Gap equations of Peierls transitions

The gap equations for the electronic Peierls transition, the $\mathrm{XY}$ spin-Peierls transition and the Heisenberg spin-Peierls transition are investigated. Conditions under which the dependence of the energy gap on temperature is of the BCS type are established. For realistic parameters, the BCS law is a good description for the electronic and the $\mathrm{XY}$ Peierls transitions. In the case of the Heisenberg spin-Peierls transition, we find that the approach of Beni is very different from the BCS-type behavior of Pytte. With the first approach, a first-order transition for any values of the parameters is obtained. Large fluctuations are expected near the critical point. The changes in the gap equation with magnetic field are also calculated and we show that the ordered phase can disappear for large enough fields. Finally, our results are compared with those of the recent theory of Cross and Fisher.