Triple nodal points characterized by their nodal-line structure in all magnetic space groups

Abstract

We analyze triply degenerate nodal points [or triple points (TPs) for short] in energy bands of crystalline solids. Specifically, we focus on spinless band structures, i.e., when spin-orbit coupling is negligible, and consider TPs formed along high-symmetry lines in the momentum space by a crossing of three bands transforming according to a 1D and a 2D irreducible corepresentation (ICR) of the little co-group. The result is a complete classification of such TPs in all magnetic space groups, including the non-symmorphic ones, according to several characteristics of the nodal-line structure at and near the TP. We show that the classification of the presently studied TPs is exhausted by 13 magnetic point groups (MPGs) that can arise as the little co-group of a high-symmetry line and which support both 1D and 2D spinless ICRs. For 10 of the identified MPGs, the TP characteristics are uniquely determined without further information; in contrast, for the 3 MPGs containing sixfold rotation symmetry, two types of TPs are possible, depending on the choice of the crossing ICRs. The classification result for each of the 13 MPGs is illustrated with first-principles calculations of a concrete material candidate.

Topological phase transitions of non-Abelian charged nodal lines in spring-mass systems

“…( 1) is symmetric and positive definite for any wave vector k. Thus the three eigenstates u 1 k , u 2 k , and u 3 k form an SO(3) orthonormal frame. Unless the degeneracies form an accidental triple point [27,75], as enforced, e.g., by the Goldstone modes [36,37], the generically stable band crossings in PT -symmetric systems are nodal lines [76] formed by only two adjacent bands. The non-Abelian frame charge for a nodal line is determined by which bands are connected by the nodal line and which band is gapped [23,24,[26][27][28]37].…”

Emerging research landscape of altermagnetism

“…The relativistic magnetic groups [51][52][53][54][55][56] consider transformations in coupled real physical space and the space of magnetic moment vectors. They have represented a primary symmetry tool for describing magnetic structures in materials' databases [55,56], and have been broadly applied in the research of equilibrium and nonequilibrium magnetic phenomena, including their modern topological variants [57][58][59][60][61][62]. The non-relativistic spin group formalism [26][27][28] is a generalization of the relativistic magnetic groups because it allows for different transformations to act simultaneously in the decoupled spin and real space.…”

“…So far we discussed the electron quasiparticles from the symmetry perspective limited to the spin-group transformations acting on the spin and momentum-dependent band structure. Additional rich quasiparticle physics, including higher-order degeneracy quasiparticles, can emerge from the analysis of spin-group transformations acting on the electron wavefunctions (spin-group representations) [16,24,62,64].…”

Phase transitions of non-Abelian charged nodal links in a spring-mass system

“…( 1) is symmetric and positive definite for any wavevector k. Thus, the three eigenstates u 1 k , u 2 k , and u 3 k form an SO(3) orthonormal frame. Unless the degeneracies form an accidental triple point [27,76], as enforced e.g. by the Goldstone modes [36,37], the generically stable band crossings in PT -symmetric systems are nodal lines [77] formed by only two adjacent bands.…”