Incommensurate charge-density waves in the adiabatic Hubbard-Holstein model

The adiabatic Holstein-Hubbard model describes electrons on a chain with step a interacting with themselves (with coupling U) and with a classical phonon field phi(x) (with coupling lambda). There is Peierls instability if the electronic ground-state energy F( p) as a functional of phi(x) has a minimum which corresponds to a periodic function with period pi/p(F), where p(F) is the Fermi momentum. We consider p(F)/pia irrational so that the charge-density wave is incommensurate with the chain. We prove in a rigorous way in the spinless case, when lambda, U are small and U/lambda large, that (a) when the electronic interaction is attractive U < 0 there is no Peierls instability and (b) when the interaction is repulsive U > 0 there is Peierls instability in the sense that our convergent expansion for F(phi), truncated at second order has a minimum which corresponds to an analytical and pi/p(F) periodic phi(x). Such a minimum is found solving an infinite set of coupled self-consistent equations, one for each of the infinite Fourier modes of phi(x).