Magnetic Response In Topological Semimetal And (d-2)-dimensional Edge State In Topological Insulator

Topological insulator and topological semimetal are new quantum states beyond the Landau-Ginzburg theory and the Fermi liquid theory.In the LandauGinzburg theory,quantum states are classified by their symmetries.However,the topological band theory says that two states with identical symmetries may differ from each other by their topologies,and on the boundary of distinct topological phases lives topological protected surface state.Moreover,low energy excitation in topological state usually can not be described by the Fermi liquid theory.In the Fermi liquid theory,quasi-particle dynamics is only determined by a few parameters like effective mass and interaction function.While in topological state,quasi-particle dynamics is also affected by the Berry's curvature.This effect significantly change the transport property of carriers.In this thesis,I try to extend the topological band theory from the two aspects of topological classification and quasi-particle dynamics.Berry's curvature can be thought as magnetic field in momentum space.Topological state usually has a nontrivial distribution of Berry's curvature in momentum space.For example,Berry's curvature of Weyl semimetal forms magnetic monopoles.An interesting question is that: what if an external magnetic field is present,and how the quasi-particle dynamics is influenced under the interactive action of magnetic fields in both real space and momentum space? Lots of theoretical and experimental researches have been devoted into this field.Now the chiral anomaly,weak anti-localization,.etc,have been well known.However,there are still lots of experimental phenomenons have not been understood.Towards the explanation of these phenomenons,I have completed three works in this field.(1)We build a first-principle method to calculate the Zeeman's effect on band structure and apply it into some well known topological materials.By this method,we can get the impact of Zeeman's coupling on the distribution of Berry's curvature.(2)We study the stability of Dirac semimetal phase under strong magnetic field.Our calculation indicates that Dirac point becomes instable under strong magnetic field due to the high degeneracy of Landau level.It will break into to either a charge-density-wave phase or a nematic phase,depending on the bands structure.(3)We predict a chiral-magnetic-effect induced anomalous electron-phonon coupling.The resonance between Weyl Fermion and phonon is reflected directly in the reflectivity spectrum.If detected,this effect will be stronger evidence of the chiral-magnetic-effect than the transport experiment.If quasi-particle dynamics can be thought as a �first order� study on topological state,then the �zeroth order� study should be topological classification.Since the birth of field of topological insulator,classification under different symmetries is one of the most important issues.The topological insulator protected by crystalline symmetry is called topological crystalline insulator.Due to the variety of crystalline symmetries,the classification of topological crystalline insulator has not been completed up to now.In my PHD period,I have one work in this field.We study a new type of topological crystalline insulator protected by 4-fold rotation.The state is 3D,but it has a 1D topological surface state.This extends the understanding of bulk-edge correspondence.Similar states can also be protected by 2-fold and 6-fold rotations.By studying the band representation,we give a�Fu-Kane-like� formula to diagnose this new topology directly from symmetry eigenvalues.