Low-energy Critical Behaviors In Two-dimensional Semi-dirac And Quadratic Band-touching Semimetals

Semimetals with intermediate properties between metals and insulators have been extensively studied in condensed matter physics.Accompanying with Dirac system development,two-dimensional(2D)semi-Dirac(SD)semimetal and electronic system with a quadratic band crossing point(QBCP)have attracted intense interest and become one of the hottest topics in modern condensed matter physics.Different from Dirac materials,2D SD system's dispersion is anisotropic,namely parabolic in one direction and linear in the other.As for 2D QBCP materials,the reduced Fermi surfaces are no longer the Dirac points but replaced by discrete QBCPs that are formed by the crossings of up and down quadratic bands.Studying the low-energy critical behaviors of 2D SD and QBCP semimetals is instructive to understand their characteristics and provide helpful clues for exploring the applications of these two materials.In this thesis,we firstly study the generation of Cooper instability by attractive fermion-fermion interaction in the 2D SD semimetals.Performing the standard renor-malization group analysis shows that the Cooper theorem is violated at zero chemical potential but instead Cooper instability can be generated only if the absolute strength of fermion-fermion coupling exceeds certain critical value and transfer momentum is restricted to a confined region,which is determined by the initial conditions.Rather,the Cooper theorem would be instantly restored once a finite chemical potential is in-troduced and thus a chemical potential-tuned phase transition is expected.Additionally,impurities can also affect the Cooper instability in 2D SD systems.Subsequently,we study the effects of competition between fermion-fermion inter-actions and impurity scatterings on the low-energy physical behaviors in 2D QBCP sys-tem.After carrying out several tedious but straightforward calculations,we realize that the quadratic band touching structure is particularly robust once the fermionic couplings flow towards the Gaussian fixed point.Otherwise,it can either be stable or broken down against the impurity scattering in quantum anomalous Hall(QAH)or nematic-spin-nematic(NSN)fixed point.In addition,we find rotational and particle-hole asymmetric parameters can be remarkably increased or heavily reduced in the low-energy regime.It means asymmetric system exhibits a sensational phenomenon that transition either from rotational or particle-hole asymmetry to symmetric situation would be activated under certain restricted condition.