Theoretical Studies On Local Evolutions Of Topological Semi-metals

Topological band theory is among the most successful fields of condensed matter physics in the past two decades.In traditional band theory descriptions,electrons in solid state exist as quasi-particles,which means though electrons inter-act with environment,they can be handled as single particles with re-normalized mass and enery-momentum relations in mean field level.The quasi-particles are described by wave-functions in quantum mechanics,while if we deal with their quasi-classical behaviours,such as transports,spectra and thermal-dynamically statistics,the quantum natures are usually ignored.Nevertheless,wave-functions play central roles in topological band theory.The wave-functions of valence band electrons in insulating states can hold topological quantum numbers,and non-zero topological numbers result in topological protected conducting edge states that have been observed by different kinds of modern experiments.Topological numbers are only well-defined in fully gapped systems until peo-ple find new phases of quantum matter which are referred as topological protected semi-metals.In such systems,conduction bands and valence bands touch at some discrete points or continuous lines in the Brillouin zone,and the degeneracies are protected by topologies instead of symmetries.These new phases include Weyl semi-metals,Dirac semi-metals,type-II Weyl semi-metals,nodal-line semi-metals,nodal-link semi-metals,nodal-chain semi-metals,nodal-knot semi-metals and so on.For example,Weyl semi-metals can hold Z₂chiral charges around each of the band touching points.locally,non-zero chiral charges can not abruptly change and thus the Weyl nodes are stable.However,the fully topological classicizations of semi-metals are still open issues.We focus on the local behaviors of nodal-points and nodal-lines in topolog-ical protected semi-metals which are nether protected by symmetries nor well-described by topological band theories.Our results are summarized as below,?1?We study local evolutions of nodal-points in two-dimensional lattice model with chiral symmetry.We find three kinds of trajectories of the nodal point under symmetry-preserving perturbations,which means the point may move in the Brillouin zone,vanish and gap out the system,or split into multi-points in the Brillouin zone without meeting other nodal-points.We define stableness as the point can only move but cannot vanish or split under external perturbations,and find out that only points with linear dispersion in both directions are stable.Our methods exhaust all the possible cases as long as the Hamiltonians are convergent in Taylor expansions around the nodal-points.The results may be generalized to systems with other kinds of symmetry,multi-band systems,and BdG formalisms of superconductors or superfluids.?2?We generalize the methods to study evolutions of topological nodal line semi-metals.By introducing space-time inversion symmetry,we construct Bloch Hamiltonians with different kinds of nodal lines.When two nodal loops touching at a point,we get a nodal chain locating at the touching point.The nodal chain phase can be viewed as intermediate state of nodal link and separate loops semi-metals.Furthermore,we deduce some sufficient conditions for the nodal link transition which lead to observable physical consequences such as energy level shifting of Landau levels when subjected to external perpendicular magnetic field.Exotic nodal line structures attract more interests in superconducting systems than in electron systems which we will study in the future.