Magnetic Topological Band Theory And The Searching For Topological Materials

In the past twenty years,the field of topological state of matter in condensed matter is developing rapidly.Novel topological states have been raised constantly,and thanks to the development of first-principles calculation,realistic materials have also been discovered.In recent years,people can classify the topological states of matter in space groups systematically,and propose topological quantum chemistry theory as an efficient way to search for topological materials,based on which,the topological materials databases have been established.In this thesis,first we review the development of topological state of matter and introduce some commonly used theoretical and numerical methods.Second we review the first-principles calculation theory.In the main part of the thesis,we introduce three research works on topological state of matter.Chapter 3 is about the work on magnetic topological quantum chemistry.Topological quantum chemistry and symmetry indicator theories were originally proposed in 230 space groups for nonmagnetic materials,thus it’s a natural generalization to study them in 1651 magnetic space groups.We analyzed how the anti-unitary symmetries work on atomic orbitals in real space and wave-function representations in k-space.Based on the results of topological quantum chemistry,we derived the band representations for all magnetic space groups.By doing Smith decomposition on the band representation matrix,one can get the symmetry indicators and corresponding calculation formulas.By calculating the symmetry indicators,we searched in the materials database and found some new magnetic topological materials.Chapter 4 is about the work on unconventional materials.The basic idea of topological quantum chemistry is to decompose the band representations in realistic materials into elementary band representations.A successful decomposition indicates the band structure of the materials is topological equivalent to atomic insulators,which are topological trivial.We found even in the topologically trivial insulators,due to the special decomposition results,one can pick up some materials whose average charge center are mismatched with atomic positions.These materials are called unconventional materials.The mismatched charge center can be described by the real space invariant.By doing band representations decomposition and calculating real space invariant,we discover hundreds of unconventional materials.Chapter 5 is about the work on Weyl semimetals.Symmetry protection is unnecessary for the existence of the Weyl points,which usually appear on the generic points in the Brillouin zone.As a result the searching for Weyl semimetals is not as easy as topological insulators or Dirac semimetals.In our work,we found that for systems with S4 symmetry,one can define a topological invariant on a specific plane in the Brillouin zone,and nonzero topological invariant indicate the existence of Weyl points.Generally,this topological invariant can be calculated by using Wilson loop method.We found it also has some connection to the symmetry indicator of S4 symmetry.By calculating the symmetry indicator,we can also get the parity of the topological invariant,and then determine the existence of Weyl points.Using this method,we found some undiscovered Weyl semimetals in S4 invariant materials.