Incompatibility of BCS pairing and the Peierls distortion in one-dimensional systems. I. Mean-field theory

Abstract

The relationship between the BCS superconducting and Peierls insulating phase transitions in one-dimensional systems is investigated within mean-field theory. This calculation lays the essential theoretical groundwork for a treatment of fluctuations in the two order parameters which is presented in a companion paper. The model Hamiltonian used here is equivalent to the reduced BCS Hamiltonian when the Peierls gap vanishes and also equal to the mean-field approximation to the Fr\"ohlich Hamiltonian, in the presence of electron-electron interactions, when the BCS gap is zero. The two coupled gap equations are derived and solved and it is found that the two instabilities are, in general, very incompatible. When the "bare" Peierls transition temperature ${T}_{P}$ is greater than the "bare" BCS transition temperature ${T}_{S}$, the system generally orders at ${T}_{P}$ and is a Peierls insulator at all lower temperatures. The converse is also true. Only when ${T}_{P}$ and ${T}_{S}$ are nearly equal will the ordered state be the "mixed" state, that in which both order parameters are simultaneously nonzero. This mixed state is shown to exhibit a "one-dimensional Meissner effect." As a side issue, the effect of electron-electron interactions on the Peierls transition (when the BCS gap is zero) is also investigated; it is shown that electron-electron interaction effects are important. Repulsive interactions lower ${T}_{P}$, while attractive interactions raise ${T}_{P}$. These effects are sufficiently severe so that in jellium, with only Coulombic interactions present, ${T}_{P}$ is identically zero.