  We consider a one-dimensional commensurate Peierls insulator in the presence
of spin-dependent sign-alternating potentials. In a continuum description, the
latter supply the fermions with spin-dependent "relativistic" masses
$m_{\uparrow,\downarrow}$. The ground-state phase diagram describes three
gapped phases: the CDW and SDW-like band insulator phases sandwiched by a
"mixed" phase in which the CDW and SDW superstructures coexist with a nonzero
spontaneous dimerization (SD). The critical lines separating the massive phases
belong to the Ising universality class. The Ising criticality is accompanied by
the Kohn anomaly in the renormalized phonon spectrum. We derive a Ginzburg
criterion which specifies a narrow region around the critical point where
quantum fluctuations play a dominant role, rendering the adiabatic (or
mean-field) approximation inapplicable.
  A full account of quantum effects is achieved in the anti-adiabatic limit
where the effective low-energy theory represents a massive version the N=4
Gross-Neveu model. Using Abelian bosonization we demonstrate that the
description of the SD phase, including its critical boundaries, is well
approximated by a sum of two effective double-frequency sine-Gordon (DSG)
models subject to self-consistency conditions that couple the charge and spin
sectors. Using the well-known critical properties of the DSG model we obtain
the singular parts of the dimerization order parameter and staggered charge and
spin susceptibilities near the Ising critical lines. We show that, in the
anti-adiabatic limit, on the line $m_{\downarrow} = 0$ there exists of a
Berezinskii-Kosterlitz-Thouless critical point separating a Luttinger-liquid
gapless phase from the spontaneously dimerized one. We also discuss topological
excitations of the model carrying fractional charge and spin.
  \pacs{71.10.Pm; 71.27.+a; 71.45.Lr; 75.40.Kb}
