Analytical study of fractionally charged solitons in a one-dimensional trimerized electron–phonon system

Abstract

The fractionally charged solitons in a trimerized electron–phonon system are studied by both analytical and numerical methods. To make those methods possible, an effective Lagrangian is derived by diagrammatic calculations. It is found that this Lagrangian is similar to that derived by the so-called derivative expansion. Both the phase and amplitude parts of the complex order parameter are included in the Lagrangian, and here we particularly focus on the nonlinear coupling between them. When the electron–phonon coupling is very weak, the aforementioned coupling is also weak, and so the soliton is almost considered to be a pure phase soliton with a constant amplitude. While, in the intermediate electron–phonon coupling case, the two parts of the order parameter are nonlinearly coupled. As a result, the soliton changes its pattern from that of a phase soliton to a strongly amplitude-deformed one. Our both methods, i.e., the analytical and numerical ones, succeed in giving such changes as gradual ones. Moreover, the coincidence of the two results is also good at a quantitative level.