Excited states in the adiabatic Holstein model

The adiabatic Holstein model describes interaction of electrons with classi cal phonons. Near the anti-integrable limit, where electron-phonon coupling dominates electron hopping, Aubry, Abramovici and Raimbault (1992 J. Star. Phys. 67 675-780) found many local minima of the energy, while at the oppo site limit, called integrable, there is only one equilibrium for each choic e of mean electronic density. To eliminate the excess local minima on passi ng from the anti-integrable to the integrable limit, there must be bifurcat ions with other critical points of higher index: excited states. In this pa per, we find all the critical points of the energy at the two limits. We fi nd that at the anti-integrable limit the excited states form submanifolds a nd stratified sets of various types, which we call resonances. We show that homology index theory implies that at least certain numbers of critical po ints from each resonance survive small perturbation from the anti-integrabl e limit. We calculate these numbers explicitly for some simple cases, and d erive some general rules. The complete homology calculation in the general case and the study of the bifurcations on the route to the integrable limit are left for the future. We conclude by generalizing the approach to allow electron spin, magnetic fields and electron-electron interactions.