  We consider a class of one-dimensional compass models with antisymmetric
Dzyaloshinskii-Moriya exchange interaction in an external magnetic field. Based
on the exact solution derived by means of Jordan-Wigner transformation, we
study the excitation gap, spin correlations, ground-state degeneracy, and
critical properties at phase transitions. The phase diagram at finite electric
and magnetic field consists of three phases: ferromagnetic, canted
antiferromagnetic, and chiral. Dzyaloshinskii-Moriya interaction induces an
electrical polarization in the ground state of the chiral phase, where the
nonlocal string order and special features of entanglement spectra arise, while
strong chiral correlations emerge at finite temperature in the other phases and
are controlled by a gap between the nonchiral ground state and the chiral
excitations. We further show that the magnetoelectric effects in all phases
disappear above a typical temperature corresponding to the total bandwidth of
the effective fermionic model. To this end we explore the entropy, specific
heat, magnetization, electric polarization, and the magnetoelectric tensor at
finite temperature. We identify rather peculiar specific-heat and polarization
behavior of the compass model which follows from highly frustrated
interactions.
