Wieder, Kim, Rappe, and Kane (Phys. Rev. Lett. 116, 186402 (2016)) showed that eightfold-degenerate double Dirac points (DDPs) can occur in seven space groups (130, 135, 218, 220, 222, 223, 230), with space group 135 capable of hosting an intrinsic double Dirac semimetal in which the DDP is the only symmetry-enforced degeneracy at the Fermi level. The paper left three linked questions open, which remain unresolved: (1) in space group 135, noninteracting band theory forces groups of eight bands to stick together at the A point, so a band insulator requires a filling that is a multiple of 8, whereas the interacting filling bound of Watanabe, Po, Vishwanath, and Zaletel (WPVZ) permits a symmetric insulator already at filling 4 — can strong electron interactions open a gap in space group 135 that preserves all crystal symmetries and time reversal, realizing a correlated insulating state forbidden by single-particle band theory, or can an argument be given that none exists? (2) Can band-structure engineering (chemical substitution, doping, strain) place an intrinsic, spin-orbit-coupled DDP exactly at the Fermi level in a real material without residual carriers, as first-principles candidates have so far fallen short? (3) For the five space groups where eightfold irreps coexist with fourfold ones, can materials or tuning protocols be found whose band ordering yields a DDP-dominated Fermi surface? Substantial adjacent progress — interacting filling-constraint theory, quantum band insulators, and experimental eightfold-degeneracy candidates such as EuPd3S4, LaPd3S4, NaPd3O4, TaTe4, NbTe4, and TaCo2Te2 — has sharpened but not resolved any of the three cores.

Background

A three-dimensional Dirac semimetal is a crystal whose conduction and valence bands touch at isolated fourfold-degenerate Dirac points; two classes were known before this work. 'Topological' Dirac semimetals (such as Cd3As2 and Na3Bi) place Dirac points on a rotation axis via band inversion, while 'nonsymmorphic' Dirac semimetals have Dirac points pinned at Brillouin-zone (BZ) boundary momenta by crystal symmetries that include a fractional translation. A nonsymmorphic symmetry is a glide plane or screw axis, i.e. a point-group operation combined with a translation by a fraction of a lattice vector; at BZ-boundary momenta such symmetries are represented projectively, which enlarges the possible irreducible representations (irreps) of the little group (the subgroup of crystal symmetries that leave a given momentum invariant up to reciprocal-lattice vectors). Wieder, Kim, Rappe, and Kane (2016) studied the case in which a four-dimensional irrep of the little group (a '4DIR', arising from double-valued projective representations in a spin-orbit-coupled system with time reversal T) is doubled by T to give an eightfold-degenerate crossing, the double Dirac point (DDP). They showed that seven space groups (SGs) host DDPs: the tetragonal SGs 130 and 135, with a single 8DIR at the A point, and the cubic SGs 218, 220, 222, 223, 230, with 8DIRs at the R or H points, in all cases with generically linear (anisotropic in the tetragonal, isotropic in the cubic groups) dispersion, checked by projecting the tensor product of the 8DIR with its conjugate onto the vector representations. In SGs 130 and 135 the DDP is guaranteed whenever the band filling (the number of electrons per primitive cell) is an odd multiple of four: band connectivity forces groups of eight bands to 'stick together' at A, so a noninteracting band insulator is possible only at filling a multiple of 8. In SG 130 additional fourfold Dirac points generically coexist, producing electron and hole pockets, so only SG 135 can host an intrinsic double Dirac semimetal with the DDP as the sole degeneracy at the Fermi level. The same paper showed, via an explicit tight-binding model, that uniaxial compressive strain along different directions drives the DDP into topologically distinct insulating phases, and that line defects in the double Dirac semimetal can bind one-dimensional helical modes. The interaction side of the problem rests on the filling constraints of Watanabe, Po, Vishwanath, and Zaletel (WPVZ, PNAS 2015): by an Oshikawa/Lieb-Schultz-Mattis-type flux-insertion argument on flat 'Bieberbach' manifolds — compact flat spaces built from the real-space unit cell by identifying opposite faces, on which inserting magnetic flux is equivalent to twisted boundary conditions — they bounded the minimal filling of a symmetric, short-range-entangled ('sym-SRE') insulator, i.e. a gapped state with neither spontaneous symmetry breaking nor fractionalized excitations, allowing strong interactions. Their Table II gives a minimal filling of 8 for SG 130, agreeing with band theory, but of 4 for SG 135, which disagrees with the band-theory band sticking; their footnote states that filling 8n−48n-4 is prohibited noninteractively and that no interacting sym-SRE model is known at these fillings either. Later band-theory work systematized the noninteracting side: Watanabe, Po, Zaletel, and Vishwanath (PRL 2016) enumerated filling-enforced gaplessness for all 230 SGs, and Po, Watanabe, Zaletel, and Vishwanath (Sci. Adv. 2016) and Krishnan, Vishwanath, and Po (PRB 2021) introduced and analyzed 'filling-enforced quantum band insulators' — noninteracting band insulators at fillings that forbid an atomic-orbital description, realized in other nonsymmorphic groups. Bradlyn et al. (Science 2016) independently catalogued such high-degeneracy 'multifold' fermions, including the eightfold fermions of SG 223 compounds. Candidate materials identified in the source paper include Bi2AuO5 (SG 130, DDP near the Fermi level plus extra degeneracies), the Pb3O4-structure oxides Sn(PbO2)2, Pb3O4, and Mg(PbO2)2 (SG 135, DDP about 0.5 eV below the Fermi level, reachable at filling 4 by substituting odd-valence atoms such as In or Sb for Sn), and GaMo3 (SG 223).

Problem Statement

The three unresolved questions share one objective: to establish whether the double Dirac semimetal phase predicted by Wieder, Kim, Rappe, and Kane can survive, be gapped, and be realized in physical systems, i.e. to determine its interaction physics and its materials existence. (1) Interaction-driven insulating state in space group 135: in a spin-orbit-coupled, time-reversal-symmetric crystal of space group 135 at band filling 4 (an odd multiple of four), where noninteracting band theory enforces eightfold band sticking at the A point and hence forbids any band insulator, determine whether strong electron interactions can open a fully gapped insulating state that preserves all space-group symmetries and time reversal without fractionalized excitations (a symmetric short-range-entangled, or sym-SRE, state allowed by the WPVZ filling bound of 4), or alternatively prove that no such symmetry-preserving insulating state exists at filling 4, so that any interacting insulator must either break a crystal symmetry or develop topological order. In either case, identify the mechanism or the obstruction and characterize the resulting phase or the enforced gaplessness. (2) Experimental realization of an intrinsic double Dirac semimetal: identify or engineer a material in one of the seven DDP space groups (130, 135, 218, 220, 222, 223, 230), by chemical substitution, doping, or strain, in which the spin-orbit-coupled, time-reversal-symmetric eightfold DDP sits exactly at the Fermi level with no other carriers at the Fermi energy, and demonstrate its stability and expected phenomenology (e.g. linear dispersion at the DDP, strain-driven transitions between topologically distinct insulating phases, and helical modes bound to line defects). (3) DDP-dominated Fermi surfaces in the mixed space groups: for the five space groups in which eightfold irreps coexist with fourfold irreps (218, 220, 222, 223, 230), where the presence of a DDP at the Fermi level depends on band ordering, identify materials or tuning protocols whose band ordering yields a Fermi surface in which the Fermi-surface area of the pockets surrounding the eightfold degeneracy exceeds that of the pockets surrounding the coexisting fourfold degeneracies.

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