Exact solutions for electron pairing models with spin-orbit interactions and Zeeman coupling

Abstract

Although a number of methods with appropriate approximations, such as mean-field theory, local density approximation, and tight-binding method have been well developed and widely used in solid state physics, they possess strong limitations, and thus physicists never stop trying to find methods that could rigorously solve the models of condensed matter systems. This thesis presents several new exact solutions for electron pairing models with spin-orbit interactions and Zeeman coupling, which have not been studied before. First, a type of electron pairing model with spin-orbit interactions or Zeeman coupling is solved exactly in the framework of the Richardson’s previous work for 2D cases. Based on the exact solutions for the electron pairing model with spin-orbit interactions, it is shown rigorously that the pairing symmetry is of the p+ip wave and the ground state possesses time-reversal symmetry, which are expected by the meanfield theory. And the difference is that such peroration from our framework is valid for any strength of the pairing interactions. Intriguingly, how Majorana fermions can emerge is also elaborated in a ribbon system as well. Condensation energy and critical magnetic field are calculated in two systems with the exact solutions, and compared with the relevant results achieved by the mean-field theory, the differences between our results and the mean-field theory show the significance of the work for exact solutions. Secondly, we generalize our scenario to 3D cases. Several remarks of the 3D case are given following the significant results from the 2D cases. And an unconventional type of Fulde-Ferrel-Larkin-Ovchinnikov ground state is revealed exactly, in which the center-of-mass momentum of the fermion pair is proportional to the Zeeman field. As a by-product, a similar Fulde-Ferrel-Larkin-Ovchinnikov state is also disclosed when the magnetic field is in the same plane of k for 2D case. In addition, applying the transformative Richardson ansatz in bosonic system, we elaborate on the drifting effect of the Zeeman field on the spin-orbit-coupled Bose-Einstein condensed matter as well. Finally, we discuss the application of the exact solutions in quantum entanglement quantification. The entanglement monotone concurrence is calculated with exact solutions for two models. It is found to be a smooth function of pairing interactions, as expected.