Multifold degenerate fermions in electronic structures of topological materials give rise to many interesting physics properties. Among them, three-, six-, and eightfold degenerate fermions in condensed matter systems are considered important because these fermions are not allowed in high-energy physics due to restriction posed by Poincáre symmetry. Phonons are the basic emergent boson of the crystalline lattice. Moreover, topological phonons also exist in crystalline solids, like fermionic electrons, due to the crystal symmetry constraints. Two-, three-, and fourfold degenerate phonons were predicted previously. This study proposes degenerate phonons with the maximum fold, i.e., sixfold; we find them in five space groups (with numbers 218, 220, 222, 223, and 230) through a detailed symmetry analysis. We also propose a series of realistic materials with above-mentioned space group numbers that host sixfold degenerate nodal-point phonons at the high-symmetry points $H$ (or $R$) point using first-principles calculations. Finally, as examples, we investigate the surface phonon spectrum of ${\mathrm{C}}_3{\mathrm{N}}_4$, ${\mathrm{Sc}}_4{\mathrm{C}}_3$, ${\mathrm{Y}}_4{\mathrm{Sb}}_3$, and ${\mathrm{K}}_8{\mathrm{Si}}_{46}$ materials and find two obvious projected phonon surface states on (110) or (100) surfaces.

In kagome metal ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$, multiple intertwined orders are accompanied by both electronic and structural instabilities. These exotic orders have attracted much recent attention, but their origins remain elusive. The newly discovered ${\mathrm{CsTi}}_3{\mathrm{Bi}}_5$ is a Ti-based kagome metal to parallel ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$. Here, we report angle-resolved photoemission experiments and first-principles calculations on pristine and Cs-doped ${\mathrm{CsTi}}_3{\mathrm{Bi}}_5$ samples. Our results reveal that the van Hove singularity (vHS) in ${\mathrm{CsTi}}_3{\mathrm{Bi}}_5$ can be tuned in a large energy range without structural instability, different from that in ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$. As such, ${\mathrm{CsTi}}_3{\mathrm{Bi}}_5$ provides a complementary platform to disentangle and investigate the electronic instability with a tunable vHS in kagome metals.

Nodal line states in electronic systems can be classified based on the order of dispersion around a band degeneracy (i.e., linear, quadratic, or cubic) and the various slopes of crossing bands (i.e., type I, type II, critical type, and hybrid type). The concept of topology has been extended to phonons, promoting the birth of topological phonons. A series of nodal line phonons have been predicted in solid-state materials both theoretically and experimentally. However, the proposed nodal line phonons have a type-I or type-II linear phonon band dispersion. This study proves, based on first principles, that the hybrid-type linear nodal ring phonons and the higher-order quadratic nodal line (QNL) phonons coexist in one realistic solid-state material—AgZr with P4/nmm structure. Moreover, an effective Hamiltonian characterizing the hybrid nodal ring (HNR) phonons and the QNL phonons was established to explore the relevant physics. The AgZr material can be viewed as an ideal material to study the entanglement between HNR and QNL phonons.

Topological physics of phonon spectra has been attracting great interest. Using the first-principle calculations and symmetry analysis, we show the realistic ternary nitride ${\mathrm{Li}}_6{\mathrm{WN}}_4$ features ideal (nearly flat) nodal-surface and nodal-line structures in its phonon spectra. These nodal degeneracies are shaped like lanterns, and their existence is guaranteed by nonsymmorphic symmetry. The corresponding topological phonon surface state covers exactly half the surface Brillouin zone and can thereby be distinguished from those of conventional nodal-line and nodal-surface semimetals. Our study enriches the classification of topological quantum phases, predicts ideal material candidates, and provides a good platform for investigating the interaction between nodal-line and nodal-surface phonons.

In this work, two kinds of competition between different Heusler structure types are considered, one is the competition between XA and L2₁ structures based on the cubic system of full-Heusler alloys, Pd₂ YZ (Y = Co, Fe, Mn; Z = B, Al, Ga, In, Tl, Si, Ge, Sn, Pb, P, As, Sb). Most alloys prefer the L2₁ structure; that is, Pd atoms tend to occupy the a (0, 0, 0) and c (0.5, 0.5, 0.5) Wyckoff sites, the Y atom is generally located at site b (0.25, 0.25, 0.25), and the main group element Z has a preference for site d (0.75, 0.75, 0.75), meeting the well known site-preference rule. The difference between these two cubic structures in terms of their magnetic and electronic properties is illustrated further by their phonon dispersion and density-of-states curves. The second type of competition that was subjected to systematic study was the competitive mechanism between the L2₁ cubic system and its L1₀ tetragonal system. A series of potential tetragonal distortions in cubic full-Heusler alloys (Pd₂ YZ) have been predicted in this work. The valley-and-peak structure at, or in the vicinity of, the Fermi level in both spin channels is mainly attributed to the tetragonal ground states according to the density-of-states analysis. ΔE _M is defined as the difference between the most stable energy values of the cubic and tetragonal states; the larger the value, the easier the occurrence of tetragonal distortion, and the corresponding tetragonal structure is stable. Compared with the ΔE _M values of classic Mn₂-based tetragonal Heusler alloys, the ΔE _M values of most Pd₂CoZ alloys in this study indicate that they can overcome the energy barriers between cubic and tetragonal states, and possess possible tetragonal transformations. The uniform strain has also been taken into consideration to further investigate the tetragonal distortion of these alloys in detail. This work aims to provide guidance for researchers to further explore and study new magnetic functional tetragonal materials among the full-Heusler alloys.