Collective Modes Of Weyl Fermionic Gases With Repulsive S-wave Interaction

The invention of laser is undoubtedly the golden key to opening the door of the world of ultra-cold atoms,and the experimental realization of Bose-Einstein Condensate(BEC)paves the way for discovering more interesting and novel phenomena in ultra-cold physics.Based on the recent discovery of Weyl semimetal,we construct a theoretical model of Weyl fermionic gases in ultra-cold atom physics to investigate collective modes of the system.In this thesis,we study the collective behavior of the two types of Weyl fermionic system by response functions.We consider the case without the tilted term for the type-Ⅰ Weyl fermionic system.We start from the Weyl fermionic Hamiltonian with the repulsive S-wave interaction in the ultra-cold system,and calculate the spin and density susceptibility in the long wave length limit.According to the poles of the bare susceptibility,we obtain the particle-hole continuum.Considering the repulsive S-wave interaction,within the scheme of random-phase approximation,we get two decoupled poles equations of the susceptibility.We find that the collective modes depend on the repulsive S-wave interaction strength,and obtain the phase diagram with respect to the interaction strength in the type-Ⅰ Weyl fermionic system.Then we choose three typical interaction strength values g=0.218,g=0.46 and g=10 to solve the collective modes by numerical calculation method,respectively.The momentum distributions have non-spherical symmetry since the Hamiltonian model has the over-tilted term,and the Fermi surface is an open hyperboloid in the type-Ⅱ Weyl fermionic system.Because of the anisotropy in momentum space,we have to obtain the particle-hole continuum with q along the tilted momentum direction and perpendicular to the tilted momentum direction by poles equations of the bare susceptibility,respectively.We obtain the dispersion relations beyond the long wave length limit by using the numerical method to solve the poles equation in the case of q along the tilted direction.We find that there is one zero sound mode at g=0.1,g=1 as well as g=5 for the tilted parameter a=1.5,and the collective modes fall into the particle–hole continuum at some interaction strengths,such as g=0.2 and g=0.5.Finally,we construct an one dimensional infinite long chain model.The offdiagonal term of the Hamiltonian of the model is parameterized by a function f(x)defined on [0,∞),its spectrum depends only on the end values of f,i.e.,f and f(∞).The wave equation of wave function is a Dirac-type equation in the momentum representation.