Topological Hourglass Dirac Semimetal in the Nonpolar Phase of ${\mathrm{Ag}}_2{\mathrm{BiO}}_3$

Materials with tunable charge and lattice degrees of freedom provide excellent platforms for investigating multiple phases that can be controlled via external stimuli. We show how the charge-ordered ferroelectric oxide ${\mathrm{Ag}}_2{\mathrm{BiO}}_3$, which has been realized experimentally, presents a unique exemplar of a metal-insulator transition under an external electric field. Our first-principles calculations, combined with a symmetry analysis, reveal the presence of a nearly ideal hourglass-Dirac-semimetal state in the nonpolar structure of ${\mathrm{Ag}}_2{\mathrm{BiO}}_3$. The low-energy band structure consists of two hourglasslike nodal lines located on two mutually orthogonal glide-mirror planes in the absence of spin-orbit coupling (SOC) effects. These lines cross at a common point and form an interlinked chainlike structure, which extends beyond the first Brillouin zone. Inclusion of the SOC opens a small gap in the nodal lines and results in two symmetry-enforced hourglasslike Dirac points on the ${\tilde{\mathcal{C}}}_{2y}$ screw rotation axis. Our results indicate that ${\mathrm{Ag}}_2{\mathrm{BiO}}_3$ will provide an ideal platform for exploring the ferroelectric-semiconductor to Dirac-semimetal transition by the application of an external electric field.

Figure 1

Schematic of the hourglasslike energy dispersion (a) on a glide mirror plane between two high-symmetry lines (without SOC), and (b) on a screw rotation axis between two time-reversal invariant momentum points (with SOC). The mirror eigenvalues in (a) and the rotation eigenvalues in (b) are shown. Hourglass band crossings trace a protected nodal line on the mirror plane and form a Dirac point on the screw rotation axis. (a) and (b) represent ${\mathrm{Ag}}_2{\mathrm{BiO}}_3$ without and with SOC, respectively (see text for details). (c) Crystal structure of nonpolar ${\mathrm{Ag}}_2{\mathrm{BiO}}_3$. (d) Bulk and projected (001) and (100) surface BZs. Various high symmetry points are marked.

Figure 2

(a) Band structure of $Pnna$ ${\mathrm{Ag}}_2{\mathrm{BiO}}_3$ (without SOC). The linear band crossings are evident near the Fermi level (horizontal dashed line). (b) The Fermi surface depicting the electron (cyan) and hole (violet) pockets. (c) Band structure on the $k_z=0$ plane along selected directions, which are marked by cyan lines in (d). Band crossings are linear and show a very small energy spread. (d) The momentum space distribution of hourglass nodal lines in the BZ. Red and green colors identify nodal lines on the $k_z=0$ and $k_x=\pi$ plane, respectively. Blue and orange thick lines show the high-symmetry directions with double band degeneracy of same and opposite mirror eigenvalues, respectively.

Figure 3

(a) Band structure of $Pnna$ ${\mathrm{Ag}}_2{\mathrm{BiO}}_3$ with SOC. (b) Closeup of the energy dispersion along the $XSY$ direction. Inset schematically shows the bands along the $XS$ line. (c) Energy dispersion in the vicinity of the $S$ point. Two hourglass Dirac points are seen to appear along the $XS$ direction. (d) Schematic of the hourglass Dirac semimetal state in ${\mathrm{Ag}}_2{\mathrm{BiO}}_3$ with four Dirac cones on the $k_z=0$ plane in the first BZ.

Figure 4

Topological electronic structure of ${\mathrm{Ag}}_2{\mathrm{BiO}}_3$ for (a) (100) and (b) (001) surfaces without SOC. (c) and (d) are the same as (a) and (b), but with the inclusion of SOC. Sharp yellow lines identify the surface states.