Effects of Umklapp Scattering on Electronic States in One Dimension

TL;DR

This paper investigates how Umklapp scattering influences electronic states in one-dimensional systems, revealing a phase diagram with coexisting ordered states and pairing phenomena at zero temperature.

Contribution

It introduces a model treating normal and Umklapp scattering independently and analyzes their effects on the density of states and phase stability in 1D.

Findings

Umklapp scattering causes splitting of the density of states.

Strong Umklapp scattering leads to an instability towards a '$G/2$-pairing' state.

The phase diagram shows coexistence of ordered states and pairing at various fillings.

Abstract

The effects of Umklapp scattering on electronic states are studied in one spatial dimension at absolute zero. The model is basically the Hubbard model, where parameters characterizing the normal ($U$) and Umklapp ($V$) scattering are treated independently. The density of states is calculated in the t-matrix approximation by taking only the forward and Umklapp scattering into account. It is found that the Umklapp scattering causes the global splitting of the density of states. In the presence of sufficiently strong Umklapp scattering, a pole in the t-matrix appears in the upper half plane, signalling an instability towards the '$G/2-$pairing' ordered state ($G$ is the reciprocal lattice vector), whose consequences are studied in the mean field approximation. It turns out that this ordered state coexists with spin-density-wave state and also brings about Cooper-pairs. A phase diagram is determined in the plane of $V$ and electron filling $n$.