To this date, various topological quasiparticles in three-dimensional (3D) crystalline solids, such as nodal points [1–10], nodal lines [11–20], and nodal surfaces [21–27], have attracted widespread attention because of their unique physical properties and potential applications. As prominent examples, Dirac/Weyl nodal point materials refer to a class of solid materials that feature topology-/symmetry-protected band degeneracies around the Fermi level, such that the Dirac/Weyl equations can describe the low-energy fermionic excitations around the band crossings in high-energy region.

Recently, the searching of nodal point states has been extended to spinless phonon systems [28–45]. Phonons can be viewed as a perfect platform for realizing the nodal point states [46–58] due to their unique device applications and the advantages of whole frequency range observation. For example, Wang et al. [53] proposed a topological triangular Weyl complex composed of one double Weyl point and two single Weyl points in the phonon dispersion of three-dimensional α-SiO₂. Liu et al. [52] reported charge-four Weyl point phonons in three-dimensional realistic materials with space group numbers 195–199, and 207–214. Xie et al. [46] reported sixfold degenerate nodal-point phonons in three-dimensional materials, including C₃N₄, Sc₄C₃, Y₄Sb₃, and K₈Si₄₆. In 2021, Chen et al. [51] systematically investigated three-dimensional Dirac phonons in all space groups with inversion symmetry. Some realistic three-dimensional materials are also proposed in their work [22] to be candidate materials with Dirac point phonons. In 2022, Ding et al. [54] reported that three-dimensional BaZnO₂ has a type-III charge-two Weyl point phonon and double-helicoid phonon surface states. In the same year, Yang et al. [47] proposed the appearance of phononic nodal points with quadratic dispersion and multifold degeneracy in the three-dimensional Ta₃Sn. Experimentally, the double Weyl points in three-dimensional FeSi [56] were detected by inelastic x-ray scattering, which provided a strong driving force for the field.

However, to our knowledge, studying phononic nodal points in two dimensions is still relatively primitive. Only a handful of two-dimensional materials have been predicted to host phononic nodal points [59–61]. Hence, a highlight or summary of research in the emerging area of phononic nodal points in two dimensions covering approximately the last two-three year is highly desired. This mini-review highlights recent and vital developments in the phononic nodal points in two dimensions. The proposed phononic Dirac point and higher-order nodal point in two dimensions will be summarized, and some typical material candidates, including graphene, CrI₃ monolayer, YGaI monolayer, TiB₄ monolayer, Ti₂P monolayer, and Cu₂Si monolayer, are concluded in this mini-review. The author will also summarize these material candidates’ related topological properties and possible applications in this mini-review.

In 2020, Li et al. [61] proposed the topological phonons in graphene based on the first-principle calculation and symmetry analysis. The phonon dispersion of graphene is collected in Figure 1A, one finds that there exist four types of Dirac points (DPs), named DP1-DP4, respectively. From Figure 1A, the DP1 and DP2 locate at K and K′ high-symmetry points. The DP3 appears on Γ-M path and the DP4 appears on Γ-M surface path, respectively. Moreover, Li et al. [61] examined the topological signatures for these DPs by calculating the Berry phases of the DP1-DP4. The Berry phases of DP1-DP4 are highlighted by “+” and “-“ for π and -π, respectively. Hence, DP1-DP4 appear in pairs and are topologically nontrivial.

Moreover, two phononic band crossing points around 24 THz are obvious along Γ-K and Γ-K′ paths (see Figure 1A). These two points are not isolated and should form a closed ring (i.e., phononic nodal ring (PNR)). Li et al. [61] also investigated the edge states for the two-dimensional graphene with the help of Green’s function iteration method. The results of the edge states are collected in Figures 1C,D. The top figure of Figure 1D shows the edge states arising from the projections of DP3 phonons. The middle figure of Figure 1D shows the edge states arising from the projections of PNR phonons, and the bottom figure of Figure 1D shows the edge states arising from the projections of DP1 phonons. The appearance of the phononic nodal points is essential for graphenes, providing an excellent direction to investigate the interesting topological phonons in two dimensions. Moreover, the predictions of the phononic nodal point in two dimensions can pave a new way to study the related topological properties, such as destructive interference and quantum (anomalous/spin) Hall-like topological effects.

Interestingly, Jin, Wang, and Xu [60] proposed a method to generate Dirac phonon states with a quantized valley Berry phase in a two-dimensional hexagonal lattice. Using this method, they [60] proved that candidates with C₃ symmetry at corners of the hexagonal Brillouin zone could host robust valley Dirac phonons. With the help of first-principle calculations, the phonon dispersions of two typical examples, i.e., CrI₃ monolayer and YGaI monolayer, are calculated by Jin, Wang, and Xu [60]. The results are collected in Figures 1E, F, in which multiple Dirac points can be observed at K and K′ high-symmetry points. Note that the CrI₃ monolayer is a magnetic semiconductor with a Curie temperature of 42.8 K. In 2018, Jiang et al. [62] proposed that the magnetism of two-dimensional CrI₃ can be controlled by electrostatic doping. Hence, Jin, Wang, and Xu [60] paved a new way to study topological phonons in two-dimensional magnetic materials. Similar to the DP1 and DP2 located at K and K′ high-symmetry points in graphene, the Dirac point phonons appear at K and K′ in the CrI₃ monolayer and YGaI monolayer. As shown in Figures 1G,H, the quantized Berry phase of π and -π are verified at K′ and K valleys by calculating the Berry curvature distributions. The edge states for the CrI₃ monolayer are visibly terminated at the projections of the Dirac points at K and K'. Their work not only provides a broad application of topological phonons in two dimensions but also extends the aspect of valley physics.

In 2022, Yu et al. [59] searched through 80-layer groups and found that the phononic higher-order nodal point can appear in two dimensions. The appearance of the phononic higher-order nodal points is protected by rotation (except the twofold one) and time-reversal symmetries. They also stated that the highest order of momentum in a two-dimensional system is the second order, named quadratic order. From Figure 2A, Yu et al. [59] pointed out that the phononic higher-order nodal points can appear in layer groups of 49–80. The high-symmetry points where the phononic higher-order nodal points appear, the PG, and the Generators are also exhibited in Figure 2A.

As shown in Figures 2B–D, they [59] proposed some two-dimensional material candidates, including TiB₄, Ti₂P, and Cu₂Si monolayers, hosting the quadratic nodal point phonons at high-symmetry points. The structural models and the calculated phonon dispersions for these three monolayers are collected in Figures 2B–D, respectively. A   Z − v a l u e d   topolo g i c a l invariant can show as N = 1 4 π i ∮ C T r σ z H − 1 ( k ) ∇ k H ( k ) ⋅ d k . Yu et al. [59] also stated that the two-dimensional quadratic nodal point could be characterized by an integer topological invariant, reflecting the appearance of the edge states. The visible edge states arising from the projections of the quadratic nodal points for these two-dimensional material candidates are collected in Figures 2B–D. Note that the edge states are very clean, benefitting the experimental detections.

In this mini-review, the author summarized the recent advances in the phononic nodal point in two dimensions covering approximately the last two-three years. Typical two-dimensional material candidates, such as the CrI₃ monolayer, YGaI monolayer, TiB₄ monolayer, Ti₂P monolayer, and Cu₂Si monolayer, are concluded in this mini-review. Their topological signatures and possible properties are also summarized. This mini-review is hoped to help study phononic nodal point phonons in two dimensions.