# Charge density wave induced nodal lines in LaTe3

Shuvam Sarkar1, Joydipto Bhattacharya2,3, Pampa Sadhukhan1, Davide Curcio4,
Rajeev Dutt2,3, Vipin Kumar Singh1, Marco Bianchi4, Arnab Pariari5, Shubhankar
Roy6, Prabhat Mandal5, Tanmoy Das7, Philip Hofmann4, Aparna Chakrabarti2,3,
Sudipta Roy Barman1 1UGC-DAE Consortium for Scientific Research, Khandwa Road,
Indore 452001, Madhya Pradesh, India  2Theory and Simulations Laboratory, Raja
Ramanna Centre for Advanced Technology, Indore 452013, Madhya Pradesh, India
3Homi Bhabha National Institute, Training School Complex, Anushakti Nagar,
Mumbai 400094, Maharashtra, India  4Department of Physics and Astronomy,
Interdisciplinary Nanoscience Center (iNANO), Aarhus University, 8000 Aarhus
C, Denmark  5Saha Institute of Nuclear Physics, HBNI, 1/AF Bidhannagar,
Kolkata 700 064, India  6Vidyasagar Metropolitan College, 39, Sankar Ghosh
Lane, Kolkata 700006, India  7Department of Physics, Indian Institute of
Science, Bangalore, 560012, India

###### Abstract

LaTe3 is a noncentrosymmetric (NC) material with time reversal (TR) symmetry
in which the charge density wave (CDW) is hosted by the Te bilayers. Here, we
show that LaTe3 hosts a Kramers nodal line (KNL), a twofold degenerate nodal
line that connects the TR invariant momenta in NC achiral systems, using angle
resolved photoemission spectroscopy (ARPES), density functional theory (DFT),
effective band structure (EBS) calculated by band unfolding, and symmetry
arguments. DFT incorporating spin-orbit coupling (SOC) reveals that the KNL –
protected by the TR and lattice symmetries – imposes gapless crossings between
the bilayer-split CDW-induced shadow bands and the main bands. In excellent
agreement with the EBS, ARPES data corroborate the presence of the KNL and
show that the crossings traverse the Fermi level. Furthermore, spinless nodal
lines – entirely gapped out by the SOC – are formed by the linear crossings of
the shadow and main bands with a high Fermi velocity.

##  I Introduction

Recent years have witnessed a rapid development in the understanding of the
physics of cooperative charge density wave (CDW) electronic state [1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12]. In particular, interplay of the CDW electronic
state with the non-trivial topological phases provides an interesting platform
for discovery of novel quasiparticles such as, axionic insulator [3, 13],
quantum spin-hall insulator [14], fractional Chern insulator states [15] and
manipulation of topologically protected states [16, 17]. CDW can drive
topological phase transitions by modifying the symmetry of the lattice, such
as breaking the inversion symmetry [18]. Interesting topological phases are
frequently found in noncentrosymmetric materials, such as, nodal chain
fermions [19], Dirac and Weyl fermions [20, 21, 22], hourglass fermions
protected by glide reflection [23], Kramers Weyl semimetal (KWS) [24] and
recently predicted Kramers nodal line (KNL) metal [25]. KNLs differ from the
Weyl nodal lines because they join two time reversal invariant momenta (TRIM)
points and should appear in all achiral noncentrosymmetric time reversal
symmetry (TRS) preserving systems [25]. For the subclass of nonsymmorphic
symmetry, KNLs emerge from the $\Gamma$ points. Unlike the previously known
nodal lines manifested by band inversion[26], the KNLs are robust under spin-
orbit coupling (SOC) unless the protecting lattice symmetries such as TRS,
mirror, or roto-inversion symmetries are broken. KNL fermions have been
predicted to exhibit novel physical properties such as quantized optical
conductivity [25]. However, in this emerging field, to the best of our
knowledge, the experimental evidence of KNL is limited to the work by Shang et
al. [27] who reported that transition metal ruthenium silicides belong to this
class and exhibit unconventional superconductivity based on muon spin
spectroscopy and density functional theory (DFT).

In recent years, multiple fascinating findings in LaTe3 [1, 7, 28, 4, 29, 30]
– a member of the RTe3 (R represents a rare earth element) family with highest
CDW transition temperature of 670 K [31, 32] – have rekindled the scientific
interest of the community in this noncentrosymmetric TRS preserving material.
The detection of an axial Higgs boson mode in LaTe3 from Raman spectroscopy
has been related to unconventional CDW excitation [1]. Photoinduced CDW state
with topological defects in this material has been discovered from ultrafast
electron diffraction and related studies [7, 28]. Transport studies have
revealed that LaTe3 possesses an unusually high non-saturated longitudinal
magnetoresistance [30], which is similar to that of nodal line material [33].
In addition, LaTe3 possesses very high carrier mobility [30], which, in
conjunction with its high transition temperature, makes it a promising
contender for next-generation electronics. Theoretical calculation of the
electronic susceptibility as well as experiments have shown that $q$ dependent
electron phonon coupling plays an important role in stabilizing the CDW state
in RTe3 [34, 35]. While there are no DFT investigations, angle resolved
photoemission spectroscopy (ARPES) measurements on LaTe3 are also scarce in
literature [36, 7]. Using time resolved ARPES, Zong et al. [7] examined light-
induced melting of the CDW state of LaTe3. A previous research by Brouet et
al. [36] demonstrated that the CDW-induced shadow bands hybridize with the
main bands existing in the non-CDW state creating a CDW gap along a particular
direction of the Brillouin zone (BZ).

In light of the fact that LaTe3 is a noncentrosymmetric achiral material with
TRS intact, topological phases brought on by inversion symmetry breaking in
conjunction with other symmetries may be anticipated in the CDW state. Here,
from an in-depth study of the band structure of LaTe3 in the CDW state by
combining ARPES and $ab-initio$ DFT using a realistic experiment-based
structure, we establish the existence of a KNL in a CDW material for the first
time. It originates from the interaction of the shadow band and the main band,
and is hosted by the TRS and the lattice symmetries. Furthermore, spinless
nodal lines that are entirely gapped out by SOC are also identified.

##  II Results

###  II.1 Modulated structure of LaTe3 in the CDW state

Fig. 1a shows a 7-fold modulated (1$\times$1$\times$7) supercell structure of
LaTe3. It is derived from the experimental structure obtained from single
crystal x-ray diffraction (XRD) at 100 K [37] (see Methods). Its space group
is $C2cm$ ($SG$ #40), which is same as that reported for the average structure
[37]. LaTe3 is made up of two main structural units: La-Te1 corrugated slabs
and Te2-Te3 bilayer. The latter, highlighted by blue double-sided arrows, are
weakly coupled by van der Waals interaction. The Te bilayer of the 7-fold
modulated structure hosts the CDW with $q_{{}_{\rm{CDW}}}$ =
$\frac{2}{7}$$c^{*}$= 0.286$c^{*}$, where c∗ is the reciprocal lattice vector
along $k_{z}$ in the non-CDW state. This value is close to the experimental
values determined from ARPES (subsection II.2) and reported in previous
literature [37, 38]. Moreover, the amplitude of the CDW modulation is nearly
similar to that from XRD [37], see Supplementary Figs. S1a,b.

The CDW modulation has been directly observed in the LaTe3 crystal studied
here from a high resolution scanning tunneling microscopy (STM) topography
image. In Fig. 1b, the modulated Te net formed by connecting the Te atoms
(orange circles) is shown by white lines. Low energy electron diffraction
(LEED) pattern also shows CDW related satellite spots (encircled) besides the
(1$\times$1) spots (Fig. 1c). Satellite spots are also observed in the Fourier
transform of the STM image, see Supplementary Note 1 and Fig. S2.

The experiment based 7 fold modulated structure discussed above has been used
for the DFT calculations. The corresponding BZ is shown in Fig. 1d inscribed
within the non-CDW BZ. Since the BZ is related to the ordering of the lattice
constants of the conventional cell, its comparison with the primitive unit
cell of the 7-fold structure is shown in Supplementary Figs. S3a,b. In our
notation, the horizontal plane in the BZ is represented by $k_{x}$-$k_{z}$ and
$q_{{}_{\rm{CDW}}}$ is oriented along $k_{z}$. The CDW BZ, containing features
such as the primitive reciprocal lattice vectors and all the pertinent high
symmetry points and directions as well as their coordinates, is depicted in
Supplementary Fig. S3c.

Figure 1: Structure, Brillouin zone and the Fermi surface of LaTe3 in the CDW
state. (a) The 7-fold (1$\times$1$\times$7) modulated primitive unit cell of
LaTe3 comprising of 56 atoms with $q_{{}_{\rm{CDW}}}$= $\frac{2}{7}$$c^{*}$
viewed perpendicular (left) and parallel (right) to the c′ direction. The
lattice constants are $a^{\prime}$= $b^{\prime}$= 13.256 Å, $c^{\prime}$=
30.778 Å with $\alpha^{\prime}$= $\beta^{\prime}$= 90∘, and
$\gamma^{\prime}$=160.99∘. The cleavage plane (light blue) occurs in between
the two weakly interacting Te layers. (b) High resolution scanning tunneling
microscopy topography image obtained with bias voltage of 0.2 V and a
tunneling current of 0.4 nA. (c) Low energy electron diffraction pattern
measured with 57 eV primary beam energy in inverted gray scale. (d) The CDW
Brillouin zone (BZ) (black) is shown within that of the non-CDW state (light
blue). The high symmetry points [39] are indicated in these respective colors.
$\Gamma$, $Y$ and $X$ points coincide for both. $\Gamma_{2}X_{2}$ (dashed
line) is in the 2nd BZ along $k_{z}$. (e) The Fermi surface (FS) in the CDW
state measured by ARPES, where the length of the white dashed arrows (#1, #2)
that join the shadow branches with the main branches of the FS represents
$q_{{}_{\rm{CDW}}}$.

###  II.2 Crossing of the bilayer-split shadow and main bands from ARPES

An $E(k_{z})$ ARPES intensity plot in a generic direction parallel to $\Gamma
Z$ at $k_{x}$= 0.68 Å-1 i.e., near the $X$ point (the length of $\Gamma X$
being 0.737 Å-1) is shown in Fig. 2a (see Methods for the experimental
details). The direction of the measurement is shown by a red line denoted by
“a” in Fig. 2g . It shows two main bands (inner and outer) centered around
$\Gamma$ that cross $E_{F}$ at $k_{z}$= $\pm$0.15 and $\pm$0.21 Å-1,
respectively. The outer main band disperses down to binding energy ($E$) of
about 1.25 eV, while the inner band has a nearly flat bottom at ∼0.8 eV. It is
interesting to note that a relatively weaker band centered around $k_{z}$=
$\pm$0.41 Å-1 is a replica of the main band shifted by $q_{{}_{\rm{CDW}}}$(=
0.28$c^{*}$= 0.41 Å-1), as shown by two horizontal white dashed arrows. This
replica band crosses $E_{F}$ at $k_{z}$= 0.21 and 0.61 Å-1 and disperses down
to $E$∼1.25 eV. No shift in its position along $E$ compared to the main band
is observed, which indicates that it is related to the initial state CDW
superlattice [40]. It has been referred to in literature as the shadow band
[36, 40, 41]. The signature of the shadow band is also evident in the Fermi
surface (FS) shown in Fig. 1e, where shadow FS branches appear at a separation
of $q_{{}_{\rm{CDW}}}$ from the main branches, as shown by white dashed arrows
in the metallic region around the $X$ point parallel to the $\Gamma Z$
direction. In the Supplementary Note 2, a discussion about the FS and
$q_{{}_{\rm{CDW}}}$ obtained from the average separation of the shadow and
main branches are provided. $q_{{}_{\rm{CDW}}}$= 0.28$\pm$005$c^{*}$
determined in this way is close to $\frac{2}{7}$$c^{*}$ and is in excellent
agreement with the values obtained from STM and LEED, see Supplementary Note
1.

In Fig. 2a, The shadow and the main bands resemble an inverted “V” and meet
each other close to $E_{F}$ at $k_{z}$= $\pm$0.21 Å-1(highlighted by a green
dashed oval). This region is shown in an expanded scale in Fig. 2e, where the
white dashed lines suggest a possible crossing of nearly linear bands at
$E_{F}$. The Fermi velocities of these bands, determined using the expression
($\frac{1}{\hslash}$$\frac{dE}{d\vec{k_{z}}}$), turn out to be
1.2$\pm$0.05$\times 10^{6}$ and 1$\pm$0.05 $\times 10^{6}$ m/s for the shadow
and main bands, respectively. These values are comparable to graphene
(1$\times 10^{6}$ m/s [42]), indicating large mobility of LaTe3 in agreement
with recent report from Hall conductivity measurements [30].

Figure 2: Crossings between the bilayer-split shadow and the main bands.
$E(k_{z})$ ARPES intensity plots measured at 100 K for $k_{x}$= (a) 0.68, (b)
0.63, (c) 0.59, and (d) 0.56 Å-1. The crossing regions are highlighted by
green dashed ovals. (e) A zoomed view of the near $E_{F}$ region of panel a
around $k_{z}$= 0.21 Å-1, the white dashed lines are obtained by curve fitting
(see Methods). (f) Curvature plot of a part of panel c shown in an expanded
scale: the crossings indicated by $L$, $B$, $R$ and $T$ are shown by blue,
green, yellow and red circles, respectively. The $T$ and $B$ crossings occur
at the $X_{2}$ point. The dashed red (black) line representing the main
(shadow) band is obtained by curve fitting as in panel e. (g) Cut A at
$k_{z}$= 0.216 of an $E$-$k_{z}$-$k_{x}$ ARPES intensity plot shows the
dispersion of the $R$ crossing and the $mu$ and $sd$ bands within the yellow
rectangle. A shadow band is observed in the cut B. (h-k) The effective band
structure (EBS) along $k_{z}$ obtained by band unfolding at similar $k_{x}$
values as in panels a-d, respectively. The band crossing regions are
highlighted by green dashed ovals.

To further investigate these bands and their possible crossing, ARPES was
performed over a range of $k_{x}$ out of which three representative plots
(along the red lines parallel to “a” up to “d” in Fig. 2g) are shown in Figs.
2b-d. Interestingly, as $k_{x}$ decreases, both the main and the shadow bands
spread out in $k_{z}$ and the crossings – highlighted by green dashed ovals –
shift below $E_{F}$ to larger $E$. However, note that their $k_{z}$ position
remains unchanged around $X_{2}$ $\approx$
$\pm$$\frac{1}{2}$$q_{{}_{\rm{CDW}}}$. In Figs. 2b-d, splittings in both the
shadow and the mains bands are observed, which are related to the coupling
between the two adjacent weakly coupled Te layers. This is the bilayer
splitting that has been reported in other RTe3 members [43, 44, 36], bilayer
graphene [45], and cuprate superconductors [46]. Here, we find that the
splitting increases with $E$, e.g., it varies from 0.016 to 0.03 Å-1 as $E$
increases from 0.1 to 0.5 eV (Fig. 2c). Evidence of varying bilayer splitting
is also observed in the FS (see Supplementary Note 2).

Both the bilayer-split shadow bands - denoted by su (sd) for smaller (larger)
$E$, see Fig. 2f - are shifted by $q_{{}_{\rm{CDW}}}$ from the corresponding
main bands (denoted by mu and md). This is shown by the white dashed
horizontal arrows in Figs. 2a,b. This leads to formation of four crossings
between the main and the shadow bands, as shown in Fig. 2f. These are also
visible in a constant energy $k_{z}$-$k_{x}$ isosurface through the band
crossing e.g. at $E$= 0.5 eV and a stack of momentum distribution curves
(MDCs), see Supplementary Figs. S8a,b.

The crossings are denoted by left (L, blue circle, crossing of mu and sd i.e.,
$mu\otimes sd$), right (R, orange circle, $md\otimes su$), top ($T$, red
circle, $mu\otimes$$su$), and bottom (B, green circle, $md\otimes sd$) (Fig.
2f). While Figs. 2a-d show the band dispersion at discrete $k_{x}$ values, we
show a continuous dispersion of the crossings, for example the $R$ in cut A of
Fig. 2g. Here, the $E(k_{x})$ at $k_{z}$=0.216 Å-1 shows the loci of the $R$
crossing within the yellow rectangle.

ARPES with different photon energies shows negligible $k_{y}$ dependence of
the crossings. For example at kx= 0.58 Å-1, Supplementary Figs. S9a-d show
that their position (highlighted by green dashed ovals) remains almost
unchanged with $k_{y}$. This is summarized in Fig. S9e through a
$k_{y}$-$k_{z}$ map at $E$= 0.39 eV, where the two crossings at $k_{z}$=
$\pm$0.2 Å-1 (marked by yellow arrows) show almost no change with $k_{y}$,
indicating the quasi-2D nature of LaTe3.

###  II.3 Effective band structure from DFT compared to ARPES

Multiple crossings indicated by ARPES is a surprising result since
hybridization gap is generally expected at the band touching points between
the Bloch states connected by $q_{{}_{\rm{CDW}}}$ [47, 44]. To understand
this, we have performed DFT calculations using the modulated structure
discussed in subsection II.1.

The $E(k_{z})$ bands calculated at different $k_{x}$ as in the experiment are
shown in an extended zone scheme spread over the non-CDW BZ in Supplementary
Fig. S10. Many bands are observed in the CDW state due to band folding. The
complexity of these bands hides the influence of the CDW on the electronic
bands and impedes their interpretation and direct comparison with ARPES. So,
an effective band structure (EBS) has been calculated by unfolding the bands
in the non-CDW BZ [48, 49, 50]. In Figs. 2h-k, EBS shows the distribution of
states as a function of their energy and momenta with appropriate spectral
weight, where broadening similar to the experiment has been applied.
Variations in the spectral weight is evident resulting in dissimilar EBS in
the different CDW BZs. This complexity in a CDW system is discussed further in
subsection II.6.

The importance of the EBS calculation is that despite the CDW effect being
small with the modulation amplitude only ∼4.5% of the average Te2-Te3
distances, the EBS reveals the shadow bands that are shifted from the main
bands by $q_{{}_{\rm{CDW}}}$ (blue dashed arrows) in excellent agreement with
ARPES (compare Figs. 2h-k with Figs. 2a-d). In contrast, the bands in the non-
CDW state, where obviously the CDW amplitude is zero, shows only the main
bands (Supplementary Fig. S11a). An analysis of the orbital character
establishes that the shadow bands (as well as the main bands) are of
predominantly in-plane Te $p_{x}$-$p_{z}$ character, see the Supplementary
Figs. S12a-f. The out-of-plane $p_{y}$ character becomes slightly significant
only for $E>$0.6 eV, and the contributions from the La-Te1 block is
negligible. However, transfer of electrons from La to the Te net determines
the band filling in the Te2-Te3 layer and this has been calculated using the
Bader charge analysis (see Supplementary Note 3 including Refs. 51, 52, 53).

From Fig. 2h, we find that the outer branch of the shadow band disperses
towards $E_{F}$ and crosses the main band around $k_{z}$= $\pm$0.21 Å-1
(highlighted by a dashed green oval). A bunch of bands that disperse weakly
between 1.15 to 1.6 eV are observed in both theory and experiment at similar
$E$. A bilayer splitting of 0.17-0.2 eV (red double arrow in Fig. 2h) is
observed in the EBS with $L$ and $R$ crossings close to $E_{F}$, while $T$
($B$) appears above (below) $E_{F}$. However, these are not visible separately
in Fig. 2a, indicating that possibly at this $k_{x}$ the experimental bilayer
splitting is smaller compared to the DFT value. The inner branch of the shadow
band that disperses to about 0.6 eV is more prominent in the EBS, but also has
its counterpart in the curvature plot of ARPES (see Supplementary Fig. S11b).
In Figs. 2i-k, for progressively smaller $k_{x}$ values, EBS portray all the
four crossings at similar $E$-$k_{z}$ observed in ARPES (compare Figs. 2i-k
with Figs. 2b-d). The agreement is also good at larger $E$: the main bands
around 0.8 eV become flatter and move to lower $E$.

Important to note is that from ARPES the crossings appear to be gapless (Fig.
2a-f) within the experimental and lifetime broadening. Although this is
supported by the EBS (Fig. 2h-k), it should be noted that the calculations
have been performed with a $k$ step size ($\delta$$k_{z}$) of
6$\times$10-3Å-1, and with broadening comparable to ARPES, which might conceal
the potential presence of minigaps. In the subsequent subsections II.4 and
II.5, we show the DFT bands calculated with smaller $\delta$$k_{z}$ to probe
these crossings further.

###  II.4 Spinless nodal lines formed by the $L$ and $R$ crossings

Figure 3: Band crossings and the Kramers nodal line. (a) $E(k_{z})$ band
dispersion from DFT at $k_{x}$= 0.59 Å-1 without (w/o) SOC. The irreducible
representations (irreps) are shown. The vertical dashed line represents the
$k_{z}$ point on $\Gamma_{2}X_{2}$ i.e., the $\Sigma$ line. Zoomed colored
rectangles of panel a show the bands around (b) $L$ ($mu\otimes sd$), (c) $R$
($md\otimes su$), (d) $T$ ($mu\otimes$$su$), and (e) $B$ ($md\otimes sd$).
Comparison of the $E(k_{x})$ ARPES intensity plot with DFT for the following
crossings: (f) $R$ at $k_{z}$= 0.229 Å-1and (g) $L$ at $k_{z}$= 0.196 Å-1. The
positions of these crossings obtained from DFT (orange and blue filled circles
for $R$ and $L$, respectively) are superimposed. (h) A schematic
representation of the gapless $L$ and $R$ crossings in the $E$-$k_{z}$-$k_{x}$
space (red dashed lines) and their projection on the $k_{x}$-$k_{z}$ plane
showing the spinless nodal lines (thick red lines on both sides of the
$\Sigma$ line). (i-m) Same as panels a-e except that the calculations are
performed with SOC. (n) $E(k_{x})$ ARPES intensity plot at $k_{z}$= 0.204 Å-1
(dashed curves here and in panels f and g serve as guide to the eye) compared
with the positions of the crossings enforced by the KNL obtained from DFT for
$T$ (red, light red circles) and $B$ (black, gray circles). (o) A schematic
representation of the four crossings (green dashed lines) related to the upper
and lower branches of $T$ and $B$. The Kramers nodal line appears along the
$\Sigma$ line and is denoted by a green thick line on the $k_{x}$-$k_{z}$
plane. The gapped out spinless nodal lines at $L$ and $R$ are also shown. The
energy dispersion along $k_{x}$ is not shown here as well as in panel h.

The $E(k_{z})$ DFT bands in the crossing region with small $\delta$$k_{z}$ (=
5$\times$10-4Å-1) in absence of SOC indeed indicate that the $L$ and $R$
crossings are gapless (Fig. 3a). This is reconfirmed in Figs. 3b,c by the
bands calculated with even smaller $\delta$$k_{z}$ (= 1$\times$10-5Å-1). It is
interesting to note that these crossings occur at generic points of the BZ in
the $k_{x}$-$k_{z}$ plane and the bands involved are linear (Figs. 3a-c).
ARPES intensity plots in Figs. 2c,f at this $k_{x}$ also demonstrate the
linearity of the bands around $L$ and $R$ in excellent agreement with DFT. The
velocities calculated from their slopes are similar to that obtained at
$E_{F}$ from Fig. 2e. Although these bands originate from the in-plane $p$
orbitals with small difference in contributions from the $p_{x}$ and $p_{z}$
orbitals as shown in Supplementary Figs. S12a-f, md and sd bands belong to
$M_{1}$ irreducible band representation (irrep), while mu and su belong to
$M_{2}$ irrep. Thus both $L$ and $R$ crossings are formed by bands belonging
to different irreps (Figs. 3a-c).

The crossings disperse with $k_{x}$ and at larger $k_{x}$= 0.685 Å-1 compared
to 0.59 Å-1, $L$ and $R$ crossings traverse the $E_{F}$ (Supplementary Fig.
S13a-c). In fact, calculations for a series of $k_{x}$ values establish that
the gapless linear crossings occur over an extended range of the $E$-$k$ space
(Supplementary Fig. S14). The loci of each crossing form a continuous curve in
the momentum space that has been referred to as a nodal line. A direct
comparison of the ($E,k_{x}$) cuts from ARPES (Figs. 3f,g for $R$ and $L$,
respectively) show that the positions of both the nodal lines are in excellent
agreement with DFT. Both disperse between ($E$, $k_{x}$) = (0 eV, $\sim$0.7
Å-1) to (∼0.6 eV, $\sim$0.5 Å-1) with $k_{z}$ at 0.195 and 0.22 Å-1 for $L$
and $R$, respectively. Thus, the crossings appear within an energy window of
$E$∼0.6 eV to the $E_{F}$. Their projections in the $k_{x}$-$k_{z}$ plane form
a pair of approximately parallel nodal lines that are 0.2 Å-1 in length and
appear at a separation of $k_{z}$∼0.02 Å-1 in a general direction on this
plane (red solid lines in Fig. 3h). These appear parallel to but on either
sides of $\Gamma_{2}X_{2}$ that occurs at kz= 0.204 Å-1. These nodal lines are
formed by two fold crossings of nondegenerate bands in the absence of spin and
so are referred to spinless nodal lines.

With inclusion of SOC, it is intriguing to find that both the $L$ and $R$
crossings (and hence the nodal lines) are entirely gapped out by minigap of
14-17 meV into upper and lower branches, as shown in Figs. 3i-k and
Supplementary Figs. S13g,h. A schematic is shown in Fig. 3o. The gap is formed
by hybridization of bands belonging to same one dimensional double group irrep
involving both $\overline{M_{3}}$ and $\overline{M_{4}}$.

Note that noncentrosymmetry should lift the SU(2) spin degeneracy. This is
visible through the spin splitting of both the upper and lower branches in
Figs. 3j,k. These spin-split bands belong to different irreps
$\overline{M_{3}}$ and $\overline{M_{4}}$ and do not cross each other. The
splittings for both $L$ and $R$ are $k$-dependent, it is maximum at the
extrema (∼3 meV for lower and 1 meV for the upper branch in Figs. 3j,k) and
decreases away from it.

###  II.5 Evidence of Kramers nodal line from the $T$ and $B$ crossings

In the case of $T$ and $B$, the shadow and main bands belong to the same
irrep, leading to hybridization-related minigaps without SOC and the creation
of upper and lower branches (Figs. 3a,d,e). The inclusion of SOC provides a
fascinating outcome: the spin-split bands in both the upper and lower branches
of $T$ and $B$ exhibit gapless crossings (Figs. 3l,m), which contrasts the
bands at $L$ and $R$ (Figs. 3j,k). Each branch of $T$ and $B$ crosses at the
same $k_{z}$= 0.204 Å-1 (dashed lines in Figs. 3i,l,m) resulting in four
crossings. This value of $k_{z}$ is special, since it falls on the
$\Gamma_{2}X_{2}$ line of the CDW BZ (i.e., $\Gamma X$ or the $\Sigma$ line)
(Fig. 1d). At $k_{x}$= 0.59 Å-1, the four crossings appear at different $E$:
0.256 (0.267) eV for the upper (lower) branch of $T$, and 0.448 (0.469) eV for
upper (lower) branches of $B$.

These gapless crossings are observed over a range of $k_{x}$, as in the case
of $L$ and $R$ (subsection II.4). For example, at $k_{x}$= 0.685 Å-1, the
crossings disperse to smaller $E$ and in this case both the branches of $T$
($B$) are above (below) the $E_{F}$ (Supplementary Fig. S13f-h).
Significantly, the crossings always appear at same $k_{z}$ i.e., along the
$\Sigma$ line, suggesting that the crossings may be enforced by the lattice
symmetries along this direction.

The band irreps with SOC shown in Fig. 3l,m are as follows: the crossing
belongs to a double valued irrep that is two dimensional
($\overline{\Sigma}_{5}$), while away from it, the spin-split bands have one
dimensional ($\overline{M}_{3}$ or $\overline{M}_{4}$) representation.
Significantly, the $\Sigma$ line that emerges from the $\Gamma$ point has the
little group that is isomorphic to $C_{2v}$ point group and has symmetries
such as the two fold rotation about the $k_{x}$\- axis denoted by
$\\{$2100$|$0,0,0$\\}$, glide reflection perpendicular to the $y$ axis in the
$k_{x}$-$k_{z}$ plane followed by a translation of $\frac{1}{2}c$ given by
$\\{$$m_{010}$$|$0,0,$\frac{1}{2}$$\\}$ and an off-centered mirror
perpendicular to the $k_{z}$ axis $\\{$$m_{001}$$|$0,0,$\frac{1}{2}$$\\}$.
$\Gamma$ is a TRIM point, where according to the Kramers theorem, each band is
at least doubly degenerate. The little group of $\Sigma$ is related to that of
$\Gamma$ by the compatibility relations. We find that $\Gamma$ and $\Sigma$
are represented by two dimensional double-valued irreps:
$\overline{\Gamma}_{5}$ and $\overline{\Sigma}_{5}$, respectively. These
representations are similar, as shown in Supplementary Table S2 [54]. The
$\Sigma$ line passes through the $X$ point with co-ordinates (0.257, 0.257,0)
and meets the TRIM point $Y_{2}$ (0.5, 0.5, 0) in the next BZ (see
Supplementary Fig. S15). $Y_{2}$ also belongs to double-valued two dimensional
irrep $\overline{Y_{2}}_{5}$ that is same as $\Gamma$ and $\Sigma$
(Supplementary Table S2). The condition that at the TRIM points the
representations should be time reversal invariant is satisfied since both
$\overline{\Gamma}_{5}$ and $\overline{Y_{2}}_{5}$ are pseudo-real [54]. Thus,
$\overline{\Gamma}_{5}$-$\overline{\Sigma}_{5}$-$\overline{Y_{2}}_{5}$ is able
to support two fold degeneracy of the bands along $\Gamma XY_{2}$ i.e., the
$\Sigma$ line. This is a Kramers nodal line (KNL) that occurs along the
$\Sigma$ line in the mirror-invariant $k$-plane in presence of TRS, and the
additional rotational symmetry constrains the KNL along a high symmetry
direction [25]. Xie et al. [25] have proved the existence of KNL along the
$C_{2}$ rotational axis ($\\{$2100$|$0,0,0$\\}$), which in our case is along
$k_{x}$ that lies in the $k_{x}$-$k_{y}$ mirror plane
($\\{$$m_{001}$$|$0,0,$\frac{1}{2}$$\\}$). In Fig. 3o, the projection of the
loci of the crossings of the four pairs of bands in the $E$-$k_{z}$-$k_{x}$
space on the $k_{x}$-$k_{z}$ plane shows the KNL (thick green line) that
enforces the crossings.

The bands in a plane that cuts the KNL perpendicularly – as is the case in
Fig. 3i – have been described in the literature as two-dimensional massless
Dirac Hamiltonian with the Berry curvature concentrated at the crossing [25].
These authors demonstrated that the Berry phase around a KNL is quantized as
m$\pi$ mod 2$\pi$. In case of quadratic and cubic dispersion of these bands,
the crossing has been dubbed as a higher-order Dirac point [25]. In the
present case, both the upper and lower branches of $T$ and $B$ exhibit
quadratic dispersion very close to the crossings, as is evident in Figs. 3l,m.
Consequently, the gapless crossings in LaTe3 that are associated with the KNL
are higher-order Dirac points.

###  II.6 ARPES and DFT along the KNL and other directions

In Fig. 4a, the $E(k_{x})$ bands calculated along the KNL (i.e., $\Gamma X$)
show two pairs of degenerate bands related to the $T$ and $B$ crossings,
highlighted by green shading in the insets i, ii. The ARPES intensity plot in
Fig. 3n represents these bands that are also identified by the degenerate
crossings of bands in the perpendicular $E(k_{z})$ direction (Figs. 3l,m)
enforced by the KNL. The existence of the KNL between 0.5 to 0.7 Å-1 along
$k_{x}$ is affirmed by ARPES through an excellent agreement of the positions
of the crossings obtained from DFT (filled circles) that are overlaid on the
experimental data in Fig. 3n. The agreement is also evident in the
Supplementary Fig. S16, where Fig. 3i is superposed on Fig. 2f. However, the
energy separations between the upper and lower branches of both $T$ and $B$
are too small to be resolved by ARPES. Similarly, the quadratic nature visible
in a tiny $k_{z}$ range shown in Figs. 3l,m is not distinguished, see the
regions enclosed by the red and black rectangles in Supplementary Fig. S16.
Nonetheless, ARPES is consistent with DFT, and both show that the crossings
disperse from $E$$\sim$0.6 eV at $k_{x}$$\sim$0.55 Å-1 and traverse the
$E_{F}$ at $k_{x}$= 0.65-0.7 Å-1.

Figure 4: DFT and ARPES along and perpendicular to the KNL. (a) The band
structure of LaTe3 along $X\Gamma Z^{\prime}Z$ in the CDW state. On the left,
zoomed regions from the blue rectangles show the two fold degeneracy (dashed-
green and black) of the bands along the KNL. On the right, zoomed regions from
the red rectangles show the spin-splitting along the $\Gamma Z$ direction in
contrast to $\Gamma X$. (b) ARPES intensity plot measured using photon energy
of 24.4 eV and (c) the EBS obtained from DFT towards $\Gamma X$ and $\Gamma
Z$.

We find that every band along $\Gamma X$ is degenerate due to the double
degeneracy enforced by the KNL (see for example the insets i-v of Fig. 4a) and
belong to two dimensional $\overline{\Sigma}_{5}$ irrep. Besides the $T$ and
$B$ related bands discussed above, there are other bands that cross $E_{F}$ in
Fig. 4a. These bands – numbered as 1, 2 and 3 and highlighted by orange
shading – have not been observed in the ARPES along $\Gamma_{2}X_{2}$
($k_{z}$= 0.204 Å-1) due to their low spectral weight at this $k_{z}$.
However, these three bands are clearly visible in the ARPES intensity plot
along $\Gamma X$ ($k_{z}$= 0 Å-1) in Fig. 4b and disperse across the $E_{F}$
in splendid agreement with the EBS along the same direction in Fig. 4c. At
larger $E$ other bands (numbered as 4-12) in the EBS along $\Gamma X$ are in
very good agreement with ARPES (Fig. 4b). On the other hand, the bands around
the $T$ and $B$ crossings both along and perpendicular to the KNL have
negligible spectral weight around $\Gamma X$ and so are observed neither in
EBS nor ARPES in Figs. 4b,c. Similarly, in Fig. 3a,i parabolic bands around
$E$= 0.3 eV and $E$= 0.5-0.6 eV are not visible in the EBS or ARPES. The is
caused by the variation of the spectral weight mentioned in subsection II.3.
On the other hand, the band structure shows the crossings formed by band
folding in every BZ, see Supplementary Fig. S10, where the crossing region is
highlighted by red ovals. This shows the importance of performing ARPES over
multiple CDW BZs in the direction of $q_{{}_{\rm{CDW}}}$ to decipher the
influence of the CDW on the electronic band structure.

Perpendicular to the KNL along $\Gamma Z$ ($\Lambda$), the bands are
represented by one dimensional irreps (Supplementary Fig. S15 and
Supplementary Table S3). Since the little group along this direction has
lesser symmetry than C2v, degeneracy is not enforced. This is shown by the
spin-splitting along $\Gamma Z$ in the zoomed insets iv,v of Fig. 4a. This is
also true for the bands calculated along various other high symmetry
directions (Supplementary Fig. S17), where the insets show degeneracy along
$\Gamma X$ and splitting along other directions. In the $\Gamma Z$ direction
no bands are found to cross the $E_{F}$, and this is corroborated by ARPES as
well as EBS. Here, a hybridization related gap is observed with bands 13,14
being the highest occupied ones (red double arrow in Figs. 4b,c). This gap has
been referred to in previous literature as the CDW gap in LaTe3 [36] and other
RTe3 systems [44, 43, 40, 55, 56]. A discussion about the variation of the CDW
gap of LaTe3 with $k_{x}$ and $k_{y}$ is provided in the Supplementary Note 4.

##  III Discussions

In this work, we show the existence of the recently predicted [25] KNL in the
CDW state of LaTe3, along the $\Sigma$ direction with $C_{2v}$ point group
(Fig. 3o) based on the results of ARPES, ab-initio DFT, EBS obtained by band
unfolding and symmetry arguments. The origin of the KNL is associated with the
time reversal symmetry (LaTe3 being non-magnetic in contrast to the other RTe3
materials) as well as the symmetries of the noncentrosymmetric and
nonsymmorphic lattice ($C_{2}$ rotation along $k_{x}$ and mirror in the
$k_{x}$-$k_{z}$ plane).

The crossings of the CDW-induced shadow and the main bands enforced by the KNL
occur from $E$∼0.6 eV, and disperse in $E$ to traverse the $E_{F}$ as $k_{x}$
increases (Fig. 3n). The dispersion in $E$ is related to the dispersion of the
main band (the crossing of the main band with $E_{F}$ moves to larger $k_{z}$
as $k_{x}$ decreases, see Figs. 2a-d, h-k) coupled with the constraint that
the shadow band is always separated from it by $q_{{}_{\rm{CDW}}}$. Other spin
degenerate bands also cross the $E_{F}$ along the KNL. We characterize LaTe3
as a KNL metal in the CDW state based on multiple bands crossing the $E_{F}$
both along and perpendicular to the KNL.

We have furthermore identified spinless nodal lines in a general direction on
the $k_{x}$-$k_{z}$ plane in the limit of negligible SOC (Fig. 3h). These
nondegenerate bands with distinct irreps involved in the crossings are linear
and traverse the $E_{F}$ with high Fermi velocity. The nodal lines are
entirely gapped with SOC. These characteristics resemble those of
noncentrosymmetric topological nodal line semi-metals such as pinictides e.g.,
CaAgAs [57], in which the nodal line is entirely gapped by SOC resulting in a
topological insulator [57, 58]. Whether the SOC-induced gap in LaTe3 has a
topological character is an open question that would require further
investigation.

The appearance of these two distinct types of nodal lines can be attributed to
the bilayer splitting, that splits both the shadow and the main bands to bands
belonging to different irreps. Thus, the foundation of our investigation is
the unambiguous observation of the bilayer-split shadow bands by ARPES with
which the theory is in impressive agreement. This occurred due to the 7 fold
modulated structure with $q_{{}_{\rm{CDW}}}$= $\frac{2}{7}$$c^{*}$=
0.286$c^{*}$, employing which DFT was done for the first time with
$q_{{}_{\rm{CDW}}}$ and the amplitude of the CDW modulation close to the
experimental value. However, the experimental values differ depending on the
method: for example, electron diffraction showed $q_{{}_{\rm{CDW}}}$=
0.28$\pm$0.01$c^{*}$ [38], whereas XRD reported $q_{{}_{\rm{CDW}}}$=
0.2757(4)$c^{*}$ [37] in the temperature range of 90-100 K. Here, from ARPES
we find $q_{{}_{\rm{CDW}}}$ to be 0.28$\pm$0.005$c^{*}$. Considering these
values, it is possible that there might be a minor deviation of experimental
$q_{{}_{\rm{CDW}}}$ from $\frac{2}{7}$$c^{*}$; hence, is often considered as
“incommensurate” [38, 37]. In order to answer the question whether the KNL
would exist if $q_{{}_{\rm{CDW}}}$ deviates, we find that it is possible to
construct a 29-fold (1$\times$1$\times$29) structure with $q_{{}_{\rm{CDW}}}$=
$\frac{8}{29}$$c^{*}$= 0.2759$c^{*}$ from the experimental XRD structure [59].
Note that it has the same space group (SG #40) as the 7-fold structure. Its
$q_{{}_{\rm{CDW}}}$ equals the experimental XRD value, which is most precise
among the distinct methods, rounded to the third decimal place. Moreover, the
required displacement of the 232 atoms in the primitive unit cell from the
atomic positions from XRD [37] is zero (0.0000) for virtually all the atoms,
with the exception of only a couple of Te atoms where it is 0.0001 Å. Since
the symmetries of SG #40 as well as the TRS are retained in the 29-fold
structure, it is expected that the KNL will remain intact.

In conclusion, the discovery of CDW-induced KNL in LaTe3 – a non-
centrosymmetric, quasi-2D, TRS-preserving material – is the cornerstone of our
work, which we feel will stimulate further investigation into the fascinating
realm of CDW materials.

##  IV Methods

Experimental: Single crystals of LaTe3 with residual resistivity ratio (RRR,
$\rho$(300 K)/$\rho$(2 K)) of ∼270 were grown by tellurium flux technique, as
discussed in our previous work [30].

The ARPES measurements presented here were performed at the SGM3 beamline at
the ASTRID2 synchrotron facility [60]. ARPES data at SGM3 beamline were
collected with an energy resolution of 18 meV at h$\nu$= 24.4 eV. The angular
resolution was 0.2∘ (0.008 Å-1). The measurements were performed at 100 K with
different photon energies ranging from 13 eV to 140 eV. All the measurements
were done on freshly cleaved surfaces at a chamber base pressure better than
2$\times$10-10 mbar. The ARPES intensity plots measured as a function of
photon energy (h$\nu$) have been converted to $k_{y}$ assuming the free
electron final states [61]. The raw data ($E_{kin}$ vs h$\nu$) have been
converted to $E$ vs $k_{y}$ by utilizing the expression $k_{y}$=
$\frac{1}{\hbar}$$\sqrt{2m(E_{kin}cos^{2}\theta+V_{0})}$, where $E_{kin}$ is
the kinetic energy of the photoelectrons, $\theta$ is the emission angle and
the inner potential $V_{0}$ has been assumed to be 7.5 eV based on the best
matching with the ARPES data. The curvature plots have been obtained as in our
earlier work [62, 63] with respect to both E and k axes following the method
proposed by Zhang $et.al.$[64]. This method improves the visibility of the
weaker bands in the ARPES intensity plots and is better than the second
derivative approach. The data analysis has been performed using IGOR pro
(version 9). The dashed curves in Figs. 2e,f are obtained by a curve fitting
the maxima of the MDCs. The maxima are identified by fitting the MDCs with
Lorentzian functions. The least-square error method has been used for the
curve fitting.

The STM measurements were carried out at a base pressure of 2$\times$10-11
mbar using a variable temperature STM from Omicron Nanotechnology GmbH, while
LEED and preliminary ARPES measurements were carried out in a workstation from
Prevac sp. z o.o. STM was performed in the constant current mode using a
tungsten tip that was cleaned by sputtering and voltage pulse method. The tip
was biased and sample was kept at the ground potential. LEED is performed
using a four grid rear view optics from OCI Vacuum Microengineering. Both STM
and LEED were performed in the CDW state at room temperature.

Density functional theory: The DFT calculations have been performed for the
7-fold modulated structure with C2cm space group (SG #40). This structure has
been derived from the experimental atomic positions at 100 K reported in Ref.
37 using the PSEUDO program with average displacement of the atoms being
$\leq$0.01Å [59]. VESTA software has been used for Crystal structure
visualization [65].

We have employed the DFT-based Vienna Ab-initio Simulation Package(VASP) [66,
67] within the framework of the projector augmented wave method(PAW) [66, 67]
to investigate the electronic structure of LaTe3. The exchange-correlation
functional is treated under the generalized gradient approximation(GGA) given
by Perdew, Burke, and Ernzerhof [68]. We have considered 11 valence electrons
of the La atom ($5s^{2}5p^{6}5d^{1}6s^{2}$) and 6 valence electrons of Te atom
($5s^{2}5p^{4}$) in the PAW pseudopotential. The energy cut-off is set to 500
eV for the expansion of the planewaves. The convergence criterion for energy
in the self-consistent-field cycle and total force tolerance on each atom are
taken to be 10-6 eV and 0.02 eV/Å, respectively. The SOC is employed by a
second-variation method as implemented in the VASP code [67]. To calculate the
EBS, we have unfolded the band structure of the CDW state into the primitive
BZ of the non-CDW state, using the PyProcar python code [50]. All the DFT
bands (and consequently the EBS) are rigidly shifted to larger $E$ by 0.1 eV
with respect to the $E_{F}$ for comparison with the ARPES data.

Furthermore, to reconfirm the band crossings and to identify their irreps, DFT
calculations have been performed using the all electron WIEN2k programme
package [69] and Quantum Espresso software package [70]. Supplementary Fig.
S18 shows that the bands are in agreement between the different methods.
WIEN2k was performed with energy cut-off of 16 Ry, where the RMTKmax value is
taken to be 9.5. Further, we have used 10 for the maximum value of angular
momentum for the (l,m) expansion of wave function or density. Convergence
criteria for energy and charge have been taken to be 10-5 Ry and 0.001 e-,
respectively. Quantum Espresso calculations have been carried out using fully
relativistic PAW pseudopotentials for La and Te atoms. A planewave cutoff of
80 Ry and a $6\times 6\times 1$ k-grid were taken along with the energy
accuracy of 10-8 Ry.

##  V Acknowledgment

S.S., P.S. and S.R.B. gratefully acknowledge the financial support from
Department of Science and Technology, Government of India within the framework
of the DST-Synchrotron-Neutron Project to perform experiments at ASTRID2
synchrotron facility. A part of this work was supported by VILLUM FONDEN via
the Centre of Excellence for Dirac Materials (Grant No. 11744). The Computer
division of Raja Ramanna Centre for Advanced Technology is thanked for
installing the DFT codes and providing support throughout.

##  VI Author contributions

S.S., P.S., D.C., and S.R.B. conducted the ARPES measurements with assistance
and support from M.B. and P.H. LEED was carried out by S.S. and P.S., while
STM was performed by V.K.S. J.B. and R.D. did the DFT calculations under the
supervision of A.C. The explanation of the results was provided by S.S., T.D.,
A.C., and S.R.B. The single crystals of LaTe3 were grown by A.P., S.R. and
P.M, the latter introduced us to this system. S.S. analyzed the experimental
data with initial help from D.C., performed the post-analysis of the DFT
results with some inputs from J.B., and prepared the figures. The project was
planned and led by S.R.B. who wrote the paper with significant contributions
from S.S., P.H., and A.C.

##  VII Competing interests

The authors declare no competing interests.

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##  Appendix A ISM Phase Diagrams

We present an analysis of the physical state of the interstellar medium on the
full global galactic scale, visualized in the phase plots presented in Figures
24 and 25. Phase plots allow us to visualize where the bulk of the gas by mass
resides within a certain parameter space in pressure (or density) and
temperature. In the pressure-temperature space in the left panel of Figure 24,
we plot the total mass in gas cells residing at a particular pressure and
temperature. The mass is color coded. These plots are often used to describe
gas phase information in simulations, especially large galactic-scale
simulations (Reina-Campos et al., 2022). As such, the areas in which the mass
(represented by the colour bars) is highest show the typical state of the gas
in our simulations. As noted in the text, our phase diagrams resemble those in
galactic simulations (Kim et al., 2016; Reina-Campos et al., 2022).

Similarly, in Figure 25 we plot the magnetic field strength versus the density
for gas in our global ISM. It shows the magnetic field strength associated
with the majority of the gas (again, see color bar). In this plot we also
superimpose two popularly used power law scaling of the magnetic field with
density, as discussed in Crutcher & Kemball (2019). Recall that our initial
conditions used a very weak toroidal field with a 2/3 power law scaling.
Turbulent dynamo action would have erased initial memory of such a weak field
very quickly, so the fact that our result finds a roughly 2/3 power law for
the denser gas is significant.

Figure 24: Pressure-temperature (left) and Temperature-density (right) phase
plots. Colourbars represent mass per bin such that the yellow colours on both
plots depict where the majority of the gas sits. Figure 25: Magnetic field-
density phase plot. Colourbars represent mass per bin, similar to figure 24.
Scaling relations for $B\propto\rho^{1/2}$ and $B\propto\rho^{2/3}$ are given
in red and black lines, respectively.

##  Appendix B Fourier Analysis and Turbulence Spectra

The turbulence kinetic power spectrum $P_{i}(\bm{k})$ for the velocity
components $\varv_{x}$, $\varv_{y}$, and $\varv_{z}$ are defined as,

$P_{i}(\bm{k})=\frac{1}{L_{x}L_{y}L_{z}}|\bar{V}_{i}(\bm{k})|^{2},~{}i=x,y,z~{},$
(B1)

where $L_{x}$, $L_{y}$, and $L_{z}$ are the lengths of the box along the three
directions, and $\bar{V}_{i}(\bm{k})$ is the Fourier transform of the
respective velocity component:

$\bar{V}_{i}(\bm{k})=\frac{1}{(2\pi)^{3/2}}\int\varv_{i}(\bm{x})exp(-i\bm{k}\cdot\bm{x}){\rm
d}^{3}\bm{x},~{}i=x,y,z~{}.$ (B2)

The total turbulence power spectrum $P(k)=P_{x}(k)+P_{y}(k)+P_{z}(k)$ can be
obtained by averaging the corresponding $P_{i}(\bm{k})$ at a constant
magnitude $k=|\bm{k}|$. The energy spectrum is thus computed as $E(k)=4\pi
k^{2}P(k)$.

Figure 26: Turbulence power $P(k)$ and energy spectrum $E(k)$ in the 3 kpc
zoomed-in regions. Left panels show these spectra for the active region at two
different times in the simulation; the initial time (after 200-300 Myr of
galactic evolution), and at the end of 10 Myr evolution in the 3 kpc box. The
right hand panels show the same except for the quiescent region.

We note from the Figure that there is a marked change in the amplitudes of
energy and power in the turbulent spectra between the active and quiescent
regions. In particular, at $t=0$ the total amplitude of $P(k)$ and $E(k)$ in
the active region exceed those of their counterparts in the quiescent region
by two and one order of magnitude, respectively. The spectral shapes are
unchanged, however. Over the course of 10 Myr the turbulent power $P(k)$
decreases by $\sim$1 order of magnitude and the turbulent energy $E(k)$
decreases by nearly $\sim$2 orders of magnitude to levels similar to those in
the quiescent region. In comparison, both $P(k)$ and $E(k)$ in the quiescent
region decrease only by a factor of a few.

One explanation for the effect is the difference in losses in the mass
reservoir of the active region as the bubbles from the previous generation of
OB star formation expand. Mass is pushed into the quiescent region so that
turbulent decay is somewhat offset by the inflow. The existence of a large
feedback bubble in the active region also provides a large initial power of
the vertical turbulence motions ($V_{z}$) as compared to that in the quiescent
region; however, the vertical turbulence is likely damped after 10 Myr. It is
also possible that the decay time scale for the turbulence in the active
region is shorter than that in the quiescent one.

##  Appendix C Filament Tracing Method

Our method to trace the filamentary structures are based on the visualization
of the 3D structures. Any prominent filaments can be represented by a starting
position ($x_{0}$, $y_{0}$, $z_{0}$) and an initial normal direction
$\bm{n}_{0}$. A 3D plane perpendicular to $\bm{n}_{0}$ can be constructed. We
then scan with a given spatial increment $\delta n$ along the normal direction
and move the plane forward in space. A maximum density point along the new
plane can be located (should also in the vicinity of the density maximum point
in the previous step). By differencing the position of the density maximum
point on the new plane and that of the previous plane, one can obtain a new
normal vector for the next scanning progression. In this fashion, a serious of
sampling points can be located along the dense filament in space. The method
is computational fast and accurate to the grid resolution of the visualization
data.

###  C.1 Velocity Profiles

To create the velocity profiles of our filaments, we use the normal directions
from our filament tracer, and create two rays perpendicular to that vector,
such that we cut through the filament in two perpendicular directions. This
gives two rays, intersecting different sides of the filament, in order to
depict the asymmetry of filaments.

Figure 27: Top: Density slices depicting filamentary areas of interest in the
quiet 3 kpc region (left) and our zoomed in 200 pc (right) region. The rays
traced for flows shown via black lines, labelled 1 and 2 for first and second
ray, respectively. Bottom: Density and velocity profiles for each ray in both
the kpc and zoomed pc scale. Black line gives the density profile of each ray,
red lines show velocity along the filament, and green lines show the velocity
onto the filament, respective to the direction of the ray.

In Fig. 27, we show velocity and density profiles of the featured filaments in
the quiescent region. For each of these rays we profile the gas density and
all three components of the gas velocity. We plot density as is, but separate
our velocities into parallel and perpendicular components via cross and dot
products. We note here that the use of parallel and perpendicular is in
reference to the direction of the filament, not the direction of the ray, such
that parallel flows are the velocity flow along the filament. To ascertain the
flow along the filament, we take the cross product of the velocities with the
direction of the ray and take its magnitude. Perpendicular flow onto the
filament is ascertained via the dot product of the velocity and direction of
the ray. An important aspect here is the significance of the sign of the
perpendicular flow. Since the sign is with respect to the direction of the
ray, then matching signs between distance and perpendicular velocity indicate
flow onto the filament. If the signs are opposite from each other, the gas is
flowing out of the filament, indicating the filament is being torn apart.

##  Appendix D Transition Scale from Galactic Shear to Local Gravity and
Rotation

We show the density probability distribution function (PDF) for the 3 kpc
region and the 200 pc disk region at $t$=6.2 Myr in Fig. 28. The PDF at large
scale can be fitted by a log normal distribution up to number densities of a
few 100 cm-3, where overdensed condensations along the filamentary structures
start to develop. The lack of bins are related to the limited re-grid
resolution at 3 kpc scale. As we enlarge the central disk GMC structure, its
density PDF clearly shows a “kink” at number density of $\sim$103 cm-3,
turning away from the log-normal curve. The high density bins corresponds to
the clumps formed at the center of the spiral and disk structures (see Fig.
17). Therefore, in terms of density, the transition between the two scales
appears to take place between a few 100 to $\sim$1000 cm-3.

Figure 28: Density PDF of the 3 kpc box (left) and the 200 pc box (right). The
green curves are best fit for the distribution function.

Figure 29: Turbulence power spectrum of 3 kpc box (top panels) and the 200 pc
box (botom panels) of the quiescent region (left panels) and the active region
(right panels) at a time frame of 6.4 Myr.

To further understand the spatial scale at which this transition occurs, we
show the turbulence spectrum in Fig. 29. A consistent power law in turbulence
spectrum indicates how kinetic energy is propagated and diffused from large to
small scales, which is the case for the 3 kpc power spectrums (minimum re-grid
resolution of $\sim$25 pc for the smallest $k$) in both the active and the
quiescent regions. The spectrum follows closely the Burgers $k^{-4}$ law
downto $k/2\pi$ of $\sim$10, which corresponds to $\sim$50 pc in spatial
scale. As it is expensive to carry out Fourier transformation across many
scales for the AMR grid, we show a separate power spectrum for the 200 pc box
region instead (bottom panels). For both the power and energy spectrum, the
Fourier components start to deviate at $k/2\pi$ $\sim$10–20, which also
corresponds to a spatial scale of $\sim$50–100 pc. At scales smaller than
this, turbulence cascade is likely dominated by the local processes in the
GMCs, such as gravity and rotational motion.

In the case of the disk-like GMC (first panel in Fig. 17) formed in the
quiescent region, the $y$ component spectrum is enhanced downto a few pc
scale, whereas the $x$ and $z$ components are somewhat suppressed. Recall that
the disk normal is more aligned with $y$-direction (see Fig. 20), it is likely
that the turbulence power along $y$-component is more or less preserved from
100 to 10 pc scale, while the turbulence energy along $x$ and $z$ starts to
transform into rotation motions. In comparison, the cluster-like complex in
the active region (second panel in Fig. 13) show different power law profiles
along different spatial directions. It seems that vertical shear motions along
the ($y$, $z$) plane is stronger in this region, so that the turbulence power
in the $x$-direction experience a lesser cascade. Such an anisotropy in the
turbulence property can also contribute to the cluster-like morphology in this
region, in which multiple substructures evolve differently.