Recently intensive research effort has been devoted to finding novel topological states of phonons, including the quantum anomalous Hall-like [1–15] and quantum spin Hall-like states [16–21]. These new quantum states of phonons are characterized by topologically protected, gapless boundary modes within the bulk gap, which are useful for various applications like high-efficiency phononic circuits/diodes and offer new paradigms for future phononics [12–14]. However, experimental realization of two-dimensional (2D) topological states is quite challenging for phonon systems. Specifically, the quantum anomalous Hall-like states require breaking time reversal symmetry of phonons, which remains experimentally illusive. The quantum spin Hall-like states rely on the pseudospin degeneracy protected by crystalline symmetries that typically get broken at the one-dimensional (1D) edges [21]. In contrast, crystalline symmetries of three-dimensional (3D) systems can preserve simultaneously in the bulk and on the surface, enabling the topological protection. Importantly, most solid materials are crystalized in 3D lattices. Nevertheless, despite a few preliminary works on phononic topological semimetals [22–25], 3D phononic topological insulators (TIs) have rarely been reported before, as far as we know. This is possibly because the spin (Kramers) degeneracy and spin-orbit coupling (SOC), which are essential to TIs, are natively missing for phonons. On the other hand, phonons are elementary excitations of lattice vibrations with zero charge and spin. The lacking of tunable degrees of freedom considerably limits their device applications. In this context, it is of critical importance to develop new quantum degrees of freedom for phonons. In light of the great success of spintronics, future research of phononics would be greatly enriched if one could establish any correlations between phonon pseudospins and electron spins [26].

In this Article, we provided a guiding principle to design 3D phononic TIs as well as topological semimetals by utilizing crystalline symmetry-protected pseudospins characterized by Kramers-like degeneracy, quantized pseudoangular momenta, and nonzero Berry curvature. Remarkably, we revealed pseudospin counterparts of the intrinsic and Rashba-Dresselhaus SOC, namely, the pseudo-SOC of phonons, which builds a duality between phonon pseudospins and electron spins. The duality feature enables exploring the physics and applications of phonon pseudospins by borrowing ideas from spintronics, thus opening new opportunities for “pseudospin phononics”.

An essential requirement of TIs is band degeneracies at no less than two high symmetry momenta (HSM) in the boundary Brillouin zone (BZ) [27–29]. The requirement is satisfied for electrons with spin degeneracies protected by time reversal symmetry. However, phonons do not have real spins, invoking different strategies for building phononic TIs. Naturally one could apply crystalline symmetries that are prevalent in solid materials to realize Kramers-like degeneracies. Such symmetries should also be preserved when projected onto the surface. However, for 2D spinless cases, no such kind of crystalline symmetry has higher than 1D irreducible representations at more than one HSM in the 1D edge BZ, implying that 2D phononic TIs protected by crystalline symmetries are forbidden [21]. The constraint is released for a variety of 3D lattices, where multiple band degenerate HSM can exist in the 2D surface BZ [28–31]. Thus the construction of 3D phononic TIs is feasible in principle.

The Hamiltonian of phonons resembles a tight-binding Hamiltonian of spinless electrons with fixed p_{x,y,z} orbitals [13]. The symmetry representation of phonons is {\Gamma }_{\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{n}}={\Gamma }^{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{v}.}\otimes {\Gamma }_{\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}}, where {\Gamma }^{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{v}.} is the equivalence representation of atomic sites and {\Gamma }_{\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}} is the representation of a 3D polar vector [32]. Here we will not thoroughly discuss all possible 3D crystalline symmetries, but focus on C_{nv}\hspace{0.17em}\hspace{0.17em}(n=\mathrm{3,4},\mathrm{6}) symmetries that show interesting topological physics for electrons [30, 31]. Take C_{\mathrm{6}v} lattices as an example. {\Gamma }^{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{v}.} is a N dimensional representation, where N is the number of atomic sites in a unit cell. When all the atomic sites are inequivalent, {\Gamma }^{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{v}.} is decomposed into N 1D irreducible representations A_{\mathrm{1}}. N\ge \mathrm{2} is necessary to build a phononic TI that requires at least four bands. For simplicity, we choose N=\mathrm{2} as displayed in Figure 1(a), which gives {\Gamma }^{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{v}.}=\mathrm{2}{A}_{\mathrm{1}}. In the momentum space, the high-symmetry lines \Gamma-A and K-H in the bulk BZ are projected onto the HSM \overline{\Gamma } and \overline{K} in the surface BZ [Figure 1(a)], which have C_{\mathrm{6}v} and C_{\mathrm{3}v} symmetries, respectively. Under C_{\mathrm{6}(\mathrm{3})v} symmetry, {\Gamma }_{\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}}=A_{\mathrm{1}}\oplus E, where A_{\mathrm{1}} and E are 1D and 2D irreducible representations for basis functions of p_z and p_x\pm i{p}_y, respectively. Therefore, {\Gamma }_{\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{n}}=\mathrm{2}{A}_{\mathrm{1}}\oplus \mathrm{2}E, implying that p_x\pm i{p}_y (p_z) are doubly degenerate (nondegenerate) along the high-symmetry lines. Note that frequencies of in-plane vibrations (p_x\pm i{p}_y) are typically higher than those of out-of-plane ones (p_z), leading to weak hybridization between the doublet and singlet states. We thus focus on in-plane vibrations [33], for which a {\mathbb{Z}}_{\mathrm{2}} classification of band topology is permitted [30].

Considering that the double degeneracy of p_x\pm i{p}_y resembles the Kramers degeneracy of spins, we introduce a pseudospin index to label the Kramers-like states. There exist two types of phonon modes, including in-phase (I) and out-of-phase (O) vibrations between the two atomic sites A and B. Their pseudospin states [Figure 1(b)] are defined in coordinates of (x_A,y_A,x_B,y_B{)}^T as

where {\epsilon }_{\pm }=(\mathrm{1},\pm i)/\sqrt{\mathrm{2}} and the vibrational magnitude of each atomic site is determined by {\theta }_{\mathbf{k}}\in [\mathrm{0},\pi /\mathrm{2}] that is \mathbf{k}-dependent. They are orthonormal and form a complete basis of in-plane vibrations.

The pseudospin states are featured by well-defined, quantized pseudoangular momenta about z-axis for \mathbf{k} along the high-symmetry lines due to the C_{\mathrm{6}(\mathrm{3})} rotational symmetry. The pseudoangular momentum operator is expressed as J_{\mathrm{p}\mathrm{h}}={\sigma }_z{s}_z [33], where {\sigma }_z and s_z are Pauli matrices with {\sigma }_z=\pm \mathrm{1} and s_z=\pm \mathrm{1} refer to pseudospin up (down) and I (O) vibrational modes, respectively. The phonon pseudoangular momentum j_{\mathrm{p}\mathrm{h}} is composed of two parts, including a local part determined by the on-site orbital and a nonlocal part contributed by the inter-site Bloch phase change [34]. Herein the nonlocal part is zero and j_{\mathrm{p}\mathrm{h}} is fully determined by the local part, giving j_{\mathrm{p}\mathrm{h}}=\mathrm{1} for |{\mathrm{I}}_{\uparrow }\rangle and |{\mathrm{O}}_{\downarrow }\rangle and j_{\mathrm{p}\mathrm{h}}=-\mathrm{1} for |{\mathrm{I}}_{\downarrow }\rangle and |{\mathrm{O}}_{\uparrow }\rangle, as depicted in Figure 1(b).

The symmetry-adapted k·p method was applied to derive effective Hamiltonians of phonons near HSM by using the basis set \{|{\mathrm{I}}_{\uparrow }\rangle ,|{\mathrm{O}}_{\uparrow }\rangle ,|{\mathrm{I}}_{\downarrow }\rangle ,|{\mathrm{O}}_{\downarrow }\rangle \} [33]. Specifically, the Bloch Hamiltonian matrix H(\mathbf{k}) was first expanded by the lowest orders of \mathbf{k} (referenced to HSM). Then, by writing down the corresponding matrices of symmetry operators and requiring H(\mathbf{k}) commute with them, some matrix elements of H(\mathbf{k}) are enforced to be zero. The remaining nonzero part gives the effective Hamiltonian. The HSM are classified into type-I (K/K^{\prime } and H/H^{\prime }) and type-II (\Gamma and A), which have C_{\mathrm{3}v} and C_{\mathrm{6}v} symmetries, respectively, and will be discussed separately. Importantly, the conservation of pseudoangular momentum requires that basis states of the same j_{\mathrm{p}\mathrm{h}} (e.g., |{\mathrm{I}}_{\uparrow }\rangle and |{\mathrm{O}}_{\downarrow }\rangle) are coupled only by terms like k_z, k_x^{\mathrm{2}}+k_y^{\mathrm{2}}, and k_z^{\mathrm{2}}, while basis states of different j_{\mathrm{p}\mathrm{h}} (e.g., |{\mathrm{I}}_{\uparrow }\rangle and |{\mathrm{O}}_{\uparrow }\rangle) are coupled only by terms like k_{\pm } and k_{\pm }^{\mathrm{2}}. The effective Hamiltonian near the type-I HSM is written as H=H_{\mathrm{0}}+H_I+H_{RD}, where H_{\mathrm{0}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(M_{\mathbf{k}},-M_{\mathbf{k}},M_{\mathbf{k}},-M_{\mathbf{k}}),

M_{\mathbf{k}}=M-B_{\mathrm{1}}(k_x^{\mathrm{2}}+k_y^{\mathrm{2}})-B_{\mathrm{2}}{k}_z^{\mathrm{2}}, k_{\pm }=k_x\pm i{k}_y. The curvature parameters B_{\mathrm{1}} and B_{\mathrm{2}} typically have the same sign.

Remarkably, this effective Hamiltonian resembles the 3D Bernevig-Hughes-Zhang (BHZ) model with broken inversion symmetry for electrons. H_{\mathrm{0}}+H_I is exactly the same as the typical 3D BHZ model [35–37], where H_I is attributed to the intrinsic SOC. For electrons, H_{RD} arises in conditions of broken inversion symmetry, which includes \pm i{C}_{\mathrm{2}}{k}_{\mp } and \mp i{C}_{\mathrm{1}}{k}_{\pm } terms, corresponding to the Rashba (k_y{\sigma }_x-k_x{\sigma }_y) and Dresselhaus (k_y{\sigma }_x+k_x{\sigma }_y) SOC, respectively. In analogy, H_I and H_{RD} are called the intrinsic and Rashba-Dresselhaus pseudo-SOC for phonon pseudospins. In this sense, phonons and electrons share essentially the same Hamiltonian, though their underlying physics is distinctly different. This implies a “duality” between phonon pseudospins and electron spins. Therefore, we can study the topological and quantum physics of phonon pseudospins by borrowing ideas from spintronics. This key result could lay the foundation for an emerging field of “pseudospin phononics”.

Then we will discuss phononic topological properties by the effective Hamiltonian. When excluding H_{RD} and selecting k_z=\mathrm{0}, H reduces to the 2D BHZ model, which gives quantized pseudospin-resolved Chern numbers {\mathcal{C}}_{\uparrow (\downarrow )} [38]. As type-I HSM exist in pairs, {\mathcal{C}}_{\uparrow (\downarrow )}=\pm \mathrm{2} when a band inversion occurs (i.e., M/B_{\mathrm{1}}>\mathrm{0}). The sum of {\mathcal{C}}^s=({\mathcal{C}}_{\uparrow }-{\mathcal{C}}_{\downarrow })/\mathrm{4} contributed by all HSM mod 2 gives a topological invariant {\mathbb{Z}}_{\mathrm{2}}. The 3D phononic TI phase is characterized by {\mathbb{Z}}_{\mathrm{2}}=\mathrm{1}. The inclusion of H_{RD} introduces intraband coupling between opposite pseudospins, which removes the pseudospin degeneracy except at the HSM. Then {\mathcal{C}}_{\uparrow (\downarrow )} gets ill defined, but the {\mathbb{Z}}_{\mathrm{2}} topological classification remains valid [30]. The {\mathbb{Z}}_{\mathrm{2}} topological invariant will not be affected by H_{RD} as far as the bulk band gap keeps open when adiabatically turning on H_{RD}.

To demonstrate the nontrivial topological states, we explicitly studied lattice vibrations in a C_{\mathrm{6}v} lattice [Figure 1(a)]. The interatomic interactions between the nearest and next-nearest neighbors were described by longitudinal and transverse force constants, as done previously [34]. The out-of-plane vibrations typically have minor influence on topological properties of in-plane vibrations [33], which are thus neglected for simplicity. We systematically searched the whole space of interatomic coupling parameters and found that the required band inversions can be obtained by a wide range of coupling parameters. Details of calculation methods and parameters related to the following discussions were described in Supplemental Material [33].

Figure 1(c) presents phonon dispersion curves with a band inversion between I and O vibrational modes at K. This band inversion leads to a nontrivial band topology {\mathbb{Z}}_{\mathrm{2}}=\mathrm{1}, as confirmed by our calculations of hybrid Wannier centers [39, 40] that display partner switching between Kramers-like pairs [33]. Moreover, there is a frequency gap between the two kinds of bands. The system is thus a 3D phononic TI. A hallmark of phononic TIs is the existence of gapless surface state within the bulk gap, which is topologically protected when the corresponding symmetry is preserved. On the (001) surface where the C_{\mathrm{6}v} symmetry preserves, we indeed observed a single pair of gapless Dirac-cone-shaped surface bands located near \overline{K} and \overline{K^{\prime }} [Figure 1(d)], as warranted by the bulk-boundary correspondence [41].

Figure 1(e) displays schematic pseudospin textures of bulk bands in the k_z=\mathrm{0} plane. There would be no net pseudospin polarization if excluding the Rashba-Dresselhause pseudo-SOC interaction H_{RD}. Interestingly, when including H_{RD}, Rashba-like and Dresshause-like pseudospin textures evolve in the O and I bands, respectively. We further considered the topological surface states (TSSs) near the Dirac point, which are described by the effective Hamiltonian (referenced to the Dirac frequency) H_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}=v_D(k_x{\sigma }_x{\tau }_z+k_y{\sigma }_y), where {\tau }_z=\pm \mathrm{1} refers to K (K^{\prime }) valley; v_D is the group velocity at the Dirac point. Noticeably, H_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}} of each valley has the same form as for TSSs of electrons [35], whose pseudospin textures are schematically displayed in Figure 3(a). By adiabatically varying \mathbf{k} along the loop enclosing K (K^{\prime }), the pseudospin vectors wind \pm \mathrm{1} times, giving quantized Berry phases of \pm \pi. The similarity between phonon pseudospins and electron spins is thus well demonstrated for both bulk and surface bands.

Type-II phononic TIs are characterized by the existence of band inversions at type-II HSM (\Gamma or A). Importantly, due to the C_{\mathrm{6}} rotation symmetry, all the linear terms of k_{\pm } are forbidden in the effective Hamiltonian near type-II HSM, which is expressed as follows [33]:

A new kind of intrinsic and Rashba-Dresselhaus pseudo-SOC (H_I^{\prime } and H_{RD}^{\prime }) is thus introduced.

Figure 2(a) presents dispersion curves of a type-II phononic TI, which is characterized by a band inversion at A, a finite frequency gap, and a nontrivial band topology {\mathbb{Z}}_{\mathrm{2}}=\mathrm{1} verified by the calculations of hybrid Wannier centers [33]. We also calculated the surface states of the (001) termination [Figure 2(b)], which shows a quadratic band crossing at \overline{\Gamma }, in contrast to a pair of linear band crossings at type-I HSM. The type-II TSSs are described by the effective Hamiltonian (referenced to the degenerate frequency) H_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^{\prime }=D[(k_x^{\mathrm{2}}-k_y^{\mathrm{2}}){\sigma }_x-\mathrm{2}{k}_x{k}_y{\sigma }_y], where D is a coefficient. Pseudospin textures of bulk bands [Figure 2(c)] are neither typical Rashba-like nor typical Dresselhaus-like, but display new pseudospin-momentum locked features. Pseudospin textures of type-II TSSs are significantly different from those of type-I TSSs [Figure 3(a)], which are characterized by winding numbers of \pm \mathrm{2} and quantized Berry phases of \pm \mathrm{2}\pi.

One prominent feature caused by the pseudo-SOC H_{RD} or H_{RD}^{\prime } is that the pseudospin and momentum of phonons are locked, which plays an important role in determining transport properties. Moreover, type-I and type-II bulk/surface bands are featured by different kinds of pseudospin-momentum locking, leading to distinct transport behaviors. To demonstrate the concept, we simulated phonon transport of TSSs in tunneling junctions [33], where a tunneling barrier is introduced into the gate region as schematically displayed in Figure 3(a). The height of tunneling barrier, in principle, could be controlled by using a piezoelectric gate that changes the on-site potential, strain, or interatomic coupling of surface atoms [26]. Figure 3(b) shows phonon transmissions as a function of incident angle \theta for type-I and type-II TSSs. Normal surface states with a linear band dispersion and no pseudospin-momentum locking were also studied for comparison. Noticeably, there exists resonant phonon transport (transmission equal to 1) along some particular directions, which satisfy the constructive interference condition of Fabry-Pérot resonances, q_{x}L=n\pi, where q_x is the x-component of wavevector in the gate region (Supplemental Figure S3), L is the barrier width, and n is an integer number. Moreover, phonon transmission of normal surface states oscillates with varying L, which is also expected by the Fabry-Pérot physics.

However, substantially different results are found for type-I and type-II TSSs. Take \theta =\mathrm{0} for example. \theta =\mathrm{0} corresponds to transport along x-axis. Transport of type-I TSSs keeps ballistic, while phonon transmission of type-II TSSs decays exponentially with increasing L [Figure 3(c)]. These unconventional transport behaviors are insensitive to material details, indicating a topological origin. Importantly, these physical effects can be well understood by pseudospin physics. Specifically, for type-I (type-II) TSSs, the forward-moving phonon modes have the same (opposite) pseudospins between the upper and lower Dirac cones, which correspond to transport channels of the source/probe and gate, respectively [Figure 3(a)]. Because of the perfectly matched (mismatched) pseudospins, phonons are able to transport across the barrier with no (full) backscattering. These quantum transport phenomena are inherently related to the quantized Berry phase \pi (\mathrm{2}\pi) of type-I (type-II) TSSs, which induces destructive (constructive) interferences between incident and backscattered waves.

Our findings suggest some potential applications of TSSs. For instance, the excellent transport ability of type-I TSSs can be utilized for low-dissipation phononic devices. The strongly angle-dependent transmission of type-II TSSs, which are confined to transport along some specific directions, can be used to design directional phononic antenna. Moreover, type-II TSSs are promising for building efficient phononic transistors, because their phonon conduction in the tunneling junction can be switched off (on) by a finite (zero) barrier.

In addition to phononic TIs, topological band inversions can introduce other novel topological phases, including topological nodal-ring semimetals and topological Weyl semimetals, which are collectively called phononic topological semimetals. The essential physics is illustrated in Figure 4(a). Let us start from H=H_{\mathrm{0}} and take two pairs of bands inverted at K for example. Generally, the pseudospin degeneracy is split by H_{RD}, and a full band gap is induced by H_I, leading to phononic TIs. However, phononic topological semimetals would emerge if the opening of band gap was forbidden by symmetry. This is possible here, providing that band splittings of H_{RD} have opposite signs between I and O vibrational modes (i.e., C_{\mathrm{1}}{C}_{\mathrm{2}}>\mathrm{0}). Then, when in the presence of mirror symmetry M_z (with the same atoms at A and B sites), the crossing rings in the k_z=\mathrm{0} plane are protected to be gapless, introducing topological nodal-ring semimetals. In contrast, under broken M_z, the nodal rings are gapped out except for some gapless points that are protected to exist in the \Gamma-K line by mirror symmetry M_x. These gapless points are Weyl points, and the resulting phase is topological Weyl semimetal. A unified description of these topological phases is provided by the phase diagram presented in Figure 4(b).

Figure 4(c) presents dispersion curves of a phononic topological nodal-ring semimetal protected by M_z, for which equivalent A and B sites were selected [33]. The gapless feature in the k_z=\mathrm{0} plane is consistent with the symmetry analysis. Moreover, we calculated the Zak phase {\theta }_{\mathrm{Z}\mathrm{a}\mathrm{k}} along the (001) direction for the 2D surface BZ, where {\theta }_{\mathrm{Z}\mathrm{a}\mathrm{k}} has quantized values of 0 or \pi [Figure 4(d)]. The nodal rings appear at boundaries between regions of {\theta }_{\mathrm{Z}\mathrm{a}\mathrm{k}}=\mathrm{0} and {\theta }_{\mathrm{Z}\mathrm{a}\mathrm{k}}=\pi, implying their topological nature. Furthermore, phononic drumhead surface states were observed by our surface-state calculations [Figure 4(e)], which is a hallmark feature of topological nodal-ring semimetals. Their evolution with varying phonon frequencies are displayed in Supplemental Figure S5.

We would like to provide guiding principles for searching candidate materials by first-principles calculations: (i) For a family of materials having a specified crystalline symmetry, use group theory to determine band degeneracies at high symmetry momenta (HSM). Topological phononic states would be allowed, if having band degeneracies at more than one HSM. (ii) For a specific material, calculate the phonon dispersion and eigenstates at HSM. Sort degenerate bands at each HSM according to their phonon frequencies. Then check whether the order of degenerate bands with different irreducible representations varies between different HSM. If so, a topological band inversion usually exists. (iii) For the material with a topological band inversion, calculate the phonon dispersion in the whole momentum space. Thus, whether it is a topological insulator or semimetal can be defined. (iv) Determine the topological nature explicitly by computing hybrid Wannier centers and topological surface states. The simple guiding principles could be applied for high throughput discovery of phononic topological materials.

Hallmarks of a topological phononic material include (i) the existence of topological band inversion for bulk phonon modes at HSM and (ii) the existence of topologically protected, gapless phononic states on the boundary. One can use infrared spectroscopy, Raman techniques, or inelastic neutron scattering to detect bulk phonon modes at HSM. The measured band degeneracy and band order at HSM could be used to determine topological band inversion. Besides, one can use angle-resolved electron energy loss spectroscopy that is a surface-sensitive technique to measure phonon dispersion of surfaces. Thus topological boundary states of phonons can be directly probed. These experimental signatures in combination with first-principles calculations could be used to determine topological nature of specific materials.

Our work sheds lights on future study of topological phononics. A few promising research directions are opened: (i) To search for new symmetry-protected phononic topological states. In addition to C_{nv}, there are many other symmetries, like (magnetic) space group symmetry, time reversal symmetry, particle-hole symmetry, etc., and their combinations, which could result in rich topological phases [28, 42]. (ii) To explore novel physical effects by breaking symmetry locally or globally. For instance, the quantum anomalous Hall-like states would emerge if time reversal symmetry breaking effects were introduced to 2D TSSs of phononic TIs. (iii) To investigate unconventional electron-phonon coupling and superconductivity caused by the pseudospin- and topology-related physics of phonons. Many material systems could simultaneously host nontrivial topological states of electrons and phonons. Thus, TSSs of electrons and phonons are able to coexist on the material surfaces. Their interactions might be exotic, for instance, due to the (pseudo-)spin-momentum locking. (iv) To find realistic candidate materials for experiment and application. Very few realistic materials of chiral phonons [21, 34, 43] and phononic topological semimetals [22–25] have been reported, and plenty of unknown candidate materials of various phononic topological phases are awaiting to be discovered. It is helpful to develop machine learning methods for high throughput discovery of phononic topological materials.