arX

iv:2

010.

0022

4v1

[co

nd-m

at.m

trl-

sci]

1 O

ct 2

020

Ubiquitous topological states of phonons in solids: Silicon as a model material

Yizhou Liu1,2#, Nianlong Zou1#, Le Zhao1, Xiaobin Chen3,4,5,∗ Yong Xu1,6,7,† and Wenhui Duan1,7,8

1State Key Laboratory of Low Dimensional Quantum Physics,

Department of Physics, Tsinghua University, Beijing, 100084 China2Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel

3School of Science, Harbin Institute of Technology, Shenzhen, 518055 China4State Key Laboratory on Tunable laser Technology and Ministry of Industry and

Information Technology Key Lab of Micro-Nano Optoelectronic Information System,

Harbin Institute of Technology, Shenzhen, 518055 China5Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, 030006 China

6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama, 351-0198 Japan7Collaborative Innovation Center of Quantum Matter, Beijing 100084, People’s Republic of China

8Institute for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China

Topological physics of phonons has attracted enormous research interest, but an appropriate model materialis still lacking. By first-principles calculations we suggest the widely used material silicon as an ideal candi-date. In silicon, all the phonon bands form topological nodal-lines protected by nonsymmorphic symmetriesand characterized by a quantized Berry phase of π. These topological nodal-lines display a periodic cage-likenetwork in the momentum space, giving drumhead surface states observable along any surface orientations.Moreover, we find a new type of spin-1 Weyl phonon featured by Fermi-arc-like surface states, which distin-guishes from other Weyl phonons by requiring no inversion and time-reversal symmetry breaking. A possibleexperimental evidence on topological surface phonon modes of Si(111) will be discussed. Furthermore, we gen-eralize the concept of symmetry and topology from silicon to other systems and identify numerous candidatematerials, demonstrating the ubiquitous existence of topological phononic states in solids. Our work opens newopportunities to study topological phonons in realistic materials.

Introduction.—Exploring novel quantum degrees of free-dom to control phonons, which play a key role in a vari-ety of physical phenomena (e.g., superconductivity, electri-cal resistivity, and heat dissipation), is of crucial importanceto both fundamental science and practical applications. Re-cent discoveries of Berry-phase and topological physics shednew lights on this subject, leading to an emerging field oftopological phononics [1–4]. So far various kinds of topo-logical phases of phonons have been proposed by theoreticalmodels, including topological gapped and gapless phases inboth two dimensions (2D) and 3D [5–42]; exotic topologicaland Berry-phase effects have also been predicted, which couldfind promising applications in thermal diodes [23], transistors[31], antennas [43], etc. However, very few experimental pro-gresses have been achieved in the field, mainly due to the lackof suitable candidate materials. A key issue is thus to find eas-ily obtainable realistic materials that are simple in both atomicstructure and topological states for the investigation of topo-logical phononics.

Generally there exist two routes to search for topologi-cal phononic materials. One requires time-reversal symme-try (TRS) breaking, the other not. While the breaking ofTRS can lead to novel topological states such as the quantum-anomalous-Hall-like states, it is difficult to realize strong TRSbreaking effects of phonons in solid materials. The secondroute is thus more favorable. To induce 2D topological statessuch as the quantum-spin-Hall-like states, pseudospin degen-eracies protected by crystalline symmetries are generally de-manded. The relevant crystalline symmetry, however, usuallygets broken at the 1D boundary, which destroys the topologi-cal protection of edge states. Such a problem can be avoided

in systems with higher dimensions. For instance, rotationaland mirror symmetries can simultaneously exist in both 3Dbulk and 2D surface. Much effort thus has been devoted tosearching for 3D topological phononic materials with or with-out a frequency gap. As phonon dispersions of most ma-terials do not have a full frequency gap, the realization oftopological gapless phase is relatively easier. Recent worksproposed a few candidate materials of phononic topologicalnodal-lines and various kinds of Weyl phonons, such as dou-ble Weyl phonons [27, 28], mixed single and double Weylphonons [38, 40], and quadruple Weyl phonons [42]. Theseprevious works mainly focused on noncentrosymmetric mate-rials. The large class of centrosymmetric materials, however,remains largely unexplored.

Playing a central role in modern semiconductor devices,silicon is both structurally simple and centrosymmetric withproperties that are greatly concerned by semiconductor in-dustry and (opto-)electronic applications [44]. Despite inten-sive and comprehensive investigation on this classical mate-rial, topological properties of phonons in silicon, as far as weknow, have not been studied before. In this work we find twodifferent kinds of topological phononic gapless states in sil-icon. Specifically we find that all the phonon branches ofbulk Si has topological nodal-lines characterized by a quan-tized Berry phase of π, which are enforced by the nonsymmor-phic glide mirror symmetries. The nodal-lines form a periodiccage-like structure in k space, leading to drumhead surfacestates observable for any surface orientations. The drumheadsurface states between longitudinal acoustic (LA) and opti-cal (LO) branches lie in a “clean” frequency range where fewbulk states are present. This feature makes the surface states

2

FIG. 1. Topological phononic gapless states in Si. (a) Atomic struc-ture of Si with three glide mirror symmetries Gx = {Mx|

14 (0, a, a)},

Gy = {My|14 (a, 0, a)}, and Gz = {Mz |

14 (a, a, 0)}, whose mirror planes

are mutually perpendicular to each other. (b) Phonon dispersion re-lation of Si from density functional theory (DFT) calculations (bluelines) and experiment (black dots) [45]. The red parts on XW and ΓL

are topological nodal-lines with quantized Berry phase π. The insetshows the dispersion of LA and LO phonons, which are the 3rd and4th branches, in the kx − kz plane across X. The red triangle labels atopological triple point formed by the band crossing between the lon-gitudinal optical (LO) and the doubly degenerate transverse optical(TO) branches. (c) Distribution of topological nodal-lines (red lines)and nodal-points (red triangles) in k space.

easily observable through neutron scattering and electron en-ergy loss spectroscopy techniques. In addition to the topologi-cal nodal-lines we also find phononic topological triple pointsformed by the band crossing between LO and doubly degen-erate TO branches. The triple points which can be viewed ascombinations of pairs of Weyl points with opposite chiralitiessupport Fermi-arc-like surface states. We also discuss the con-nection between our work and previous experimental investi-gations. Our work provides a new explanation to the originof surface phonons of Si from viewpoint of topology. Finallywe generalize the symmetry enforced topological phonons toother centrosymmetric materials. A full classification of topo-logical phononic nodal states in space groups (SGs) with in-version symmetry is presented.

Symmetry enforced topological nodal-lines.—Bulk Si has adiamond structure with space group Fd3m (No. 227), whichhas three glide mirror symmetries [Fig. 1(a)]. It has sixphonon branches, which form three doubly degenerate pairsalong XW [Fig. 1(b)]. This degeneracy originates from thefact that the little group along XW has only irreducible rep-resentations of dimensions two. Specifically, this is becausethe two glide mirror symmetries Gy = {My|

14 (a, 0, a)} and

Gz = {Mz|14 (a, a, 0)} anticommute with each other (a = 5.43 Å

is the edge length of conventional unit cell of Si). For aphonon Bloch state |ψk〉 on XW which has Gz eigenvalue

±ei(kxa4+ky

a4 ), Gy will change its sign by switching ky from 2π

a

to − 2πa

. Therefore, the nonsymmorphic glide symmetries en-force all bands to be doubly degenerate along XW, forming a3D cage-like structure in k space [Fig. 1(c)].

We next demonstrate that all the nodal-lines related to XW

are topologically nontrivial. Away from the nodal-lines thephonon bands are generally nondegenerate. Along a closedloop C encircling XW, the Berry phase of each branch can bedefined as: θn =

∮C−iu

†

nk∇kunk · dk, where unk is the phonon

polarization vector. The presence of inversion (P) and TRS(T ) guarantees that θn can only take quantized value 0 or π[46]. Numerical calculation shows that all six branches of Sialong the XW line have nonzero Berry phases of π, whichindicates a topological line node threading the integrationloop. The line nodes cannot be removed by adding small in-version symmetric perturbations because otherwise the Berryflux would become zero.

It should be noted that not all the doubly degenerate bandsare topological. For example, both the transverse acoustic(TA) and optical (TO) branches are degenerate along ΓX butthey are not topological because the Berry phase on a smallloop encircling the ΓX line equals 0 (modulo 2π). However,the degenerated bands along ΓL do have π Berry phase [47].The main difference lies in the fact that the little group alongΓX is isomorphic to C4v which has a two-fold rotation (C2y)symmetry, but ΓL only has three-fold rotation (C3[111]) sym-metry. The two-fold rotation which transforms kx,z to −kx,z

forbids linear-order terms in the effective model of kx,z plane.Thus, both the TA and TO bands must quadratically touchat ΓX, leading to Berry phases of 0 (modulo 2π), but thethree-fold rotation allows linear crossing across ΓL and givesa Berry phase of π [47, 48].

The 3D structure of the topological nodal-lines in Si makesthe surface states observable for arbitrary surface orientations.For an arbitrary surface direction, the Zak phase ϕZ definedas the Berry phase of the Wilson loop parallel to the surfacenorm are quantized to 0 or π. Therefore the ν = mod(ϕZ/π, 2)defines a Z2 topological invariant for any k‖ (k‖ is the com-ponent perpendicular to the surface norm). By continuouslychanging k‖ the value of ν keeps unchanged unless the loopmeets a topological nodal-line.

To analyze the topological surface states, we calculated theZak phase of all branches, ϕZ(k‖). Figures 2(a)-(c) show thedistribution of ϕZ(k‖) in the surface Brillouin zone for (001),(111), and (110) surfaces, respectively, with the regions ofv = 1 (0) marked in purple (cyan). The distribution of Zakphase respects all bulk symmetries of Si. For example, projec-tions of the bulk topological nodal-lines onto the (001) surfacedivide the surface BZ into four parts, each of which transformto the adjacent one under the four-fold skew rotation (S 4z) orglide mirror symmetry Gx and Gy. The S 4 or Gx,y relatedpoints have their ϕZ changed by π relative to each other i.e.their Z2 invariant ν changed by 1 [47], ϕZ(αk⊥) = ϕZ(k⊥) ± πfor α = S 4z or Gx,y. Although the line nodes of TA and TOdegeneracies on ΓL are also topological with Berry phase π,they have no contribution to the Zak phase or Z2 invariant on

3

FIG. 2. Zak phase and surface states. (a)-(c) Projection of bulk BZ onto (001), (111), and (110) surfaces, respectively, and the correspondingZak phases ϕZ as function of k‖ in the surface BZ (red dashed lines). ϕZ = π (0) regions are marked purple (cyan). (d)-(f) Surface local densityof states (LDOS) of (001), (111), and (110) surfaces, respectively. Bulk states are colored grey while surface states are colored green (fixedboundary condition [18]) and purple (free boundary condition).

the (001) surface because two distinct nodal-lines ΓL and ΓL′

are projected onto the same k‖ points, leading to a change of2π in Zak phase. However, when projecting onto some otherdirections, the nonzero Berry phase of ΓL makes a difference.

According to the bulk-boundary correspondence, a topo-logical boundary state exists in the ν , 0 region. The exis-tence of ν , 0 regions in Fig. 2 (a)-(c) is consistent with thek‖-resolved surface local density of states (LDOS) calculationin Fig. 2(d)-(f) (green lines), where a fixed boundary condi-tion [18] was used. The surface states look like a drumheadspanning within the ν = 1 region. Specifically the surfacestates arising from the degeneracy of LA and LO branchesare prominent due to the fewer bulk states in the phonon en-ergy window of 40–50 meV. We note that for (001) and (111)surfaces there are two different kinds of surface terminationsfor which the Z2 invariant ν is opposite. This can be under-stood through the gauge choice of the bulk unit cell [47].It is interesting to note that for (001) surface the ν = 0 re-gions also have surface states. Specifically for the topologi-cal nodal-line formed by the 3rd and 4th bands (i.e., LA andLO branches), there are two rather than zero surface states inthe ν = 0 regions, indicating that the topology of LA and LOband crossing should be described by a Z topological invariantrather than Z2. We conjecture that this is possibly because the

physics of LA and LO band crossing are better described bya two band model. For a two band model the topological in-variant is indeed Z rather than Z2 while a more than two bandmodel is described by Z2 based on the homotopy argument[49].

Symmetry breaking of surface and bulk states.—So far ouranalysis is based on the full space group of diamond-structureSi. In reality, some of the symmetries may be broken due tofactors such as surfaces, defects, and strain. How robust isthe band topology under various external perturbations? Atfirst we used fixed boundary condition for the surface statesin Fig. 2. However, for a free surface the boundary is notfixed and this induces Rayleigh modes on the surface [Figs.3 (d)-(f) purple lines]. Rayleigh modes are another kind ofsurface states, which is intrinsically different from the topo-logical surface states. It usually emerges on a free surfaceand has relatively lower frequencies and velocities than thoseof bulk states. Rayleigh modes are quite sensitive to surfaceconditions as one compares the surface LDOSs between fixedand open boundary conditions in Fig. 2(d)-(f). For the topo-logical surface states, however, although dispersion of topo-logical surface states change a lot under different boundaryconditions, their existence is ensured by the topology of bulkbands. No matter how surface details change, the topological

4

FIG. 3. Topological triple points (TPs) and their surface states. (a) 2D Band structure of the TP in Fig. 1(b). Chern numbers C1,2,3 from lowerto higher bands are calculated on a sphere enclosing the TP. The dashed lines are the projection of the band structure in ky = 0 and kx,z = 0planes. (b) Chern numbers of the six TPs marked by red and blue triangles. (c) (111) Surface LDOS for different phonon energies. Openboundary condition with augmented surface atomic mass msurf = 1.2mSi is used (mSi is the atomic mass of Si atom).

surface states always exist near the bulk nodal-lines as long asthe bulk symmetries are preserved.

When bulk symmetries are broken, however, topologicalsurface states may vanish due to the disappearance of nodal-lines in bulk states. The relevant symmetries here includeTRS, inversion symmetry, and three glide mirror symmetries.Since we always assume TRS is preserved, we consider threedifferent cases: (i) both inversion and glide mirror symme-tries are broken, (ii) glide symmetries are broken but inversionsymmetry is still present, and (iii) two of the glide mirror sym-metries are broken and one remains. Physically the case (i)can be realized by atomic defects where the two atoms withina unit cell become different. This corresponds to the case ofGaAs where all the nodal-lines are gapped out. In case (ii)since all the glide mirror symmetries are broken, the band de-generacies at X point are lifted because the little group at X

only has 1D irreducible representations. However the nodal-lines do not disappear since the π Berry phases defined by thephonon modes on a loop away from the nodal-line keep un-changed as long as the perturbation does not change the bandorder on the loop. Thus a topological nodal-line must existinside the closed loop although it moves away from XW. Incase (iii) because of the glide mirror symmetry, all the nodal-lines not only exist but also lie within or perpendicular to themirror plane [47].

Topological triple-point phonon.—In addition to the topo-logical nodal-lines enforced by the glide mirror symme-tries, bulk Si also has topological nodal-point states inducedby band crossing between LO and doubly degenerate TObranches on ΓX thus forming topological triple points (TPs)[red triangles in Fig. 1(b), (c)]. Near a TP the dispersion re-lation of LO and TO branches in the kykz,x plane is shown inFig. 3(a). The Chern numbers C1,2,3 for each band is definedon a small sphere enclosing the TP. On the sphere the Berrycurvature vanishes everywhere except the two points on ΓX

where the Berry phase (flux) through the two points is ±2π.

Based on the effective Hamiltonian model analysis we find theChern numbers of three bands are 1, −2, and 1 with increasingphonon energy [47]. For other TPs Fig. 3(b) shows the Chernnumbers of all six TPs. The nonzero Chern numbers give riseto the fermi-arc-like surface states like phononic Weyl points.A fundamental difference between phononic topological TPsand Weyl phonons lies in the fact that TPs do not require in-version symmetry breaking. Figure 3(c) shows the (111) sur-face LDOS. Each TP gives rise to two fermi-arc-like states, ofwhich one results from the nonzero Chern number C1 of low-est band while the other results from the nonzero sum of thelowest two bands C1 +C2.

Experimental signatures.—The robustness of topologicalsurface states is evidenced by previous observations on sur-face phonons of Si, which have been extensively studied [50–59]. In these studies, various approaches focused on the de-tails of how surface reconstructions affects surface oscilla-tion frequencies. Surface phonon modes of Si were found onthe (111),[50, 53, 54, 56] (001),[60] and (110)[61] surfaces,which is in good agreement with the inference that surfacestates of Si generally exist on all surfaces due to nodal-linesand TPs of bulk bands. This feature can be further verifiedby the low energy electron loss spectroscopy. We also pointout that the observed low frequency (Rayleigh) modes (about10.5 meV in acoustic branches [54]) are fundamentally dif-ferent from the higher frequency modes (about 50–55 meV inLO branch [50, 52]) due to the lack of bulk topological origin.

Generalization to other inversion symmetric systems.—Sofar we have discussed the topological phononic states in Siwith the specific space group Fd3m, where the nonsymmor-phic symmetries play an important role. In 3D space, thereare 157 nonsymmorphic SGs in total, within which 68 SGsare also inversion symmetric. The nonsymmorphic SGs canenforce degeneracy of points, lines, or even planes at bound-aries of the Brillouin zones when TRS is also taken into ac-count. Here we give a full classification of the symmetry-

5

TABLE I. List of inversion symmetric space groups (SGs) that pro-tect topological point-, line-, or plane- degeneracies.

SGs No.point nodes 130, 133, 135, 137, 138, 163, 165, 167, 176,

192, 193, 201, 205, 206, 222, 223, 224, 226–228, 230

topological linenodes

13–15, 48–50, 52–54, 56, 58, 60–62, 64, 67, 68,70, 72–74, 85, 86, 88, 124–126, 128, 129, 131–142, 163, 165, 167, 176, 192, 201, 203, 206,222–224, 226–228, 230

plane nodes 11, 14, 52–64, 66, 68, 127, 130, 135–138, 176,193, 194, 205

enforced topological nodal-points, -lines, or planes in all 3Dinversion symmetric SGs to benefit future search for topolog-ical phononic materials.

For an arbitrary space group we first check those k pointson high-symmetry-points (HSPts), -lines (HSLs), or -planes(HSPls) whose little group (with TRS taken into account) con-tain 2D or higher dimensional irreducible representations. Forthe nodal-points, we find 21 SGs having isolated nodal-pointsat HSPts and all the nodal-points are 3-fold degenerate orhigher. Since the topological invariant of nodal-points andnodal-planes with inversion symmetry is still unclear, we fo-cus on the topological nodal-lines below. 56 SGs can protectnodal-lines but not all of them are topological. If the littlegroup of nodal-line contains a 4-fold or 6-fold rotation thenodal-line is trivial since under 4-fold or 6-fold rotation, onlyquadratic k terms are allowed in the effective-model Hamil-tonian (k referenced to the nodal-line) [48], leading to zeroBerry phase of the nodal-line. Among the 56 SGs only one ofthem (i.e., SG No. 84) is trivial while all others 55 SGs haveat least one nonsymmorphic-symmetry-enforced topologicalnodal-line. The number of SGs enforcing nodal-points, -lines,and -planes are shown in Tab. I. For each of the SGs, we listthe k space positions of the enforced HSPts, HSLs, and HSPlsexplicitly in the Supplemental Material [47].

Discussions.—Many SGs can enforce topological gaplessstates of phonons. Thus the topological phonon states mayexist ubiquitously among various solid state materials [62]. Itis of great interest to explore the Berry phase and topologi-cal states of phonons, which may induce phonon (valley) Halleffect etc. In addition, the topological phononic drumheadsurface states result in a large phonon local density of stateson the surface. Therefore, the topological surface states mayplay an important role in electron-phonon coupling and evensuperconductivity in some systems.

Conclusions.—In this work, we investigatednonsymmorphic-symmetry-enforced topological phononicnodal-lines and triple points (TPs) in bulk silicon. Dueto glide mirror symmetries and TRS, phonon branches ofSi along the XW line are all doubly-degenerated. Thesenodal-lines form a 3D cage-like structure in k space, leadingto drumhead topological surface states on an arbitrary surfaceof Si. Such surface states are protected by the topology

of bulk bands as long as bulk symmetries are preserved.The TPs are induced by the band crossing between LO andTO branches which can be characterized by nonzero Chernnumbers. The TPs give rise to fermi-arc-like surface states.

Acknowledgement.—We thank Prof. Atsushi Togo fromNational Institute for Materials Science (NIMS) JAPAN forsharing the raw data of phonon database.# Both authors contributed equally to this work.

∗ chenxiaobin@hit.edu.cn† yongxu@mail.tsinghua.edu.cn

[1] Sebastian D Huber, “Topological mechanics,”Nat. Phys. 12, 621–623 (2016).

[2] Yizhou Liu, Yong Xu, and Wenhui Duan,“Berry phase and topological effects of phonons,”Natl. Sci. Rev. 5, 314–316 (2017).

[3] Guancong Ma, Meng Xiao, and Che Ting Chan, “Topo-logical phases in acoustic and mechanical systems,”Nat. Rev. Phys. 1, 281–294 (2019).

[4] Yizhou Liu, Xiaobin Chen, and Yong Xu, “Topologicalphononics: from fundamental models to real materials,”Adv. Funct. Mater. 30, 1904784 (2020).

[5] Emil Prodan and Camelia Prodan, “Topological phononmodes and their role in dynamic instability of microtubules,”Phys. Rev. Lett. 103, 248101 (2009).

[6] Lifa Zhang, Jie Ren, Jian-Sheng Wang, and BaowenLi, “Topological nature of the phonon Hall effect,”Phys. Rev. Lett. 105, 225901 (2010).

[7] CL Kane and TC Lubensky, “Topological boundary modes inisostatic lattices,” Nat. Phys. 10, 39–45 (2014).

[8] Zhaoju Yang, Fei Gao, Xihang Shi, Xiao Lin, Zhen Gao,Yidong Chong, and Baile Zhang, “Topological acoustics,”Phys. Rev. Lett. 114, 114301 (2015).

[9] Xu Ni, Cheng He, Xiao-Chen Sun, Xiao ping Liu, Ming-HuiLu, Liang Feng, and Yan-Feng Chen, “Topologically protectedone-way edge mode in networks of acoustic resonators with cir-culating air flow,” New J. of Phys. 17, 053016 (2015).

[10] V. Peano, C. Brendel, M. Schmidt, and F. Mar-quardt, “Topological phases of sound and light,”Phys. Rev. X 5, 031011 (2015).

[11] Yao-Ting Wang, Pi-Gang Luan, and Shuang Zhang, “Coriolisforce induced topological order for classical mechanical vibra-tions,” New J Phys. 17, 073031 (2015).

[12] Pai Wang, Ling Lu, and Katia Bertoldi, “Topologi-cal phononic crystals with one-way elastic edge waves,”Phys. Rev. Lett. 115, 104302 (2015).

[13] Alexander B Khanikaev, Romain Fleury, S Hossein Mousavi,and Andrea Alu, “Topologically robust sound propagation inan angular-momentum-biased graphene-like resonator lattice,”Nat. Commun. 6, 1–7 (2015).

[14] Roman Susstrunk and Sebastian D Huber, “Observation ofphononic helical edge states in a mechanical topological insu-lator,” Science 349, 47–50 (2015).

[15] S Hossein Mousavi, Alexander B Khanikaev, and Zheng Wang,“Topologically protected elastic waves in phononic metamate-rials,” Nat. Commun. 6, 1–7 (2015).

[16] Roman Susstrunk and Sebastian D Huber, “Classificationof topological phonons in linear mechanical metamaterials,”Proc. Nat. Acad. Sci. 113, E4767–E4775 (2016).

6

[17] Romain Fleury, Alexander B Khanikaev, and An-drea Alu, “Floquet topological insulators for sound,”Nat. Commun. 7, 1–11 (2016).

[18] Toshikaze Kariyado and Yasuhiro Hatsugai, “Ma-nipulation of Dirac cones in mechanical graphene,”Sci. Rep. 5, 18107 (2015).

[19] Raj Kumar Pal, Marshall Schaeffer, and MassimoRuzzene, “Helical edge states and topological phase tran-sitions in phononic systems using bi-layered lattices,”J Appl. Phys. 119, 084305 (2016).

[20] Cheng He, Xu Ni, Hao Ge, Xiao-Chen Sun, Yan-Bin Chen,Ming-Hui Lu, Xiao-Ping Liu, and Yan-Feng Chen, “Acous-tic topological insulator and robust one-way sound transport,”Nat. Phys. 12, 1124–1129 (2016).

[21] Zhaoju Yang and Baile Zhang, “Acoustic type-ii Weyl nodes from stacking dimerized chains,”Phys. Rev. Lett. 117, 224301 (2016).

[22] Hamed Abbaszadeh, Anton Souslov, Jayson Paulose, Hen-ning Schomerus, and Vincenzo Vitelli, “Sonic landau lev-els and synthetic gauge fields in mechanical metamaterials,”Phys. Rev. Lett. 119, 195502 (2017).

[23] Yizhou Liu, Yong Xu, Shou-Cheng Zhang, and WenhuiDuan, “Model for topological phononics and phonon diode,”Phys. Rev. B 96, 064106 (2017).

[24] Yizhou Liu, Chao-Sheng Lian, Yang Li, Yong Xu, and Wen-hui Duan, “Pseudospins and topological effects of phonons in aKekule lattice,” Phys. Rev. Lett. 119, 255901 (2017).

[25] Leyou Zhang and Xiaoming Mao, “Fracturing of topologicalMaxwell lattices,” New J Phys. 20, 063034 (2018).

[26] Feng Li, Xueqin Huang, Jiuyang Lu, Jiahong Ma, andZhengyou Liu, “Weyl points and fermi arcs in a chiral phononiccrystal,” Nat. Phys. 14, 30–34 (2018).

[27] Tiantian Zhang, Zhida Song, A. Alexandradinata, Hong-ming Weng, Chen Fang, Ling Lu, and Zhong Fang,“Double-Weyl phonons in transition-metal monosilicides,”Phys. Rev. Lett. 120, 016401 (2018).

[28] H. Miao, T. T. Zhang, L. Wang, D. Meyers, A. H. Said, Y. L.Wang, Y. G. Shi, H. M. Weng, Z. Fang, and M. P. M. Dean,“Observation of double Weyl phonons in parity-breaking FeSi,”Phys. Rev. Lett. 121, 035302 (2018).

[29] Jiangxu Li, Qing Xie, Sami Ullah, Ronghan Li, Hui Ma, Di-anzhong Li, Yiyi Li, and Xing-Qiu Chen, “Coexistent three-component and two-component weyl phonons in TiS, ZrSe, andHfTe,” Phys. Rev. B 97, 054305 (2018).

[30] Zhan Xiong, Hai-Xiao Wang, Hao Ge, Jinjie Shi, Jie Luo, YunLai, Ming-Hui Lu, and Jian-Hua Jiang, “Topological node linesin mechanical metacrystals,” Phys. Rev. B 97, 180101 (2018).

[31] Yizhou Liu, Yong Xu, Wenhui Duan, et al., “Three-dimensionaltopological states of phonons with tunable pseudospin physics,”Research 2019, 5173580 (2019).

[32] Yujiang Ding, Yugui Peng, Yifan Zhu, Xudong Fan, Jing Yang,Bin Liang, Xuefeng Zhu, Xiangang Wan, and Jianchun Cheng,“Experimental demonstration of acoustic Chern insulators,”Phys. Rev. Lett. 122, 014302 (2019).

[33] Boyang Xie, Hui Liu, Hua Cheng, Zhengyou Liu, ShuqiChen, and Jianguo Tian, “Experimental realization oftype-ii Weyl points and fermi arcs in phononic crystal,”Phys. Rev. Lett. 122, 104302 (2019).

[34] Yihao Yang, Hong-xiang Sun, Jian-ping Xia, Haoran Xue, ZhenGao, Yong Ge, Ding Jia, Shou-qi Yuan, Yidong Chong, andBaile Zhang, “Topological triply degenerate point with doublefermi arcs,” Nature Physics 15, 645–649 (2019).

[35] Qing-Bo Liu, Hua-Hua Fu, Gang Xu, Rui Yu, andRuqian Wu, “Categories of phononic topological Weyl open

nodal lines and a potential material candidate: Rb2Sn2O3,”J Phys. Chem. Lett. 10, 4045–4050 (2019).

[36] B. W. Xia, R. Wang, Z. J. Chen, Y. J. Zhao, and H. Xu,“Symmetry-protected ideal type-ii Weyl phonons in CdTe,”Phys. Rev. Lett. 123, 065501 (2019).

[37] T. T. Zhang, H. Miao, Q. Wang, J. Q. Lin, Y. Cao, G. Fab-bris, A. H. Said, X. Liu, H. C. Lei, Z. Fang, H. M. Weng, andM. P. M. Dean, “Phononic helical nodal lines with PT protec-tion in mob2,” Phys. Rev. Lett. 123, 245302 (2019).

[38] R. Wang, B. W. Xia, Z. J. Chen, B. B. Zheng, Y. J. Zhao, andH. Xu, “Symmetry-protected topological triangular weyl com-plex,” Phys. Rev. Lett. 124, 105303 (2020).

[39] Jiangxu Li, Qing Xie, Jiaxi Liu, Ronghan Li, MinLiu, Lei Wang, Dianzhong Li, Yiyi Li, and Xing-Qiu Chen, “Phononic Weyl nodal straight lines in MgB2,”Phys. Rev. B 101, 024301 (2020).

[40] Qing-Bo Liu, Yuting Qian, Hua-Hua Fu, and Zhi-jun Wang, “Symmetry-enforced Weyl phonons,”npj Comput. Mater. 6, 1–6 (2020).

[41] Tiantian Zhang, Ling Lu, Shuichi Murakami, Zhong Fang,Hongming Weng, and Chen Fang, “Diagnosis schemefor topological degeneracies crossing high-symmetry lines,”Phys. Rev. Research 2, 022066 (2020).

[42] Tiantian Zhang, Ryo Takahashi, Chen Fang, and Shuichi Mu-rakami, “Twofold quadruple weyl nodes in chiral cubic crys-tals,” Phys. Rev. B 102, 125148 (2020).

[43] Zhiwang Zhang, Ye Tian, Yihe Wang, Shuxiang Gao, YingCheng, Xiaojun Liu, and Johan Christensen, “Directionalacoustic antennas based on valley-Hall topological insulators,”Adv. Mater. 30, 1803229 (2018).

[44] Xiaobin Chen, Yizhou Liu, and Wenhui Duan, “Ther-mal engineering in low-dimensional quantum devices: A tu-torial review of nonequilibrium green’s function methods,”Small Methods 2, 1700343 (2018).

[45] G Dolling, Inelastic Scattering of Neutrons in Solids and Liq-

uids. V. II. Proceedings of the Symposium on Inelastic Scatter-

ing of Neurons in Solids and Liquids (1963).[46] Youngkuk Kim, Benjamin J Wieder, CL Kane, and An-

drew M Rappe, “Dirac line nodes in inversion-symmetric crys-tals,” Phys. Rev. Lett. 115, 036806 (2015).

[47] Supplemental materials.[48] Y. D. Chong, Xiao-Gang Wen, and Marin

Soljacic, “Effective theory of quadratic degeneracies,”Phys. Rev. B 77, 235125 (2008).

[49] Chen Fang, Hongming Weng, Xi Dai, andZhong Fang, “Topological nodal line semimetals,”Chin. Phys. B 25, 117106 (2016).

[50] H. Ibach, “Surface vibrations of silicon detected by low-energyelectron spectroscopy,” Phys. Rev. Lett. 27, 253–256 (1971).

[51] H. Froitzheim, H. Lammering, and H. L. Gunter, “Energy-loss-spectroscopy studies on the adsorption of hydrogen on cleavedSi(111)-(2×1) surfaces,” Phys. Rev. B 27, 2278–2284 (1983).

[52] O. L. Alerhand, D. C. Allan, and E. J. Mele,“Dipole activity of surface phonons on Si(111)2×1,”Phys. Rev. Lett. 55, 2700–2703 (1985).

[53] N. J. DiNardo, W. A. Thompson, A. J. Schell-Sorokin,and J. E. Demuth, “Polarization dependence of Si(111)-2×1 surface-phonon and surface-state excitations,”Phys. Rev. B 34, 3007–3010 (1986).

[54] U. Harten, J. P. Toennies, and Ch. Woll, “Observation of a10-mev einstein oscillator mode on the Si(111) (2×1) surface,”Phys. Rev. Lett. 57, 2947–2950 (1986).

[55] O. L. Alerhand and E. J. Mele, “Renormalized acoustic branchin the vibrational spectrum of the π-bonded–chain model of

7

Si(111)21,” Phys. Rev. Lett. 59, 657–660 (1987).[56] U. Harten, J. P. Toennies, Ch. Woll, L. Miglio, P. Ruggerone,

L. Colombo, and G. Benedek, “Surface phonons in Si(111) +H(1×1),” Phys. Rev. B 38, 3305–3310 (1988).

[57] L. Miglio, P. Santini, P. Ruggerone, and G. Benedek, “Dy-namics of extensively reconstructed surfaces: Si(111) 2×1,”Phys. Rev. Lett. 62, 3070–3073 (1989).

[58] F. Ancilotto, W. Andreoni, A. Selloni, R. Car, andM. Parrinello, “Structural, electronic, and vibrational prop-erties of Si(111)-2×1 from ab initio molecular dynamics,”Phys. Rev. Lett. 65, 3148–3151 (1990).

[59] C. H. Patterson, “Validity of the HREELS surface dipole

selection rule at semiconductor surfaces: Si(001)-(2×1)h/d,”Europhys. Lett. 98, 66001 (2012).

[60] N. Takagi, S. Shimonaka, T. Aruga, and M. Nishi-jima, “Surface phonons of the Si(001)(2 × 1) surface,”Phys. Rev. B 60, 10919–10925 (1999).

[61] Stephane Yu Matsushita, Kazuki Matsui, Hiroki Kato, Taro Ya-mada, and Shozo Suto, “Anisotropic surface phonon disper-sion of the hydrogen-terminated Si(110)-(1×1) surface: One-dimensional phonons propagating along the glide planes,”The Journal of Chemical Physics 140, 104709 (2014).

[62] Atsushi Togo and Isao Tanaka, “First principles phonon calcula-tions in materials science,” Scripta Materialia 108, 1–5 (2015).