| The investigation of the effects of disorder on various semimetals attracts broad attention in recent years.In semimetal systems,the conduction band and valence band touch to form degenerate nodal points,because of the divergence of the wavelength as the Fermi energy closes to the nods,as exemplified by graphene,the effects of the disorder are drastically different from that in conventional metals with a finite Fer-mi surface.Those Fermion systems easily enter into the strong scattering limit,the usual perturbative calculation is not controllable and the important multiple scattering process should be considered.One way to tackle this problem is the renormalization group(RG)approach,in spite of the perturbative nature of the method,part of the high order multi-scattering process can be summed up by the RG approach.The obtained results are meaningful and provide a further understanding of the properties of disor-dered semimetal systems.On the other hand,the non-perturbative approaches(such as Bosonization)are frequently used to find the exact solution of the Hamiltonian.Indeed,the exact solution can be found for some special systems which possess certain types of symmetries,for example,the two dimensional Dirac Fermion in the presence of random gauge potential.In general cases,the exact analytical solution of the disorder system is hard to obtain.Another non-perturbative approach is to conduct numerical simulation,the Hamiltonian can be iteratively diagonalized by recursion method and calculate the physical quantities.The advantage of the numerical method is the inclusion of all the high order scattering process,however,it is difficult to probe the large scale physical process due to the limited computer memory.In this thesis,we use the Lanczos method to calculate self energy of single particle Green's function and analyze the quasi-particle and transport properties.In Chapter 1,we briefly introduce the concept of topological materials and the semimetal-insulator transition phenomenon.We start from the three dimensional Weyl semimetal and point out the low energy quasi-particle dispersion which is responsible for the physical properties of the system is determined by the topological charge of the Weyl nodes.The merging of the nodes of semimetal leads to semimetal-insulator tran-sition where the gapless low energy quasi-particle excitation can emerge at the quan-tum transition point.We use the two dimensional honeycomb lattice with anisotropic hopping integral to elaborate the transition process and derive the low energy effective Hamiltonian of anisotropic Weyl Fermion.Then,we present the introduction of the disorder effects on the semimetal systems.In Chapter 2,we introduce some important theoretical and numerical method which is frequently used in literature and this thesis.We calculate the self energy of the Schrodinger electron in conventional metal using the diagrammatic perturbation theo-ry.We use the linear response theory to calculate the conductivity of disorder metal and also discuss the quantum interference correction to the classical conductivity induced by different symmetry types of disorder.We derive the low energy effective model of a d dimensional system with anisotropic dispersion(pa),we present the calculation based on the mean field theory and discuss the behavior of long wavelength mode.Then,we turn to the renormalization group analysis of the system based on the derivation and solution of the RG flow equations and discuss the possible disorder induced quantum phase transition.At last,we introduce the non-perturbative Lanczos method and its way to compute the density of states and single particle Green's function.In Chapter 3,Exploiting the Lanczos method in momentum space,we determine accurately the quasiparticle properties of two dimensional(2D)semi-Dirac system in the presence of sublattice uncorrelated disorder,the system possesses anisotropic dis-persion which is linear in one direction and quadratic along with the other.Meanwhile,we also present the perturbative analytical analysis based on the self-consistent Born approximation(SCBA)and RG methods.We obtain the analytical solution of self ener-gy from the equation of SCB A and compare with the numerically exact results obtained by the Lanczos method,we find that even in the presence of weak disorder,the multi-scattering effect gives non-negligible influence on the system.The RG analysis on the tree level shows that disorder is a relevant perturbation in the system,any amount of disorder will generate a characteristic energy scale,below which the system is in the regime of diffusive metal.We calculate the single particle Green's function based on the Lanczos method and obtain the quasiparticle self energy.We reveal that the low-energy quasiparticle properties are substantially corrected by multiple scattering events and manifested by the power-law function of self-energy.Near the nodal point,the quasiparticle residue is considerably reduced and vanishes as ZE oc Er with disorder dependent exponent r.To highlight the importance of such unconventional quasiparticle residue behavior,we compute the classical diffusive conductivity via Kubo formalism,We show that the sharp change of ZE in the vicinity of the nodal point gives rise to the strong temperature dependence of classical conductivity,which can be directly tested by transport measurements in the future.In Chapter 4,we investigate the quasiparticle and transport properties of the dis-ordered Weyl semimetals.Using the RG analysis,we find the double and triple Weyl semimetal(WSM)are unstable against disorder in contrast to the single WSM where there exist disorder driven quantum phase transition.The calculation based on SCBA shows that any finite amount of disorder can generate finite density of states at the Weyl nodes and destroy the properties of WSM,and therefore there exists a characteristic en-ergy scale,below which the system is in diffusive metal phase.Then,we focus on the discussion of disordered double WSM.The perturbative calculation shows the existence of a logarithmic singularity in the correction terms of the quasiparticle properties which is similar to the disordered 2D Dirac Fermion.Then we use the Lanczos approach to further calculate the quasiparticle self energy and find the real and imaginary part of self-energy obeys a common power law behavior.The logarithmic function is just the weak disorder limit of the power law function.Based on this non-trivial self energy,we reveal that the low-energy quasiparticle properties are substantially modified near the Weyl point which is responsible for the unconventional transport behavior.Specifical-ly,the conductivity along the direction of linear dispersion possesses similar behavior of disordered 2D Dirac systems.The conductivity displays a sharp dip as the Fermi energy close to the nodal point and shows strong temperature dependence.On the other hand,we find the conductivity at the nodal point is independent of the disorder strength which is also an analogy with the universal minimum conductivity of graphene at the Dirac point.The presence of anisotropic dispersion effectively reduces the dimension of the system,and therefore both the density of states and conductivity show similarity between double WSM and graphene.And then,we see the unconventional transport behavior also exists in the triple WSM.We also discuss the behavior of the optical con-ductivity of disordered WSM using the non-trivial quasiparticle self energy.In this thesis,we investigate the quasiparticle properties and transport behavior of two types of semimetal systems with anisotropic dispersion in the presence of disorder.This theoretical research deepens the understanding of the physical properties of the semimetal materials and provides theoretical guidance for future related experimental observations and applications. |
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