Hourglass phonons jointly protected by symmorphic and nonsymmorphic symmetries

Hourglass dispersion is generally believed to be solely protected by nonsymmorphic symmetries, because these symmetries can introduce high-dimensional projective representations. Here, based on symmetry arguments, we propose that the hourglass dispersion can be jointly protected by symmorphic and nonsymmorphic symmetries, instead of the conventional nonsymmorphic symmetry. Moreover, using first-principles calculations, we realize our proposal in the phonon spectra of realistic materials that share an antiperovskite structure with space group $P4$/nmm. Importantly, the neck points of these hourglass dispersions trace out two nodal rings tangential to four nodal lines, forming a unique hourglass nodal cage in the bulk Brillouin zone. The Berry phase analysis reveals the nontrivial topology of these nodal rings and nodal lines. Furthermore, nontrivial surface states and isofrequency surface arcs are visible, facilitating their experimental confirmation of such exotic quasiparticles. work not only offers different insights into the hourglass dispersion, but also expands aspects for studying promising topological quasiparticles in condensed-matter systems.

Figure 1

Schematic of hourglass dispersions along the (a) nonsymmorphic or (b) symmorphic symmetry-invariant path, in which $\tilde{\mathcal{G}}$ and $G$ represent a nonsymmorphic operation and a symmorphic operation, respectively. The orange and purple bands correspond to the positive and negative eigenvalues of $\tilde{\mathcal{G}}$ or $G$, respectively. (c) Bulk BZ and the projected surface BZs of space group $P4$/nmm. The black box denotes the bulk BZ, and the orange square and blue rectangle represent the (001) and (010) surface BZs, respectively. (d) Schematic of the hourglass nodal line in the $k_x=0$ plane.

Figure 2

(a) Crystal structure of ${\mathrm{SrPt}}_3\mathrm{P}$, in which the green, blue, and orange spheres represent strontium, platinum, and phosphorus atoms, respectively. The Pt(1) atoms lie on the basal plane of the ${\mathrm{PPt}}_6$ octahedra, while the Pt(2) atoms are located at its apices. (b) Phonon spectra (left panel) and DOS (right panel) of ${\mathrm{SrPt}}_3\mathrm{P}$. The contributions of Sr, Pt, and P for phonon DOS are respectively marked with green, blue, and orange curves. Hourglass dispersions related to the (c) hourglass nodal lines and (d) hourglass nodal rings. The path in (c) is terminated by the arbitrarily selected points $\mathrm{\Lambda } (0,0,0.2)$ on $\mathrm{\Gamma }\text{−}Z$ and $W (0,0.5,0.4)$ on $X\text{−}R$, while the path in (d) connects the $\mathrm{\Gamma }$ point and the arbitrarily chosen point $Y (0.3,0.5,0)$ on $X\text{−}M$.

Figure 3

(a) Schematic of the hourglass dispersions in the $k_z=0$ ($k_z=\pi$) plane, in which the orange lines denote the phonon branches with positive eigenvalues, while the purple lines represent the phonon branches with negative eigenvalues. $k_c$ is the neck point, and the labels in parentheses correspond to the hourglass dispersions in the $k_z=\pi$ plane. (b) Schematic of the hourglass nodal ring in the $k_z=0$ ($k_z=\pi$) plane, in which the point with the same ${\tilde{M}}_z$ eigenvalues is marked with the green dot, while the boundary of the $k_z=0$ ($k_z=\pi$) plane with opposite ${\tilde{M}}_z$ eigenvalues is denoted as blue, as well as the high-symmetry points. The red loop represents the hourglass nodal ring. (c) Hourglass nodal cage composed of two hourglass nodal rings and fourfold-symmetry hourglass nodal lines. (d) Variation of the Berry phase when the closed path goes across the hourglass nodal ring in the $k_z=0$ plane.

Figure 4

Phonon surface states and isofrequency surface arcs of ${\mathrm{SrPt}}_3\mathrm{P}$. Surface states along (a) $\tilde{X}\text{−}\tilde{\mathrm{\Gamma }}\text{−}\tilde{M}$ on the (001) plane and (b) $\widehat{\mathrm{\Gamma }}\text{−}\widehat{A}\text{−}\widehat{M}$ on the (100) plane, in which the red dots are the hourglass nodal points on the nodal ring in the $k_z=\pi$ plane and nodal point on the nodal line. Projected isofrequency surface arcs (c) on the (001) plane and (d) on the (100) plane, in which the high-symmetry points are marked with orange dots and blue dots, respectively. The frequencies in (c) and (d) correspond to the orange and blue dashed lines in (a) and (b), respectively. The red dots in (d) denote the nodal points with a frequency of 8.9 THz in the nodal lines.