Unconventional Charge‐Two Weyl Phonons in High‐Symmetry Lines

Abstract

Weyl phonons have long been regarded as an important kind of topological bosonic quasiparticle states. Previous investigations in this field mainly focus on Weyl phonons located at high‐symmetry points (HSPs) in Brillouin zone (BZ), while those located at high‐symmetry lines (HSLs, not including HSPs) are still rarely studied. Considering that their symmetry‐protected conditions are much different, we name them conventional and unconventional Weyl phonons, respectively. In this work, taking charge‐two Weyl phonons (CTWPs) as examples, the complete classifications of all unconventional CTWPs is summarized by performing symmetry analysis in all 230 space groups (SGs). Moreover, due to their different phononic dispersions, the unconventional CTWPs can be classified as Type‐I, Type‐II, and Type‐III ones. Particularly, the k · p model of unconventional CTWPs shows a k‐type feature along the invariant lines while a k²‐type feature along other directions. Additionally, we uncover that a real chiral crystal material, that is, CsBe₂F₅ in SG 213, can exhibit well‐defined Type‐I and Type‐II unconventional CTWPs characterized by multiple double‐helicoid surface states. Our theoretical work not only puts forward an effective way for seeking for unconventional CTWPs, but also uncover the unique advantages of this kind of CTWPs toward realistic device applications.

Introduction

To seek for new kinds of topological quasiparticles with novel nontrivial properties has long been regarded as one of central topics in topology physics. Up to date, a series of topological excitations including 0D (Weyl,^[1–5] Dirac,^[6–8] high-folder Weyl points, and others^[9–11]), 1D (nodal chains, links, nets, and others^[12–20]), and 2D (nodal surfaces and nodal walls^[21–26]) with different geometrical configurations have been proposed in solid-state materials, and some of them have already been observed successfully in experiments.^[27–31] Among them, Weyl-type excitations are of particular importance, and have been attracted intense studies by researchers in the fields of topological semimetals and topological bosonic crystals,^[32–36] because this kind of basic topological excitations provide the physical foundations to construct Weyl nodal-line^[25,26] and Weyl nodal-surface excitations.^[37,38] We well know that Weyl point is characterized by a quantized chiral charge or Chern number $$\mathcal {C}$$,^[39–41] and usually appears in a system with breaking either the time-reversal ($$\mathcal {T}$$) or space-inversion ($$\mathcal {I}$$) symmetry. In previous literatures, however, investigations on Weyl point mainly focus on that located at high-symmetry points (HSPs) in the first Brillouin zone (BZ).^[42,43] The main reason is that at HSPs, the required conditions for Weyl point, including various screw rotation symmetries and breaking $$\mathcal {C}$$ or $$\mathcal {I}$$, can be ensured conveniently. Moreover, highly-degenerated band crossings characterized by multi-fold rotation symmetries tend to also appear at HSPs, which provide the possibilities to realize intriguing Weyl-type excitations such as quadratic-double, cubic-triple and high-order Weyl points with higher Chern number $$\mathcal {C}$$.^[44] Comparatively speaking, at high-symmetry lines (HSLs, not including HSPs), the highly-degenerated band crossings and the required screw rotation symmetries for Weyl point are much more difficult to ensure simultaneously. As a result, topological characters of Weyl point at HSLs are still rarely studied, making that some shortcomings still remain in the topic of Weyl-type excitations both in electronic and bosonic systems.

It is obvious that as we further consider the specific crystal structures of real materials, the required $$\mathcal {I}$$, $$\mathcal {T}$$ symmetries, especially the multi-fold screw rotation symmetries, required for the Weyl points localized at HSPs and HSLs are much different, leading to different topologically nontrivial features. For convenience to perform our studies, we name the Weyl points at HSPs and HSLs as conventional and unconventional ones, respectively. Moreover, according to their different band dispersions, we uncover that the unconventional Weyl point can be classified as Type-I, Type-II, and Type-III one,^[23,45,46] as illustrated in Figure 1, where the linear (k-type) dispersion appears along the k_{x, y, z} axis, while the k²- or k³-type dispersion at any other directions and meanwhile, the unconventional Weyl points with $$\mathcal {C}=\pm 1$$, ±2, and ±3 are also illustrated. Considering that charge-three Weyl point^[47] is generally uncommon and that charge-four Weyl point^[48] requires both $$\mathcal {T}$$ and high degeneracies on nontrivial bands, in this work, we take unconventional charge-two Weyl phonons (CTWPs) in crystalline solids as examples to study their topological properties. Then by checking the symmetries of all 230 space groups (SGs), we uncover further nearly all unconventional CTWPs existing in the 3D Brillouin Zone (BZ) of bosonic systems and enumerate them in Table 1. Particulary, when a HSL possesses screw rotation or helical axis symmetries, the associated unconventional CTWP localized at this HSL is protected by the above symmetries. More interestingly, when the HSL possesses C₄ or C₆ screw rotation symmetry, the unconventional CTWPs at this line would display a k-type feature along the HSL, while a k²-type feature for any other directions. Additionally, the unconventional CTWPs are still characterized by nontrivial surface arc states such as double-helical surface arcs.

Table 1 The completed list of unconventional CTWPs and charge-one Weyl phonons. The first column indicates the SG number, the second column indicates the SG symbol, the third column shows the Irreps, the fourth column shows the Generators, and the fifth column shows the location information of Weyl points localized at the HSLs

SG No.	SG Symbol	Irreps	Generators	Lines
75	P4	R1R3; R2R4	$$C_{4z}^+$$	Γ-Z;M-A
76	P41	R1R3; R2R4	$$C_{4z}^+$$	Γ-Z;M-A
77	P42	R1R3; R2R4	$$C_{4z}^+$$	Γ-Z;M-A
78	P43	R1R3; R2R4	$$C_{4z}^+$$	Γ-Z;M-A
79	I4	R1R3; R2R4	$$C_{4z}^+$$	Γ-Z;Z-V
80	I41	R1R3; R2R4	$$C_{4z}^+$$	Γ-Z;Z-V
89	P422	R1R3; R2R4	$$C_{4z}^+$$	Γ-Z;M-A
90	P4212	R1R3; R2R4	$$C_{4z}^+$$	Γ-Z
91	P4122	R1R3; R2R4	$$C_{4z}^+$$	Γ-Z;M-A
92	P41212	R1R3; R2R4	$$C_{4z}^+$$	Γ-Z;M-A
93	P4222	R1R3; R2R4	$$C_{4z}^+$$	Γ-Z;M-A
94	P42212	R1R3; R2R4	$$C_{4z}^+$$	Γ-Z
95	P4322	R1R3; R2R4	$$C_{4z}^+$$	Γ-Z;M-A
96	P43212	R1R3; R2R4	$$C_{4z}^+$$	Γ-Z;M-A
97	I422	R1R3; R2R4	$$C_{4z}^+$$	Γ-Z;Z-V
98	I4122	R1R3; R2R4	$$C_{4z}^+$$	Γ-Z;Z-V
207	P432	R1R3; R2R4	$$C_{4y}^+$$	Γ-X;M-A
208	P4232	R1R3; R2R4	$$C_{4y}^+$$	Γ-X;M-A
209	F432	R1R3; R2R4	$$C_{4y}^+$$	Γ-X;M-A
210	F4132	R1R3; R2R4	$$C_{4y}^+$$	Γ-X
211	I432	R1R3; R2R4	$$C_{4y}^+$$	Γ-H
212	P4332	R1R3; R2R4	$$C_{4y}^+$$	Γ-X
213	P4132	R1R3; R2R4	$$C_{4y}^+$$	Γ-X
214	I4132	R1R3; R2R4	$$C_{4y}^+$$	Γ-H
168	P6	R1R3; R2R4; R3R5; R4R6	$$C_{6z}^+$$	Γ-A
169	P61	R1R3; R2R4; R3R5; R4R6	$$C_{6z}^+$$	Γ-A
170	P65	R1R3; R2R4; R3R5; R4R6	$$C_{6z}^+$$	Γ-A
171	P62	R1R3; R2R4; R3R5; R4R6	$$C_{6z}^+$$	Γ-A
172	P64	R1R3; R2R4; R3R5; R4R6	$$C_{6z}^+$$	Γ-A
173	P63	R1R3; R2R4; R3R5; R4R6	$$C_{6z}^+$$	Γ-A
177	P622	R1R3; R2R4; R3R5; R4R6	$$C_{6z}^+$$	Γ-A
178	$$P\overline{6}$$	R1R3; R2R4; R3R5; R4R6	$$C_{6z}^+$$	Γ-A
179	P6522	R1R3; R2R4; R3R5; R4R6	$$C_{6z}^+$$	Γ-A
180	P6222	R1R3; R2R4; R3R5; R4R6	$$C_{6z}^+$$	Γ-A
181	P6422	R1R3; R2R4; R3R5; R4R6	$$C_{6z}^+$$	Γ-A
182	P6322	R1R3; R2R4; R3R5; R4R6	$$C_{6z}^+$$	Γ-A

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In this work, we focus our investigations on the topological characters and the related nontrivial surface arc states of unconventional CTWPs in bosonic systems. First, by using symmetry analysis, we establish an effective low-energy k · p model Hamiltonian in SG 213 to study the basic properties of phononic dispersions. Then, by using first-principles calculation, we adopt a real material sample, that is, the crystal CsBe₂F₅ in SG P4₁32 (No. 213), to demonstrate the coexistence of Type-I and Type-II unconventional CTWPs with different Chern numbers.^[45] It is interesting that an ideal Weyl complex composed by unconventional Weyl phonons with different topological charges exist in this material and thus, multiple double-helicoid surface states display clearly. Our theoretical results make up for the studies on topology physics of unconventional Weyl point located at HSLs and offer opportunities to understand the unconventional Weyl complex in a real material with SG P4₁32. More importantly, the unique advantages of these unconventional Weyl points and the associated surface arc states toward realistic device applications are also discussed. Note that our findings on the topology physics of unconventional CTWPs can be extended to the Weyl-type quasiparticles in Fermi systems.

Results and Discussion

Symmetry Analysis and Phononic Band Calculations on A Real Material

To examine the topological nature of unconventional CTWPs localized at HSLs, we consider a two-band model on a HSL along the path Γ–X in the 3D BZ with SG 213. It is noted that the spatial symmetries for this line are determined mainly by the C₄-fold rotation symmetry.^[49] As two phononic branches are very close to each other around a frequency value of k_wp at this screw rotation axis. Therefore, we may adopt a 2 × 2 effective low-energy model Hamiltonian to describe the Weyl-type quasiparticle excitations as below1 $$$\begin{eqnarray} \mathcal {H}_{kp}=f_x(q)\sigma _x+f_y(q)\sigma _y+f_z(q)\sigma _z, \end{eqnarray}$$$where q = k − k_wp denotes the wave vector relative to k_wp. Moreover, under the C₄-fold rotation operator and according to the constraint of C_4y, the effective Hamiltonian should give the following relation 2 $$$\begin{eqnarray} C_{4y}\mathcal {H}(q) C_{4y}^{-1}=\mathcal {H}(-q_z,q_y,q_x). \end{eqnarray}$$$Taking the eigenvalues of $$C_{4y}^+$$ operator for those two bands, and considering the 2D irreps (R1R3) along this HSL, the matrix $$C_{4y}^+$$ for the CTWP at this line can be written as 3 $$$\begin{eqnarray} &C_{4y}^+&={\left[ \def\eqcellsep{&}\begin{array}{cc}e^{i \pi v/2} & 0 \\[3pt] 0 & e^{-i\pi (1-v)/2} \end{array} \right]}. \end{eqnarray}$$$Then, the final k · p-invariant Hamiltonian for the unconventional CTWP with the topological charges ±2 under the C₄-rotation symmetry, can be expressed as 4 $$$\begin{eqnarray} \mathcal {H}_{k p}=a_{3} q_{z}^2\sigma _{x}+b_{3} q_{z}\sigma _{y}, \end{eqnarray}$$$Furthermore, considering the C₆ symmetry operator, the screw rotation symmetry C_6z constrains the Hamiltonian as described below 5 $$$\begin{eqnarray} C_{6z}\mathcal {H}(q) C_{6z}^{-1}=\mathcal {H}(\frac{1}{2} q_{x}-\frac{\sqrt {3}}{2} q_{y},\frac{\sqrt {3}}{2} q_{x}+\frac{1}{2} q_{y}, q_{z}) \end{eqnarray}$$$Additionally, we apply the similar treatment on the Hamiltonian by the operator $$C_{6z}^+$$ along this HSL. It is noted that the related $$C_{6z}^+$$ matrices can be described as 6 $$$\begin{eqnarray} &C_{6z}^+&={\left[ \def\eqcellsep{&}\begin{array}{cc}e^{i \pi v/3} & 0 \\[3pt] 0 & e^{i \pi (2-v)/3} \end{array} \right]}. \end{eqnarray}$$$Finally, the symmetry-permitted expression for the unconventional CTWP as a function of q to the lowest orders can be written as 7 $$$\begin{eqnarray} \mathcal {H}_{k p}&=&{\left(a(q_x^2-q_y^2)-2bq_xq_y \right)} \sigma _x\nonumber\\ &&+{\left(b(q_x^2-q_y^2)-2aq_xq_y \right)}\sigma _y+cq_z\sigma _z. \end{eqnarray}$$$The above final k · p model Hamiltonian shows that the low-energy phononic dispersions of unconventional CTWP display a k²-type feature in the k_{x, y}-directions while a k-type one in the k_z-direction.

To further explore the topological characters of the unconventional CTWP, the phononic dispersions of a real chiral crystalline solid, that is, CsBe₂F₅ in cubic structure with SG 213 (P4₁32), are studied in detail by using first-principles calculation. Our theoretical calculations show that the optimized lattice parameter of the crystal CsBe₂F₅ is 8.06Å, which is well consistent with the value 7.93Å obtained from experimental observation.^[50] It is noted that the chiral crystal CsBe₂F₅ was successfully fabricated by evaporating excess BeF₂ hydrofluoride solution with Cs₂CO₂ carbonate dissolved in it at 55°C.^[50] The crystal CsBe₂F₅ is composed by four structural units, and every unit consists of 1 Cs, 2 Be, and 5 F atoms, which are denoted by green, dark-green and light-purple balls, respectively, as illustrated in Figure 2(a). The Wyckoff positions for Cs, Be, and F atoms are 4a (0.875, 0.125, 0.625), 8c (0.099726, 0.099726, 0.099726), and 12d (0.675, 0.466696, 0.716696). Moreover, the real-space crystal structure, the bulk BZ, the (001) and (110) surfaces BZ are drawn in Figure 2(b), respectively. It is stressed that the SG P4₁32 holds the tetrahedron (O) point-group symmetry, which provides four two-fold rotation symmetries $$\lbrace C_{2z}|\frac{1}{2}0\frac{1}{2}\rbrace$$, $$\lbrace C_{2y}|0\frac{1}{2}\frac{1}{2}\rbrace$$, $$\lbrace C_{2a}|\frac{3}{4}\frac{1}{4}\frac{1}{4}\rbrace \text{,}$$ and (C_2a:$$xyz \mapsto yx\bar{z}$$), and one three-fold rotation symmetry $$\lbrace C_{31}^-|000\rbrace$$ ($$C_{31}^-$$:xyz↦yzx) along (111) direction as generators at the point Γ. Particularly, along the k_x-axis, $$\lbrace C_{4x}|\frac{3}{4}\frac{1}{4}\frac{1}{4}\rbrace$$ gives its main generator and $$\mathcal {T}$$ is also ensured.

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Topological Features of Type-I Unconventional CTWPs

Since every unit cell of the crystal of CsBe₂F₅ is composed by 32 atoms, its phononic dispersions display 96 phonon bands, including three acoustic and 93 optical phonon branches. The disappearance of virtual frequency in the phonon spectrum ensures the dynamical stability of material sample. In Figure 2c, we plot the phonon bands of CsBe₂F₅ along high-symmetry paths and the corresponding phonon density of states (DOSs) at the frequency region 0–30 THz. From symmetry analysis, charge-one Weyl phonons, Type-I CTWP and Type-II CTWPs should appear in SG 213. To determine the existence of unconventional Weyl points, we should further examine the nontrivial band crossing nodes localized at the related HSLs. To the end, we focus our attention on the crossing phononic bands in the frequency range from 15.38 to 15.46 THz and the range from 8.00 to 8.30 THz. First, we find that a typical Weyl phonon with the charge of +1 (−1) appear in the HSL Γ-R (Γ-M), as drawn in Figure 2c. More importantly, the band crossing at the HSL Γ-X, formed by the 68th and 69th phonon branches, possesses the topological charge of -2, indicating that this crossing node displays a Type-I CTWP as drawn in Figure 2c. Moreover, the crossing node contributed by the 48th and 49th phonon branches at the HSL Γ-X shows the topological charge of +2. The topological natures of the above charge-one Weyl phonon WP1 (WP2) at the frequency f = 15.436 (15.387) THz and the Type-I CTWP are confirmed further by their Chern numbers calculated by Wilson loop method,^[51] as illustrated in Figure 2d–f. It is noted that as the phonon frequency approaches 15.387 THz, the phonon dispersion of CTWP along the high-symmetry path k_{x(y, z)}-WP4-k_other displays a k-type feature along the k_{x(y, z)} directions while a k²-type feature along any other directions at the point WP3 (see Figure 2f), which are well consistent with the results drawn from the k · p model Hamiltonian. These unique properties of low-energy phonon dispersions confirm further their topological classifications.

It is obvious that the phononic Weyl complex composed of three different kinds of Weyl nodes possessing topological charges of ±1 and −2 is the basic topological character of material sample studied here. To highlight this finding, we turn to examine all Weyl nodes appearing in the first BZ. Due to the special screw rotation symmetries in the crystal CsBe₂F₅, 12 charge-one Weyl nodes with the topological charge of +1 (denoted by red dots, WP1) and 24 charge-one Weyl nodes with the charge of −1 (denoted by green dots, WP2) exist in the 3D BZ, as drawn in Figure 3a. According to the compensation effect restrained by the no-go theorem,^[52,53] there should exist six Type-I CTWPs with the total charges of -12 in the 3D BZ. Actually, by performing Chern number calculations and symmetry analysis,^[54] six type-I CTWPs, denoted by blue dots (WP3), are confirmed to exist as plotted in Figure 3a. Moreover, these six nonconventional CTWPs are distributed at the HSLs along the k_x-, k_y- and k_z-directions (see Figure 3a). Importantly, the phononic dispersions of these CTWPs along the high-symmetry path k_{(x, y, z)}-WP3-k_other show a k-type feature while along any other directions, a k²-type one displays clearly.

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To gain more details of these three different kinds of unconventional Weyl nodes appearing in the crystal CsBe₂F₅, their positions, energy frequencies, charge numbers, band degeneracies and other messages of charge-one Weyl phonons and Type-I CTWPs are listed in Table 2. Moreover, to further explore the topologically nontrivial features of Type-I CTWPs and other Weyl nodes, we plotted the evolutions of the average position of Wannier centers for both charge-one Weyl phonons with opposite chirality in Figure 3b and for a Type-I unconventional CTWP (WP3) in Figure 3c. One may find that their Chern numbers are characterized by ±1 and -2, which are well consistent with the results drawn from the previous symmetry analysis. To examine their compensation effect, we also calculated the distributions of Berry curvature fields around these three kinds of Weyl nodes as illustrated in Figure 3d–f. As expected, the charge-one Weyl phonon with the topological charge of +1 (WP1) acts as a source point (see Figure 3d), whereas the charge-one Weyl phonon with the charge of -1 (WP2) and the Type-I unconventional CTWP with the charge of -2 (WP3) can be viewed as sink points (see Figure 3e,f). To confirm further their low-energy phononic features, in Figure 3g,h, we plotted the 3D configurations of phononic dispersions around these two charge-one Weyl phonons with opposite chirality in the k_x–k_y plane. One may find the basic features low-energy phononic dispersion of these different Weyl nodes are well consistent with the results drawn from the k · p Hamiltonian model as discussed above. More importantly, a quadratic phononic dispersion appear obviously around the Type-I unconventional CTWP in the k_x-k_y plane as drawn in Figure 3i.

Table 2 The completed list of Type-I and Type-II unconventional CTWPs and charge-one Weyl phonons in the crystal CsBe₂F₅. The first column indicates the Weyl points, the second column indicates the phononic band numbers, the third column shows the Positions of Weyl points, the fourth column shows Frequency (THz) value, the fifth column shows the topological Charge, the sixth column shows their Type, and the seventh column shows their Degeneracy

Weyl points	Phonon bands	Position	Frequency(THz)	Charge	Type	Degeneracy
WP1	68-69	(0.00,0.28,0.12)	15.436	+1	type-I	24
WP2	68-69	(0.20,0.20,0.00)	15.446	−1	type-I	12
WP3	68-69	(0.00,0.00,0.35)	15.387	−2	type-I	6
WP4	48-49	(0.00,0.00,0.31)	8.17	+2	type-II	6
WP5	48-49	(0.00,0.00,0.06)	8.07	−2	type-I	6
WP6	48-49	(0.17,0.17,0.30)	8.37	+1	type-I	24
WP7	48-49	(0.45,0.08,0.00)	8.38	−1	type-I	24

In what follows, we turn to study the nontrivial surface states induced by the unconventional CTWPs in the crystal CsBe₂F₅. First, the surface states in the frequency range from 15.38 to 15.46 THz and the corresponding local DOSs projected on the (001) and (100) surfaces BZ are shown in Figure 4a,b. One may find that there exist two visible phononic surface states, and both of them start from the projection of Type-I CTWP at the point Γ. To study further their nontrivial properties, the associated isofrequency surface projections at the frequency 15.446 THz on the (001) and (110) surfaces BZ are also calculated as drawn in Figure 4b,d. Moreover, the phononic surface states on the (001) surface BZ at the isofrequency f = 15.446 THz display as double-helicoid surface arcs in the k_x-k_y plane. Particularly, two long surface arcs connecting the projections of WP1, WP2, and WP3 appear clearly as drawn in Figure 4b, in which the surface states starting from the projection of Type-I CTWPs (blue dots) tend to form a long surface arc by ending at two charge-one Weyl phonons (red dots). To confirm the above findings, we plot the phononic surface states at the isofrequency with f = 15.446 THz on the (110) surface BZ, which display double-helicoid surface arcs due to the projections of two charge-one Weyl phonons coincided at the point $$\widetilde{{\Gamma }}$$. To further demonstrate the topological features of the surface phononic states induced by the unconventional CTWPs, the isofrequency surface projections with f = 15.446 THz on the (001) and (110) surfaces BZ are illustrated in Figure 4c,d. One may see that the phononic surface projections with the isofrequency f = 15.446 THz on the (001) surface state display double-helicoid surface arc states in the k_x-k_y plane and two long surface arcs connecting the projections of WP1, WP2, and WP3 (see Figure 4c). Similarly, long surface arcs starting from the type-I CTWP (blue dots) also display by ending at two charge-one Weyl phonons and meanwhile, the phononic surface projections on the isofrequency with f = 15.446 THz on the (110) surface BZ also display long double-helicoid surface arc states.

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Topological Features of type-II Unconventional CTWPs

It is fortunate that another kind of special CTWP, i.e., Type-II unconventional CTWP, also exists in this material sample CsBe₂F₅. To highlight this inspiring finding, we plotted the 48th and 49th phononic bands along the high-symmetry path Γ–Z in Figure 5a. One can find that two crossing nodes, namely, WP4 and WP5 contributed by two nontrivial bands, appear at this HSL, but not at HSPs. Our Wilson-loop calculations demonstrate that the crossing points WP4 and WP5 host the topological charge +2 and −2, respectively, confirming their topologically nontrivial nature. Moreover, their band dispersion relations demonstrate that the crossing node WP4 belongs to Type-II unconventional CTWP, while the crossing node WP5 belongs to Type-I unconventional one. One can see that the coexistence of different kinds of unconventional CTWPs with the charges ±2 and charge-one Weyl phonons with the topological charges ±1 is an important and unique topological character of the crystal CsBe₂F₅, which provides us an ideal material plateau to study the compensation effect between two different kinds of unconventional CTWPs. Therefore, up to date, the all Weyl nodes in the first BZ of the crystal CsBe₂F₅ have been uncovered, as listed in Table 2. More interesting, due to the unique minor symmetry in this material, both Type-I (WP5) and Type-II (WP4) unconventional CTWPs appear symmetrically in the k_x − k_y plane to construct a unique Weyl-type phononic complex, which are composed by two Type-I and two Type-II unconventional CTWPs as shown in Figure 5b.

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Furthermore, the evolutions of the average position of Wannier centers for Type-I CTWP (blue squares) and Type-II CTWP (yellow balls) with positive chirality are shown in Figure 5c, confirming further the topologically nontrivial nature of these two different kinds of CTWPs. The projections of local phononic DOSs on the (001) surface BZ are given in Figure 5d. We find that there clearly exist four surface phononic arc states, and both of them start from the projections of Type-II CTWP (WP4) by ending at the projections of Type-I CTWPs at the point $$\overline{{\Gamma }}$$, displaying double-helicoid surface arc states. Additionally, on the (001) surface BZ, Type-I and Type-II CTWPs located as the k_x-, k_y- and k_z-axis are just projected to the k_x- and k_y-lines, and the projection point $$\overline{{\Gamma }}$$ as drawn in Figure 5e,f. As expected, both isofrequency surfaces at the frequency f = 8.07 and 8.17 THz also display clear double-helical surface states.

To Summary the Differences between the Conventional and Unconventional WPs

From the above systematic investigations on the topological features of unconventional CTWPs, we may understand fully the differences on topological characters between the conventional and unconventional Weyl phonons. First, as for the conventional Weyl phonons, their phononic dispersion relations, including a linear, k²- and k³-dependent ones, could be easily obtained, because they are localized at the HSPs and the associated symmetry-protected symmetry conditions are ensured fully. Therefore, their topologically nontrivial features exhibit richly in the 3D BZ. However, the protection symmetries of unconventional Weyl phonons are reduced remarkably, because the HSLs may lose some spatial symmetries in comparison with those at the HSPs. As a consequence, their phononic dispersion relations should be determined further by the Irreps and Chern numbers of nontrivial bands, as listed in Table 1. Second, in a given SG, all HSPs are locked by the crystal structure, thus the conventional Weyl points at the HSPs can not be changed arbitrarily. While in the same SG, the positions of band crossings in HSLs may be unfixed and thus, they can be regulated and controlled by an external operator, such as a mechanical strain or an external electric field. This unique feature supports that the spatial positions of unconventional Weyl phonons, and the geometries and lengthens of nontrivial surface arc states may be adjusted more easily in experiments comparing with the conventional Weyl phonons. This property is helpful for us to design and realize some special device applications based on unconventional topological phonons.

Conclusion

In summary, by performing symmetry analysis in all 230 SGs, we uncovered that there exist unconventional Weyl phonons as listed in Table 1, which are localized at HSLs but not at HSPs, and can be classified into Type-I, Type-II, and Type-III classifications, according to their different phononic band relations. Moreover, taking the unconventional CTWPs as samples, we find that these special CTWPs are protected by crew rotation symmetries while breaking the time-reversal symmetry in HSLs. By constructing the k · p model Hamiltonian of unconventional CTWPs, we find that the phononic dispersions of Type-I and Type-II unconventional CTWPs display a k-type feature along the high-symmetry paths while a k²-type feature along any other directions, which is much different from the conventional Weyl phonons. To confirm the results drawn from symmetry analysis, we chose a real material sample, that is, the crystal CsBe₂F₅ in SG 213, to study their topological characters. The first-principles calculations show the existence of Type-I and Type-II unconventional CTWPs at the HSLs, and the related surface states display clear double-helical surface arc states. Moreover, our theoretical results show that the unconventional CTWPs may appear also in SGs 75-80, 89-98, 168-173, 177-182, and 207-214. This theoretical work not only puts forward an effective way for seeking for Type-I, Type-II, and Type-III unconventional CTWPs, but also uncovers their topological characters and unique advantages toward topological device applications.

Experimental Section

Symmetry Analysis and Computational Details

The phononic dispersions of real materials were calculated by the density functional theory (DFT) using the Vienna ab initio Simulation Package (VASP) with the generalized gradient approximation (GGA) in the form of Perdew–Burke–Ernzerhof (PBE) function for the exchange-correlation potential.^[55–57] An accurate optimization of structural parameters was employed by minimizing the interionic forces less than 0.001 eV Å⁻¹ and an energy cutoff at 520 eV. The first BZ gridded with 3× 3 ×3 k points was adopted. Then the phononic spectra were gained using the density functional perturbation theory (DFPT), implemented in the PHONOPY package.^[58,59] The force constants were calculated using a 2 × 2 × 1 supercell. To reveal the topological nature of phonons, the phononic Hamiltonian of the tight-binding (TB) model and the surface local DOSs were constructed using the open-source software WANNIERTOOLS code^[60] and surface Green's functions.^[61] The Chern numbers or topological charges of charge-one WPs, Type-I, and Type-II CTWPs are calculated by the Wilson loop method,^[62] and the crystal structures were obtained from the Materials Project (MP).^[63]

Acknowledgements

This work was supported by the National Science Foundation of China with grants nos: 11774104, 12147113, and U20A2077, and project funded by the China Postdoctoral Science Foundation with grant no. 2021M691149.

Conflict of Interest

The authors declare no conflict of interest.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

References

S.‐Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C.‐L. Zhang, R. Sankar, G.‐Q. Chang, Z.‐J. Yuan, C.‐C. Lee, S.‐M. Huang, H. Zheng, J. Ma, D.‐S. Sanchez, B.‐K. Wang, A. Bansil, F.‐C. Chou, P.‐P. Shibayev, H. Lin, S. Jia, M. Z. Hansan, Science 2015, 349, 6248.

N. P. Armitage, E. J. Mele, A. Vishwanath, Rev. Mod. Phys. 2018, 90, [eLocator: 015001].

X. C. Huang, L.‐X. Zhao, Y.‐J. Long, P.‐P. Wang, D. Chen, Z.‐H. Yang, H. Liang, M.‐Q. Xue, H.‐M. Weng, Z. Fang, X. Dai, G.‐F. Chen, Phys. Rev. X 2015, 5, [eLocator: 031023].

A. A. Burkov, L. Balents, Phys. Rev. Lett. 2011, 107, [eLocator: 127205].

C. Fang, M. J. Gilbert, X. Dai, B. A. Bernevig, Phys. Rev. Lett. 2012, 108, [eLocator: 266802].

H.‐J. Zhang, C.‐X. Liu, X.‐L. Qi, X. Dai, Z. Fang, S.‐C. Zhang, Nat. Phys. 2009, 5, 438.

Z. K. Liu, B. Zhou, Y. Zhang, Z.‐J. Wang, H.‐M. Weng, D. Prabhakaran, S.‐K. Mo, Z.‐X. Shen, Z. Fang, X. Dai, Z. Hussain, Y. L. Chen, Science 2014, 343, 864.

S. M. Young, C. L. Kane, Phys. Rev. Lett. 2015, 115, [eLocator: 126803].

M. Z. Hasan, G.‐Q. Chang, I. Belopolski, G. Bian, S.‐Y. Xu, J.‐X. Yin, Nat. Rev. Mater. 2021, 6, 784.

H.‐X. Wang, Z.‐K. Lin, B. Jiang, G.‐Y. Guo, J.‐H. Jiang, Phys. Rev. Lett. 2020, 125, [eLocator: 146401].

B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory, C. Felser, R. J. Cava, B. A. Bernevig, Science 2016, 353, 6299.

Q.‐B. Liu, Z.‐Q. Wang, H.‐H. Fu, New J. Phys. 2022, 24, [eLocator: 113040].

Q.‐B. Liu, H.‐H. Fu, G. Xu, R. Yu, R. Wu, J. Phys. Chem. Lett. 2019, 10, 4045.

G.‐Q. Chang, S.‐Y. Xu, X.‐T. Zhou, S.‐M. Huang, B. Singh, B.‐K. Wang, I. Belopolski, J.‐X. Yin, S.‐T. Zhang, A. Bansil, H. Lin, M. Z. Hasan, Phys. Rev. Lett. 2017, 119, [eLocator: 156401].

X.‐T. Wang, T. Yang, Z. Cheng, G. Surucu, J. Wang, F. Zhou, Z. Zhang, G. Zhang, Appl. Phys. Rev. 2022, 9, [eLocator: 041304].

S. Li, Y. Liu, S.‐S. Wang, Z.‐M. Yu, S. Guan, X.‐L. Sheng, Y. G. Yao, S. A. Yang, Phys. Rev. B 2018, 97, [eLocator: 045131].

S. Li, Z.‐M. Yu, Y. Liu, S. Guan, S.‐S. Wang, X. Zhang, Y. Yao, S. Y. Yang, Phys. Rev. B 2017, 96, [eLocator: 081106].

C. Fang, H. Weng, X. Dai, Z. Fang, Chin. Phys. B 2016, 25, [eLocator: 117106].

J.‐Y. You, X.‐L. Sheng, G. Su, Phys. Rev. B 2021, 103, [eLocator: 165143].

Q.‐B. Liu, H.‐H. Fu, R. Wu, Phys. Rev. B 2021, 104, [eLocator: 045409].

B.‐B. Fu, C.‐J. Yi, T.‐T. Zhang, M. Caputo, J.‐Z. Ma, X. Gao, B. Lv, L.‐Y. Kong, Y.‐B. Huang, P. Richard, M. Shi, V. N. Strocov, C. Fang, H.‐M. Weng, Y.‐G. Shi, T. Qian, H. Ding, Sci. Adv. 2019, 5, [eLocator: eaau6459].

J.‐Z. Ma, Q.‐S. Wu, M. Song, S.‐N. Zhang, E. B. Guedes, S. A. Ekahana, M. Krivenkov, M. Y. Yao, S.‐Y. Gao, W.‐H. Fan, T. Qian, H. Ding, N. C. Plumb, M. Radovic, J. H. Dil, Y.‐M. Xiong, K. Manna, C. Felser, O. V. Yazyev, M. Shi, Nat. Commun. 2021, 12, 1.

T. Yang, L. Jin, Y. Liu, Z. Zhang, X. Wang, Phys. Rev. B 2021, 103, [eLocator: 235140].

W. Wu, Y. Liu, S. Li, C. Zhong, Z.‐M. Yu, X.‐L. Sheng, Y. X. Zhao, S. A. Yang, Phys. Rev. B 2018, 97, [eLocator: 115125].

C. Xie, H. Yuan, Y. Liu, X. Wang, G. Zhang, Phys. Rev. B 2021, 104, [eLocator: 134303].

Q.‐B. Liu, Z.‐Q. Wang, H.‐H. Fu, Phys. Rev. B 2021, 104, [eLocator: L041405].

H. Miao, T.‐T. Zhang, L. Wang, D. Meyers, A.‐H. Said, Y.‐L. Wang, Y.‐G. Shi, H.‐M. Weng, Z. Fang, M. P. M. Dean, Phys. Rev. Lett. 2018, 121, [eLocator: 035302].

K. Deng, G. Wan, P. Deng, K. Zhang, S. Ding, E. Wang, M. Yan, H. Huang, H. Zhang, Z. Xu, J. Denlinger, A. Fedorov, H. Yang, W. Duan, H. Yao, Y. Wu, S. Fan, H. Zhang, X. Chen, S. Zhou, Nat. Phys. 2016, 12, 1105.

L. Lu, Z.‐Y. Wang, D.‐X. Ye, L.‐X. Ran, L. Fu, J. D. Joannopoulos, M. Soljaci, Science 2015, 349, 622.

J.‐Z. Ma, Q.‐S. Wu, M. Song, S.‐N. Zhang, E. B. Guedes, S. A. Ekahana, M. Krivenkov, M. Y. Yao, S.‐Y. Gao, W.‐H. Fan, T. Qian, H. Ding, N. C. Plumb, M. Radovic, J. H. Dil, Y.‐M. Xiong, K. Manna, C. Felser, O. V. Yazyev, M. Shi, Nat. Commun. 2021, 12, 1.

X. Huang, L. Zhang, Y. Long, P. Wang, D. Chen, Z. Yang, H. Liang, M. Xue, H. Weng, Z. Fang, X. Dai, G. Chen, Phys. Rev. X 2015, 5, [eLocator: 031023].

W.‐Y. Deng, J.‐Y. Lu, F. Li, X.‐Q. Huang, M. Yan, J.‐H. Ma, Z.‐Y. Liu, Nat. Commun. 2019, 10, 1769.

F. Zhou, Z.‐Y. Zhang, H. Chen, M.‐Q. Kuang, T. Yang, X.‐T. Wang, Phys. Rev. B 2021, 104, [eLocator: 174108].

B. W. Xia, R. Wang, Z. J. Chen, Y. J. Zhao, H. Xu, Phys. Rev. Lett. 2019, 123, [eLocator: 065501].

Q.‐B. Liu, Y.‐T. Qian, H.‐H. Fu, Z.‐J. Wang, npj Comput. Mater. 2020, 6, 95.

J.‐X. Li, Q. Xie, S. Ullah, R.‐H. Li, H. Ma, D.‐Z. Li, Y.‐Y. Li, X.‐Q. Chen, Phys. Rev. B 2018, 97, [eLocator: 054305].

J.‐X. Li, Q. Xie, J.‐X. Liu, R.‐H. Li, M. Liu, L. Wang, D.‐Z. Li, Y.‐Y. Li, X.‐Q. Chen, Phys. Rev. B. 2020, 101, [eLocator: 024301].

Z.‐M. Yu, W. Wu, Y.‐Z. Zhao, S. A. Yang, Phys. Rev. B. 2019, 104, [eLocator: 041118(R)].

G. Xu, H.‐M. Weng, Z.‐J. Wang, X. Dai, Z. Fang, Phys. Rev. Lett. 2011, 107, [eLocator: 186806].

C. L. Kan, E. J. Mele, Phys. Rev. Lett. 2005, 94, [eLocator: 146802].

Y. Hatsugai, Phys. Rev. Lett. 1993, 71, 3697.

R. Wang, B. W. Xia, Z. J. Chen, B. B. Zheng, Y. J. Zhao, H. Xu, Phys. Rev. Lett. 2020, 124, [eLocator: 105303].

T.‐T. Zhang, R. Takahashi, C. Fang, S. Murakami, Phys. Rev. B 2020, 102, [eLocator: 125148].

Z.‐Q. Wang, Q.‐B. Liu, X.‐F. Yang, H.‐H. Fu, Phys. Rev. B 2022, 106, [eLocator: L161302].

A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, B. A. Bernevig, Nature 2015, 527, 495.

B.‐B. Zheng, B. W. Xia, R. Wang, Z.‐J. Chen, J.‐Z. Zhao, Y.‐J. Zhao, H. Xu, Phys. Rev. B 2020, 101, [eLocator: 100303(R)].

X.‐T. Wang, F. Zhou, Z.‐Y. Zhang, Z.‐M. Yu, Y.‐G. Yao, Phys. Rev. B 2022, 106, [eLocator: 214309].

Q.‐B. Liu, Z.‐J. Wang, H.‐H. Fu, Phys. Rev. B 2021, 103, [eLocator: L161303].

C. J. Bradley, A. P. Cracknell, The mathematical theory of symmetry in solids: representation theory for point groups and space groups, Oxford University Press, Oxford 2009.

Y. L. Far, S. Aléonard, Acta Crys. 1972, B28, 2115.

A. Bouhon, A. M. Black‐Schaffer, R. J. Slager, Phys. Rev. B 2019, 100, [eLocator: 195135].

I. T. Grodsky, R. F. Streater, Phys. Rev. Lett. 1968, 20, 695.

M. Bruni, C. Germani, R. Maartensa, Phys. Rev. Lett. 2001, 87, [eLocator: 231302].

R. Yu, X.‐L. Qi, A. Bernevig, Z. Fang, X. Dai, Phys. Rev. B 2011, 84, [eLocator: 075119].

W. Kohn, L. J. Sham, Phys. Rev. B 1964, 136, [eLocator: A1133].

P. E. Blöchl, Phys. Rev. B 1994, 50, [eLocator: 17953].

J. P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 1996, 77, 3865.

A. Carreras, A. Togo, I. Tanaka, Comput. Phys. Commun. 2017, 221, 221.

A. Togo, I. Tanaka, Scr. Mater. 2015, 108, 1.

Q.‐S. Wu, S.‐N. Zhang, H.‐F. Song, M. Troyer, A. A. Soluyanov, Comput. Phys. Commun. 2018, 224, 405.

M. P. L. Sancho, J. M. L. Sancho, J. M. L. Sancho, J. Rubio, J. Phys. F: Metal Phys. 1985, 15, 851.

H. Weng, R. Yu, X. Hu, X. Di, Z. Fang, Adv. Phys. 2015, 64, 227.

H. Geoffroy, F. Chris, E. Virginie, J. Anubhav, C. Gerbrand, Inorg. Chem. 2011, 50, 656.