Bipolaronic charge density waves and polaronic spin density waves in the Holstein-Hubbard model

At large electron-phonon coupling, the phonon of the Holstein-Hubbard model can be (generally) treated classically. Exact results can be obtained for this model at any band filling and any finite dimension where the Hubbard term with coefficient U is treated rigorously. The existence of chaotic bipolaronic states is proven when U is smaller than a certain value Uc and for an electron-phonon coupling k sufficiently large. The existence of magnetic polaronic states is also proven for positive U and k sufficiently large. For 0, o describing a single electronic band with bandwidth 2 Td where T is the transfer integral of the electrons between the neighbouring sites of a d-dimensional square lattice. c. and c. are the standard creation and annihilation fermion operators of an electron at site i with spin o=T or 4 2a dispersionless phonon Hamiltonian 2 P = £ 2m+ 20i>= X *>O. h, K and U are classical constants which are not directly sensitive to isotope effects). It is now convenient to redefine new variables