Effects of saddle-point singularities on the electron lifetime.

Abstract

We calculate the imaginary part of the response function (Im \ensuremath{\chi}) and the imaginary part of the electron self-energy (Im \ensuremath{\Sigma}) for a two-dimensional energy band with a saddle point close to the Fermi energy ${\mathit{E}}_{\mathit{F}}$, using lowest-order perturbation theory in the screened Coulomb interaction. We find that Im \ensuremath{\chi}(q,\ensuremath{\omega}) remains finite for arbitrarily small frequencies \ensuremath{\omega} for certain directions of q if the saddle point is at ${\mathit{E}}_{\mathit{F}}$, leading to a rich behavior of Im\ensuremath{\Sigma}. For an electron far from the saddle point, Im\ensuremath{\Sigma}(k,E(k))\ensuremath{\sim}[E(k)-${\mathit{E}}_{\mathit{F}}$${]}^{1.5}$, except in specific directions of k where the power is only 4/3. For an electron close to a saddle point, a linear dependence on the energy is obtained. These results are compared with numerical calculations for the Emery model of the ${\mathrm{CuO}}_{2}$ planes in high-${\mathit{T}}_{\mathit{c}}$ superconductors.