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

Ground state phases and quantum criticalities of one-dimensional Peierls model with spin-dependent sign-alternating potentials