# Physics of Phonons in Systems with Approximate Screw Symmetry

Hisayoshi Komiyama  Tiantian Zhang  Shuichi Murakami  Department of Physics,
Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551,
Japan

###### Abstract

Properties of systems with exact $n$-fold screw symmetry $(n=2,3,4,6)$ have
been well studied because they can be understood in terms of space groups. On
the other hand, existence of materials with approximate screw symmetries, such
as 7-fold and 10-fold screw symmetries, has been predicted. In this paper, we
study properties of phonons in crystals with approximate screw symmetries,
which will lead to unique and new physical phenomena. In a crystal with an
approximate screw symmetry, we propose a method to extract information of
pseudoangular momentum of phonons, which is a quantum number characterizing
the properties of phonon modes under screw symmetry, based on the fact that
the information of the quantum numbers defined under exact screw symmetry
partially remains in the eigenvectors of approximate screw symmetric systems.
As a preparation, we study a one-dimensional crystal with partially broken
translation symmetry to have an enlarged unit cell, and we show how to extract
information of a quantum number corresponding to the pseudoangular momentum,
by studying a relative phase between neighboring atoms. We also extend this
method to systems with an approximate screw symmetry, and discuss properties
of the pseudoangular momentum. We apply this method to results of our first-
principle calculations on candidate materials with an approximate
translational symmetry or with an approximate screw symmetry, and show how
this approximate symmetry is reflected in the phonon wavefunctions.

##  I Introduction

A phonon is a quasiparticle of quantized lattice vibrations and has long been
studied to account for various physical properties such as specific heat,
transport and electric resistance. Recently, chiral phonons, in which atomic
motions have rotational components with chirality, have been predicted
theoretically [1] and observed experimentally [2]. In addition, chiral phonons
are expected to have a variety of applications and have been studied
extensively [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20].
Furthermore, the concept of chiral phonons has been studied in crystals with
screw symmetry, and interactions with other particles with chirality and
experimental methods have been proposed [21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39].

So far, properties of chiral phonons have been studied in materials with exact
rotational and screw symmetries. In this case, the chiral phonons are
characteristic by a quantity called pseudoangular momentum [40, 1, 25], which
is used to discuss interactions with other particles. The definition of the
pseudoangular momentum is based on rotation or screw symmetry. As
crystallographic symmetries, those symmetries are limited to $n$-fold rotation
and screw rotation symmetries with $n=2,3,4,6$. On the other hand, materials
with approximate screw symmetries such as $7_{2}$ screw and $10_{7}$ screw
symmetries are predicted. Since pseudoangular momentum cannot be defined for
the approximate screw symmetry, pseudoangular momentum cannot be defined
directly in such systems. However, we can expect that materials with
approximate symmetries have the same pseudoangular momentum information as
those with strict screw symmetries. Therefore, in this paper, we discuss how
to extract information of pseudoangular momentum for the phonons in systems
with approximate symmetry, starting with approximate translation symmetry, and
then extending to approximate screw rotation symmetries. Furthermore, we also
discuss characteristic behaviors of the phonon eigenmodes in these materials
with approximate symmetries. Such properties under approximate symmetry affect
physical processes involving various particles/quasiparticles, such as
electronic processes involving multi-phonons/photons and exciton scattering
processes. In systems with exact symmetry, symmetry restricts such processes
as selection rules, and even when the symmetry is not exact but approximate,
selection rules remain valid. While exact screw rotation symmetries are
limited in crystals to twofold, threefold, fourfold and sixfold ones due to
the restriction of translational symmetry, an approximate screw symmetry leads
to a richer variety of multifold screw symmetries such as 7- and 10-fold ones,
and they will lead to selection rules that are absent for exact screw
symmetries. Furthermore, anisotropy in transport properties involving chiral
phonons will also be reflected the approximate screw symmetry. Thus, the
approximate screw symmetries affect phonon-related transport phenomena.

This paper is organized as follows. In Sec. II, we study phonons in a one-
dimensional crystal with approximate translation symmetry, i.e., slightly
broken translation symmetry to have an enlarged unit cell. From the relative
phases between neighboring atoms in the case with the exact translation
symmetry, we show how to obtain the quantum number, which correponds to the
pseudoangular momentum in 3D systems, for crystals with approximate
translation symmetry In Sec. III, we show how to extract the phonon
pseudoangular momenta for crystals with approximate screw symmetries, which is
also obtained from the exact screw symmetries, and discuss their properties.
In Sec. IV, we apply this method to first-principle calculations on phonon
modes in candidate materials and show that the approximate symmetries are well
reflected in phonon wavefunctions. In Sec. V, we summarize this paper.

##  II  One-dimensional phonon

Figure 1:  Phase differences between neighboring atoms for the one-dimensional
phonon system. (a) Schematic picture of the system (a-1) with and (a-2)
without $\hat{T}_{a/n}$ translation symmetry. (b-1), (b-2) Band dispersions
$\omega_{k}$ for the systems with (b-1) exact translational symmetry
$\hat{T}_{a/3}$ ($c_{1}=c_{2}=c_{3}=1$) and (b-2) broken translational system
$\hat{T}_{a/3}$ ($c_{1}=1$, $c_{2}=1.1$, $c_{3}=1.2$). (c-1), (c-2) Relative
phase $\theta_{i}^{(\sigma=2)}(k)$ of the second band from the bottom and
(d-1), (d-2) Trajectory of the complex number $z_{i}(k)$ of the second band
from the bottom ($\sigma=2$). (c-1), (d-1) correspond to the case with exact
translational symmetry in (b-1), and (c-2), (d-2) to the case with broken
translation symmetry $\hat{T}_{a/3}$ in (b-2).

In this section, we study phonon quantum number in one-dimensional crystals
with partially broken translational symmetry, i.e., approximate translation
symmetry, such that the unit cell is enlarged by a factor $n$ ($n$ : integer),
and discuss phase properties of the phonon eigenvectors. We consider a one-
dimensional chain of atoms with equal spacing $a/n$ ($a$: constant, $n$:
integer), and the atoms can oscillate only along the chain direction. We begin
with the case with all the atoms being equivalent, leading to a translation
symmetry by $a/n$ (Fig. 1 (a-1)).

Then we perturb the system while preserving a translation symmetry by $a$
(Fig. 1 (a-2)). Thus, the size of the resulting unit cell becomes $a$, having
$n$ atoms. The Hamiltonian of the one-dimensional phonon system is

$\displaystyle
H=\sum_{l,l^{\prime}}\left(\frac{1}{2}\bm{p}_{l}^{\mathrm{T}}\bm{p}_{l}\delta_{l.l^{\prime}}+\frac{1}{2}\bm{u}_{l}^{\mathrm{T}}K_{l,l^{\prime}}\bm{u}_{l^{\prime}}\right),$
(1)

where $\bm{u}_{l}=\left(u_{l,1},u_{l,2},\cdots,u_{l,n}\right)^{\mathrm{T}}$,
$u_{l,i}$ is a displacement of the $i$th atom in the $l$th unit cell
multiplied by the square root of the mass of the atom, $p_{l,i}$ is a
conjugate momentum corresponding to $u_{l,i}$. The atoms are mutually
connected by springs, and $K_{l,l^{\prime}}$ is a mass-weighted force constant
matrix.

Because of the translation symmetry by $a$, the Hamiltonian $H$ commutes with
a translation operator $\hat{T}_{a}$ by length $a$:
$\left[H,\hat{T}_{a}\right]=0$. Therefore, from the Bloch’s theorem, the
eigenfunction $\psi(x)$ of the Hamiltonian $H$ satisfies
$\hat{T}_{a}\psi(x)=e^{-ika}\psi(x)$ ($\frac{-\pi}{a}<k\leq\frac{\pi}{a}$),
where $k$ is the Bloch wavenumber corresponding to the translation
$\hat{T}_{a}$. Furthermore, if the system preserves the translation symmetry
by length $a/n$ represented by $\hat{T}_{a/n}$,
$\left[H,\hat{T}_{a/n}\right]=0$ holds and the eigenfunction $\psi(x)$
satisfies $\hat{T}_{a/n}\psi(x)=e^{-ik^{\prime}a/n}\psi(x)$
$\left(-\frac{n\pi}{a}<k^{\prime}\leq\frac{n\pi}{a}\right)$ where $k^{\prime}$
is the Bloch wavenumber corresponding to the translation $\hat{T}_{a/n}$. By
using $\left(\hat{T}_{a/n}\right)^{n}=\hat{T}_{a}$, we can relate $k$ and
$k^{\prime}$ as $e^{ika}=e^{ik^{\prime}a}$, i.e
$k^{\prime}=k+\frac{2m\pi}{a}$, ($m$: integer). Therefore, we have

$\displaystyle T_{a/n}\psi(x)=e^{-i(ka+2\pi m)/n}\psi(x)\
\left(\frac{-\pi}{a}<k\leq\frac{\pi}{a}\right)$ (2)

where $m$ is given by

$\displaystyle m\equiv\left\\{\begin{array}[]{ll}0,\pm
1,\cdots,\pm(n-1)/2&\text{for}\ n\ \text{odd}\\\ 0,\pm
1,\cdots,\pm(n-2)/2,n/2&\text{for}\ n\ \text{even}\end{array}\right.\pmod{n}.$
(5)

This integer $m$ is regarded as a quantum number having information of the
chain with exact $\hat{T}_{a/n}$ symmetry and is defined in terms of modulo
$n$, as we discuss in the following. From now on, in this section, we discuss
how the information of the quantum number $m$ remains in the eigenfunction
when the translational symmetry $\hat{T}_{a/n}$ is slightly broken while
keeping the translational symmetry $\hat{T}_{a}$.

We label the atoms in the unit cell of the one-dimensional system to be
$1,2,\cdots,n$, and we assume that the springs exist only between neighboring
atoms (Fig. 1 (a-2)). Then, the phonon dynamical matrix $D(k)$ for the system
is written as

$\displaystyle
D(k)=\begin{pmatrix}c_{1}+c_{n}&-c_{1}&\cdots&0&-c_{n}e^{-ika}\\\
-c_{1}&c_{1}+c_{2}&\cdots&0&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&\cdots&c_{n-2}+c_{n-1}&-c_{n-1}\\\
-c_{n}e^{ika}&0&\cdots&-c_{n-1}&c_{n-1}+c_{n}\end{pmatrix},$ (6)

where $k$ is the Bloch wavenumber corresponding to the translation
$\hat{T}_{a}$ and has a value in range $-\frac{\pi}{a}<k\leq\frac{\pi}{a}$.
Let $c_{i}$ $(i=1,\cdots,n-1)$ denote the spring constant between atom $i$ and
atom $i+1$ and $c_{n}$ denote the spring constant between atom $n$ and atom
$1$ of the neighboring unit cell. The crystal has the translational symmetry
$T_{a/n}$ when the spring constants $c_{i}$ $(i=1,\cdots,n)$ are all equal, in
which case the eigenvector of the dynamical matrix $D(k)$ for the $\sigma$-th
eigenmode is

$\displaystyle\bm{\epsilon}_{k}^{(\sigma)}=\begin{pmatrix}1\\\
e^{i(ka+2m\pi)/n}\\\ \vdots\\\ e^{i(n-1)(ka+2m\pi)/n}\end{pmatrix}.$ (7)

Here the $i$-th component of the eigenvector $\bm{\epsilon}_{k}^{(\sigma)}$ of
the dynamical matrix $D(k)$ represents the displacement of atom $i$.
Therefore, to exact the information of the quantum number $m$ from this
eigenvector for the mode $\sigma$, we define a relative phase
$\theta_{i}^{(\sigma)}(k)$ between atom $i$ and atom $i+1$ as

$\displaystyle z_{i}^{(\sigma)}(k)$
$\displaystyle=\left\\{\begin{array}[]{ll}\epsilon_{k,i+1}^{(\sigma)}/\epsilon_{k,i}^{(\sigma)}&i=1,\cdots,n-1\\\
\epsilon_{k,1}^{(\sigma)}e^{ika}/\epsilon_{k,n}^{(\sigma)}&i=n\end{array}\right.,$
(10) $\displaystyle\theta_{i}^{(\sigma)}(k)$
$\displaystyle=\arg{z_{i}^{(\sigma)}(k)},$ (11)

where we take the branch $-\pi<\theta_{i}^{(\sigma)}(k)\leq\pi$. When the
system has the exact translational symmetry $\hat{T}_{a/n}$, from the
eigenvector Eq (7), the values of $z_{i}^{(\sigma)}(k)$ and
$\theta_{i}^{(\sigma)}(k)$ are

$\displaystyle z_{i}^{(\sigma)}(k)$ $\displaystyle=e^{i\frac{ka+2m\pi}{n}},$
(12) $\displaystyle\theta_{i}^{(\sigma)}(k)$
$\displaystyle=\frac{ka+2m\pi}{n}$ (13)

for $i=1,\cdots,n$, and the relative phase is a linear function with an
intecept equal to $2m\pi/n$. On the other hand, when the translational
symmetry $\hat{T}_{a/n}$ is broken by slightly changing the value of the
spring constant $c_{i}$, we can also define the relative phases for the
eigenvector. Therefore, we can calculate the relative phases for the case with
broken symmetry and extract the information of the quantum number $m$.

As an example, we calculate the relative phase $\theta_{i}^{(\sigma)}(k)$ in a
specific model when we break the translational symmetry $T_{a/n}$ while
keeping the translational symmetry $T_{a}$, corresponding to the change from
Fig. 1 (a-1) to (a-2). We take the number of atoms in the unit cell to be
$n=3$. In this calculation, the spring constants are varied from
$c_{1}=c_{2}=c_{3}=1$ to $c_{1}=1,c_{2}=1.1,c_{3}=1.2$, in order to break the
translational symmetry $T_{a/3}$ while keeping the translational symmetry
$T_{a}$. Then, the phonon dispersion changes from Fig. 1 (b-1) to Fig. 1 (b-2)
and the relative phase $\theta_{i}^{(\sigma)}(k)$ of the $\sigma=2$ band (the
blue band in Figs. 1 (b-1)(b-2)) changes from Fig. 1 (c-1) to (c-2), and
$z_{i}^{(\sigma)}$ changes from Fig. 1 (d-1) to (d-2). From the results of
Fig. 1, in the case with the translational symmetry $T_{a/3}$, the relative
phases $\theta_{i}^{(\sigma)}(i=1,2,3)$ are independent of $i$ and linear in
$k$ and the complex number $z_{i}^{(\sigma)}(k)$ is on the unit circle in the
complex plane (see Eq. (12)). Next, when the translational symmetry
$\hat{T}_{a/3}$ is slightly broken, the relative phase
$\theta_{i}^{(\sigma)}(k)$ and the complex number $z_{i}^{(\sigma)}(k)$ remain
almost the same with the case with strict translational symmetry
$\hat{T}_{a/3}$. Therefore, by calculating the relative phase for the case of
slight symmetry breaking and comparing it with that for the case of strict
symmetry, we can extract the information of the quantum number $m$ for each
mode. We show the results of the values of $m$ in Fig. 1 (b-1) and (b-2).

Figure 2:  Phonon dispersions and relative phases between neighboring atoms in
the extended Brillouin zone. (a-1) Band dispersion $\omega_{k,\sigma}$ and
(a-2) Relative phase $\theta_{i}(k)$ for the system with translation symmetry
$\hat{T}_{a/3}$ ($c_{1}=c_{2}=c_{3}=1$). In the extended Brillouin zone (BZ),
there is only one band with the quantum number $m=0$. By folding the BZ into
the region $-\pi/a\leq k\pi/a$ (light yellow region), the wavenumber $k$ is
shifted and the quantum number $m$ changes accordingly. The wavenumber $k$ in
the region ”BZ(1)” (light green region) is shifted by $-2\pi$, and the quantum
number $m$ changes by $+1$. Similarly, the wavenumber $k$ in the region
”BZ(-1)” (light blue region) is shifted by $2\pi$, and the quantum number $m$
changes by $-1$. (b-1) Band dispersion $\omega_{k,\sigma}$ and (b-2) Relative
phase $\theta_{i}(k)$ for the system without translation symmetry
$\hat{T}_{a/3}$ ($c_{1}=1,c_{2}=1.1,c_{3}=1.2$). The relative phase holds the
linear tendency before breaking the symmetry $\hat{T}_{a/3}$, but changes
significantly at the time-reversal invariant momenta $k=l\pi/a$ ($l$:
integer).

We can naturally understand the behaviors of the quantum number $m$ using the
extended Brillouin zone (BZ). When the system has exact translational symmetry
of $a/3$, the Brillouin zone is extended to $-3\pi/a\leq k\leq 3\pi/a$. The
band dispersion in the extended Brillouin zone for the systems with exact
translation symmetry $\hat{T}_{a/3}$ ($c_{1}=c_{2}=c_{3}=1$) is shown in Fig.
2 (a-1). In this zone scheme, there is only the phonon branch (solid line in
Fig. 2 (a-1)), and its relative phase is given by

$\displaystyle\theta_{i}(k)=\frac{ka}{n},$ (14)

which corresponds to $m=0$ in Eq. (13). It is seen from our numerical result
for the relative phase shown as a black line in Fig. 2 (a-2). When the unit
cell size is enlarged to $a$, the Brillouin zone becomes $-\pi/a\leq
k\leq\pi/a$ and the phonon bands are folded into this new Brillouin zone.
Copies of the original phonon band appear as shown in broken lines in Fig. 2
(a-1). In this case, the right-neighboring Brillouin zone (BZ(1)) moves to the
center Brillouin zone (BZ(0)) which adds $+1$ to the quantum number $m$, and
the left-neighboring Brillouin zone (BZ(-1)) moves to BZ(0), adding $-1$ to
the quantum number $m$. As a result, three branches appear within the new BZ.
From this argument, it shows that when $k$ increases and goes across the
boundary of the Brillouin zone from $k=\pi/a$ to $k=-\pi/a$, the quantum
number $m\pmod{n}$ increases by $+1$, which is consistent with the discussions
in Ref. [25]. When the $\hat{T}_{a/3}$ symmetry is broken, the crossings in
Fig. 2 (a-1) will become anticrossings as shown in Fig. 2 (b-1) for
$c_{1}=1,c_{2}=1.1,c_{3}=1.2$. Within the extended zone scheme, the relative
phases for the system with slightly broken translation symmetry
$\hat{T}_{a/3}$ behave similarly to Eq. (14), as shown in Fig. 2 (b-2) expect
for $k\sim 0$ and $k\sim\pi$. From this we can interpret that the band has
quantum numbers like Fig. 1 (b-1) and (b-2).

On the other hand, near the time-reversal invariant momenta $k=0,\pi/a$, the
relative phase $\theta_{i}^{(\sigma)}(k)$ and the complex number
$z_{i}^{(\sigma)}(k)$ largely deviate from those in the system with $T_{a/n}$
symmetry. In particular, at $k=0,\pi/a$, the relative phase
$\theta_{i}^{(\sigma)}(k)$ quantizes to $0$ or $\pi$ because of the time-
reversal symmetry. To quantify this behavior, we define three quantum numbers
from the relative phase $\theta_{i}^{(\sigma)}(k)$:

$\displaystyle p_{0,i}^{(\sigma)}$
$\displaystyle\equiv\theta_{i}^{(\sigma)}(0)/\pi\ \pmod{2},$ (15)
$\displaystyle p_{\pi,i}^{(\sigma)}$
$\displaystyle\equiv\theta_{i}^{(\sigma)}(\pi/a)/\pi\ \pmod{2},$ (16)
$\displaystyle w_{i}^{(\sigma)}$
$\displaystyle=\frac{1}{2\pi}\int_{k=-\pi/a}^{k=\pi/a}\frac{d\theta_{i}^{(\sigma)}}{dk}dk.$
(17)

Here, $p_{0,i}^{(\sigma)},p_{\pi,i}^{(\sigma)}$ take the values $0,1$ modulo
$2$ because $\theta_{i}^{(\sigma)}(0)$ and $\theta_{i}^{(\sigma)}(\pi/a)$
quantize to $0$ or $\pi$ due to the time-reversal symmetry. On the other hand,
$w_{i}^{(\sigma)}$ represents the number of times the curve
$z_{i}^{(\sigma)}(k)$ $\left(-\pi/a<k\leq\pi/a\right)$ travels
counterclockwise around the origin in the complex plane, and is an integer.
The results of the calculation of these quantum numbers in the model of Fig. 1
are shown in Table 1. It is straightforward to show

$\displaystyle w_{i}^{(\sigma)}\equiv
p_{\pi,i}^{(\sigma)}-p_{0,i}^{(\sigma)}\pmod{2},$ (18)

because $\theta_{i}^{(\sigma)}$ is an odd function of $k$ (i.e.
$\theta_{i}^{(\sigma)}(-k)=-\theta_{i}^{(\sigma)}(k)$), and indeed it holds in
our results in Table 1.

Table 1:  Quantum numbers $p_{0,i}^{(\sigma)}$, $p_{\pi,i}^{(\sigma)}$ and $w_{i}^{(\sigma)}$ of the model of Fig. 1.  $\sigma$ | $p_{0,1}^{(\sigma)}$ | $p_{0,2}^{(\sigma)}$ | $p_{0,3}^{(\sigma)}$ | $p_{\pi,1}^{(\sigma)}$ | $p_{\pi,2}^{(\sigma)}$ | $p_{\pi,3}^{(\sigma)}$ | $w_{1}^{(\sigma)}$ | $w_{2}^{(\sigma)}$ | $w_{3}^{(\sigma)}$
---|---|---|---|---|---|---|---|---|---
1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0
2 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | -1
3 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0

We discuss properties of $p_{0,i}^{(\sigma)}$ and $p_{\pi,i}^{(\sigma)}$ with
slightly broken translational symmetry $T_{a/n}$ in the following. First, the
eigenvector $\bm{\epsilon}_{k}^{(\sigma)}$ with the translational symmetry
$T_{a/n}$ is specified by the quantum number $m$ and is given by

$\displaystyle\bm{\epsilon}_{k}^{(\sigma)}=\left(\begin{array}[]{c}1\\\
e^{i(ka+2m\pi)/n}\\\ \vdots\\\ e^{i(n-1)(ka+2m\pi)/n}\end{array}\right).$ (23)

Meanwhile when $T_{a/n}$ is slightly broken, the eigenvectors should be real
at time-reversal invariant momenta $(k=0,\pi/a)$. Therefore, a complex
eigenvectors should be degenerate with its complex conjugate. Let $m_{1}$ and
$m_{2}$ be the quantum numbers for the two bands degenerate at the time-
reversal invariant momentum $\left(k=0,\ \pi/a\right)$;
$\bm{\epsilon}_{k,m_{1}}=\bm{\epsilon}_{k,m_{2}}^{*}$. As seen in Fig. 2
(a-1), this degeneracy exists in the systems with exact $T_{a/n}$ translation
symmetry, and is due to the zone folding from the larger Brillouin zone
($-n\pi/a\leq k\leq n\pi/a$), and it is lifted by slightly breaking the
translational symmetry $T_{a/n}$. Let $\bm{\epsilon}_{1}$ and
$\bm{\epsilon}_{2}$ be the eigenvectors at the time-reversal invariant
momentum when the translational symmetry $T_{a/n}$ is broken. Within the
zeroth order perturbation, they are linear combinations of
$\bm{\epsilon}_{k,m_{1}}$ and
$\bm{\epsilon}_{k,m_{2}}(=\bm{\epsilon}_{k,m_{1}}^{*})$. Since the
eigenvectors $\bm{\epsilon}_{1}$ , $\bm{\epsilon}_{2}$ are real vectors due to
time-reversal symmetry, they are given by

$\displaystyle\bm{\epsilon}_{1}=\left(\begin{array}[]{c}\cos{(\phi)}\\\
\cos{(\phi+\xi)}\\\ \vdots\\\
\cos{\left(\phi+(n-1)\xi\right)}\end{array}\right),\
\bm{\epsilon}_{2}=\left(\begin{array}[]{c}\sin{(\phi)}\\\ \sin{(\phi+\xi)}\\\
\vdots\\\ \sin{\left(\phi+(n-1)\xi\right)}\end{array}\right),$ (32)

within the zeroth order perturbation, where $\phi$ is a constant determined
from the spring constant matrix $K_{l,l^{\prime}}$ and
$\xi=(m_{1}-m_{2})\pi/n$. Therefore, for both the eigenvectors
$\bm{\epsilon}_{1}$ and $\bm{\epsilon}_{2}$ at $k(=0,\pi/a)$, among $n$ values
of $p_{ka,i}^{(\sigma)}(=0,1)$ ($i=1,\cdots,n$), the number of
$p_{ka,i}^{(\sigma)}=1$ is $|m_{1}-m_{2}|$, which is equal to the number of
times the sequence
$\left(\cos{(\phi)},\cdots,\cos{\left(\phi+n\xi\right)}\right)$ (or
$\left(\sin{(\phi)},\cdots,\sin{\left(\phi+n\xi\right)}\right)$) changes sign.
One can easily check this conclusion in Table 1. For example, the anticrossing
at $k=0$, $\omega\sim 2$ in Fig. 1 (b-2) is between the bands with
$\sigma=2,3$, having $m=\pm 1$. It corresponds to $m_{1}=1$ and $m_{2}=-1$
leading to $|m_{1}-m_{2}|=2$. Thus, among the three values of $p_{0,i}^{(2)}$,
two (i.e. $i=1$ and $i=2$) have the value of $1$, in accordance with Table 1.
The similar conclusion holds also for $p_{0,i}^{(3)}$. We explain another
example on the anticrossing at $k=\pi$, $\omega\sim 1.1$ in Fig. 1 (b-2)
between the bands with $\sigma=1$, and $\sigma=2$, corresponding to $m_{1}=0$
and $m_{2}=\pm 1$ i.e. with $|m_{1}-m_{2}|=1$. Thus, among the three values of
$p_{\pi,i}^{(1)}$ (and similarly among the three values of $p_{\pi,i}^{(2)}$),
only one has the value of $1$, in accordance with Table 1.

To confirm the theory in this section, we calculate the relative phase
$\theta(k)$ for the material $\ce{Li6B5}$ with approximate translational
symmetry $\hat{T}_{a/5}$, which are shown in Sec. IV.

##  III  Phonons in 3D systems with approximate screw symmetry

In this section, we extend the result in Sec. II for 1D systems with
approximate translation symmetry to 3D systems with approximate screw
symmetry. In such 3D systems, the pseudoangular momentum $m$ plays the role of
the quantum number in Sec. II. Namely, we discuss a method to extract the
information of the pseudoangular momentum by calculating the relative phases
in phonon systems with approximate screw symmetry in 3D systems.

We suppose the system has the $n_{l}$ screw symmetry $C_{n}^{l}$ in the $z$
direction, where $n$ and $l$ are positive integers with $1\leq l<n$. The screw
operation $C_{n}^{l}$ is a combination of a rotation by an angle $2\pi/n$
around the $z$-axis, and a translation by $(l/n)a$ along the $z$ axis, where
$a$ denotes the lattice constant along the $z$ axis. Then its $n$th power is
equal to the the translation $T_{la}$ by $la$:
$\hat{T}_{la}=\left(\hat{C}_{n}^{l}\right)^{n}$. From the Bloch’s theorem, an
eigenfunction $\psi(x)$ satisfies $\hat{T}_{a}\psi(x)=e^{-ika}\psi(x)$, where
$k$ is the Bloch wavenumber $\left(\frac{-\pi}{a}<k\leq\frac{\pi}{a}\right)$
in the $z$ direction. Since we need to consider eigenstates under
$\hat{C}_{n}^{l}$, the wavevector $\vec{k}=(k_{x},k_{y},k_{z})$ should be
invariant under this screw symmetry. Here we choose $(k_{x},k_{y})=(0,0)$, and
write $\vec{k}=(0,0,k)$. Therefore, when the $n_{l}$ screw symmetry
$\hat{C}_{n}^{l}$ is preserved, the eigenfunction $\psi(x)$ can be
characterized by a integer $m^{\prime}$ and satisfies
$\hat{C}_{n}^{l}\psi(\vec{r})=e^{i(kla+2m^{\prime}\pi)/n}\psi(\vec{r})$, where

$\displaystyle m^{\prime}\equiv\left\\{\begin{array}[]{ll}0,\pm
1,\cdots,\pm(n-1)/2&\text{for}\ n\ \text{odd}\\\ 0,\pm
1,\cdots,\pm(n-2)/2,n/2&\text{for}\ n\ \text{even}\end{array}\right.\pmod{n}.$
(35)

In the following, we call $m^{\prime}$ a pseudoangular momentum. We note that
this $m^{\prime}$ corresponds to the quantized integer part of the
pseudoangular momentum defined in Ref. [25].

Next, we extend our theory in the previous section to 3D systems with
approximate screw symmetry, which needs to define relative phases between
neighboring atoms. In this section, we restrict ourselves to the case where
$n$ and $l$ are coprime for simplicity. The cases where $n$ and $l$ are not
coprime can be studied similarly, as shown in Appendix A. When $n$ and $l$ are
coprime, the $n$ atoms in the unit cell are periodically located with a period
$a/n$ along the screw axis, which we call $z$ axis. Now, we show that there
exists a symmetry operation
$\hat{\mathcal{O}}\equiv(\hat{C}_{n}^{l})^{p}(\hat{T}_{a})^{q}$ ($p$, $q$:
integer) which relates between neighboring atoms. Since the difference in the
$z$-component between the neighboring atoms is $\frac{a}{n}$, the integers $p$
and $q$ satisfy $p\frac{la}{n}+qa=\frac{a}{n}$, i.e. $lp+nq=1$. Thanks to the
Bézout’s lemma this equation always has integer solutions for $p$ and $q$.
Specifically, for $l=2$ and $n=7$, this equation has a solution $p=-3$ and
$q=1$. Hence $\hat{\mathcal{O}}=(\hat{C}_{n}^{l})^{p}(\hat{T}_{a})^{q}$ is the
operator that rotates the system by $\frac{2p\pi}{n}$ and translates it by
$a/n$. Therefore,

$\displaystyle\hat{\mathcal{O}}\psi(\vec{r})$
$\displaystyle=(\hat{C}_{n}^{l})^{p}(\hat{T}_{a})^{q}\psi(\vec{r})$
$\displaystyle=\left(e^{-i(lka+2m^{\prime}\pi)/n}\right)^{p}\left(e^{ika}\right)^{q}\psi(\vec{r})$
(36)

holds. Then, similarly to the definition of the pseudoangular momentum, we put
the eigenvalue of $\hat{\mathcal{O}}$ to be equal to $e^{i(ka+2m\pi)/n}$
because $\hat{\mathcal{O}}^{n}=e^{ika}$, where $m$ is an integer representing
the relative phase between neighboring atoms displaced by $a/n$:

$\displaystyle\hat{\mathcal{O}}\psi(\vec{r})=e^{-i(ka+2m\pi)/n}\psi(\vec{r}).$
(37)

Then, from (36) and (37), $m$ is given by $m\equiv m^{\prime}p\pmod{n}$. By
using $lp+nq=1$, we get

$\displaystyle m^{\prime}\equiv ml\pmod{n}.$ (38)

In particular for $n_{1}$ screw symmetry (i.e. $l=1$), $m^{\prime}\equiv m$
holds. Therefore, from this correspondence, the wave function information
possessed by the integers $m$ and $m^{\prime}$ is the same. In the following
discussion, we consider assigning the integer $m$ instead of the pseudoangular
momentum $m^{\prime}$ for phonon eigenmodes.

Let
$\bm{\epsilon}^{(\sigma)}(k)=(\vec{\epsilon}_{1}^{(\sigma)}(k),\cdots,\vec{\epsilon}_{n}^{(\sigma)}(k))^{\mathrm{T}}$
denote the eigenvector of the dynamical matrix $D(\vec{k})$, where
$\vec{\epsilon}_{j}^{(\sigma)}(k)$ is the displacement of the $j$-th atom in
the $\sigma$-th phonon mode at the wavenumber $k$ along the $z$ axis. Here the
$n$ atoms are numbered in the increasing order of the $z$ coordinate. This
means that the eigenvector of the dynamical matrix $D(k)$ with the screw
symmetry $C_{n}^{l}$ can be expressed in terms of the pseudo angular momentum
$m$ as

$\displaystyle\bm{\epsilon}^{(\sigma)}(k)$
$\displaystyle=\left(\begin{array}[]{c}\vec{v}^{(\sigma)}(k)\\\
e^{i(ka+2m\pi)/n}A_{p}\vec{v}^{(\sigma)}(k)\\\ \vdots\\\
e^{i(n-1)(ka+2m\pi)/n}A_{p}^{n-1}\vec{v}^{(\sigma)}(k)\end{array}\right),$
(43) $\displaystyle A_{p}$
$\displaystyle=\left(\begin{array}[]{ccc}\cos{2p\pi/n}&-\sin{2p\pi/n}&0\\\
\sin{2p\pi/n}&\cos{2p\pi/n}&0\\\ 0&0&1\end{array}\right),$ (47)

where $\vec{v}^{(\sigma)}(k)$ is a three-dimensional complex vector. Thus,
when the system has exact screw symmetry, the eigenvector is characterized by
the quantum number $m$. On the other hand, without the screw symmetry, we
cannot define the quantum number $m$ for the eigenvector. However, in the case
with approximate screw symmetry, the information of the quantum number $m$
defined under exact screw symmetry partially remains in the eigenvector.

Based on Eq. (37), under the exact screw symmetry, we define the relative
phase to be

$\displaystyle z_{i,\alpha}^{(\sigma)}(k)$
$\displaystyle=\left\\{\begin{array}[]{ll}u_{i+1,\alpha}^{(\sigma)}(k)/u_{i,\alpha}^{(\sigma)}(k)&i=1,\cdots,n-1\\\
u_{1,\alpha}^{(\sigma)}(k)e^{ika}/u_{n,\alpha}^{(\sigma)}(k)&i=n\end{array}\right.,$
(50) $\displaystyle\theta_{i,\alpha}^{(\sigma)}(k)$ $\displaystyle=\arg
z_{i,\alpha}^{(\sigma)}(k),$ (51)

for the $\sigma$-th eigenvectors $\bm{\epsilon}^{(\sigma)}(k)$ of the
dynamical matrix $D(k)$, where $\alpha=x,y,z$ and
$\vec{u}_{i}^{(\sigma)}(k)=A_{p}^{-(i-1)}\vec{\epsilon}_{i}^{(\sigma)}(k),\
(i=1,2,\cdots,n)$. In the case of the exact screw symmetry, we can use Eq.
(43) to calculate the relative phase as

$\displaystyle\theta_{i,\alpha}^{(\sigma)}(k)=(ka+2m\pi)/n,(i=1,\cdots,n\
,\alpha=x,y,z);$ (52)

namely, $\theta_{i,\alpha}^{(\sigma)}(k)$ is a linear function with an
intercept of $2m\pi/n$ independent of the direction $\alpha$.

Figure 3:  Phonon system with three-fold $3_{1}$ screw symmetry in the $z$
direction. (a), (b) Schematic picture of this model, and red dots denote
atoms. (a) shows that this is the $3_{1}$ screw symmetry. (b) shows the
positions of the atoms within the $xy$ plane. (c-1) (c-2) Band dispersions
$\omega_{k,\sigma}$ with (c-1) exact screw symmetry and (c-2) broken screw
symmetry. (d-1) (d-2) Relative phases $\theta_{i,x}^{\sigma}(k)$ for the
$\sigma=8$ band (blue line) in the band dispersions (c-1) and (c-2). (d-1)
corresponds to the case with exact screw symmetry, and (d-2) corresponds to
the case with broken screw symmetry.

As an example, we calculate phonon modes in a model that slightly breaks
threefold screw symmetry $C^{1}_{3}$. Here, because $n=3$ and $l=1$, the
quantum number $m$ is equal to the pseudoangular momentum $m^{\prime}$. We
consider a system with three atoms 1, 2 and 3 in a unit cell shown in Figs. 3
(a) and (b), arranged along a helix. The result of the phonon calculation is
shown in Figs. 3 (c) and (d). In this calculation, let $K_{1}$, $K_{2}$ and
$K_{3}$ denote the spring constant matrices between atom $1$ and atom $2$,
between atom $2$ and atom $3$, and between atom $3$ and atom $1$ of the
neighboring unit cell, respectively. The spring constant matrix $K_{i}$
consists of longitudinal components with a spring constant $K_{L,i}=c_{i}$ and
transverse ones with a spring constant $K_{T,i}=c_{i}/4$. In our calculation,
we change the parameter values from those with the exact screw symmetry
$\left(c_{1}=c_{2}=c_{3}=1\right)$ to those with approximate screw symmetry
$\left(c_{1}=1,c_{2}=1.01,c_{3}=1.02\right)$. When the screw symmetry is
slightly broken in this way, the band dispersion relation changes slightly
from Fig. 3 (c-1) to (c-2), with a tiny gap opened at $k=0$. Here, if we take
the $\sigma=8$ band (blue lines in (c-1) and (c-2)) as an example, the
relative phase change from Fig. 3 (d-1) to (d-2). In the case of exact screw
symmetry, the relative phase is a linear function, Eq. (52). Since the value
of the relative phase $\theta_{i,\alpha}^{(\sigma)}(k)$ hardly changes even
when the screw symmetry is slightly broken, we can use this relative phase to
assign the pseudoangular momentum $m(=m^{\prime})$ to all the bands, except
for the vicinity of band anticrossing. As shown in Fig. 3 (c-1), when the
wavenumber $k$ along the $z$ axis increases across the Brillouin zone boundary
from $k=\pi/a$ to $k=-\pi/a$, the quantum number $m\pmod{n}$ increases by
$+1$. This result is similar to that of the one-dimensional model in Fig. 1,
and can be interpreted by using the extended Brillouin zone method [25]. In
addition, the relative phase $\theta_{i,\alpha}^{(\sigma)}(k)$ with broken
symmetry is significantly different from that with exact symmetry near the
time-reversal invariant momenta $k=0,\pi/a$, ant it is quantized to $0$ or
$\pi$ on $k=0,\pi/a$ because its eigenvector is real. In the one-dimensional
system with broken $T_{a/n}$ translation symmetry (Fig. 1), the property that
the number of $\theta_{i,\alpha}^{(\sigma)}(k)$ ($i=1,\cdots,n$) at
$k=0,\pi/a$ going to $\pi$ is equal to the difference $|m_{1}-m_{2}|$ of
pseudoangular momentum $m=m_{1},m_{2}$ of the crossing bands holds as in the
one-dimensional case. However, this property is not generally valid in three-
dimensional systems with slightly broken screw symmetry.

##  IV  First-principles calculation

In this section, we calculate the relative phases between atoms in phonon
modes in a material $\ce{Li6B5}$[41] with approximate translational symmetry
(discussed in Sec. II) and two materials $\ce{SnIP}$ [42] and $\ce{S10}$ [43]
with approximate screw symmetries by first-principle calculations (discussed
in Sec. III). By comparing their relative phases with the ones with exact
symmetry, we can extract the information of the quantum number $m$.
$\ce{Li6B5}$, $\ce{SnIP}$ and $\ce{S10}$ are fully relaxed before the phonon
calculations. We calculate the force constants of $\ce{Li6B5}$, $\ce{SnIP}$
and $\ce{S10}$ with density functional perturbation theory (DFPT) in a
$3\times 3\times 3$, $2\times 2\times 2$ and $2\times 2\times 3$ supercell
with an equivalent $k$-mesh of $9\times 9\times 9$, $6\times 6\times 4$ and
$4\times 4\times 6$ via VASP [44, 45], respectively.

###  IV.1 Li6B5: one-dimensional chain with approximate translational symmetry

Figure 4:  Relative phase of the material $\ce{Li6B5}$. (a) Schematic picture
of the atomic positions. The boron atoms have the approximate translational
symmetry $T_{c/5}$. (b) Phonon band dispersion. The bands
($\sigma=30,31,32,33$) shown in color in the figure are phonon modes with
displacements along the direction
$\vec{a}_{c}=\vec{a}_{1}+\vec{a}_{2}+\vec{a}_{3}$. The band structure is
classified into two groups, the one with a wider bandwidth ($\sim 40$THz,
shown in the light blue background) and the other with a narrower bandwidth
($\sim 15$THz, shown in the light yellow background). (c)-(f) Relative phases
between neighboring atoms of the bands $\sigma=$ (c) 31, (d) 30, (e) 32 and
(f) 33, respectively.

We discuss the results of the calculations for the material $\ce{Li6B5}$ shown
in Fig. 4. $\ce{Li6B5}$ contains a one-dimensional chain of boron atoms with
approximate translational symmetry $\hat{T}_{c/5}$, where $c$ is the lattice
constant along the $z$ axis, as shown in Fig. 4 (a). The lattice is
rhombohedral and its lattice vectors are

$\displaystyle\vec{a}_{1}=(\sqrt{3}a/2,a/2,c/3),$ (53)
$\displaystyle\vec{a}_{2}=(-\sqrt{3}a/2,a/2,c/3),$ (54)
$\displaystyle\vec{a}_{3}=(0,-a,c/3),$ (55)

where $a=3.973$Å and $c=7.945$Å. The corresponding primitive reciprocal
vectors are

$\displaystyle\vec{b}_{1}=(2\pi/(\sqrt{3}a),2\pi/(3a),2\pi/c),$ (56)
$\displaystyle\vec{b}_{2}=(-2\pi/(\sqrt{3}a),2\pi/(3a),2\pi/c),$ (57)
$\displaystyle\vec{b}_{3}=(0,-4\pi/3a,2\pi/c).$ (58)

The unit cell contains five boron atoms $\ce{B}$ almost equally spaced along
the vector $\vec{a}_{c}=\vec{a}_{1}+\vec{a}_{2}+\vec{a}_{3}=(0,0,c)$. Thus, in
this material the boron atoms have an approximate translational symmetry
$\hat{T}_{c/5}$ by the vector $(0,0,c/5)$. Since this lattice has a
translational symmetry $\hat{T}_{i}$ $(i=1,2,3)$ in the direction of the
lattice vectors $\vec{a}_{i}$, the eigenfunctions $\psi(x)$ satisfy
$\hat{T}_{i}\psi(x)=e^{-i\vec{k}\cdot\vec{a}_{i}}\psi(x)$ where
$-\pi<\vec{k}\cdot\vec{a}_{i}\leq\pi$ due to the Bloch’s theorem. By using
this, under the translation symmetry $\hat{T}_{a}$ in the direction of
$\vec{a}_{c}$, $\hat{T}_{c}=\hat{T}_{1}\hat{T}_{2}\hat{T}_{3}$, $\psi(x)$
satisfies

$\displaystyle\hat{T}_{c}\psi(x)=e^{-i\vec{k}\cdot\vec{a}_{c}}\psi(x),\
\left(-3\pi<\vec{k}\cdot\vec{a}_{c}\leq 3\pi\right).$ (59)

Furthermore, if the boron atoms preserve the translational symmetry
$\hat{T}_{c/5}$,

$\displaystyle\hat{T}_{c/5}\psi(x)=e^{-i(\vec{k}\cdot\vec{a}_{c}+2m\pi)/5}\psi(x)$
(60)

holds with the quantum number $m=0,\pm 1,\pm 2$.

Figure 4 (b) shows the phonon band structure along the $k_{z}$ direction. The
band structure is classified into two groups, the one with a wider bandwidth
($\sim 40$THz) and the other with a narrower bandwidth ($\sim 15$THz). These
two groups of bands correspond to the phonon modes with displacements along
$z$ axis and these along the $xy$ plane, respectively. In the former group
with displacements along $z$ axis, the motions of the $\ce{B}$ atoms are well
decoupled from those of the $\ce{Li}$ atoms, which is ideal for our purpose.
Henceforth, we consider the bands $\sigma=30,31,32,33$, having boron atoms
vibrating only in the $z$ direction. Thus, when $\hat{T}_{a/5}$ translation
symmetry is preserved, we can write the eigenvector
$\bm{\epsilon}_{k}^{(\sigma)}$ for the phonon eigenvectors with displacements
along the $z$ direction under this assumption, as

$\displaystyle\bm{\epsilon}_{k}^{(\sigma)}=\left(\begin{array}[]{c}1\\\
e^{i(\vec{k}\cdot\vec{a}_{c}+2m\pi)/5}\\\
e^{2i(\vec{k}\cdot\vec{a}_{c}+2m\pi)/5}\\\
e^{3i(\vec{k}\cdot\vec{a}_{c}+2m\pi)/5}\\\
e^{4i(\vec{k}\cdot\vec{a}_{c}+2m\pi)/5}\end{array}\right),$ (66)

where $m$ is the quantum number $m=0,\pm 1,\pm 2\pmod{5}$. Here
$\vec{k}\cdot\vec{a}_{c}$ takes values within the range
$-3\pi<\vec{k}\cdot\vec{a}_{c}\leq 3\pi$, so the wavenumber
$k=\vec{k}\cdot\vec{a}_{c}/c$ in the $z$ direction takes the range
$\frac{-3\pi}{c}<k\leq\frac{-3\pi}{c}$. Therefore, as we showed in Sec. II,
the relative phase from this eigenvector is

$\displaystyle\theta(\vec{k})=(kc+2m\pi)/5.$ (67)

However, the translational symmetry ${T}_{c/5}$ is slightly broken in this
material, so the relative phase is slightly deviated from Eq. (67). But the
deviation is small, and we can still extract the information of the quantum
number $m$. In particular, the relative phase of the band $\sigma=31$ is shown
in Fig. 4 (c), which agrees well with the relative phase of the case with
exact symmetry (Eq. (67)). Thus, we can extract the quantum number $m$ as
shown in the figure. The results of the relative phase calculations for the
bands $\sigma=30,32,34$ are shown in Fig. 4 (d)-(f). Since the relative phase
of $\sigma=30$ agrees well with the relative phase of the case with exact
symmetry, we can extract the quantum number $m$. On the other hand, the
relative phases of $\sigma=32$ and $33$ do not match those of the exact
symmetric case near $k=0$. This is because the bands are close to each other
and thus hybridized. The results for the quantum number $m$ is summarized in
in Fig. 4(b). The behavior of $m$ for the bands $\sigma=32,33$ will be
discussed later. .

Figure 5:  Extended Brillouin zone of $\ce{Li6B5}$. (a) Comparison of
Brillouin zones with and without the translational symmetry $\hat{T}_{c/5}$.
The blue line is the virtual Brillouin zone with $\hat{T}_{c/5}$, and the red
line is the actual Brillouin zone without $\hat{T}_{c/5}$. (b) Extended
Brillouin zone with actual Brillouin zones on $k_{z}$ axis in the reciprocal
lattice space. (c) Band dispersion on $k_{z}$ axis. The band shown by the
thick green line is the phonon mode with the quantum number $m=0$.

We can assign the quantum number $m$ from the relative phases as shown in Fig.
4 (b). In this figure, when the wavenumber increases across the boundary of
the Brillouin zone from $k=3\pi/c$ to $k=-3\pi/c$, the quantum number
$m\pmod{5}$ increases by $+3$. This is quite different from the increase by
$+1$ across the boundary of the Brillouin zone in Fig. 1. In the following, we
explain this difference using the extended band scheme shown in Fig. 5.

Let us assume that the system is invariant under the translation by the vector
$\vec{a}_{c}/5=(0,0,c/5)$, meaning that the five atoms within the unit cell
are equivalent under this symmetry. Thus, the new primitive translation
vectors become

$\displaystyle\vec{a}^{\prime}_{1}=\vec{a}_{1}-2\vec{a}_{c}/5=(\sqrt{3}a/2,a/2,-c/15),$
(68)
$\displaystyle\vec{a}^{\prime}_{2}=\vec{a}_{2}-2\vec{a}_{c}/5=(\sqrt{3}a/2,a/2,-c/15),$
(69)
$\displaystyle\vec{a}^{\prime}_{3}=\vec{a}_{3}-2\vec{a}_{c}/5=(0,-a,-c/15).$
(70)

Their sum is equal to $-\vec{a}_{c}/5$, representing the $\hat{T}_{c/5}$
translation symmetry. Because the translation vector along the $z$ axis
becomes shorter, the Brillouin zone becomes longer. These tranlation vectors
lead to a virtual Brillouin zone given by corresponding primitive reciprocal
vectors

$\displaystyle\vec{b}^{\prime}_{1}=(2\pi/(\sqrt{3}a),2\pi/(3a),-10\pi/c),$
(71)
$\displaystyle\vec{b}^{\prime}_{2}=(-2\pi/(\sqrt{3}a),2\pi/(3a),-10\pi/c),$
(72) $\displaystyle\vec{b}^{\prime}_{3}=(0,-4\pi/3a,-10\pi/c).$ (73)

Figure 5 (a) is a figure comparing the virtual Brillouin zone (blue line) when
the system is assumed to have translational symmetry $\hat{T}_{c/5}$ and the
actual Brillouin zone (red lines) when it is not assumed. These Brillouin
zones are of the similar shape, consisting of 12 faces. Namely, the virtual
Brillouin zone (blue lines) for the case with the translational symmetry
$\hat{T}_{c/5}$ is narrowed to the actual Brillouin zone (red line) by
breaking the symmetry. Figure 5 (b) shows five virtual Brillouin zones along
$k_{z}$ axis and the accompanying actual Brillouin zones in Fig. 5 (a).
Furthermore, we show the band dispersion relation along the $k_{z}$ axis in
Fig. 5 (c), to compare with the Brillouin zones in Fig. 5 (b). As a one-
dimensional system, the translation vector $(0,0,c/5)$, gives rise to the one-
dimensional reciprocal vector $10\pi/c$, and the complex structure of the
colored bands in Fig. 4(b) is then mapped to a single band in the extended BZ
with the size $10\pi/c$, shown as the light green line in Fig. 5 (c). This
single band corresponds to the quantum number $m=0$, as we discussed in Sec.
II. In this case, the size of the actual Brillouin zone along $k_{z}$ is
$6\pi/c$ because
$\vec{b}_{c}=\vec{b}_{1}+\vec{b}_{2}+\vec{b}_{3}=(0,0,6\pi/c)$, and the
quantum number $m\pmod{5}$ increases by $+3$ from Eq. (67) if we shift the
wavenumber by $-6\pi/c$. In this way, in folding the virtural BZ to the actual
BZ, if the phonon bands are shifted by $(-6\pi/c)\times\bar{m}$, ($\bar{m}$:
integer), the quantum number $m$ characterizing the relative phases between
neighboring atoms, is equal to $\bar{m}$. The value of $m$ for the phonon
bands completely agrees with that obtained in Fig. 4 (b). Thus the phonon
wavefunctions retains information for the case with $T_{c/5}$ symmetry.

Now we discuss in which cases the quantum number $m$ is well defined and the
screw symmetry is regarded as a relatively good symmetry. In Figs. 4(c)-(f),
we can compare the result for the quantum number $m$ for the bands
$\sigma=30,31,32,33$. We see that the quantum number $m$ is well defined in
most cases, except for the bands $\sigma=32,33$ near $k=0$ (Fig. 4(e)-(f)).
From the phonon dispersion (Fig. 4(b)) the bands $\sigma=32,33$ are close to
each other, making them to hybridize and to make the quantum number $m$
relatively ill-defined. One may wonder that the bands $\sigma=32,33$ are
mutually close not only near $k=0$ but also around $k=\pm 2\pi/c$, but even
near $k=\pm 2\pi/c$ the quantum number $m$ is well defined for $\sigma=32,33$
as seen in Fig. 4(e)-(f). One can understand this difference between $k\sim 0$
and $k\sim\pm 2\pi/c$ by the above scheme of the virtual Brillouin zone. As
mentioned in the present subsection, the size of the reciprocal lattice vector
along the $c$-axis is $6\pi/c$, which gives rise to the folding of the
Brillouin zone. This folding originates from the Umklapp scattering of phonons
by the wavenumber $\pm 6\pi/c$. From Fig. 5 (b), the near degeneracy at $k\sim
0$ between the $\sigma=32,33$ bands is between the bands originally at $k\sim
6\pi/c$ and $k\sim-6\pi/c$, via the second-order effect in the Umklapp
scattering. On the other hand, the near degeneracy at $k\sim 2\pi/c$ between
the bands $\sigma=32,33$ is between the bands originally at $k\sim 14\pi/c$
and $k\sim-4\pi/c$, via the third-order effect in the Umklapp scattering. The
near degeneracy at $k\sim-2\pi/c$ is treated similarly as the third order in
the Umklapp scattering. Thus the near degeneracy around $k\sim\pm 2\pi/c$ is
of the higher order in the Umklapp scattering than that around $k\sim 0$, and
the hybridization is weaker. It is the reason why the bands $\sigma=32,33$
near $k\sim\pm 2\pi/c$ do not hybridize so much (and the quantum number $m$ is
well defined), compared with those near $k\sim 0$, in agreement with the
result in Fig. 4(e)-(f).

###  IV.2 SnIP and S10: materials with approximate screw symmetry

Figure 6:  Approximate screw symmetry of phonons in the material $\ce{SnIP}$
(a), (b) Schematic pictures of atomic positions. (a) show that one helical
chains has approximate $7_{2}$ screw symmetry in the $\vec{a}_{1}$ direction.
(b) shows the atomic positions as seen from the $\vec{a}_{1}$ direction. (c)
(d) Band dispersions. In (c) we highlight the focused bands
($\sigma=85,\cdots,92$) in blue. The bands with $\sigma=85,\cdots,92$. are
shown in (d). We show the value of the quantum number $m$ for each band from
the relative phase discussion. (e) (f) Relative phases of the bands
$\sigma=85$ and $91$, respectively.

First, we explain the calculations for $\ce{SnIP}$ shown in Fig. 6. The atomic
position in the unit cell is shown in Figs. 6 (a)-(b), where the unit cell has
two helical chains with opposite chiralities and $42$ atoms. One helical chain
has approximate $7_{2}$ screw symmetry shown in Fig. 6 (b), and this chain is
transformed to the other one by glide reflection. Figure 6 (c) shows the
phonon dispersions in $\ce{SnIP}$, which has $126$ bands. In this paper, we
analyze the approximate screw symmetry of the helical chains formed by
$\ce{P}$ atoms, and we focus on the bands with indices $\sigma=85,\cdots,98$
shown in Fig. 6 (d). All of these bands ($\sigma=85,\cdots,98$) are almost
doubly degenerate because phonon modes in the two helical chains in the unit
cell are almost decoupled, and these two chains have the same phonon
eigenfrequencies due to the glide symmetry [46]. Therefore, we estimate the
quantum number $m$ by extracting one of the helical chains and calculating the
relative phase. Figures 6 (e) and (f) show the calculations of the relative
phases for the $\sigma=85$ and $\sigma=91$ bands, respectively. The
calculations for the other bands are given in Appendix B. By comparing them
with the relative phases for the case with exact screw symmetry (Eq. (67)), we
can assign the value of the quantum number $m$ for each band as shown in Fig.
6 (d). The value of the pseudoangular momentum $m^{\prime}$ (associated with
$7_{2}$ symmetry) is calculated by $m^{\prime}\equiv 2m\pmod{7}$, which
follows from Eq. (38) for the case with mutually coprime $n$ and $l$. We can
see that the quantum number $m$ of the band $\sigma=85$ is $0$ and that of the
band $\sigma=91$ changes from $2$ to $-2$ at the $\Gamma$ point. Similarly, we
can assign the values of $m$ for each band as presented in Appendix B without
ambiguity, and the result is shown in Fig. 6 (d). In agreement with our
discussion in Sec. III, the quantum number $m\pmod{7}$ increases by $+1$
across the Brillouin zone boundary. Thus, even when the $7_{2}$ screw symmetry
is broken in SnIP, the eigenvectors of the system still have properties that
well reflect the approximate screw symmetry.

Figure 7:  Approximate screw symmetry of phonons in the material $\ce{S10}$
(a), (b) Schematic pictures of atomic positions. (a) show that one helical
chains has approximate $10_{7}$ screw symmetry in the $\vec{a}_{1}$ direction.
(b) shows the atomic positions as seen from the $\vec{a}_{1}$ direction. (c)
(d) Band dispersions. In (c) we highlight the focused bands
($\sigma=41,\cdots,60$) in blue. The bands with $\sigma=41,\cdots,60$. are
shown in (d). We show the value of the quantum number $m$ for each band from
the relative phase discussion. (e) (f) Relative phases of the bands
$\sigma=41$ and $48$, respectively.

Next, we explain the calculations for $\ce{S10}$ shown in Fig. 7. The atomic
positions in the unit cell are shown in Figs. 7 (a)(b) and this material has
approximate $10_{7}$ screw symmetry, with two helical chains (A and B) and
$20$ atoms in one unit cell. Thus, phonon spectra of $\ce{S10}$ have 60 bands,
as shown in Fig. 7 (c). In this paper, we focus on the bands with indices
$\sigma=41,\cdots,60$ shown in Fig. 7 (d). As seen in the band structure,
every mode is nearly doubly degenerate, reflecting the two chains in the unit
cell. Figures 7 (e) and (f) show the calculations of the relative phases for
the $\sigma=41$ and the $\sigma=48$ bands, respectively. The results for the
other bands are shown in Appendix B. In these calculations we use the
wavefunctions in the chain A.

This material has been synthesized before, while the calculated phonon spectra
have an imaginary part, suggesting that the material is unstable. In fact, our
phonon calculation has the following two limitations, by which the imaginary
part is unavoidable. First, the experimental structure is a high-pressure and
high-temperature one, while DFPT calculation can only obtain the phonon
spectra for atmospheric pressure and zero temperature. Second, the $\ce{S10}$
crystal structure used in our calculation is just 1/4 of the experimentally
reported one, which have 80 atoms per unit cell. Since each experimental unit
cell have 4 left-hand and 4 right-hand helical sulfur chains, with each chain
having 10 sulfur atoms, the calculation on the phonon spectra will be very
heavy. Here, the crystal structure is approximately described by a smaller
unit cell containing only 1 left-handed chain and 1 right-handed chain in our
calculation, which has the same symmetry and the same pseudoangular momentum
as the experimental structure. Thus we took the smaller unit cell, which does
not affect our analysis on the pseudoangular momentum.

By comparing with the case of exact symmetry, we can assign the value of the
quantum number $m$ for the bands $\sigma=41,\ 42,\cdots,\ 54$ as shown in Fig.
7 (d), while the bands with $\sigma=55,\cdots,60$, the quantum number $m$ is
ill-defined. The value of the pseudoangular momentum $m^{\prime}$ (associated
with $10_{7}$ symmetry) is calculated by $m^{\prime}\equiv 7m\pmod{10}$, which
follows from Eq. (38) for the case with mutually coprime $n$ and $l$. These
figures show that the quantum number is $m=5$ for the $\sigma=40$ band and
changes from $m=-3$ to $m=3$ at $\Gamma$ point for the $\sigma=47$ band.
Similarly, we can assign the values of $m$ for each band as presented in
Appendix B without ambiguity, and the result is shown in Fig. 7 (d). In
agreement with our discussion in Sec. III, the quantum number $m\pmod{10}$
increases by $+1$ across the Brillouin zone boundary. Thus, even when the
$10_{7}$ screw symmetry is broken, the eigenvectors have properties that well
reflect the approximate screw symmetry.

##  V  Conclusion and discussion

In this paper, we studied how to extract the pseudoangular momentum from the
phonon eigenfunction with approximate screw symmetry. In preparation for this,
we considered a one-dimensional system with partially broken translational
symmetry and show how to assign the quantum number to each band, which
characterizes the relative phase between neighboring atoms. We showed that the
behavior of this quantum number is naturally understood in the extended
Brillouin zone, and showed several key properties of this quantum number and
the relative phase. We performed model calculations for a simple case, and
extended this method to systems with approximate screw symmetry to extract
information about the pseudoangular momentum. By applying the relative phase
defined in this way to the material $\ce{Li6B5}$ with approximate
translational symmetry and materials $\ce{SnIP}$ and $\ce{S10}$ with
approximate screw symmetry, we confirmed that the information of the quantum
number and the pseudoangular momentum can be extracted by exact screw
symmetries. We showed that the assigned pseudoangular momentum is naturally
understood in the extended scheme.

Through our theory and its applications to real materials in this paper, we
discuss when the screw symmetry becomes a good symmetry and the pseudoangular
momentum becomes well-defined. One can say that the screw symmetry is a good
symmetry, when the energy (or frequency) scale of the symmetry-breaking
perturbation is much smaller than the energy scale of the original band
structure under exact screw symmetry, which can be a gap size near the band
gap, or be a bandwidth. Meanwhile, the calculations in this paper show that it
is not so simple whether the screw symmetry is a good symmetry, and the
condition for the approximate screw symmetry depends even on the wavenumber
and the band considered. It is because depending on the band and the
wavenumber, the effect of the symmetry-breaking term works differently, as we
discussed in Sec. IV. Whether or not the screw symmetry becomes a good
symmetry and the pseudoangular momentum becomes well-defined is not a simple
question in general, and that is why the method to calculate the pseudoangular
momentum in this paper is relevant for understanding phonon physics in systems
with approximate screw symmetry. From the analysis in this paper,
hybridization between bands with different values of the quantum number $m$
(or the pseudoangular momentum) breaks the screw symmetry and makes $m$ ill-
defined, especially at the band crossings. The screw symmetry is largely
broken when the hybridization occurs at a lower degree in the Umklapp
scattering, and it may depend on the wavevector.

As shown in this study, approximate symmetries can exist in crystals and they
affect eigenmodes of the crystals, in addition to exact symmetries indicated
by space groups. For example, in the case of screw symmetry, while only $2$,
$3$, $4$, and $6$-fold screw symmetries are allowed as exact symmetries in
crystals, we show that approximate $7$-fold and $10$-fold screw symmetries can
be realized, which are also reflected in the phonon wave function. Since such
approximate symmetries are not included in the studies on physical properties,
the physics of such approximate symmetries is an interesting topic for further
study. For example, such an approximate symmetry is expected to affect
physical processes involving various particles/quasiparticles, such as
electronic processes involving multi-phonons/photons and exciton scattering
processes. In systems with exact symmetry, symmetry restricts such processes
as selection rules, and even when the symmetry is not exact but approximate,
selection rules remain valid. While exact screw rotation symmetries are
limited in crystals due to the restriction of translational symmetry to
twofold, threefold, fourfold and sixfold, an approximate screw symmetry leads
to a more variety of multifold screw symmetries such as 7- and 10-fold ones,
and they will lead to selection rules that is absent for exact screw
symmetries, as has been studied in the context of chiral phonons under exact
symmetries, Furthermore, anisotropy in transport properties involving chiral
phonons will also reflect the approximate screw symmetry. Angular dependence
of phonon transport will lead to $n$-fold anisotropic behavior where $n$ is
different from 2, 3, 4 or 6.

###### Acknowledgements.

This work was partly supported by JSPS KAKENHI Grants No. JP20H04633,
JP22H00108 and No. JP21K13865.

##  Appendix A  Screw symmetry $n_{l}$ where $n$ and $l$ are not coprime

In this appendix, we discuss definitions of the quantum numbers $m$ and
$m^{\prime}$ and their relationship under $n_{l}$ screw symmetry
$\hat{C}_{n}^{l}$, when $n$ and $l$ are not coprime. Here $m$ represents a
relative phase between neighboring atoms, while $m^{\prime}$ is associated
with an eigenvalue of the $\hat{C}_{n}^{l}$ screw operation. We consider the
system with screw symmetry $\hat{C}_{n}^{l}$ with respect to the $z$ axis and
translational symmetry $\hat{T}_{a}$ in the $z$ direction, where $a$ denotes
the lattice constant along $z$.

We here consider the case with $n$ and $l$ not being coprime. Let $g(\geq 2)$
denote the greatest common divisor of $n$ and $l$, and we write

$\displaystyle n=gn^{\prime},\ \ l=gl^{\prime},$ (74)

where $n^{\prime}$ and $l^{\prime}$ are coprime integers. Then the system
automatically has $g$-fold rotation symmetry $\hat{C}_{g}$
($=(\hat{C}_{n}^{l})^{n^{\prime}}(\hat{T}_{a})^{-l^{\prime}}$). The
translation part of the screw operation $\hat{C}_{n}^{l}$ is by
$al/n=al^{\prime}/n^{\prime}$, which means that the atoms are placed with a
period of $a/n^{\prime}$ along the $z$ direction. Namely, in systems with
$n_{l}$ symmetry, $g$ atoms related by $\hat{C}_{g}$ symmetry share the same
$z$ coordinate, and these groups of $g$ atoms are placed along the $z$ axis
with a spacing $a/n^{\prime}$. Thus the number of atoms within the unit cell
related by symmetry is $gn^{\prime}=n$, as expected.

In the main text, we study the case with $n$ and $l$ being coprime, and we
show how to relate two quantum numbers $m$ and $m^{\prime}$. In this Appendix
we estabish the similar relationship between them, when $n$ and $l$ are not
coprime. To this end, we show that there exists a symmetry operation
$\hat{\mathcal{O}}\equiv(\hat{C}_{n}^{l})^{p}(\hat{T}_{a})^{q}$ ($p$, $q$:
integer) which relates between neighboring atoms, mutually displaced by
$\frac{a}{n^{\prime}}$ along the $z$ direction. Therefore, integers $p$ and
$q$ satisfy $p\frac{la}{n}+qa=\frac{a}{n^{\prime}}$, i.e.
$l^{\prime}p+n^{\prime}q=1$. Since $l^{\prime}$ and $n^{\prime}$ are coprime,
thanks to the Bézout’s lemma, this equation always has integer solutions for
$p$ and $q$. Hence, $\hat{\mathcal{O}}=(\hat{C}_{n}^{l})^{p}(\hat{T}_{a})^{q}$
is an operator that rotates the system by $\frac{2p\pi}{n}$ and translates it
by $a/n^{\prime}$.

We here note one important point; there are various choices for integers $p$
and $q$, and not all the choices are appropriate for our purpose. In the cases
with coprime $n$ and $l$, discussed in the main text, there exists one-to-one
correspondence between $m$ and $m^{\prime}$. In contrast, if they are not
coprime, we need to choose $p$ and $q$ properly to guarantee their one-to-one
correspondence. For this purpose, we impose a condition that
$(\hat{\mathcal{O}})^{r}$ be equal to $\hat{C}_{n}^{l}$ modulo translation
symmetry, i.e.

$\displaystyle(\hat{\mathcal{O}})^{r}(\hat{T}_{a})^{s}=\hat{C}_{n}^{l}\ \
(r,s:\ \mbox{integer}),$ (75)

which guarantees that the operators $\\{\hat{\mathcal{O}},\hat{T}_{a}\\}$ are
generators of the screw symmetry $n_{l}$. This means

$\displaystyle\frac{r}{n^{\prime}}+s=\frac{l}{n},\ pr\equiv 1\pmod{n}.$ (76)

The second equation leads to $pr+nt=1$ for integers $r$ and $t$. The condition
for this equation to have solutions for $r$ and $t$ is that $p$ and $n$ are
coprime. As noted earlier there are various choices for $p$, and not all the
possible values of $p$ is coprime with $n$. Meanwhile, we can show that among
the various possible values of $p$, there always exist a value of $p$ that is
coprime with $n$. To show this, we note that $p$ and $q$ are solutions of
$l^{\prime}p+n^{\prime}q=1$, which has solutions of the form
$(p,q)=(p_{0}+n^{\prime}\kappa,\ q_{0}-l^{\prime}\kappa)$ ($\kappa$: integer).
Here we note that $p_{0}$ and $n^{\prime}$ are coprime because
$l^{\prime}p_{0}+n^{\prime}q_{0}=1$. Thus the set of possible solutions for
$p$ forms an arithmetic series, $p=p_{0}+n^{\prime}\kappa$ ($p_{0}$,
$n^{\prime}$: coprime integers), and it contains prime numbers from the
Dirichlet’s theorem on arithmetic progressions. Thus by choosing $p$ to be a
prime number, $p$ and $n$ are coprime. We can also show that the first
equation of (76) is satisfied then.

With such a choice of the integers $p$ and $q$, our theory is similar to the
case with coprime $n$ and $l$ in the main text. The pseudoangular momentum
$m^{\prime}$ is defined as
$\hat{C}_{n}^{l}\psi(\vec{r})=e^{i(kla+2m^{\prime}\pi)/n}\psi(\vec{r})$ for
the wave function $\psi(\vec{r})$ of this system,where

$\displaystyle m^{\prime}\equiv\left\\{\begin{array}[]{ll}0,\pm
1,\cdots,\pm(n-1)/2&\text{:}\ n\ \text{odd}\\\ 0,\pm
1,\cdots,\pm(n-2)/2,n/2&\text{:}\ n\ \text{even}\end{array}\right.\pmod{n}.$
(79)

Then we have

$\displaystyle\hat{\mathcal{O}}\psi(\vec{r})$
$\displaystyle=(\hat{C}_{n}^{l})^{p}(\hat{T}_{a})^{q}\psi(\vec{r})$
$\displaystyle=\left(e^{i(lka+2m^{\prime}\pi)/n}\right)^{p}\left(e^{ika}\right)^{q}\psi(\vec{r})$
(80)

holds. Meanwhile, we put the eigenvalue of $\hat{\mathcal{O}}$ to be equal to
$e^{i(gka+2m\pi)/n}$ because
$(\hat{\mathcal{O}})^{n}=(\hat{T}_{a})^{g}=e^{igka}$, where $m$ is an integer
representing the relative phase between neighboring atoms displaced by
$a/n^{\prime}$ along the $z$ axis. Then, from (80), $m$ is given by $m\equiv
m^{\prime}p\pmod{n}$. By using $pr\equiv 1\pmod{n}$, we get $m^{\prime}\equiv
mr\pmod{n}$. Therefore, we have successfuly established a one-to-one
correspondence between the integers $m$ and $m^{\prime}$. In this case, the
operators $\\{\hat{\mathcal{O}},\hat{T}_{a}\\}$ are generators of the screw
symmetry, and in particular, $g$-fold rotation symmetry $C_{g}$ is expressed
in terms of $\hat{\mathcal{O}}$ as
$(\hat{\mathcal{O}})^{rn^{\prime}}(\hat{T}_{a})^{-r}=\hat{C}_{g}$. We
summarize properties of these operators in Table 2.

Figure 8:  Schematic figure for the case with $6_{4}$ screw symmetry, which
corresponds to $n=6$ and $l=4$ being non-coprime. The orange dots represent
positions of atoms, and the red arrow represents the screw operation
$\hat{C}_{6}^{4}$. The green arrow represents the operation
$\hat{\mathcal{O}}$, which connects between atoms in the neighboring layers.
It seems that there are two choices for the operator $\hat{\mathcal{O}}$ that
relates neighboring atoms displaced by $a/3$ along the $z$ axis, but we cannot
choose the blue arrow as the operator $\hat{\mathcal{O}}$, because it does not
generate all the screw operations; in other words, the operation by the blue
arrow does not connect all the atoms in the unit cell.

We show an example of $n=6$, $l=4$ in Fig. 8. Here we have $n^{\prime}=3$ and
$l^{\prime}=2$, and two solutions of $l^{\prime}p+n^{\prime}q=1$ are
$(p,q)=(-1,1),(2,-1)$, and the operator $\hat{\mathcal{O}}$ defined by these
choices of $(p,q)$ are shown by the green arrow and the blue arrow,
respectively. In our discussion above, we need to take $p$ to be coprime with
$n$, so only the former choice of $(p,q)=(-1,1)$ (green arrow) is allowed. By
this proper choice of $\hat{\mathcal{O}}$, all the atoms are related by
$\hat{\mathcal{O}}$. On the other hand, if we take the other choice of
$(p,q)=(2,-1)$, not all the atoms are related by $\hat{\mathcal{O}}$. Thus the
eigenvalue of $\hat{\mathcal{O}}$ does not possess the same infomation as that
of $\hat{C}_{n}^{l}$.

With such a proper choice of $\hat{\mathcal{O}}$, we name the atoms in the
unit cell like $(i,j)$ ($i,j$: integer) in the following way. Within the unit
cell (e.g. Fig. 8), we begin with the $g$ atoms with the smallest $z$
coordinate, and we name them $(1,1)$, $(1,2)$, $\cdots,(1,g)$, in the way that
the $(1,j)$ atom is transformed to $(1,j+1)$ by the $g$-fold rotation
$\hat{C}_{g}$. Then we name other atoms so that the operator
$\hat{\mathcal{O}}$ transforms the $(i,j)$ atom to $(i+1,j)$. Then, we express
the eigenvectors $\bm{\epsilon}$ of the dynamical matrix in the form
$\bm{\epsilon}=(\vec{\epsilon}_{1,1},\cdots,\vec{\epsilon}_{1,g},\cdots,\vec{\epsilon}_{n^{\prime},1},\cdots,\vec{\epsilon}_{n^{\prime},g})^{\mathrm{T}}$,
where $\vec{\epsilon}_{i,j}$ is the displacement of the atom $(i,j)$ $(1\leq
i\leq n^{\prime},1\leq j\leq g)$. We define te relative phase for this
eigenvector $\bm{\epsilon}$ and explain how to use it to extract the
pseudoangular momentum $m^{\prime}$.

In this case, for the eigenvector $\bm{\epsilon}^{(\sigma)}$ for the
$\sigma$-th band, we define the relative phase $\theta_{i,j,\alpha}$ as

$\displaystyle z_{i,j,\alpha}^{(\sigma)}(k)$
$\displaystyle=\left\\{\begin{array}[]{ll}u_{i+1,j,\alpha}^{(\sigma)}/u_{i,j,\alpha}^{(\sigma)}&(i=1,\cdots
n^{\prime}-1)\\\
u_{1,j+p,\alpha}^{\prime(\sigma)}e^{ika}/u_{n^{\prime},j,\alpha}^{(\sigma)}&(i=n^{\prime})\end{array}\right.,$
(83) $\displaystyle\theta_{i,j,\alpha}^{(\sigma)}(k)$ $\displaystyle=\arg
z_{i,j,\alpha}^{(\sigma)}(k),$ (84)

where $\alpha=x,y,z$,
$\vec{u}_{i,j}^{(\sigma)}=A^{i-1}_{p}\vec{\epsilon}_{i,j}$,
$\vec{u}_{1,j,\alpha}^{\prime(\sigma)}=A^{n^{\prime}}_{p}\vec{\epsilon}_{1,j}$
and $A_{p}$ is a $2p\pi/n$ rotation matrix (47). This relative phase is given
by $\theta_{i}(k)=(gka+2m\pi)/n$ when the system has exact $n_{l}$ screw
symmetry. Then, even in the case where the $n_{l}$ screw symmetry is slightly
broken, we can extract the information of the quantum number $m$ and the
pseudoangular momentum $m^{\prime}$ from the eigenvector
$\bm{\epsilon}^{(\sigma)}$.

Operator | Rotation | Translation |
---|---|---|---
|  angle |  |
$\hat{C}_{n}^{l}$ | $\frac{2\pi}{n}$ | $\frac{l^{\prime}}{n^{\prime}}a$ | $=(\hat{\mathcal{O}})^{r}(\hat{T})^{s}$
$\hat{\mathcal{O}}$ | $\frac{2\pi p}{n}$ | $\frac{1}{n^{\prime}}a$ | $=(\hat{C}_{n}^{l})^{p}(\hat{T})^{q}$
$\hat{C}_{g}$ | $\frac{2\pi}{g}$ | $0$ | $=(\hat{\mathcal{O}})^{rn^{\prime}}(\hat{T})^{-r}=(\hat{C}_{n}^{l})^{n^{\prime}}(\hat{T})^{l}$
Table 2: Relationships between various operations in the screw symmetry. The
operators $\\{\hat{C}_{n}^{l},\hat{T}_{a}\\}$ generate the screw symmetry
$n_{l}$, and so do the operators $\\{\hat{\mathcal{O}},\hat{T}_{a}\\}$

##  Appendix B  Relative phase of the other bands for two materials,
$\ce{SnIP}$ and $\ce{S10}$

In this appendix, we show the results of calculations for the relative phases
of $\ce{SnIP}$ and $\ce{S10}$ for bands not shown in the main text.

First, we show the relative phase of $\ce{SnIP}$ in Fig. 9. Figures 9 (a),
(b), (c), (d) and (e) correspond to the relative phases of the bands
($\sigma=87$, $89$, $93$, $95$ and $97$) in Fig. 6 (d) in the main text,
respectively. From this result, we can extract the quantum number $m$,
summarized in Fig. 6. We note that the relative phase for the $\sigma=97$ band
in Fig. 9(e) shows a crossover of the value of $m$ as
$0\rightarrow-3\rightarrow 3\rightarrow 0$. It is due to hybridization between
the bands with $m=\pm 3$ and that with $m=0$ at their anticrossings, as seen
around 8THz in Fig. 6 (c). Thus at the anticrossing between bands, the
properties of the wavefunction of the bands are exchanged.

Figure 9:  Relative phase of $\ce{SnIP}$. (a), (b), (c), (d) and (e)
correspond to the relative phases of the bands $\sigma=87$, $89$, $93$, $95$
and $97$ in Fig. 6 (d), respectively.

Finally, we show the relative phase of $\ce{S10}$ in Fig. 10. Figures 10 (a),
(b), (c), (d), (e), (f), (g), (h) and (i) correspond to the relative phases of
the bands ($\sigma=43$, $45$, $47$, $49$, $51$, $53$, $55$, $57$ and $59$) in
Fig. 7 (d) in the main text, respectively. From this result, we can extract
the quantum number $m$ for $\sigma=41,\ 42,\cdots,\ 54$, summarized in Fig. 7,
while for the bands with $\sigma=55,\cdots,60$, the quantum number $m$ is ill-
defined.

Figure 10:  Relative phase of $\ce{S10}$. (a), (b), (c), (d), (e), (f), (g),
(h) and (i) correspond to the relative phases of the bands $\sigma=43$, $45$,
$47$, $49$, $51$, $53$, $55$, $57$ and $59$ in Fig. 7 (d), respectively.

By comparing the results for the two bands $\sigma=48$ (Fig. 7(f)) and
$\sigma=47$ (Fig. 10 (c)), which are almost degenerate in the band structure,
we notice that the latter has an anomalous behavior near $k=\pm 0.6\pi$, while
the former does not. One might think that this anomaly comes from anticrossing
between bands, but it is not the case, since such an anomaly from anticrossing
should appear simultaneously in the two bands $\sigma=47,48$. Instead, this
anomaly comes from an anomalous behavior of hybridization between phonon modes
in the two chains A and B, as we explain in the following. In fact, at the
wavenumber $k=\pm 0.6\pi$, the $\sigma=47$ band is localized only in the chain
B, while the $\sigma=48$ band only in the chain A. It is seen in Fig. 11 where
the sum of the squares of the phonon amplitudes from the normalized
eigenvalues within each chain is plotted for the two bands. It is noted that
in our calculation of the quantum number $m$ for S10, we used the
wavefunctions in the chain A. Then in the phonon mode $\sigma=47$ in Fig. 10
(c), the phonon amplitudes in the chain A is almost zero, which gives rise to
the anomaly around $k=\pm 0.6\pi$, because at this point the result becomes
highly sensitive to details of the system and to numerical errors. We also
checked that if we use the phonon amplitude in the chain B instead, an anomaly
appears in the $\sigma=48$ band. Thus, this anomaly in the relative phases is
attributed to the localization of the eigenmodes into a single chain.

Here, one may wonder why this localization of eigenmodes into a single chain
happens in this case. In general, even when two chains exist within the unit
cell, the phonon modes within the two chains are hybridized and they are never
localized within a single chain. In the present case, we find that it is
allowed from the $C_{2x}T$ symmetry, where $C_{2x}$ is the twofold rotation
symmetry with respect to the axis along $\vec{a}_{2}$ perpendicular to the
chains (see Fig. 7(a)(b)), and $T$ is the time-reversal symmetry. In the
present material with two chains, A and B, this $C_{2x}T$ symmetry is
preserved within each chain, and the $N\times N$ phonon dynamical matrix
$D_{i}(k)$ satisfies

$\displaystyle MD_{i}(k)M^{\dagger}=D_{i}(k)^{*},$ (85)

where $N(=10)$ is the number of atoms within the unit cell in one chain,
$i=A,B$ indicates the chains A and B, $k$ is the wavenumber along the chain,
and $M$ is a $N\times N$ matrix representing the $C_{2x}$ operation. For
example, in the present case, for the phonons with atomic displacements along
the $z$ axis, $M$ is given by

$\displaystyle M=\begin{pmatrix}&&&&-1\\\ &&&-1&\\\ &&\cdots&&\\\ &-1&&&\\\
-1&&&&\end{pmatrix}.$ (86)

Under this symmetry, when $u_{i}(k)$ is an eigenvector at the wavenumber $k$,
the vector $(Mu_{i}(k))^{*}$ is also an eigenvector having the same
eigenfrequency, and therefore they are proportional to each other:
$u_{i}(k)\propto(Mu_{i}(k))^{*}$. By properly choosing the phase of
$u_{i}(k)$, one can always make $u_{i}(k)$ to satisfy
$u_{i}(k)=(Mu_{i}(k))^{*}$. Thereby, the dynamical matrix $D_{i}(k)$ is
diagonalized by a $N\times N$ unitary matrix $U_{i}(k)$,
$U_{i}(k)^{\dagger}D_{i}(k)U_{i}(k)=\varepsilon_{i}(k)$, satisfying
$U_{i}(k)=(MU_{i}(k))^{*}$, where $\varepsilon_{i}(k)$ is a diagonal matrix
with the eigenfrequencies as diagonal elements. Next, if we combine the two
chains without hybridization, the dynamical matrix for the entire system

$\displaystyle D(k)=\begin{pmatrix}D_{A}(k)&\\\ &D_{B}(k)\end{pmatrix}$ (87)

is diagonalized by a $2N\times 2N$ unitary matrix $U(k)$ satisfying

$\displaystyle U(k)=\begin{pmatrix}U_{A}(k)&\\\ &U_{B}(k)\end{pmatrix},$ (88)
$\displaystyle U_{i}(k)^{\dagger}D_{i}(k)U_{i}(k)=\varepsilon_{i}(k),$ (89)
$\displaystyle U_{i}(k)=(MU_{i}(k))^{*}\ \ (i=A,B),$ (90)

Next, we introduce an interchain coupling preserving $C_{2x}T$ symmetry. Then
the dynamical matrix becomes

$\displaystyle D^{\prime}(k)=\begin{pmatrix}D_{A}(k)&V(k)\\\
V(k)^{\dagger}&D_{B}(k)\end{pmatrix},$ (91)

where $V(k)$ is a $N\times N$ matrix satisfying $MV(k)M^{\dagger}=V(k)^{*}$.
Then, after unitary transformation by $U(k)$, the dynamical matrix becomes

$\displaystyle
U(k)^{\dagger}D^{\prime}(k)U(k)=\begin{pmatrix}\varepsilon_{A}(k)&V^{\prime}(k)\\\
V^{\prime}(k)^{\dagger}&\varepsilon_{B}(k)\end{pmatrix}\equiv d(k),$ (92)

where $V^{\prime}(k)=U_{A}(k)^{\dagger}V(k)U_{B}(k)$. Then from the $C_{2x}T$
symmetry, we get $V^{\prime}(k)=V^{\prime}(k)^{*}$. Therefore the matrix
$d(k)$ is a real matrix with its diagonal elements representing the
eigenfrequencies in each chain, and the off-block-diagonal elements in
$V^{\prime}(k)$ represent a hybridization between the modes in the chains A
and B. In the present case the spectra of the chains A and B are the same, and
we focus on the particular eigenmodes with the frequency $\omega_{n}$. Then we
can approximate the dynamical matrix by retaining only the matrix elements
involving these modes with the frequency $\omega_{n}$, and the reduced
dynamical matrix is

$\displaystyle\bar{d}(k)=\begin{pmatrix}\omega_{n}(k)&v^{\prime}(k)\\\
v^{\prime}(k)&\omega_{n}(k)\end{pmatrix},$ (93)

where $v^{\prime}(k)$ is a real parameter, representing the interchain
hybridization. Only when $v^{\prime}(k)=0$, the two chains are decoupled.
Because $v^{\prime}(k)$ is real, if $v^{\prime}(k)$ has a different sign
between $k=0$ and $k=\pi$, it necessarily goes across zero, where the two
chains are decoupled and the phonon modes are localized only in a single
chain, either A or B. It explains the anomalous behavior shown in Fig. 11. We
note that without this $C_{2x}T$ symmetry, the hybridization $v^{\prime}(k)$
becomes complex in general, and it cannot be zero when changing only one
parameter $k$. Thus, to summarize, under the $C_{2x}T$ symmetry, it can happen
at a certain value of $k$ that an interchain hybridization becomes zero and
the eigenmodes are localized only in a single chain. Among the bands
considered, the same behavior is seen also in the bands $\sigma=41,42$ around
$k=\pm 0.8\pi$ and in the bands $\sigma=45,46$ around $k=0.6\pi$, but not in
other bands, and this is consistent with our scenario.

Figure 11:  Norm of the phonon modes within the chains A and B for the phonon
modes with (a) $\sigma=47$ and (b) $\sigma=48$ in the material $\ce{S10}$. At
the wavenumvers $k\sim\pm 0.6\pi$ (red circles), the modes are localized only
in one chain.

## References

  * Zhang and Niu [2015] L. Zhang and Q. Niu, Chiral phonons at high-symmetry points in monolayer hexagonal lattices, Physical Review Letters 115, 115502 (2015).
  * Zhu  _et al._ [2018] H. Zhu, J. Yi, M.-Y. Li, J. Xiao, L. Zhang, C.-W. Yang, R. A. Kaindl, L.-J. Li, Y. Wang, and X. Zhang, Observation of chiral phonons, Science 359, 579 (2018).
  * Liu  _et al._ [2017] Y. Liu, C.-S. Lian, Y. Li, Y. Xu, and W. Duan, Pseudospins and topological effects of phonons in a Kekulé lattice, Physical Review Letters 119, 255901 (2017).
  * Chen  _et al._ [2018] H. Chen, W. Zhang, Q. Niu, and L. Zhang, Chiral phonons in two-dimensional materials, 2D Materials 6, 012002 (2018).
  * Gao  _et al._ [2018] M. Gao, W. Zhang, and L. Zhang, Nondegenerate chiral phonons in graphene/hexagonal boron nitride heterostructure from first-principles calculations, Nano Letters 18, 4424 (2018).
  * Xu  _et al._ [2018a] X. Xu, W. Zhang, J. Wang, and L. Zhang, Topological chiral phonons in center-stacked bilayer triangle lattices, Journal of Physics: Condensed Matter 30, 225401 (2018a).
  * Xu  _et al._ [2018b] X. Xu, H. Chen, and L. Zhang, Nondegenerate chiral phonons in the Brillouin-zone center of 3$\times$ 3 honeycomb superlattices, Physical Review B 98, 134304 (2018b).
  * Chen  _et al._ [2019a] H. Chen, W. Wu, S. A. Yang, X. Li, and L. Zhang, Chiral phonons in kagome lattices, Physical Review B 100, 094303 (2019a).
  * Li  _et al._ [2019a] Z. Li, T. Wang, C. Jin, Z. Lu, Z. Lian, Y. Meng, M. Blei, S. Gao, T. Taniguchi, K. Watanabe, T. Ren, S. Tongay, L. Yang, D. Smirnov, T. Cao, and S.-F. Shi, Emerging photoluminescence from the dark-exciton phonon replica in monolayer $\ce{WSe2}$, Nature Communications 10, 1 (2019a).
  * Li  _et al._ [2019b] Z. Li, T. Wang, C. Jin, Z. Lu, Z. Lian, Y. Meng, M. Blei, M. Gao, T. Taniguchi, K. Watanabe, T. Ren, T. Cao, S. Tongay, D. Smirnov, L. Zhang, and S.-F. Shi, Momentum-dark intervalley exciton in monolayer tungsten diselenide brightened via chiral phonon, ACS Nano 13, 14107 (2019b).
  * Liu  _et al._ [2019] E. Liu, J. van Baren, T. Taniguchi, K. Watanabe, Y.-C. Chang, and C. H. Lui, Valley-selective chiral phonon replicas of dark excitons and trions in monolayer $\ce{WSe2}$, Physical Review Research 1, 032007 (2019).
  * Chen  _et al._ [2019b] X. Chen, X. Lu, S. Dubey, Q. Yao, S. Liu, X. Wang, Q. Xiong, L. Zhang, and A. Srivastava, Entanglement of single-photons and chiral phonons in atomically thin $\ce{WSe2}$, Nature Physics 15, 221 (2019b).
  * Delhomme  _et al._ [2020] A. Delhomme, D. Vaclavkova, A. Slobodeniuk, M. Orlita, M. Potemski, D. Basko, K. Watanabe, T. Taniguchi, D. Mauro, C. Barreteau, E. Giannini, A. F. Morpurgo, N. Ubrig, and C. Faugeras, Flipping exciton angular momentum with chiral phonons in $\ce{MoSe2}$/$\ce{WSe2}$ heterobilayers, 2D Materials 7, 041002 (2020).
  * Zhang  _et al._ [2020] W. Zhang, A. Srivastava, X. Li, and L. Zhang, Chiral phonons in the indirect optical transition of a $\ce{MoS2}$/$\ce{WS2}$ heterostructure, Physical Review B 102, 174301 (2020).
  * Grissonnanche  _et al._ [2020] G. Grissonnanche, S. Thériault, A. Gourgout, M.-E. Boulanger, E. Lefrançois, A. Ataei, F. Laliberté, M. Dion, J.-S. Zhou, S. Pyon, T. Takayama, H. Takagi, N. Doiron-Leyraud, and L. Taillefer, Chiral phonons in the pseudogap phase of cuprates, Nature Physics 16, 1108 (2020).
  * Chen  _et al._ [2021a] C. Chen, X. Chen, B. Deng, K. Watanabe, T. Taniguchi, S. Huang, and F. Xia, Probing interlayer interaction via chiral phonons in layered honeycomb materials, Physical Review B 103, 035405 (2021a).
  * Suri  _et al._ [2021] N. Suri, C. Wang, Y. Zhang, and D. Xiao, Chiral phonons in Moiré superlattices, Nano Letters 21, 10026 (2021).
  * Yin  _et al._ [2021] T. Yin, K. A. Ulman, S. Liu, A. Granados del Águila, Y. Huang, L. Zhang, M. Serra, D. Sedmidubsky, Z. Sofer, S. Y. Quek, and Q. Xiong, Chiral phonons and giant magneto-optical effect in $\ce{CrBr3}$ 2D magnet, Advanced Materials 33, 2101618 (2021).
  * Maity  _et al._ [2022] I. Maity, A. A. Mostofi, and J. Lischner, Chiral valley phonons and flat phonon bands in moire materials, Physical Review B 105, L041408 (2022).
  * Zhang and Murakami [2022a] T. Zhang and S. Murakami, Chiral phonons entangled with multiple Hall effects and unified convention for pseudoangular momentum in 2D materials, arXiv preprint arXiv:2205.09950 (2022a).
  * Zhang  _et al._ [2018] T. Zhang, Z. Song, A. Alexandradinata, H. Weng, C. Fang, L. Lu, and Z. Fang, Double-Weyl phonons in transition-metal monosilicides, Physical Review Letters 120, 016401 (2018).
  * Kishine  _et al._ [2020] J. Kishine, A. Ovchinnikov, and A. Tereshchenko, Chirality-induced phonon dispersion in a noncentrosymmetric micropolar crystal, Physical Review Letters 125, 245302 (2020).
  * Ishito  _et al._ [2021] K. Ishito, H. Mao, Y. Kousaka, Y. Togawa, S. Iwasaki, T. Zhang, S. Murakami, J.-i. Kishine, and T. Satoh, Truly chiral phonons in $\alpha$-$\ce{HgS}$ observed by circularly polarised Raman spectroscopy, arXiv preprint arXiv:2110.11604 (2021).
  * Chen  _et al._ [2021b] H. Chen, W. Wu, J. Zhu, W. Gong, W. Gao, S. A. Yang, and L. Zhang, Chiral phonons in chiral materials, arXiv preprint arXiv:2109.08872 (2021b).
  * Zhang and Murakami [2022b] T. Zhang and S. Murakami, Chiral phonons and pseudoangular momentum in nonsymmorphic systems, Phys. Rev. Research 4, L012024 (2022b).
  * Chen  _et al._ [2021c] H. Chen, W. Wu, J. Zhu, S. A. Yang, and L. Zhang, Propagating chiral phonons in three-dimensional materials, Nano Letters 21, 3060 (2021c).
  * Li  _et al._ [2021] H. Li, T. Zhang, A. Said, Y. Fu, G. Fabbris, D. G. Mazzone, J. Zhang, J. Lapano, H. N. Lee, H. C. Lei, M. P. M. Dean, S. Murakami, and H. Miao, Observation of a chiral wave function in the twofold-degenerate quadruple Weyl system BaPtGe, Physical Review B 103, 184301 (2021).
  * Juneja  _et al._ [2021a] R. Juneja, S. Thebaud, T. Pandey, C. A. Polanco, D. Moseley, M. E. Manley, Y. Cheng, B. Winn, D. L. Abernathy, R. P. Hermann, and L. Lindsay, Quasiparticle twist dynamics in non-symmorphic materials, Materials Today Physics 21, 100548 (2021a).
  * Wang  _et al._ [2022] Q. Wang, S. Li, J. Zhu, H. Chen, W. Wu, W. Gao, L. Zhang, and S. A. Yang, Chiral phonons in lattices with ${C}_{4}$ symmetry, Physical Review B 105, 104301 (2022).
  * Skórka  _et al._ [2022] J. Skórka, K. J. Kapcia, P. T. Jochym, and A. Ptok, Chiral phonons in binary compounds $A$$\ce{Bi}$ ($A$ = $\ce{K}$, $\ce{Rb}$, $\ce{Cs}$) with P$2_{1}$/c structure, arXiv preprint arXiv:2203.05524 (2022).
  * Chen  _et al._ [2022] H. Chen, W. Wu, J. Zhu, Z. Yang, W. Gong, W. Gao, S. A. Yang, and L. Zhang, Chiral phonon diode effect in chiral crystals, Nano Letters 22, 1688 (2022).
  * Choi  _et al._ [2022] W. J. Choi, K. Yano, M. Cha, F. M. Colombari, J.-Y. Kim, Y. Wang, S. H. Lee, K. Sun, J. M. Kruger, A. F. de Moura, and N. A. Kotov, Chiral phonons in microcrystals and nanofibrils of biomolecules, Nature Photonics 16, 366 (2022).
  * Pandey  _et al._ [2018] T. Pandey, C. A. Polanco, V. R. Cooper, D. S. Parker, and L. Lindsay, Symmetry-driven phonon chirality and transport in one-dimensional and bulk $\mathrm{B}{\mathrm{a}}_{3}\mathrm{N}$-derived materials, Phys. Rev. B 98, 241405(R) (2018).
  * Lindsay  _et al._ [2009] L. Lindsay, D. A. Broido, and N. Mingo, Lattice thermal conductivity of single-walled carbon nanotubes: Beyond the relaxation time approximation and phonon-phonon scattering selection rules, Phys. Rev. B 80, 125407 (2009).
  * Lindsay and Broido [2012] L. Lindsay and D. A. Broido, Theory of thermal transport in multilayer hexagonal boron nitride and nanotubes, Phys. Rev. B 85, 035436 (2012).
  * Gunatilleke  _et al._ [2021] W. D. C. B. Gunatilleke, R. Juneja, O. P. Ojo, A. F. May, H. Wang, L. Lindsay, and G. S. Nolas, Intrinsic anharmonicity and thermal properties of ultralow thermal conductivity Ba6Sn6Se13, Phys. Rev. Materials 5, 085002 (2021).
  * Lindsay  _et al._ [2010] L. Lindsay, D. A. Broido, and N. Mingo, Diameter dependence of carbon nanotube thermal conductivity and extension to the graphene limit, Phys. Rev. B 82, 161402(R) (2010).
  * Juneja  _et al._ [2021b] R. Juneja, S. Thébaud, T. Pandey, C. Polanco, D. Moseley, M. Manley, Y. Cheng, B. Winn, D. Abernathy, R. Hermann, and L. Lindsay, Quasiparticle twist dynamics in non-symmorphic materials, Materials Today Physics 21, 100548 (2021b).
  * Popov  _et al._ [2000] V. N. Popov, V. E. Van Doren, and M. Balkanski, Elastic properties of single-walled carbon nanotubes, Phys. Rev. B 61, 3078 (2000).
  * Božovic [1984] I. Božovic, Possible band-structure shapes of quasi-one-dimensional solids, Physical Review B 29, 6586 (1984).
  * Wang  _et al._ [2017] D.-H. Wang, H.-Y. Zhou, C.-H. Hu, Y. Zhong, A. R. Oganov, and G.-H. Rao, Prediction of thermodynamically stable $\ce{Li}$–$\ce{B}$ compounds at ambient pressure, Physical Chemistry Chemical Physics 19, 8471 (2017).
  * Pfister  _et al._ [2016] D. Pfister, K. Schäfer, C. Ott, B. Gerke, R. Pöttgen, O. Janka, M. Baumgartner, A. Efimova, A. Hohmann, P. Schmidt, _et al._ , Inorganic double helices in semiconducting snip, Advanced Materials 28, 9783 (2016).
  * Lind and Geller [1969] M. D. Lind and S. Geller, Structure of pressure-induced fibrous sulfur, The Journal of Chemical Physics 51, 348 (1969).
  * Kresse and Furthmüller [1996] G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Physical Review B 54, 11169 (1996).
  * Gonze and Lee [1997] X. Gonze and C. Lee, Dynamical matrices, Born effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory, Physical Review B 55, 10355 (1997).
  * Peng  _et al._ [2021] B. Peng, S. Murakami, B. Monserrat, and T. Zhang, Degenerate topological line surface phonons in quasi-1D double helix crystal SnIP, npj Computational Materials 7, 195 (2021).