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Title

Truly chiral phonons in α-HgS

Authors

Kyosuke Ishito, Huiling Mao, Yusuke Kousaka, Yoshihiko Togawa,
Satoshi Iwasaki, Tiantian Zhang, Shuichi Murakami, Jun-ichiro Kishine,
Takuya Satoh

Citation

Nature Physics, Vol. 19, Issue 1, pp. 35-39

Pub. date

2023, 1note

The version of record of this article, first published in Nature Physics, isavailable online at Publisher s website:
https://dx.doi.org/10.1038/s41567-022-01790-x

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Letter

https://doi.org/10.1038/s41567-022-01790-x

Truly chiral phonons in α-HgS
Received: 30 September 2021
Accepted: 5 September 2022

Kyosuke Ishito1,7, Huiling Mao 1,7, Yusuke Kousaka 2,3, Yoshihiko Togawa
Satoshi Iwasaki 3, Tiantian Zhang 1,4, Shuichi Murakami 1,4,
Jun-ichiro Kishine 5,6 & Takuya Satoh 1,

2

Published online: 31 October 2022
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Chirality is a manifestation of the asymmetry inherent in nature. It has
been defined as the symmetry breaking of the parity of static objects, andthe definition was extended to dynamic motion such that true and false
chiralities were distinguished. Recently, rotating, yet not propagating,atomic motions were predicted and observed in two-dimensional materials,
and they were referred to as ‘chiral phonons’. A natural development would
be the discovery of truly chiral phonons that propagate while rotating inthree-dimensional materials. Here we used circularly polarized Raman
scattering and first-principles calculations to identify truly chiral phonons
in chiral bulk crystals. This approach enabled us to determine the chiralityof a crystal in a non-contact and non-destructive manner. In addition,
we demonstrated that the law of the conservation of pseudo-angularmomentum holds between circularly polarized photons and chiral phonons.
These findings are expected to help develop ways for transferring thepseudo-angular momentum from photons to electron spins via propagatingchiral phonons in opto-phononic-spintronic devices.

Chirality, the breaking of reflection and inversion symmetries1, appears
at all levels of nature and has been studied in many fields of research2.
Barron extended the concept of chirality to the dynamic regime byclassifying into true and false chiralities3. True chirality is demonstrated
by systems that exist in two different enantiomeric states that are interconverted by spatial inversion (𝒫𝒫𝒫, rather than by time inversion (𝒯𝒯𝒫
combined with appropriate spatial rotation (ℛ𝒫 (ref. 3). This concept
contrasts with false chirality, where the two states are interconverted
by 𝒫𝒫 and by 𝒯𝒯 , for example, magneto-optical Faraday rotation.
Chirality was further extended to the dynamics of quasi-particles.
Possible band structure shapes have been classified in terms ofpseudo-momentum and pseudo-angular momentum (PAM)4, which
originate from the phase factors acquired by discrete translation
and rotation symmetry operations on wave functions, respectively.
In crystals, these are different from the momentum and angularmomentum (AM), which are continuous linear and circular motionsof an object in real space, respectively. Recently, circularly rotating
atomic motion with non-zero PAM and AM at high-symmetry points

in the Brillouin zone in monolayer hexagonal lattices was theoretically predicted and named ‘chiral phonons,5,6. This prediction wasverified experimentally in the transition-metal dichalcogenide WSe2
via transient infrared spectroscopy7.
Employing Barron’s true chirality, one can extend the concept ofa chiral phonon to ‘truly chiral phonons’, which have two enantiomericchiral modes that propagate with finite group velocity while circularly
rotating in three-dimensional materials wherein PAM and AM have
opposite signs. These modes should be interconverted by 𝒫𝒫 but not byℛ𝒯𝒯 . The observation of truly chiral phonons would require the use ofchiral materials, as has been proposed theoretically8. Several Raman
experiments designed to identify phonon symmetry in chiral crystals
have been conducted9–15. Frequency-degenerated phonon modes wereobserved to undergo splitting at the centre of the Brillouin zone (Γ3doublets) in α-quartz and trigonal tellurium9,11. Moreover, theoreticalwork on phonon band dispersion has been reported on α-HgS (cinnabar), where the splitting was attributed to the linear wave vector
shifts caused by chirality16. Nonetheless, previous studies have mostly

1
Department of Physics, Tokyo Institute of Technology, Tokyo, Japan. 2Department of Physics and Electronics, Osaka Metropolitan University, Osaka,
Japan. 3Research Institute for Interdisciplinary Science, Okayama University, Okayama, Japan. 4Tokodai Institute for Element Strategy (TIES), Tokyo
Institute of Technology, Tokyo, Japan. 5Division of Natural and Environmental Sciences, The Open University of Japan, Chiba, Japan. 6Institute for
Molecular Science, Okazaki, Japan. 7These authors contributed equally: Kyosuke Ishito and Huiling Mao.
e-mail: satoh@phys.titech.ac.jp

Nature Physics | Volume 19 | January 2023 | 35–39

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https://doi.org/10.1038/s41567-022-01790-x
aThree-fold rotation
around c axis

d
c

Hg
5

S
5

c
2

PAM = –1

4

5

PAM = 0
Г

N

A

a

b

e

a

6

5

2

PAM = +1

1

1

c

6
1

b

4

3

a

b

2

3

c

6

4
3

4

c

a

1

c

3

b

2

3

4

6
1

6

5

2

a

b

f

b
A

S

∆
Г

T

H
L

S’
T’

4

5
P
K

M

c

a

4

3

6

1

c
5

1
2
b

b

2

3

6
a

Fig. 1 | Chiral properties of α-HgS. a, Right-handed helical chain along the c axisof R-HgS. Each Hg atom (red sphere) is bonded with two S atoms (blue spheres).
Chemical bonds are denoted by green lines. b, Brillouin zone of α-HgS. The bluelines indicate the reciprocal lattice vectors. The green lines and dots represent
the high-symmetry paths and points, respectively. c, Schematic diagram ofphonon dispersion from the Γ to A point. Filled red, blue and black circles at the N
point correspond to the phonon modes of d, e and f, respectively. d–f, Twodimensional projections of schematic atomic motions (Hg, gold arrows; S, blue

arrows) in R-HgS at point N for a Δ2 mode with msPAM = +1 and mAM > 0 (d), a Δ3mode with msPAM = −1 and mAM < 0 (e) and a Δ1 mode with msPAM = 0 and mAM ≃ 0(f). The black lines indicate the lattice vectors. Counter-clockwise, clockwise
circular and linear motions indicate that mAM is positive, negative and zero,
respectively. The three-fold rotation symmetry operation generates phasefactors of atomic motions, namely PAM msPAM. The phase difference at theposition of each atom after the operation is shown.

focused on examining the phonon symmetry without considering
PAM and AM.
Here we used circularly polarized Raman spectroscopy (Methods)
to observe the phonon modes and their splitting in chiral crystals of
α-HgS. Moreover, we performed first-principles calculations (Methods) to compute the dispersion, PAM and AM of phonons in α-HgS. A
comparison between the experiment and calculation shows that the
split phonons were indeed truly chiral phonons. We also confirmedthat the conservation law of PAM holds in circularly polarized Raman
scattering by taking into account the three-fold rotational symmetry.
Trigonal α-HgS crystals contain two enantiomeric cinnabar structures with right- and left-handed helices (R- and L-HgS), which belong
to the space groups P3121 (translation by c/3 in three-fold rotation) andP3221 (translation by 2c/3 in three-fold rotation), respectively17. This crystal has a three-fold helical axis along the c direction (Fig. 1a), with large
optical activity18. Our samples were single crystals of naturally grown
α-HgS with lateral dimensions of 3 mm. We selected a c-axis-oriented
as-grown surface of R-HgS, which was confirmed by X-ray analysis.
The phonon modes at the centre of the Brillouin zone (Γ point) of
(1−3)
(1,2)
α-HgS are classified into two Γ1 singlets (Raman active), three Γ2(1−5)
singlets (infrared active) and five Γ3 doublets (Raman and infrared(1,2)
active). Our measurement led to the observation of the Γ1 singlets(2−5)
and Γ3
doublets, which were assigned on the basis of the selection
rule19 for the four incident/scattered polarization configurations (RR,
LL, RL and LR), where IRR:ILL:IRL:ILR = 1:1:0:0 and 0:0:1:1 for the Γ1 and Γ3
modes, respectively (Extended Data Fig. 1). The transverse optical (TO)phonons of the Γ3 mode were generated because the incident and scattered light were propagated along the c axis. A weak signal was detected
for the longitudinal optical (LO) phonon at ~145 cm−1, presumably owing
to misalignment of the experimental setup.

The phonon frequencies obtained by the Raman experiment arecompared with the results of the first-principles calculations and withprevious results10,16 in Extended Data Table 1. The results of the Raman
experiments both of our study and of a previous study10 are identicalto within 3 cm−1. The results of our calculations at the Γ point also closely
approximate those in ref. 16, and the small differences are attributed
to the difference in the positions of the atoms. In our experiments, the(1)Γ3 mode was not observed because this mode is nearly degenerate(1)with the Γ1 mode, which has a much higher intensity. In Extended Data
(2−5)
Table 1, the frequencies of the Γ3modes obtained in our Raman
experiment are averaged for the RL and LR configurations, which weresplit as shown in the next paragraph.
The Stokes and anti-Stokes Raman spectra (Raman optical activity)(2−5)
of each Γ3mode are shown in Fig. 2a–h. All four Γ3 modes wereobserved with the opposite circularly polarized light between theincident and scattered light (RL and LR). On the anti-Stokes spectra(2)(4)(Fig. 2a–d), the absolute values of the Raman shift of the Γ3 and Γ3
modes in the RL configuration are lower than those in the LR configuration in Fig. 2a,c. In Fig. 2b,d, the absolute values of the Raman shift of(3)(5)
the Γ3 and Γ3 modes in the RL configuration are higher than those in
the LR configuration. On the Stokes spectra (Fig. 2e–h), the splittings
are mirror images of those on the anti-Stokes spectra (Fig. 2a–d). The(1,2)
Γ1 modes do not split.
The properties of phonons are discussed here on the basis of our
first-principles calculations. Figure 3a displays the phonon dispersion
curve calculated along the path Δ from the Γ to A point, which is parallel(1−5)
to the c axis (Fig. 1b). Linear splittings of Γ3
modes appear aroundthe Γ point, which is consistent with previous calculations16,20. Fromthe conservation law of pseudo-momentum, the wavenumber k ofphonons observed by backscattering Raman spectroscopy is expressed

Nature Physics | Volume 19 | January 2023 | 35–39

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Letter

https://doi.org/10.1038/s41567-022-01790-x
a

e

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

–90

–0.4 ± 0.01 cm–1
–89

–88

–87

–86

–85

–84

–83

–82

–81

–80

b

f

1.0

1.0

0.9

0.9

0.8

0.8

0.7

Normalized intensity (a.u.)

80

0.6

0.3 ± 0.04 cm

83

84

85

86

87

88

89

90

105

106

107

108

284

285

286

287

346

347

348

–0.3 ± 0.16 cm–1

0.5

–108 –107 –106 –105 –104 –103 –102 –101 –100 –99

–98

98

g

1.0

1.0

0.9

0.9

0.8

0.8

0.7

0.7

0.6

99

100

101

102

103

104

0.6
–0.4 ± 0.06 cm–1

0.4 ± 0.08 cm–1

0.5

0.4

0.4

–287 –286 –285 –284 –283 –282 –281 –280 –279 –278 –277

277

d

h

1.0

1.0

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5
0.3

82

0.6

c

0.4

81

0.7
–1

0.5

0.5

0.3 ± 0.01 cm–1

278

279

280

281

282

283

0.5
0.4

0.2 ± 0.01 cm–1

–0.3 ± 0.05 cm–1

0.3

–348 –347 –346 –345 –344 –343 –342 –341 –340 –339 –338

338

339

340

341

342

343

344

345

Raman shift (cm–1)

Fig. 2 | Raman spectra of R-HgS. a–h, Anti-Stokes (a–d) and Stokes (e–h)
spectra of the Γ3(2) (a,e), Γ3(3) (b,f), Γ3(4) (c,g) and Γ3(5) (d,h) modes, along with theΓ3 modes observed experimentally with RL- (blue line) and LR-polarized (redline) configurations and the Γ3 doublet splittings (dashed vertical lines) in a–h.
The error values take into account that multiple splitting measurements wereconducted in the experiment, owing to the roughness of the sample surface.

as k = 4πn/λ, where λ is the wavelength of the incident or scattered light,
and n = 2.77 is the absolute value of the complex refractive index of
α-HgS21. The value of k was calculated to be ~1/80 of the wavenumber
π/c at the A point. We denoted this specific point on the path Δ as ‘N’ in
Fig. 3a. The experimental splitting values of the Γ3 modes at this point(Stokes scattering) are almost consistent with the calculations(Extended Data Table 2). Therefore, we attribute the splits observed inFig. 2 to the linear k shifts at the N point.
To understand the selection rule of the incident/scattered polarization configurations on the Raman spectra in Fig. 2, we calculated thePAM of phonons in R-HgS (Methods). As shown in Fig. 3a, the Γ3 doublets
have spin PAM msPAM = ±1, whereas the Γ1 and Γ2 singlets have msPAM = 0.
The phonon bands with msPAM = −1 and msPAM = 0 intersect at the A
point. These are the common features of the 31 (right-handed) helix4,9,22.
(2−5)
From Figs. 2e–h and 3a, Γ3phonons with msPAM = +1 and −1 wereobserved in the LR and RL configurations, respectively (Extended Data
Table 2). Γ1 phonons with msPAM = 0 were observed in the RR and LL

configurations. Right- and left-handed circularly polarized light possess PAM of σ = +1 and −1, respectively23. Therefore, we confirmed aconservation law to exist between the spin PAM of phonons and thePAM of the incident/scattered photons in the Stokes spectra as

Nature Physics | Volume 19 | January 2023 | 35–39

σs − σi = −msPAM + 3p,(1)

where σs and σi represent the PAM of the scattered and incident photons, respectively. In addition, p = 0 and ±1 for the Γ1 and Γ3 modes,
respectively. The factor 3p on the right-hand side of equation (1) canbe understood by considering the three-fold rotational symmetry of
α-HgS and the Umklapp process in Raman scattering4,24–27.
In crystals with discrete rotational symmetry, a clear distinction
should be made between the PAM and AM of the phonons. The AM ofphonons arises from the circular vibration of atoms in the real space,as shown in Fig. 1d–f (see also Supplementary Videos 1 and 2). The
AM of phonons along the c axis is displayed in Fig. 3b along with the37Letter

https://doi.org/10.1038/s41567-022-01790-x

a

b
350

0
–1
+1

0
–1

+1
250 –1
0

+1
–1
0

(5)
3

300

–ħ

+1

(4)
3

Frequency (cm–1)

ħ

200
150

0

(3) –1
3
100 +1+1
(2)
3
–1
50 –1

0

ГN

0
–1
+1

0 +1

0 +1

+1
–1
0 –1
+1+1 00
–1

(∆)

A

0
–1

ГN

(∆)

A

Fig. 3 | Dispersion of R-HgS along the right-handed helical axis. a, Phonon
dispersion curves for spin PAM msPAM = 0 (black), +1 (red) and −1 (blue). The
wavenumbers at the N and A points are in the ratio of 1:80. b, Phonon AM along
the helical axis, denoted by the colour gradient. Red and blue curves correspondto the positive and negative AM, respectively.

Γ to A point (Methods). In the vicinity of the Γ point, the Γ3 modes
have clockwise and counter-clockwise rotations, corresponding to
the positive and negative AM of phonons, respectively. This is a clear
manifestation of the existence of chiral phonons. The Γ1 modes havelinear vibrations, and their phonons have zero AM. The signs of AM at
the N point are listed in Extended Data Table 2, where the spin PAM and
AM of phonons do not correspond with each other. This indicates that,
contrary to PAM, the AM of phonons is not conserved during the Ramanprocess in a crystal with discrete rotational symmetry.
We also calculated the PAM and AM of phonons in L-HgS (Extended
Data Fig. 2). The signs of the spin PAM and AM are reversed with respect
to Fig. 3, that is, msPAM (RH, k, j𝒫 = −msPAM (LH, k, j𝒫 and mAM(RH, k, j) = −mAM(LH, k, j), where RH and LH denote crystals with right- andleft-handed helices, respectively. This means that the splitting of theΓ3 modes arises from chirality. From the first-principles calculation,
the phonon eigenvector at the N point u(RH, k, j) and u(LH, k, j) areco nve r te d by 𝒫𝒫 , 𝒯𝒯 a n d 𝒞𝒞2 a s 𝒫𝒫u(RH, k, j𝒫 = u(LH, −k, j𝒫 ,𝒞𝒞2 𝒯𝒯u(RH, k, j𝒫 = 𝒞𝒞2 u(RH, −k, j𝒫 = u(RH, k, j𝒫, where 𝒞𝒞2 is a two-fold rotation around the a, b or a + b direction. These relations satisfy the definition of true chirality with two different enantiomeric phonon modes:u(RH, k, j) and u(LH, −k, j).
We also recorded the Raman spectra of an α-HgS crystal for which
the handedness (chirality) of the measured region was unknown
because the size of the chiral domain was smaller than the experimental resolution (~1 mm) of our X-ray diffractometer. Contrary to the
X-ray analysis, our Raman experiment with a spatial resolution of a few
microns succeeded in observing the splitting of the Γ3 modes (Extended
Data Fig. 3), and the directions of the splittings were opposite to
the results in Fig. 2. Considering the data in Extended Data Table 3,
the chirality of the sample at the point of measurement was found to
be left-handed.
Space group analysis enabled us to additionally assign an irreducible representation of the phonon dispersion curve in R-HgS. On path
Δ from Γ to A, three irreducible representations are known to exist: Δ1,
Δ2 and Δ3 (refs. 20,28). The labels of the irreducible representations are
based on the convention in ref. 28. The Γ1 and Γ2 modes at the Γ pointchange to the Δ1 mode, whereas the Γ3 modes split into Δ2 and Δ3 modesin accordance with the compatibility relations. In Supplementary
Note 1, we present a derivation of the Raman tensors that correspondto the Δ2 and Δ3 modes. It follows that the Raman intensities of the Δ2and Δ3 modes are in the ratio of IRR:ILL:IRL:ILR = 0:0:1:0 and 0:0:0:1, respectively. This is consistent with the results of the Raman experiments.
Nature Physics | Volume 19 | January 2023 | 35–39

In other words, spin PAM msPAM (RH𝒫 = 0, +1 and −1 correspond to Δ1, Δ2
and Δ3, respectively, because the phase change by three-fold rotationis the index of each irreducible representation. For L-HgS, msPAM (LH𝒫= 0,+1 and −1 correspond to Δ1, Δ3 and Δ2, respectively28.
Finally, we discuss the propagation of the chiral phonons. From
(1−5)
Fig. 3, the group velocities of the Γ3modes at the N point are calculated to be approximately ±(0.4–2) × 103 m s−1, which is comparable
to the sound velocity of acoustic phonons. Note that the nonreciprocal propagation of chiral phonons can be controlled by the PAM of
the photon. This implies the possibility of transferring the PAM fromphotons to electron spins via propagating chiral phonons in
opto-phononic-spintronic devices. For example, taking advantageof the long coherence of long-wavelength phonons, transferring the
PAM from phonon to electron spins may be realized on the macroscopic scale.
Weyl phonons can exist widely in chiral crystals, as has been predicted by the first-principles calculations and verified by inelastic X-ray
scattering29,30. Note that the circularly polarized Raman spectroscopy
presented in this paper may open the possibility to detect phonons carrying a non-zero Chern number by measuring the phonon PAM, insteadof detecting the eigenvalue and eigenvectors of the topological bands.
We observed chiral phonons in a three-dimensional chiral systemusing circularly polarized Raman spectroscopy and first-principles
calculations. The chiral phonons were labelled with spin PAM of {+1, −1}
corresponding to {Δ2, Δ3} and {Δ3, Δ2} for R- and L-HgS, respectively,
with opposite group velocities of ~1 km s−1. The parity and time-reversal
symmetries of the phonons satisfy the definition of truly chiral phonons, which propagate while rotating along the c axis. This is distinct
from the chiral phonons observed in two-dimensional hexagonal
systems. Our work also provides an optical method to identify thehandedness of chiral materials using PAM, and we demonstrated thatspatial imaging of chiral domains can be achieved in a non-contact andnon-destructive manner.

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acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41567-022-01790-x.

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39

Letter

https://doi.org/10.1038/s41567-022-01790-x

Methods

Raman scattering
We used a custom-built Raman spectroscopy system31 equipped witha monochromator, charge-coupled device camera and optical elements. Details of the setup can be found in Supplementary Fig. 2. The
excitation light was generated by a diode laser with a wavelength of
λ = 785 nm. The incident and scattered light were propagated along
the c axis of the crystal. All the measurements were performed at room
temperature.

which is quantized as an integer. The latter factor, exp [ik ⋅ Rl ], provides
1
the orbital PAM moPAM, which is therefore equal to k ⋅ c under a2𝜋𝜋three-fold helical rotation. Consequently, for each wavenumber k, thetotal PAM ismPAM = msPAM + moPAM .

(6)

Here we considered only the spin PAM in the phonon dispersion, as
shown in Fig. 3a.

First-principles calculations

Calculation of AM

First-principles calculations were performed using the ABINIT package32,33, which implements density functional perturbation theory34.
An anaddb code was used to obtain the phonon dispersion curve35,36.
For the calculation of the phonon dispersion, the exchange-correlationenergy of electrons is described in the generalized gradient approximation with norm-conserving pseudo-potentials37,38, and spin–orbit
coupling was not considered. We set the kinetic energy cut-off to
60 Ha, which was necessary to obtain a well-converged result, andthe wave vector k grids of 6 × 6 × 4 to describe the phonon dispersion curve. In addition, we increased the precision of the k grids to8 × 8 × 8 to obtain the converged phonon frequency at the Γ point.
The following lattice parameters of α-HgS were used: cell volume of
V0 = 160.02 Å3, cell lengths of a = 4.11 Å, b = 4.11 Å and c = 9.40 Å, andcell angles of α = 90∘, β = 90∘, and γ = 120° (refs. 39,40). R-HgS is composed of three Hg atoms at (0.699, 0.699, 0.000), (0.000, 0.301, 2/3)and (0.301, 0.000, 1/3) and three S atoms at (0.509, 0.509, 1/2),(0.000, 0.491, 1/6) and (0.491, 0.000, 5/6), whereas L-HgS is composed of three Hg atoms at (0.699, 0.699, 0.000), (0.301, 0.000, 2/3)and (0.000, 0.301, 1/3) and three S atoms at (0.509, 0.509, 1/2),(0.491, 0.000, 1/6) and (0.000, 0.491, 5/6), of which the bases are thea, b and c components. The essential part of the ABINIT results was
confirmed using the VASP package.

The AM of phonons at wavenumber k of mode number j is defined as41†mAM (k, j𝒫 = (ϵ(k, j𝒫 Mϵ(k, j𝒫) ℏ,
0 −i 0where M = ( i 0 0 ) ⊗ In×n, where the basis of the 3 × 3 matrix is repre0 0 0

sented using the orthogonal bases (ux, uy, uz), n is the number of atoms
in a unit cell and In×n is a unit matrix of n × n. The eigenvector of the
dynamical matrix ϵ(k, j) is normalized as†ϵ(k, j) ϵ(k, j𝒫 = 1.

(8)The origin of spin and orbital PAMs
Here we provide an explicit form of the spin and orbital PAMs associatedwith the helical operation {C3 | c/3}, where c/3 is the non-primitive translation. The atomic displacement vector field uk(r) associated with thephonon propagation may be generally written asuk (r𝒫 = ϵ(k𝒫 exp (ik ⋅ r) .

(9)Then, uk(r) is transformed in accordance with a general rule−1

{C3 | c/3} uk (r𝒫 = [C3 ϵ(k𝒫] exp (ik ⋅ {C3 |c/3} r) .

Calculation of PAM
In a material with three-fold rotational symmetry, PAM is defined as5,4
2𝜋𝜋{C3 | 0} u(k𝒫 = exp [−i mPAM (k𝒫] u(k𝒫,3(2)

where mPAM(k) is the PAM of phonons, u(k) is the displacement vectorand {C3 | 0} is the three-fold rotation around the c axis. However, α-HgS
has a three-fold helical rotational symmetry instead of a three-foldrotational symmetry. Therefore, we defined the PAM of phonons in
R- and L-HgS as27
{C3 | c/3} u(RH, k𝒫 = exp [−i2𝜋𝜋mPAM (RH, k𝒫] u(RH, k𝒫,
32𝜋𝜋{C3 | 2c/3} u(LH, k𝒫 = exp [−i mPAM (LH, k𝒫] u(LH, k𝒫,
3

(3)

(4)

where {C3 | c/3}and {C3 | 2c/3}are combinations of the three-fold rotation
around the c axis and the c/3 and 2c/3 translations along the c axis,
respectively.
The displacement vector of phonons can be expressed as−

1uκ (k, l, j𝒫 = mκ 2 ϵκ (k, j𝒫 exp [i {k ⋅ Rl − ω(k, j𝒫t}] ,(5)where j is the number of phonon modes, mκ is the mass of the κth atomin the lth unit cell, ϵκ(k, j) is the eigenvector of the dynamical matrix
and Rl is the position of the lth unit cell. The use of equation (5) indicatesthat the phase of the eigenvector consists of two parts: ϵ(k) andexp [ik ⋅ Rl ], both of which contribute to the calculation of the PAM. In
the former, the phase difference of ϵ(k) leads to the spin PAM msPAM,
Nature Physics

(7)

(10)

−1

= {C−1| − C−1c/3} , k⋅ r is a scalar and33m) ϵ(k𝒫, thus we have

Note that {C3 |c/3}
C3 ϵ(k𝒫 = exp (−i2𝜋𝜋3{C3 | c/3} uk (r𝒫 = exp [−i2𝜋𝜋(msPAM + moPAM )] uk (r𝒫,3

(11)

where the respective spin and orbital PAMs are introduced as
msPAM = m,

(12)

and
moPAM = −3
1{(C3 k) − k} ⋅ r +(C3 k) ⋅ c,
2𝜋𝜋
2𝜋𝜋

(13)

which yield the total PAM, mPAM = msPAM + moPAM.
In this study, we consider the path Δ from Γ to A, where C3k = kdenotes the modulo reciprocal lattice vectors. Thus, moPAM = 1 k ⋅ c.
2𝜋𝜋
The same symmetry properties apply to the photons propagating along
the helical axis. Then, we obtain the selection rule in equation (1), which
includes only msPAM, by cancelling out the moPAM terms because of the lawof conservation of momentum.

Data availability

Source data are provided with this paper.

Code availability

All custom codes used for the data processing and numerical simulations are available from the corresponding author upon reasonable
request.

Letter

References

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174432 (2020).
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and nanosystem properties. Comput. Phys. Commun. 180,
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33. Romero, A. H. et al. ABINIT: overview, and focus on selectedcapabilities. J. Chem. Phys. 152, 124102 (2020).
34. Hamann, D. R., Wu, X., Rabe, K. M. & Vanderbilt, D. Metric tensor
formulation of strain in density-functional perturbation theory.
Phys. Rev. B 71, 035117 (2005).
35. Gonze, X. First-principles responses of solids to atomic
displacements and homogeneous electric fields: implementationof a conjugate-gradient algorithm. Phys. Rev. B 55,
10337–10354 (1997).
36. Gonze, X. & Lee, C. Dynamical matrices, Born effective charges,
dielectric permittivity tensors, and interatomic force constantsfrom density-functional perturbation theory. Phys. Rev. B 55,
10355–10368 (1997).
37. van Setten, M. J. et al. The PseudoDojo: training and grading a
85 element optimized norm-conserving pseudopotential table.
Comput. Phys. Commun. 226, 39–54 (2018).
38. Hamann, D. R. Optimized norm-conserving Vanderbilt
pseudopotentials. Phys. Rev. B 88, 085117 (2013).
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and the Einstein–de Haas effect. Phys. Rev. Lett. 112,085503 (2014).

Nature Physics

https://doi.org/10.1038/s41567-022-01790-x

Acknowledgements

We thank M. Kichise, A. Kawano, K. Matsumoto, A. Koreeda, Y. Fujii,
E. Oishi and H. M. Yamamoto for valuable discussions and technicalsupport. T.S. was supported financially by the Japan Society for the
Promotion of Science KAKENHI (grant nos. JP19H01828, JP19H05618,
JP19K21854, JP21H01032 and JP22H01154) and the Frontier Photonic
Sciences Project of the National Institutes of Natural Sciences (grantnos. 01212002 and 01213004).

Author contributions

T.S. and J.K. conceived the study. K.I. and H.M. performed the Raman
experiments. Y.K., Y.T. and S.I. conducted the X-ray analyses. K.I., H.M.,
T.Z. and S.M. performed the first-principles calculations. K.I., H.M., J.K.
and T.S. wrote the manuscript. All authors discussed the results andcommented on the manuscript.

Competing interests

The authors declare no competing interests.

Additional information

Extended data is available for this paper athttps://doi.org/10.1038/s41567-022-01790-x.
Supplementary information The online version contains supplementarymaterial available at https://doi.org/10.1038/s41567-022-01790-x.
Correspondence and requests for materials should be addressed
to Takuya Satoh.
Peer review information Nature Physics thanks Przemyslaw Piekarz
and the other, anonymous, reviewer(s) for their contribution to thepeer review of this work. This article has been peer reviewed as part ofSpringer Nature’s Guided Open Access initiative.
Reprints and permissions information is available atwww.nature.com/reprints.

Letter

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Extended Data Fig. 1 | Full Raman spectra of R-HgS. Full Raman spectra were recorded at room temperature using various polarized configurations. The black,green, red and blue spectra represent the LL-, RR-, LR- and RL-polarized configurations, respectively. R and L represent the right- and left-handed helicities of circularly
polarized light, respectively.

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Extended Data Fig. 2 | Dispersion of L-HgS along the left-handed helical axis. (a) Phonon dispersion curves for spin PAM msPAM = 0 (black), +1 (red) and − 1 (blue).
(b) Phonon AM along the helical axis, denoted by the colour gradient. Red and blue curves correspond to the positive and negative AM, respectively.

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Extended Data Fig. 3 | Raman spectra of α-HgS with unknown chirality. (a–h),
Anti-Stokes (a–d) and Stokes (e–h) spectra of the Γ3(2) (a,e), Γ3(3) (b,f), Γ3(4) (c,g)
and Γ3(5) (d,h) modes, along with the Γ3 modes observed experimentally with
RL- (blue line) and LR-polarized (red line) configurations and the Γ3 doublet

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splittings (dashed vertical lines) in (a–h). The error values take into account thatmultiple splitting measurements were conducted in the experiment, owing to theroughness of the sample surface.

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Extended Data Table 1 | phonon frequencies in α -HgS. The experimental values in cm−1 were obtained at room temperature. The calculated values in cm−1 are atthe Γ point.

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Extended Data Table 2 | Chiral properties of phonons in R-HgS along the right-handed helical axis. Owing to the roughness of the sample surface, multiple
splitting measurements were conducted, with the deviation indicated by the error values.

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Extended Data Table 3 | Chiral properties of phonons in α-HgS with unknown chirality. The error values take into account that multiple splitting measurements
were conducted in the experiment, owing to the roughness of the sample surface. Calculations were performed for L-HgS.

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