Classification,Diagnosis,and First-Principle Studies Of Topological States

Topological matters are new quantum matters beyond Landau-Ginzburg theory and Landau-Fermi liquid theory,which have topologically protected edge states.Starting from the quantum Hall effects,the theory of topological matters has developed for many decades and has become one of the basic frameworks to describe condensed matter systems.In recent years,with the developments of first-principle calculations and high-efficient topological diagnosis methods based on group representation theory,topological materials have been discovered in abundance,which serves as the solid bases for further practical applications and inter-discipline research.In this thesis,I first introduce the historical developments in the field of topological matters in Chapter 1,including the topological insulators in gapped systems and topological semimetals in gapless systems.In Chapter 2,I introduce two commonly-used methods in this field,i.e.,the first-principle calculations and group theory.Chapters 3-5 are the main contents of this thesis,where I introduce three main works during my Ph.D.career.Chapter 3is the high-throughput search of topological materials in non-magnetic space groups,while Chapter 4 is the extension of topological states from non-magnetic space groups to magnetic space groups,including the topological classifications,symmetry indicators,and k·p theory.Chapter 5 is the further extension of magnetic space groups,i.e.,the enumeration of spin-space groups.The thesis is concluded in Chapter 6.In Chapter 3,the work of high-throughput search for non-magnetic topological materials is introduced.This work is based on topological quantum chemistry and symmetry-based indicator theory,where the irreducible representations of the occupied bands are used to fast-diagnose the topological states.We develop a high-throughput algorithm and searched nearly forty thousand inorganic crystalline materials,and find more than nine thousand new topological materials.We build an online database Materiae to show the results,which include crystal structures,band structures,and topological classification information.We also develop an automatic package Sym Topo that can diagnose topological properties for materials.In Chapter 4,we introduce the work of topological classifications,diagnosis,and k·p theory in magnetic space groups.Topological classification is to find how many inequivalent gapped topological states under a given symmetry group,while topological diagnosis denotes the numerical methods to determine which topological state a given system is.We generalize the real-space recipe originally developed in non-magnetic space groups to 1421 magnetic space groups and obtain their topological classifications.We also use the magnetic symmetry indicator theory to derive indicator formulas as well as their physical meanings for magnetic space groups.If an indicator corresponds to a gapped system,we map it to magnetic topological invariants.If not,considering that we already have the classifications for all gapped states,it could only correspond to a gapless system,which is Weyl semimetals in magnetic space groups.We give minimal Weyl point configurations for these Weyl indicators.We also developed a package kdotpgenerator that can compute k·p effective Hamiltonians in magnetic space groups,and pre-compute the k·p Hamiltonians for all magnetic space groups up to the third order.In Chapter 5,we introduce the work of the enumeration of spin-space groups.Spinspace groups are used to describe the symmetries of three-dimensional periodic magnetic moment fields,which are natural generalizations of magnetic space groups.In spin-space groups,the spin-space and real-space rotations are unlocked,where for example,a fourfold rotation in real space could accompany a twofold rotation in spin space.Spin-space groups can be applied to electronic systems with weak spin-orbit couplings.However,the study of spin-space groups is in the initial stage.Our work is the first systematic enumeration of spin-space groups.We start from 230 space groups and exhaust their invariant subgroups up to 12-fold supercells and use the three-dimensional real representations of the quotient group of the invariant subgroup to construct spin-space groups.We enumerate more than 150 thousand spin-space groups within a 12-fold supercell range and develop an online database to show the results.