Conversion Rules for Weyl Points and Nodal Lines in Topological Media

Abstract

According to a widely held paradigm, a pair of Weyl points with opposite chirality mutually annihilate when brought together. In contrast, we show that such a process is strictly forbidden for Weyl points related by a mirror symmetry, provided that an effective two-band description exists in terms of orbitals with opposite mirror eigenvalue. Instead, such a pair of Weyl points convert into a nodal loop inside a symmetric plane upon the collision. Similar constraints are identified for systems with multiple mirrors, facilitating previously unreported nodal-line and nodal-chain semimetals that exhibit both Fermi-arc and drumhead surface states. We further find that Weyl points in systems symmetric under a π rotation composed with time reversal are characterized by an additional integer charge that we call helicity. A pair of Weyl points with opposite chirality can annihilate only if their helicities also cancel out. We base our predictions on topological crystalline invariants derived from relative homotopy theory, and we test our predictions on simple tight-binding models. The outlined homotopy description can be directly generalized to systems with multiple bands and other choices of symmetry.

Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry

“…In general, for a two-dimensional spinless sys-tem (or a two-dimensional momentum subspace) with no inversion symmetry, we can instead use the symmetry class AI + C 2⊥ T (with the C 2⊥ -axis perpendicular to the system, here C 2z ), since C 2⊥ T similarly imposes a reality condition on the classifying space within the σ h -mirror invariant plane (σ h = C 2⊥ I) leading again to π 1 (H 1+1 AI+C 2⊥ T ) = Z. 66,67…”

Symmetry representation approach to topological invariants in C2zT -symmetric systems

“…This sequence is exact because the image of a map is the kernel of the next map, e.g., im i * p = ker j * p . It is also valid when π p (M, X) is substituted by π p (M/X) because the two homotopy groups are isomorphic [48,50].…”

“…II, we analyze the general properties of the sewing matrix for C 2z T symmetry. Our main technical tool is the exact sequences of homotopy groups that have been used to classify the space of Hamiltonians for topological insulators [46][47][48]. Using the exact sequences, in one and two dimensions, we show that the homotopy class of the sewing matrix gives the same classification of topological phases as the homotopy class of the Hamiltonian spaces.…”

Large Josephson current in Weyl nodal loop semimetals due to odd-frequency superconductivity

“…In-between a conventional metal with its two-dimensional Fermi surface and a Weyl semimetal with its 0D Fermi points, we find the Weyl and Dirac nodal loop semimetals (WNLs and DNLs), which have 1D nodal loop Fermi surfaces 1,2 . Multiple such materials have recently been both proposed [3][4][5][6][7][8][9][10][11] and experimentally observed in compounds such as PbTaSe 2 , ZrSiS, Ca 3 P 2 , and CaAgAs [12][13][14][15] . Away from the nodal loop Fermi surface the dispersion is Weyl-like, completely locking the electron momentum to the electron orbital (for DNLs) or spin (for WNLs) degrees of freedom.…”

“…While the explicitly broken spin degeneracy have so far made the experimental realization of WNLs more demanding, there already exists candidate WNLs. For example, HgCr 2 Se 4 was recently shown using ab initio calculations to be a WNL 8,9 and experimental probes have also been proposed for how to easily detect the spin polarization 10 . Moreover, spin polarization has experimentally been found in PbTaSe 2 , making it a likely WNL candidate 12 .…”