensional compass models
which stand for interacting orbital degrees of freedom in a Mott insulator. By
employing the Jordan-Wigner transformation we map these models on
noninteracting fermions and discuss how spin correlations, high degeneracy of
the ground state, and $Z_2$ symmetry in the quantum compass model are visible
in the fermionic language. Considering a zigzag chain of ions with singly
occupied $e_g$ orbitals ($e_g$ orbital model) we demonstrate that the orbital
excitations change qualitatively with increasing transverse field, and that the
excitation gap closes at the quantum phase transition to a polarized state.
This phase transition disappears in the quantum compass model with maximally
frustrated orbital interactions which resembles the Kitaev model. Here we find
that finite transverse field destabilizes the orbital-liquid ground state with
macroscopic degeneracy, and leads to peculiar behavior of the specific heat and
orbital susceptibility at finite temperature. We show that the entropy and the
cooling rate at finite temperature exhibit quite different behavior near the
critical point for these two models.
