# Fate of multipolar physics in $5d^{2}$ double perovskites

Ahmed Rayyan1, Xiaoyu Liu2, Hae-Young Kee1,3<EMAIL_ADDRESS>1Department of
Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada 2Department
of Materials Science and Engineering, University of Washington, Seattle,
Washington 98195, USA 3CIFAR Program in Quantum Materials, Canadian Institute
for Advanced Research, Toronto, Ontario M5G 1M1, Canada

(September 3, 2024)

###### Abstract

In a cubic environment, the ground state of spin-orbit coupled $5d^{2}$ ions
is a non-Kramers $E_{g}$ doublet, which hosts quadrupole and octupole moments.
A series of $5d^{2}$ osmium double perovskites Ba${}_{2}M$OsO6 (M = Mg, Ca,
Zn, Cd) have recently been proposed to exhibit multipolar orders. We
investigate the structural properties of these materials using ab-initio
calculations and find that the cubic structure is unstable for the Cd compound
while the Mg, Ca, and Zn materials retain $Fm\bar{3}m$ symmetry. We show that
Ba2CdOsO6 favours a rhombohedral $R\bar{3}$ structure characterized by
$a^{-}a^{-}a^{-}$ octahedral tiltings as indicated by unstable
$\mathcal{T}_{1g}$ phonon modes. Trigonal distortions split the excited
$T_{2g}$ triplet into an $E^{\prime}_{g}$ doublet and an $A_{g}$ singlet,
which may cross energy levels with the $E_{g}$ doublet and suppress the
multipolar physics. We find a window where $E_{g}$ remains the lowest energy
state under trigonal distortion, enabling the emergence of multipole phases in
non-cubic crystal environments.

##  I Introduction

Strongly correlated materials feature many distinct phases often classified by
different order parameters associated with their broken symmetries. A textbook
example is the magnetic order described by various arrangements of magnetic
dipole moments. The magnetic order parameter can be probed using several
experimental techniques such as neutron scattering, and its onset would
accompany thermodynamic phase transitions. Under certain conditions, the
dipole moment is absent yet higher-rank moments may transition into an ordered
state. However, due to their multipolar nature, they can be “hidden” from some
experimental probes [1]. A physical framework encompassing these cases
motivates studying the ordering mechanisms of states with non-trivial
multipolar moments.

The $f$-electron systems have been a natural platform to explore multipolar
physics, as the rare-earth ions carry total angular momentum $J$ via strong
spin-orbit coupling (SOC) [2, 3]. Examples include the $4f^{2}$ Pr materials
where the Pr3+ ions carry a $J=4$ multiplet [4, 5, 6, 7, 8, 9]. The ninefold
degeneracy is lifted by octahedral or tetrahedral crystal electric fields
(CEFs) and yields a doublet where the Kramers degeneracy does not apply due to
the even number of electrons. The resulting non-Kramers doublet lacks a dipole
moment yet carries quadrupole and octupole moments. A Landau theory for Pr3+
1-2-20 materials has been developed in recent years [10, 11, 12, 13] and it
suggests that the hidden octupole moment can be revealed within magnetoelastic
experiments by applying a $[111]$ magnetic field [14].

The situation in $d$-electron systems is more subtle. In the $3d$ Mott
insulators, the light magnetic ions carry a much weaker SOC so that the
orbital angular momentum is often quenched by CEFs. In the case where the
orbital degrees of freedom may yet fluctuate, such as for one hole or one
electron in $e_{g}$ states (e.g. $3d^{9}$ or low-spin 3$d^{7}$ respectively),
then orbital ordering is usually found via the Kugel-Khomskii mechanism [15].
The resulting orbital ordering is equivalent to a motif of charge quadrupoles
and can be accompanied by a structural transition via the cooperative Jahn-
Teller effect [16]. The ordering of higher multipoles, including octupolar
moments, is difficult to achieve in the lighter transition metals.

However, higher-rank multipoles may be relevant in the heavier transition
metal compounds such as the $5d^{1}$ and $5d^{2}$ double perovskites (DPs)
$A_{2}BB^{\prime}$O6 [17]. The combination of strong SOC, large separation
between magnetic $B^{\prime}$O6 octahedra, and high cubic symmetry satisfies
the necessary preconditions for local multipolar physics [18, 17, 19, 20, 21,
22, 23]. A promising platform for octupolar ordering lies within the $5d^{2}$
DPs, where the magnetic $B^{\prime}$ ion features a $J=2$ SOC multiplet that
is split into a low-lying non-Kramers doublet $\left(E_{g}\right)$ and an
excited triplet $\left(T_{2g}\right)$ via orbital $t_{2g}$-$e_{g}$ mixing [24,
25]. Similar to the $4f^{2}$ case, the non-Kramers $E_{g}$ doublet carries
multipolar moments with interactions obtained by projecting $d$ orbital
hopping channels onto the $E_{g}$ doublet to form an effective pseudospin-1/2
model. These include compass quadrupole and Ising octupole interactions [26]
where the strength (and sign) of these terms depends on the details of each
material under consideration [27].

The $5d^{2}$ barium osmate DPs with Ba2$M$OsO6 and $M\in\left\\{\text{Mg, Ca,
Zn, Cd}\right\\}$ have been proposed as a series of compounds which features
multipolar orderings [28, 29, 30, 31]. These materials exhibit at most a
single transition at temperature $T^{*}$ that does not coincide with a
reduction of cubic symmetry. Various candidates for the ordering observed in
these materials include antiferro-quadrupolar [26], ferri-octupolar [32], and
ferro-octupolar [33, 34, 35, 27]. The octupolar order hypothesis is favoured
for the Mg, Ca, and Zn compounds, as the ordering at $T^{*}\sim 30-50$ K
coincides with the onset of muon-spin relaxation ($\mu$SR) oscillations,
signalling a loss of time-reversal symmetry [29, 30]. On the other hand, the
Cd compound does not feature thermodynamic anomalies or $\mu$SR oscillations
down to $T^{*}=0.47$ K [30], and its low-temperature structural properties is
yet to be determined by the high-resolution synchrotron x-ray diffraction as
in Ref. [31]. Thus, a set of questions arise naturally which we aim to resolve
in this work: is the cubic structure of the $5d^{2}$ Os DPs Ba2$M$OsO6 stable
at low temperatures, and if not, what is the fate of the multipolar physics in
the low-symmetry environment?

The paper is organized as follows. In Section II we briefly review how the
$E_{g}$ doublet arises out of the $J=2$ states and discuss challenges in
identifying the crystal structure in some 5$d$ DPs. In Section III we
investigate the phonon spectrum of each compound using _ab_ -_initio_ density
functional theory (DFT) simulations, and find that the Cd compound favours a
rhombohedral structure characterized by a set of octahedral tiltings. In
Section IV we evaluate how the $E_{g}$ and $T_{2g}$ states are modified in the
rhombohedral structure where each OsO6 octahedron is trigonally compressed
along the $[111]$ direction. We conclude with a summary of our findings and
their implications, as well as avenues of future work.

##  II $E_{g}$ and $T_{2g}$ States in $5d^{2}$ Osmium Double Perovskites

Figure 1:  a) Os6+ ions (yellow) enclosed in oxygen octahedral cages (grey)
arranged in the fcc structure. The crystallographic $xyz$ coordinate system is
shown along with fcc nearest-neighbour bonds of type $x$ (green), $y$ (blue),
and $z$ (red). The $M$O6 octahedra and BaO12 cuboctahedra are not shown. b)
The single-ion level scheme for $5d^{2}$ electrons in an octahedral CEF, where
the low-lying $E_{g}$ doublet is separated from the excited $T_{2g}$ triplet
by a residual CEF splitting $\Delta\sim 10$ meV. The corresponding $E_{g}$ and
$T_{2g}$ states are shown, with regions of red and blue denoting non-zero spin
density. [Reproduced from Fig. 1b) in Ref. [36].]

The $5d^{2}$ Os DPs Ba2$M$OsO6 form a rock-salt pattern of alternating $M$O6
and OsO6 corner-sharing octahedra that enclose Ba2+ ions. The Os6+ ions carry
the relevant magnetic degrees of freedom and form an fcc lattice, see Fig.
1a). The octahedral CEF lifts the electronic $d$ orbital degeneracy into
irreps of the octahedral group $O_{h}$ as $\Gamma_{l=2}=t_{2g}\oplus e_{g}$,
with the low-lying $t_{2g}$ triplet separated from the excited $e_{g}$ doublet
by energy gap $10Dq$. The two $t_{2g}$ electrons form a high-spin
configuration with $S=L=1$ via Hund’s rules.

The strong SOC $-|\lambda|\mathbf{L}\cdot\mathbf{S}$ of the $5d$ ions with
$|\lambda|<10Dq$ favours a total $J=2$ multiplet with magnetic moment
$\mathbf{M}=-\mathbf{L}+2\mathbf{S}=\mathbf{J}/2$ with magnitude $\sim
1.25\mu_{B}$ [17]. However, heat capacity, $\mu$SR, and inelastic neutron
scattering experiments find a low-lying doublet separated from excitations by
a gap of $\Delta\sim 10$ meV with a small magnetic moment of
$\mathcal{O}(0.1\mu_{B})$, in apparent contradiction with the $J=2$ picture
[30, 31]. Some mechanisms for a residual CEF which generates a low-lying
$E_{g}$ doublet and an excited $T_{2g}$ triplet as shown in Fig. 1b) have been
proposed to lift the $J=2$ degeneracy, including $t_{2g}$-$e_{g}$ mixing via
SOC and Hund’s coupling, or non-spherical Coulomb interactions in a
$t_{2g}$-only model [24]. Defining $\ket{m}\equiv\ket{J=2;J^{z}=m}$ where
$m\in\left\\{2,1,0,-1,-2\right\\}$, the $E_{g}$ and $T_{2g}$ states in the
cubic harmonic basis are linear combinations of the states shown in Fig. 1b),
and are given by

$\displaystyle\ket{\uparrow}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{-2}+\ket{2}\right),$
$\displaystyle\ket{T_{x}}$
$\displaystyle=\frac{i}{\sqrt{2}}\left(\ket{-1}+\ket{1}\right),$
$\displaystyle\ket{\downarrow}$ $\displaystyle=\ket{0},$
$\displaystyle\ket{T_{y}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{-1}-\ket{1}\right),$ (1)
$\displaystyle\ket{T_{z}}$
$\displaystyle=\frac{i}{\sqrt{2}}\left(\ket{-2}-\ket{2}\right).$

The $E_{g}$ doublet is of non-Kramers type and has vanishing dipole moments
$\braket{\mathcal{P}_{E_{g}}^{\dagger}\mathbf{J}\mathcal{P}_{E_{g}}}=0$ where
$\mathcal{P}_{E_{g}}=\sum_{\omega\in\left\\{\uparrow,\downarrow\right\\}}\ket{\omega}\bra{\omega}$
projects onto the $E_{g}$ states. Three higher-rank multipoles retain a finite
moment: the quadrupole operators
$Q_{z^{2}}=\frac{1}{\sqrt{3}}\left(3J_{z}^{2}-\mathbf{J}^{2}\right)$ and
$Q_{x^{2}-y^{2}}=J_{x}^{2}-J_{y}^{2}$, and the octupole operator
$T_{xyz}=\frac{\sqrt{15}}{6}\overline{J_{x}J_{y}J_{z}}$ where the overline
symbol denotes symmetrization [2]. These multipolar operators have the same
matrix elements as the Pauli matrices within the $E_{g}$ doublet and can be
considered as effective pseudospin-1/2 operators. Microscopic models of the
$E_{g}$ doublets feature a variety of multipolar-ordered ground states
including antiferro-quadrupolar (AF$\mathcal{Q}$) and ferro-octupolar
(F$\mathcal{O}$) orders [32, 26, 33, 34, 35, 27]. The lack of an observed
structural transition for the Mg, Ca, and Zn compounds seems to imply that
AF$\mathcal{Q}$ is unlikely to be the low-temperature phase. On the other
hand, the predicted antiferro-distortions tends to generate tiny structural
deformations which can be missed in some diffraction experiments.

For example, consider the $5d^{1}$ DP Ba2MgReO6 [37, 38] which features a
$j_{\text{eff}}=3/2$ pseudospin on each Re6+ ion and two thermodynamic
anomalies at temperatures $T_{q}>T_{m}$ where $T_{q}=33\text{ K}$
$\left(T_{m}=18\text{ K}\right)$ corresponds to the onset of quadrupolar
(dipolar) order [39, 40, 41, 42]. In Ref. [39] the $\mu$SR data suggests that
there are two inequivalent oxygen sites which hints at an underlying
tetragonal distortion, yet no such effect is detected via neutron diffraction.
On the other hand, a very small cubic-to-tetragonal distortion below $T_{q}$
was detected using the high-resolution synchrotron x-ray diffraction on a
sample with high crystallinity, and is predominantly associated with the
ordering of alternating $Q_{x^{2}-y^{2}}$ quadrupoles near $T_{q}$ [41]. This
highlights the challenges faced in the accurate structural determination of
DPs, especially the detection of quadrupolar-induced antiferro-distortions in
$5d$ materials [21, 41]. It is also worthwhile to note that in the case of
CeB6 ($4f^{1}$) there is no change in B6 positions despite the established AF
quadrupole order [43, 44, 45].

While the lack of a structural transition in the Ba${}_{2}M$OsO6 for the $M=$
Mg, Ca, Zn compounds has been verified using synchrotron x-ray diffraction
[31], it is worthwhile to examine the stability of the high-symmetry cubic
structure. Note that the non-Kramers doublet degeneracy is sensitive to
general structural deformation which may result in the loss of multipolar
physics. Furthermore, the situation regarding the $M=$ Cd material is not yet
fully determined. In the next section we investigate the phonon spectra of the
four $5d^{2}$ Os DPs to examine the stability of the cubic structure.

##  III Structural Properties of $5d^{2}$ Osmium Double Perovskites

Figure 2:  a) Ba${}_{2}M$OsO6 in the ideal rock-salt structure with space
group $Fm\bar{3}m$, where OsO6 (light brown) and $M$O6 (fuschia) corner-
sharing octahedra alternate in an fcc pattern surrounding Ba ions (green). b)
1${}^{\text{st}}$ Brillouin zone of the fcc primitive reciprocal lattice
vectors along with points of high symmetry. Low-energy phonon spectrum for c)
Ba2CaOsO6 and d) Ba2CdOsO6 in the $Fm\bar{3}m$ structure as calculated by the
_ab_ -_initio_ approach detailed in Appendix A; the phonon irrep at the
$\Gamma$ point is indicated in grey. The k-path traversed is indicated in b).
The presence of unstable $\mathcal{T}_{1g}$ modes for the cubic $M=$ Cd
material (dashed red) signals a structural distortion instability induced by
octahedral tilting at low temperatures.

Early approaches to analyze perovskite structural properties include
Goldschmidt’s tolerance factor, which is adapted for DPs
$A_{2}BB^{\prime}\text{O}{}_{6}$ as

$t=\frac{(r_{A}+r_{\text{O}})}{\sqrt{2}\left(\frac{r_{B}+r_{B^{\prime}}}{2}+r_{\text{O}}\right)}$
(2)

where $r_{l}$ is the radius of ion $l$; $t=1$ would then correspond to the
high-symmetry cubic structure with space group symmetry $Fm\bar{3}m$ (no. 225)
for the rock-salt formation, see Fig. 2a) [46, 47]. Deviations from $t=1$
indicate a mismatch between the size of the $A$ and $B/B^{\prime}$ ions,
inducing structural distortions that lower the crystal symmetry. The
Ba${}_{2}M$OsO6 materials we consider fall into two doppelgänger pairs since
the radii of the Mg2+/Zn2+ (0.72/0.74 Å) and Ca2+/Cd2+ (1.00/0.95 Å) ions are
roughly equivalent, resulting in tolerance factors of
$t_{\text{Mg,\text{Zn}}}\sim 1.04$ and $t_{\text{Ca,\text{Cd}}}\sim 0.985$
[48]. As a result, one may expect the structural properties within each pair
of doppelgängers to be equivalent.

We test this hypothesis by investigating the phonon dispersion of each
material in its electronic ground state, which can be calculated using DFT. We
first perform a structural optimization where the ions are relaxed from the
initial $Fm\bar{3}m$ structure. The fcc conventional unit cell has Os at
representative Wyckoff position $4a=\left(0,0,0\right)$, $M$ at
$4b=\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$, Ba at
$8c=\left(\frac{1}{4},\frac{1}{4},\frac{1}{4}\right)$, and O at
$24e=\left(w,0,0\right),$ with $w\sim 0.23-0.24$ depending on the choice of
$M$. We relax the primitive unit cell which has lattice constant $a\sim
5.7-6.0\text{ \AA}$ and contains one formula unit, ie. 10 atoms. Once the
optimized structure is obtained, we calculate the interatomic force constants
using density functional perturbation theory (DFPT) [49, 50], which, after
Fourier interpolation, yields the dynamical matrix at non-zero $\mathbf{q}$.
Diagonalization of the dynamical matrix then yields the dispersion
$\omega_{s}\left(\mathbf{q}\right)$ for a given phonon branch $s=1,\ldots,30$.
Computational details of this three-step procedure are given in Appendix A.

Figure 3:  a) Crystal structure of the $a^{-}a^{-}a^{-}$ distorted Ba2CdOsO6
with space group $R\bar{3}$, where each OsO6 octahedron undergoes a uniform
tilting in size and orientation. The Cd-O-Os angles is given by
$\psi=148.3^{\circ}$. b) 1${}^{\text{st}}$ Brillouin zone of the rhombohedral
primitive reciprocal lattice vectors along with points of high symmetry; the
rhombohedral setting is used. c). Low-energy phonon spectrum for the crystal
structure shown in a) calculated by _ab_ -_initio_ techniques. The k-path
traversed is indicated in b). The lack of imaginary phonon frequencies
indicates that the $R\bar{3}$ structure is dynamically stable at low
temperatures for the $M=$ Cd compound.

The phonon dispersion for the Ca/Cd doppelgänger pair in the $Fm\bar{3}m$
structure is shown in Fig. 2; the Mg/Zn pair are given in Appendix A for
completeness 111Note added: In the process of producing this manuscript we
became aware of Ref. [81]’s study which also features the phonon dispersion
for the $M=$ Ca material in the $Fm\bar{3}m$ structure. Our results match
those of Ref. [81] and includes the $M=$ Mg, Zn, Cd materials in addition.. We
will use calligraphic font to distinguish phonon irreps from the states in Eq.
(1), ie. the $E_{g}$ spin-orbital doublet vs. the $\mathcal{E}_{g}$ phonon
mode. At the $\Gamma$ point the 27 optical phonon modes decompose into irreps
of $O_{h}$ as
$\mathcal{A}_{1g}\oplus\mathcal{E}_{g}\oplus\mathcal{T}_{1g}\oplus
2\mathcal{T}_{2g}\oplus 4\mathcal{T}_{1u}\oplus\mathcal{T}_{2u}$ [52]. The
clearest difference among the Ca/Cd pair is the $\mathcal{T}_{1g}$ mode which
takes on an imaginary frequency at the $\Gamma$ point for the Cd compound, see
Fig. 2d). This occurs when the phonon spectrum is computed about an unstable
structural equilibrium [53]; in this case, our ab-initio DFT analysis suggests
that the ideal cubic structure of Ba2CdOsO6 with $Fm\bar{3}m$ symmetry is
unstable and undergoes a structural transition to a state with lower symmetry
at low temperatures.

The correct structure for Ba2CdOsO6 is obtained by displacing the atoms
according to the three unstable eigenvectors associated with the
$\mathcal{T}_{1g}$ modes. This corresponds to fixing the positions of the Ba,
Cd, and Os atoms while rotating the oxygen ions about the [111] direction. The
resulting structure is shown in Fig. 3a) and consists of uniformly staggered
octahedral canting, ie. $a^{-}a^{-}a^{-}$ in Glazer notation with space group
$R\bar{3}$ (no. 148) [54, 55]. The tilting reduces the Cd-O-Os bond angle to
$\psi=148.3^{\circ}$ as shown in Fig. 3a). While the space group symmetry is
lowered, the primitive unit cell still contains one formula unit and maintains
its shape; ie. rhombohedral with $\alpha=60^{\circ}$. We repeat the _ab_
-_initio_ analysis for Ba2CdOsO6 in the low-symmetry structure and find that
all optical phonon modes carry non-zero frequency, see Fig. 3c), indicating
that the $R\bar{3}$ structure is stable.

##  IV $E_{g}$ and $T_{2g}$ States Under Trigonal Distortion

In the previous section we argued that the stable structure for Ba2CdOsO6 has
space group $R\bar{3}$, which breaks most $O_{h}$ point group operations at
each Os${}^{6}+$ ion yet retains the $C_{3}$ symmetry about the [111] axis.
This is compatible with trigonal deformation of the OsO6 octahedra; for
Ba2CdOsO6 we find a compression of the OsO6 cage along the [111] axis, see
Fig. 6 in Appendix B. In the trigonal environment the $t_{2g}$ orbital
degeneracy is reducible into $t_{2g}=a_{g}\oplus e^{\prime}_{g}$ irreps of
$S_{6}$. The $E_{g}$ non-Kramers doublet arising from $J=2$ is not protected
by the time-reversal symmetry and may be sensitive to the breaking of cubic
crystalline symmetries. However, as the trigonal distortion preserves $C_{3}$
symmetry, the multipolar physics of the $E_{g}$ doublet may yet survive if the
excited $T_{2g}$ states (which decompose as $T_{2g}=A_{g}\oplus
E^{\prime}_{g}$ in analogy with $t_{2g}$ orbitals) do not cross energies with
the $E_{g}$ doublet.

We investigate the fate of the multipolar physics by modelling the trigonal
CEF and using exact diagonalization (ED) to solve for the single-ion spectrum
of the electronic $5d^{2}$ configuration. The CEF Hamiltonian is given by

$H_{\text{CEF}}(\delta)=10Dq\sum_{\alpha\sigma;\beta\sigma^{\prime}}\left[\Xi^{\alpha\beta}(\delta)\otimes\mathbb{1}_{\sigma\sigma^{\prime}}\right]d^{\dagger}_{\alpha\sigma}d_{\beta\sigma^{\prime}},$
(3)

where $d_{\alpha\sigma}^{\dagger}$ creates a single electron with spin
$\sigma\in\left\\{+,-\right\\}$ in $d$ orbital $\alpha$, and $10Dq=4$ eV as is
typical for the materials of interest [27]. $\delta$ is an angle
parameterizing the degree of trigonal distortion with $\delta>0$ ($\delta<0$)
corresponding to trigonal elongation (compression), see Fig. 6; the relaxed
structure for Ba2CdOsO6 obtained in Sec. III has $\delta=-2.87^{\circ}$,
equivalent to a reduction of Os-O bonds by $\sim 2\%$. The orbital level
scheme is dictated by the $5\times 5$ matrix $\Xi$ which we estimate within
the point-charge approximation in Appendix B. The local physics is also
governed by SOC

$H_{\text{SOC}}=\xi\sum_{\alpha\sigma;\beta\sigma^{\prime}}\left[\mathbf{l}^{\alpha\beta}\cdot\mathbf{s}_{\sigma\sigma^{\prime}}\right]d^{\dagger}_{\alpha\sigma}d_{\beta\sigma^{\prime}},$
(4)

where $\mathbf{l}$ and $\mathbf{s}$ are angular momentum operators of $l=2$
and $s=1/2$ respectively and $\xi$ is the single-particle SOC strength, and
the Kanamori-Hubbard interactions

$\displaystyle H_{\text{int}}$
$\displaystyle=U\sum_{\alpha}n_{\alpha+}n_{\alpha-}+U^{\prime}\sum_{\alpha\neq\beta}n_{\alpha+}n_{\beta-}$
$\displaystyle+\left(U^{\prime}-{\color[rgb]{0,0,0}J_{H}}\right)\sum_{\alpha<\beta,\sigma}n_{\alpha\sigma}n_{\beta\sigma}$
(5)
$\displaystyle-{\color[rgb]{0,0,0}J_{H}}\sum_{\alpha\neq\beta}\left(d_{\alpha+}^{\dagger}d_{\alpha-}d^{\dagger}_{\beta-}d_{\beta+}+d_{\alpha+}^{\dagger}d_{\beta-}d_{\alpha-}^{\dagger}d_{\beta+}\right),$

where $n_{\alpha\sigma}=d_{\alpha\sigma}^{\dagger}d_{\alpha\sigma}$, $J_{H}$
is Hund’s coupling strength, and $U$
$\left(U^{\prime}=U-2{\color[rgb]{0,0,0}J_{H}}\right)$ is the intraorbital
(interorbital) Hubbard parameter. For a given $\delta$ the local Hamiltonian
$H_{\text{int}}+H_{\text{SOC}}+H_{\text{CEF}}(\delta)$ can be diagonalized
within the basis of $\binom{10}{2}=45$ possible $d^{2}$ states with $10Dq=4$
eV, $U=2.5$ eV, $\xi=0.4$ eV, and finite $J_{H}$. At $\delta=0$ the five
lowest states are the $E_{g}$ doublet and $T_{2g}$ triplet and we choose
${\color[rgb]{0,0,0}J_{H}}=0.2U$ so that the $E_{g}$-$T_{2g}$ splitting is
$\Delta\sim 10$ meV. For our choice of $U$ this corresponds to $J_{H}=$ 0.5 eV
which is a reasonable upper bound for the Hund’s coupling in the $5d$ series
[56]. The result for finite $\delta$ is given in Fig. 4; within the set of
parameters considered the five lowest eigenvalues are gapped from the rest of
the spectrum.

Unlike the tetragonal case, trigonal distortions do not split the non-Kramers
$E_{g}$ doublet due to the presence of $C_{3}$ symmetry, while the excited
triplet decomposes as $T_{2g}=A_{g}\oplus E^{\prime}_{g}$. The
$E_{g}$-$E^{\prime}_{g}$ doublets undergo level repulsion with the $E_{g}$
states lower in energy for all $\delta$. On the other hand, the $A_{g}$
singlet is the lowest eigenvalue at larger distortions, crossing the $E_{g}$
doublet and suppressing the multipolar physics within. In the case of
elongation, $A_{g}$ is lower in energy than $E^{\prime}_{g}$ and the singlet
crosses the $E_{g}$ doublet at a relatively small distortion of
$\delta\sim+0.18^{\circ}$. The case of trigonal compression is more
interesting, as the $E_{g}$ doublet remains the state of lowest energy until
$\delta\sim-2.1^{\circ}$.

The asymmetry between compression and elongation can be traced back to filling
the $a_{g}$ and $e^{\prime}_{g}$ single-electron orbital states. Trigonal
elongation lowers the $e^{\prime}_{g}$ doublet and the resulting two-electron
state has its orbital degrees of freedom quenched. On the other hand, trigonal
compression lowers the $a_{g}$ singlet, and the two-electron state is
predominantly composed of a) the filled singlet and b) the state where both
$a_{g}$ and $e^{\prime}_{g}$ are singly-occupied. Thus there is a competition
between the distortion and Hund’s coupling, with the latter seeking to
eliminate orbital fluctuations. This frustration manifests itself in the
relatively small energy difference between the $E^{\prime}_{g}$ and $A_{g}$
states when $\delta\lesssim 0$ (see Fig. 4), and is finally resolved in the
limit of strong distortions by stabilizing the $A_{g}$ singlet. The trigonal
compression in Ba2CdOsO6 is given by $\delta=-2.85^{\circ}$ which is just
beyond the $A_{g}$-$E_{g}$ crossing. This suggests that the multipolar physics
in rhombohedral Ba2CdOsO6 is likely to be revealed in pressure-tuned
experiments by approaching the cubic limit.

Figure 4:  ED calculation of the five lowest eigenvalues of
$H_{\text{int}}+H_{\text{SOC}}+H_{\text{CEF}}(\delta)$ as $\delta$ is varied
with $10Dq=4$ eV, $U=2.5$ eV, ${\color[rgb]{0,0,0}J_{H}}=0.5$ eV, and
$\xi=0.4$ eV. $\delta>0$ ($\delta<0$) corresponds to trigonal elongation
(compression). At $\delta=0$ the $T_{2g}$ triplet is separated by a gap of
$\Delta\sim 10$ meV from the $E_{g}$ doublet (red) and splits into an $A_{g}$
singlet (blue) and $E^{\prime}_{g}$ doublet (green). The shaded region
corresponds to the range of distortion where the $E_{g}$ doublet remains the
lowest energy level. The purple star corresponds to the distortion in
Ba2CdOsO6 of $\delta=-2.85^{\circ}$.

##  V Summary and Discussion

In summary, we investigated the structural properties of four $5d^{2}$ Os DPs
Ba${}_{2}M$OsO6 using an ab-initio calculation of the phonon spectrum. We
found that while the $M=$ Mg, Zn, Ca compounds have a stable cubic structure
with space group $Fm\bar{3}m$, the $M=$ Cd material forms the rhombohedral
$R\bar{3}$ structure at low temperatures. The transition to the low-symmetry
state is indicated by softening of the $\mathcal{T}_{1g}$ phonon modes which
results in out-of-phase tilting of the oxygen octahedra in each of three
crystallographic directions, ie. $a^{-}a^{-}a^{-}$. The rhombohedral structure
generally allows for trigonal distortion of the OsO6 octahedra, which modifies
the physics of the $J=2$ states by splitting the excited $T_{2g}$ triplet into
an $E^{\prime}_{g}$ doublet and $A_{g}$ singlet. There exists a finite window
where the $E_{g}$ doublet is the lowest energy level, allowing for the
ordering of various multipolar orders. A strong enough deformation stabilizes
the $A_{g}$ singlet and suppresses the $E_{g}$ doublet’s multipolar magnetism.

It is interesting to note that while $r_{\text{Ca}}>r_{\text{Cd}}$ by only
0.05 Å, Ba2CaOsO6 remains cubic at low temperatures despite having a lower
tolerance factor than Ba2CdOsO6. This is likely due to the non-ionic nature of
DPs which cannot be accounted for by the tolerance factor calculated from
tabulated ionic radii [21]. One likely explanation for the difference between
the Ca and Cd compounds is the number of $d$-electrons present on the $M^{2+}$
ion, ie. $d^{0}$ and $d^{10}$ respectively. The capacity for $\pi$-bonding in
Ba2CdOsO6 favours the reduction of the Cd-O-Os bond angles, while the lack of
such for Ba2CaOsO6 results in linear Ca-O-Os bonds [57, 58, 59]. While Eq. (2)
provides a good first approach to the classification of perovskite structural
properties, the fact that Ba2CaOsO6 and Ba2CdOsO6 form a doppelgänger pair but
have different low-temperature structures motivates the need to go beyond the
tolerance factor by considering each compound’s chemical properties.

The presence of trigonal distortion allows for new interorbital exchange
pathways including $xy$-$xz/yz$ hopping (parameterized by $t_{4}$ in Ref.
[27]) for bonds parallel to the $xy$-plane. The parameters of the $E_{g}$
pseudospin-1/2 effective model are modified in the weak hopping limit, and the
dependence on $t_{4}$ in $5d^{2}$ DPs with small trigonal distortion is an
interesting subject left for future study. We find that the distortion in
Ba2CdOsO6 is large enough to place the material outside the region where the
effective $E_{g}$ model is valid. Since the singlet crosses the doublet and
becomes the lowest state separated by a small gap from the $E_{g}$, the
intersite exchange interaction may become important in Ba2CdOsO6. While the
singlet itself has no moment, the intersite exchange interaction may induce a
magnetic moment.

Our first main conclusion is that Ba2CdOsO6 should feature a cubic-
rhombohedral structural transition at low temperatures. Investigation of this
scenario can be determined using high-resolution synchrotron x-ray
diffraction, a reliable technique for detecting antiferro-distortions in $5d$
DPs [41, 31]. The cubic-rhombohedral structural transition has not been
considered in previous studies of the $5d^{2}$ Ba${}_{2}M$OsO6 series yet is
common in other DPs, for example the strontium osmate Sr2CrOsO6 [60, 61, 62]
and a series of strontium antimony oxides Sr${}_{2}B^{\prime\prime}$SbO6 [63,
64, 65]. The structural properties of these materials has been explored using
x-ray and neutron powder diffraction measurements, and display a series of
structural transitions (monoclinic $\rightarrow$ rhombohedral $\rightarrow$
cubic) as temperature is increased. In particular, the $R\bar{3}\rightarrow
Fm\bar{3}m$ transition is observed to be of second-order as predicted by
Landau theory [55] and corresponds to the suppression of the cubic $(642)$
peak below the ordering temperature [63, 64, 65, 66]. Similar observations in
the low-temperature phase of Ba2CdOsO6 would serve as experimental evidence
for rhombohedral symmetry in this material. Another signature of the symmetry
lowering would be the splitting of triplet phonon modes $\mathcal{T}_{1g,2g}$
at the $\Gamma$ point into irreps of $S_{6}$ as
$\mathcal{T}_{1g,2g}=\mathcal{A}_{g}\oplus\mathcal{E}_{g}$. In Fig. 3c) we see
that the $\mathcal{A}_{g}$-$\mathcal{E}_{g}$ splitting is roughly 5 meV; such
an energy difference can be resolved via Raman scattering. Moreover, inelastic
scattering experiments can reveal signatures of strong vibronic coupling in
the excitation spectrum as recently demonstrated for K2IrCl6 [67]. In this
case, the spin-orbit-lattice entangled vibronic state does not result in
static structural distortions due to dynamic Jahn-Teller considerations; we
leave this scenario open for future research and analysis. Our findings
highlight the importance of considering the interplay between crystal
structure and electronic states when exploring the magnetic properties of $5d$
spin-orbit coupled magnets.

The second main conclusion is that multipolar physics can survive in $5d^{2}$
materials with trigonal distortions. The $E_{g}$ doublet does not split under
trigonal distortions and there exists a window of finite distortion where it
remains the lowest energy level, especially for the case of trigonal
compressions. This suggests that multipolar physics can be found in structures
with lower symmetry such as $R\bar{3}$. Interestingly, this space group is
common amongst several honeycomb compounds including CrI3 [68]. A recent
proposal has suggested that $5d^{2}$ honeycomb compounds could host exotic
ordered and disordered multipolar phases, including the Kitaev multipolar
liquid [36]. The identification of a $5d^{2}$ honeycomb material with, e.g.,
$R\bar{3}$ symmetry and small trigonal compression is an excellent first step
to the realization of the Kitaev multipolar liquid and forms an interesting
direction for future work.

###### Acknowledgements.

A.R. is grateful for helpful discussions with D. Churchill and S. Voleti. We
acknowledge support from the Natural Sciences and Engineering Research Council
of Canada Discovery Grant No. 2022-04601. H.Y.K. also acknowledges support
from the Canadian Institute for Advanced Research and the Canada Research
Chairs Program. Computations were performed on the Niagara supercomputer at
the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for
Innovation; the Government of Ontario; Ontario Research Fund - Research
Excellence; and the University of Toronto.

##  Appendix A _ab_ -_initio_ Calculations and Phonon Dispersions for
$M=\text{Mg/Zn}$

The DFT calculations are performed using Vienna Ab initio Simulation Package
(VASP) with the Perdew-Burke-Ernzerhof exchange-correlation functional and
cutoff energy of 600 eV [69, 70]. The crystal structures of all four materials
considered in this work are obtained from the Materials Project [71].
Structural relaxation of the primitive unit cell in the initial $Fm\bar{3}m$
structure is performed using a $\Gamma$-centered $8\times 8\times 8$ k-mesh
with maximum force/atom of $10^{-4}$ eV/Å. In the ionic relaxation all degrees
of freedom are varied including ionic positions, cell shape, and cell volume;
care is taken to eliminate artifacts associated with the Pulay stress by
repeated re-relaxation of the intermediate ionic positions. We then used VASP
to calculate interatomic force constants using DFPT simulations on a $2\times
2\times 2$ supercell with a $\Gamma$-centered $4\times 4\times 4$ k-mesh and
an energy convergence threshold of $10^{-8}$ eV. The supercell contains 8
formula units and has a (super)lattice constant of roughly $12$ Å; a large
supercell is chosen to eliminate artifacts arising from long-range forces.
Finally, phonopy performs the diagonalization of the dynamical matrix at
$\mathbf{q}$ commensurate with the supercell and the Fourier interpolation at
general $\mathbf{q}$ [72, 73]. In Fig. 5 we present the phonon dispersion for
the Mg/Zn doppelgänger pair. VASPKIT was used to prepare the VASP simulations
such as in creation of the position and ionic relaxation/DFPT input files
[74]. Crystal structures shown in Figs. 2a) and 3a) were visualized using
VESTA [75]. The Brillouin zones in Fig. 2b) and 3b) were generated using the
Atomic Simulation Environment package [76] with symmetry points labelled
according to the HPKOT convention [77].

Figure 5:  Low-energy phonon spectrum calculated via an _ab_ -_initio_
approach for the a) Ba2MgOsO6 and b) Ba2ZnOsO6. materials in the $Fm\bar{3}m$
structure, with symmetry points shown in Fig. 2. The phonon irrep at the
$\Gamma$ point is shown in grey. The lack of imaginary phonon frequencies
indicates the stability of the $Fm\bar{3}m$ structure at low temperatures for
the $M=\text{Mg},\text{Zn}$ compounds.

##  Appendix B Construction of the trigonal CEF

We can estimate the effect of trigonal distortion on electrons in the $5d$
shell by constructing a trigonal CEF Hamiltonian in the point charge limit. We
will only consider trigonal compression and elongation of the OsO6 cage, but
other deformations are allowed in principle [78]. The oxygen positions can be
parameterized as

$\displaystyle\mathbf{r}_{1}$
$\displaystyle=\frac{b}{\sqrt{2}}\hat{\mathbf{x}}+f(\delta)\hat{\mathbf{c}},$
$\displaystyle\mathbf{r}_{4}$ $\displaystyle=-\mathbf{r}_{1},$
$\displaystyle\mathbf{r}_{2}$
$\displaystyle=\frac{b}{\sqrt{2}}\hat{\mathbf{y}}+f(\delta)\hat{\mathbf{c}},$
$\displaystyle\mathbf{r}_{5}$ $\displaystyle=-\mathbf{r}_{2},$ (6)
$\displaystyle\mathbf{r}_{3}$
$\displaystyle=\frac{b}{\sqrt{2}}\hat{\mathbf{z}}+f(\delta)\hat{\mathbf{c}},$
$\displaystyle\mathbf{r}_{6}$ $\displaystyle=-\mathbf{r}_{3},$

where
$\hat{\mathbf{c}}=\frac{1}{\sqrt{3}}\left(\hat{\mathbf{x}}+\hat{\mathbf{y}}+\hat{\mathbf{z}}\right)$,
$b$ is the length of the octahedron edge in the ideal limit, and

$f(\delta)=\frac{b}{4}\left[\text{tan}\left(60^{\circ}+\frac{\delta}{2}\right)-\text{tan}\left(60^{\circ}\right)\right]$
(7)

where $\delta>0\;(\delta<0)$ corresponds to the case of trigonal elongation
(compression), see Fig. 6. For Ba2CdOsO6 we find that $b=2.94\text{ \AA}$ and
$\delta=-2.85^{\circ}$, equivalent to a reduction in Os-O bonds by $\sim 2$%.

Figure 6:  Geometry used to derive the CEF in the case of trigonal compression
$(\delta<0)$, where the $[111]$ and $[\bar{1}\bar{1}\bar{1}]$ faces (shaded)
are pressed towards each other. In Ba2CdOsO6 the three oxygen ions at
$\mathbf{r}_{1,2,6}$ form an isosceles triangle (blue) with base $b=2.94$ Å
and $\delta=-2.85^{\circ}$. The $xyz$ coordinates in this figure (ie. the
local octahedral coordinates at each Os atom) differ from the $xyz$
coordinates shown in Fig. 3a) (the crystallographic directions) by a rotation
about the [111] direction due to the octahedral canting in the $R\bar{3}$
structure.

Note that the $xyz$ coordinates in Eq. (B) refer to the local octahedral
coordinates at each osmium atom, which differ from the crystallographic $xyz$
coordinates by a rotation about the [111] direction. This is due to the
octahedral tilting within the $R\bar{3}$ structure which affects all OsO6
octahedra uniformly. With this in mind, we will continue to use the $xyz$
symbols to simplify notation.

The potential energy of the $5d^{2}$ electrons in each Os6+ ion in the
presence of the O2- point charges can be written in a multipole expansion
given by

$\displaystyle
V\left(\mathbf{r}\right)=\sum_{k=0}^{\infty}\sum_{p=-k}^{k}\;r^{k}\left[\frac{4\pi
A}{2k+1}\sum_{i=1}^{6}\frac{Y_{kp}\left(\theta_{i},\phi_{i}\right)^{*}}{r_{i}^{k+1}}\right]Y_{kp}\left(\theta,\phi\right),$
(8)

for $r\equiv|\mathbf{r}|<|\mathbf{r}_{i}|$ [79]. The proportionality constant
$A>0$ is fixed so that the $t_{2g}$-$e_{g}$ splitting at $\delta=0$ is
$10Dq=4$ eV. We then evaluate matrix elements of Eq. (8) between hydrogenic
wave functions $R_{n=5,l=2}(r)Y_{l=2,l_{z}}(\theta,\phi)$. Since $l=2$ we may
restrict the $k$ summation to $k\leq 4$ and we further ignore the $k=0$
contribution as it does not carry an angular dependence. When $\delta\neq 0$
the $t_{2g}$ degeneracy is lifted and it becomes more appropriate to write the
trigonal CEF in the following basis of single-electron creation operators [80]

$\displaystyle d^{\dagger}_{a_{g}}$
$\displaystyle=\frac{1}{\sqrt{3}}\left(d^{\dagger}_{xy}+d^{\dagger}_{yz}+d^{\dagger}_{xz}\right),$
$\displaystyle d^{\dagger}_{e_{g}^{\prime-}}$
$\displaystyle=\frac{1}{\sqrt{3}}\left(d^{\dagger}_{xy}+\omega^{2}d^{\dagger}_{yz}+\omega
d^{\dagger}_{xz}\right),$ $\displaystyle d^{\dagger}_{e_{g}^{\prime+}}$
$\displaystyle=\frac{-1}{\sqrt{3}}\left(d^{\dagger}_{xy}+\omega
d^{\dagger}_{yz}+\omega^{2}d^{\dagger}_{xz}\right),$ (9) $\displaystyle
d^{\dagger}_{e_{g}^{\mp}}$ $\displaystyle=\frac{\pm
1}{\sqrt{2}}\left(d^{\dagger}_{z^{2}}\mp id^{\dagger}_{x^{2}-y^{2}}\right),$

where $\omega=\text{exp}\left(2\pi i/3\right)$ and the spin index is
suppressed. For $\delta=-2.85^{\circ}$ as in Ba2CdOsO6 we find that the
trigonal CEF is given by

$\Xi^{\alpha\beta}(\delta)=\left(\begin{array}[]{c|cc|cc}-0.197&0&0&0&0\\\
\hline\cr 0&0.0406&0&0.145&0\\\ 0&0&0.0406&0&0.145\\\ \hline\cr
0&0.145&0&1.058&0\\\ 0&0&0.145&0&1.058\end{array}\right),$ (10)

and the numerical values are in units of $10Dq$. By diagonalizing Eq. (10) one
can check that the trigonal compression $\delta<0$ yields a low-lying $a_{g}$
singlet whereas elongation $\delta>0$ prefers the lowering of the
$e^{\prime}_{g}$ doublet. Eq. (10) demonstrates the mixing of $e_{g}$ and
$e_{g}^{\prime}$ states of the same parity under trigonal deformations.

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# Does (TaSe 4) 2I really harbor an axionic charge density wave ?

A.A. Sinchenko<EMAIL_ADDRESS>M.V. Lomonosov Moscow State University,
119991, Moscow, Russia  Kotelnikov Institute of Radioengineering and
Electronics of RAS, 125009 Moscow, Russia  R. Ballou  J.E. Lorenzo  Th. Grenet
P. Monceau<EMAIL_ADDRESS>Univ. Grenoble Alpes, Inst. Neel,
F-38042 Grenoble, France, CNRS, Inst. Neel, F-38042 Grenoble, France

###### Abstract

A recent experimental work has reported an excess of the non-linear
conductivity in the charge density wave (CDW) sliding mode of the quasi one-
dimensional compound (TaSe4)2I when a magnetic field is applied co-linearly to
the electric field (J. Gooth, et al., Nature 575, 315 (2019). This result has
opened a new conceptual approach where the CDW gap in (TaSe4)2I is opened
between Weyl fermions of opposite chirality with the assumption that this
compound is a Weyl semi-metal in its undistorted high temperature phase. We
report measurements in the sliding state of (TaSe4)2I performed in similar
conditions. We have found no increase of the magnetoconductivity. In our
attempts for understanding this unsettling discrepancy we stress the specific
nature of the Peierls transition in (TaSe4)2I and the strong electron-phonon
coupling present in this compound. Given the lack of further evidences, we
think that it is premature to assert that (TaSe4)2I is an axionic insulator.

††preprint: AIP/123-QED

Experimental and theoretical investigations in condensed matter can allow the
realization of quantum field theory predictions such as, for example, the
axion electrodynamics[1]. The axion particle has become a candidate
responsible for some or all of the dark matter of the universe[2]. The search
for experiment detection of axion has been recently reviewed [3] and new
experiment proposal is underway[4].

Since the discovery of three-dimensional (3D) topological insulators[5], a
large interest has been devoted to Weyl semi-metals (see Ref. 6 for a recent
review). They are materials in which low energy electronic quasi-particles
behave as chiral relativistic fermions without rest mass known as Weyl
fermions. Weyl fermions exist at isolated crossing points between conduction
and valence bands, called Weyl nodes. Axion electrodynamics in Weyl semi
metals arise from the separation of Weyl nodes of opposite chirality in
momentum space and energy. Turning on interactions, chiral symmetry breaking
is the phenomenon of the spontaneous generation of an effective mass of Weyl
fermions, namely a pairing between the fermions (electrons) and holes with
different chiralities which opens gaps at the Weyl points and induces a charge
density wave (CDW). The opening of this gap transforms the Weyl semi-metal
into an axion insulator and the phase of the CDW is identified to the dynamic
axion field $\theta$ [7]. The CDW phason is the axion mode and its dynamics is
therefore described by the topological term $\theta E\cdot B$. The CDW phason
mode is also known as the Goldstone mode. Applying $B$ collinearly to $E$
leads to an extra chiral flow of charges which can be detected from
magnetoelectric measurements.

Recently a gigantic negative longitudinal magnetoresistance (LMR) was observed
[8] in the CDW compound (TaSe4)2I when the magnetic field co-linear to the
electric field is applied parallel to the CDW direction. Such LMR appears only
in the state of collective CDW motion (sliding) at electric fields larger than
the characteristic threshold electric field, $E_{t}$. The authors concluded
that the LMR results from a chiral anomaly appearing in the Peierls state of
(TaSe4)2I and originating from the anomalous axionic contribution to the
phason current. This result was considered as the first determination of the
occurrence of axion electrodynamics in condensed matter [7, 9], reinforcing
the new concept of axionic charge density wave [10].

Oddly, and in contrast to the results reported in Ref. 8, the LMR measured by
Cohn et al. [11] was positive and does not exceeded a fraction of per cent in
the sliding as well as in the static regimes of the CDW in the temperature
range down to 30K. The main difference between both experiments is the sample
geometry and the position of the contacts: in Ref. 8 massive single crystals
with cross-section of order $10^{3}-10^{4}$ $\mu$m2 and the length between the
potential contacts 3-10 mm were used. While in Ref. 11 very thin samples in
the order 10 $\mu$m2 of cross-section and distances between the edges of the
contacts of order 102 $\mu$m.

In view of this strong disagreement we have repeated the measurements of the
non-linear state of (TaSe4)2I under magnetic field. To that end we have set
experimental conditions mostly identical to those in Ref. 8 using nearly the
same sample geometry, the same contact preparation method and the same
measurement setup. We have not found any significant contribution of the
magnetic field in the sliding motion of the CDW. We discuss the possible
reasons of the discrepancy including the generation of CDW dislocations under
a strong thermal gradient and we stress the role of the strong electron-phonon
coupling in (TaSe4)2I.

(TaSe4)2I is a chain-like material displaying a Peierls phase transition (for
a review on the Peierls transition and compounds, see Refs. 12, 13) at
T${}_{P}\approx$263K, this value varies rather strongly, that has been
intensively studied since several decades [15, 14]. (TaSe4)2I crystallizes
with tetragonal symmetry (space group I$422$) and consists of TaSe4 Van der
Waals chains parallel to $c$-axis separated by iodine atoms. In an TaSe4
infinite chain, each Ta is sandwiched by two nearly rectangular selenium units
forming a Se${}_{2}^{-2}$ pair. The dihedral angle between adjacent rectangles
is almost 45∘ (44∘ and 46∘), so the Ta atom is located at the center of a
rectangular anti-prism of eight Se atoms[14]. The interaction between Ta atoms
is through $d_{z^{2}}$ overlap along the chain (the shortest interchain
metal–metal distance is about 6.7Å to be compared to the intrachain average
value $d$ of about 3.2Å.

Band structure calculations [16] suggest a single $d_{z^{2}}$ electronic band
at the Fermi level. This band is 1/4 filled with one free electron per Ta+4
Ta5+ 4Se2- 2I- formula unit. Consecutive Ta atoms occupy two alternating non-
equivalent sites, but the Ta-Ta distance is unique, d${}_{Ta-Ta}=$3.206Å.
Owing to the Se4-unit rotation pattern, the crystallographic unit cell
parameter is $c=4d_{Ta-Ta}$. On that basis, one may expect (TaSe4)2 to be an
insulator, because the Fermi wave vector corresponds to a Brillouin zone
boundary: $k_{F}=1/4(\pi/d_{Ta-Ta})=\pi/c=c^{*}/2$. Assuming that Coulomb
repulsion favors an antiphase arrangement of CDWs on adjacent chains, one
expects the CDW satellites below $T_{P}$ to appear at or near $c^{*}+$ and
$-a^{*}$ or $c^{*}+$ and $-b^{*}$ close to the allowed Bragg reflections, $G$,
of the parent undistorted structure. Indeed electron and X-ray diffraction
experiments[17] below $T_{P}$ show the appearance of eight first-order
satellites around each room-temperature reflection at incommensurate positions
of the type $G+$ ($\pm\delta h$, $\pm\delta k$, $\pm\delta l$) close to each
main Bragg reflection $G=(H,K,L)$ with $\delta h=\delta k=0.045$ and $\delta
l=0.085$, effectively very near of the Brillouin zone center. These eight
satellites correspond to four degenerate domains with a single incommensurate
modulation, single $q$-domains with orthorhombic and identically stable
monoclinic symmetry [18].

The atomic displacements of the modulated CDW state consist of two parts: the
major one [17, 19] is a transverse acoustic-like wave (perpendicular to $q$)
with equal amplitudes of all atoms (0.13Å) and a new periodicity involving
mostly $z$-polarized Ta displacements that corresponds to a LLSS pattern of
long and short in-chain Ta-Ta distances (Ta-tetramerization modes) which
represents the CDW. The amplitude of the displacements of this tetramerization
is one order of magnitude smaller than that of the acoustic component, and was
only detectable by using X-ray anomalous scattering techniques [20]. Since the
electronic variables are unlikely to couple directly to such a long-wavelength
acoustic shear wave, a model has been proposed [19] in which the soft Ta-
tetramerization modes interact with the acoustic degrees of freedom and induce
the condensation of a mixed acoustic/optic ionic modulation. In this model,
the values of the satellite wave vector components are not related to the
topology of the Fermi surface (FS). They are determined by the strength of the
gradient interaction terms between optical and acoustic degrees of freedom.
Similarly to models developed in the context of incommensurate long-wavelength
modulated dielectrics such as quartz, the incommensurate structure arises from
the presence of a pseudo-Lifshitz invariant involving an optical order
parameter and the elastic deformations [19]. The presence and further
detection of the Ta-tetramerization [20] provides the key for understanding
the mechanism of the phase transition at $T_{P}$ which can be described as a
Brillouin zone center Peierls instability.

The FS of (TaSe4)2 has been mapped by angle-resolved photoelectron
spectroscopy (ARPES) [21]. It consists of parallel planes oriented
perpendicularly to the chain direction. The dispersion of the $d_{z^{2}}$ band
has also been recorded [22] at $T=300$ K for a range of wavevectors along the
1D chain direction. The band shows a strong dispersion throughout the first
Brillouin zone with minimum at $\Gamma$, the centre of the Brillouin zone,
also detectable with a weaker intensity in the second and third Brillouin
zones. The low intensity at $E_{F}$ for wave vectors close to $k_{F}$ is the
indication of a pseudogap still present above the Peierls transition
temperature. The temperature dependence of the optical conductivity exhibits a
strong absorption due to the CDW gap. That gap is already present in the
electronic excitation spectrum above the Peierls transition manifesting that,
even at high temperature free carriers are condensed into a fluctuating CDW
ground state. However, above $T_{P}$, as shown by neutron[19] or X-rays
scattering [17], the CDW does not exhibit long-range order, but has the
physical properties of a liquid. Below $T_{P}$ this “liquid” condensate
crystallizes into a superlattice structure. Optical spectra show a nearly
continuous change between 15 and 300K. The shape of the longitudinal optical
conductivity spectrum exhibits just a continuous narrowing through the Peierls
transition. Recent ARPES measurements are also in agreement with the gapless
nature of (TaSe4)2I at room temperature and reveal its characteristic
dispersion [23].

Single crystals of (TaSe4)2I were grown by direct combination of the elements
in sealed quartz tubes in a gradient furnace at a temperature ranging from
850-900 C with iodine as transport agent [15]. For the experiments we selected
single crystals with nearly the same size as those studied in Ref. 8. Contacts
for electrical transport measurements in the four-probe configuration have
been prepared using thin gold wires glued on the sample by silver paste. All
current-voltage characteristics (IVc) and resistance measurements were carried
out in the current mode. To exclude possible heating effects a part of the
sample was covered by highly heat-conducting epoxy as it was done in Ref. 8
and another part of sample was covered by apiezon grease as we usually do it
for such experiments. Magnetotransport measurements were performed at two
mutually perpendicular orientations of the magnetic field in the field range
up to 8 T using a superconducting solenoid.

Figure 1:  Logarithm of the electrical resistance, $R$, normalized to the
$R_{0}=R$ (300 K) (right axis, red) and its logarithmic derivative (left axis,
blue) as a function of the inverse temperature.

The temperature dependence of the three studied samples displayed the same
behavior: a strong monotonic grow of the resistance with decrease of
temperature (Fig. 1 right axis). A pronounced peak is observed in the
logarithmic derivative, $d[log(R/R_{0})]/d(1/T)$ versus the inverse
temperature (Fig. 1 left axis). The position of this peak determines the
transition into the CDW state [24, 25] the value of which is $T_{P}=248$ K in
our samples. It has been often argued that differences in $T_{P}$ values
reflect small deviations from stoechiometry (loss of iodine). In that case one
expects the change in the value of $T_{P}$ to be correlated with a change in
the conduction electron Fermi wavevector and consequently in the satellite
position. No such correlation was observed [19]. We shall come back to this
point below.

Measurements of IVc on the samples covered by epoxy demonstrated a strong
monotonic decrease of the resistance with increase of the current without
exhibiting a strong threshold behavior at the onset of the CDW sliding regime,
probably due to Joule heating. We only observed sharp threshold
characteristics for samples covered by apiezon. Fig. 2 (a) shows a set of such
characteristics for one of the samples in the temperature range 90-130 K. In
Fig. 2 (b) we show the temperature dependence of the Joule power, $P$,
dissipated at the threshold voltage (blue circles) and at twice the threshold
voltage (red squares). It can be seen that Joule power increases as the
temperature increases, qualitatively in the same way as it was reported in
Ref. 8 but with a difference of near three orders of magnitude.

Figure 2:  (a) Differential current-voltage characteristics, $dV/dI(V)$, in
one of the samples demonstrating the CDW collective motion in the temperature
range 90-130 K; (b) Joule power dissipated at the threshold electric field,
$P(E_{t})$, (blue circles) and at twice the threshold, $P(2E_{t})$, (red
squares) as a function of temperature.

The main experimental result of the present work is obtained from measurements
under magnetic field, $B$. Fig. 3 (a) shows differential IVc at T = 90 K at
$B=0$ T (green), at $B=7$ T (red) with the magnetic field aligned parallel to
the $c$-axis (chain direction) and collinear with the electric field, $E$, and
in the transverse orientation (blue) with $B$ and $E$ perpendicular. It can be
seen that the effect of magnetic field is negligibly small as it was also
found in Ref. 11. Moreover, the magnetoresistance is positive both for
transverse and longitudinal orientation of $B$. As an example Fig. 3 (b) shows
the result of the direct measurement of magnetoresistance,
$MR=\dfrac{R(B)-R(0)}{R(0)}$, swiping B from 0 to 7T in the case of $B$
parallel to $I$ with the applied current of $I=6\times 10^{-4}$ mA just
corresponding to the sliding state of the CDW. It can be seen that MR is
positive, very small (of order $10^{-3}$) and follows a quadratic dependence
on magnetic field.

Figure 3:  (a) Differential current-voltage characteristics, $dV/dI(V)$, at
$T=90$ K in magnetic field $B=0$ T (dark yellow symbols), magnetic field $B=7$
T parallel to the current (red symbols) and $B=7$ T perpendicular to the
current (blue symbols); (b) Magnetoresistance of the same (TaSe4)2I sample at
$T=90$ K in a magnetic field parallel to the current $I=1.2I_{t}$ where
$I_{t}$ is the threshold current for the CDW sliding.

Gooth et al. [8] have measured a longitudinal MR in the sliding state of
(TaSe4)2I which has been considered as the first experimental observation of
the axion dynamics in condensed matter. Our results, as well as those of Ref.
11, do not show any manifestation of magnetoresistance. In the following we
discuss some points that may serve to illustrate the difficulty to reconcile
the actual features of (TaSe4)2I with the Weyl semi-metal physics and the
axion electrodynamics.

(TaSe4)2I was characterized as a semi-metal which undergoes a phase transition
by breaking of the chiral symmetry between Weyl nodes (WP). From first
principles calculations (TaSe4)2I is described as a semi-metal [26] with
coupled WP by e-e interactions that yield the opening of a gap and the onset
of a CDW. It is found that the entire Fermi surface is formed from topological
bands connecting bulk chiral fermions. In the electronic structure 48 WPs
within 15 meV of $E_{F}$ are found.

The same authors performed X-ray diffraction experiments in their samples[26].
A set of 8 satellites, appearing at positions ($\pm\eta,\pm\eta,\pm\delta$),
are obtained at $\eta=0.027$, and $\delta=0.012$ and $T_{P}=248$ K. However
this CDW modulation wavevector is much shorter than the nesting vectors
between WPs. These nesting vectors can be approximated to the closest integer
number of the experimental CDW wave vector $h,h,k$ such as: $q_{WP}=mh,nk,ol$
with values of $m$, $n$ and $o$ ranging from 0-20 that yield the authors [26]
“to suggest that FWs nesting is not itself the origin of the CDW”. Thus the
suggested electronic CDW does not nest the Weyl points and therefore does not
break the chiral symmetry. The broken symmetry originates from e-ph coupling
between pieces of the Fermi surface separated by the $q$ vector. To end this
paragraph on the components of the CDW modulation wavevector, these values
contrast with those reported in Ref. 17, 19, 20: $\eta=0.045\pm 0.005$, and
$\delta=0.085\pm 0.005$ in the nominally pure compound. A smaller value agrees
with the results in Nb-doped samples [19].

More importantly, the authors describe (TaSe4)2I as an assembly of nearly
independent infinite (TaSe4) chains with a very weak interchain coupling. That
is not corroborated by elastic neutron scattering [19] which show that above
$T_{P}$ the inverse correlation lengths along the three directions $c^{*}$,
$a^{*}+b^{*}$ and $a^{*}-b^{*}$ are nearly isotropic with an anisotropy
parameter amounting to 1.8. 40K above $T_{P}$ the average number of correlated
chains was estimated at around 200. The strong interchain interactions of the
CDW, which are at variance with those typically found for other quasi-1D
conductors, is also manifested in the long wavelength acoustic character of
the atomic displacements, as has been explained above. This strong interaction
with the underlying lattice was lacking in this work [26], as well.

The possibility that the CDW can slide was suggested by Fröhlich [27] and
studied by Allender et al. [28] in a tight binding model. If the lattice wave
moves with the electrons with velocity $v_{s}$ the CDW order parameter will
vary as: $\Delta=\mid\Delta\mid exp(2ik_{F}(x-v_{s}t)$. In a 1D model the two
planes of the FS are at $(-\pi/k_{F})+q$ and $(+\pi/k_{F})+q$ with
$m^{*}v_{s}=\hbar q$ where $m^{*}$ is the effective electronic mass, that
yield in the static reference frame a current $J=nev_{s}$ where $n$ is the
number of electrons in the band affected by the CDW. The CDW velocity can be
measured from the periodic signal superposed to the static nonlinear current,
the so-called narrow band noise (NBN) or from the Shapiro steps structure when
the motion of the CDW is locked by an applied external rf field. Thus the
energy gap reduces the elastic scattering of individual electrons because
there in no state for energy relaxation. The properties of this nonlinear
state have been studied in detail for quasi-1D materials with different
crystallographic symmetries where chirality does not play any role. Note that
the $\hbar k_{F}$ term makes the difference in energies between the left- and
right-band sides of the displaced distribution.

In the case in which the ground CDW state is an axion insulator, in addition
to the Fröhlich Hamiltonian, a supplementary axion term (similarly to quantum
field theory) was added [8, 29] containing the contribution $\theta E\cdot B$
of the chiral anomaly in the undistorted Weyl semi-metal, with $\theta$ being
the dynamical axion field identified to the dynamics of the phase of the CDW.
There is thus an extra contribution to the phason current from this chiral
anomaly and an excess conductivity in the sliding state. In addition of the
electric force which, above the depinning electric field from impurities,
makes the CDW to slide, there is an extra force which increases the CDW
velocity and therefore the nonlinear current. This effect is similar to the
case where the velocity of a CDW in the sliding state is increased by the
application of an independent current on a small part of the sample[30]. This
result also illustrates the very long range coherence of the CDW motion.

Let us consider now the experimental results of Ref. 8. The non-linearity in
the IV characteristics have been essentially studied in one crystal (A). The
data do not exhibit a real sharp threshold for the onset of the CDW sliding,
but rather a continuous decrease of resistance. More surprisingly, this non-
linearity in the IV characteristics is only seen at $T<180$ K much below the
Peierls transition temperature when previous results show that the sliding
occurs immediately below $T_{P}$ with a temperature dependent $E_{t}$. When a
magnetic field is applied collinearly with the electric field a sharper
decrease of the resistivity in the non-linear state is observed at a very
delimited conditions: at a single temperature, namely $T=80$ K and under a
very large applied voltage. This effect is quantified by measuring the
magnetoconductivity as: $\Delta(dI/dV)_{B}=(dI/dV)_{B}-(dI/dV)_{0T}$. The most
disturbing result is the temperature dependence of $\Delta(dI/dV)_{B}$ in Fig.
3h in Ref. 8. It is null down to 120K and it sharply increases at 80K. The
data for samples A and B at $T=105$ K with $B$ and $E$ collinear shown in the
Extended Data Fig. 8 (a), (b) for sample A and (d), (e) for sample B are very
similar to our results shown in Fig. 3. Something else occurs between 105 K
and 80 K which is not explained in Ref. 8.

In our work (see Fig. 2), as well as in previously published studies [24, 25,
31] in (TaSe4)2I, the relative resistance drop $R(E<E_{t})/R(4E_{t})$ at
$T=80-120$ K is small, not more than a factor of 2. In contrast, in Ref. 8
this relation is more than 10 times larger at zero magnetic field and near 100
times larger at $B=9$ T. Gigantic heating effect in Ref. 8 which takes place
at high bias ought to be considered. Indeed, the authors have estimated Joule
heating only at threshold voltage determined as $V_{t}=6$ V at 80 K and
obtained for $P\approx 20$ $\mu$W that is really small enough. However, at
$V=27$ V (near $4E_{t}$) this power is already $P\approx 60$ mW that is more
than 3 order of magnitude larger. However at 105 K these values are in the
range of 200 mW. By comparison the power dissipated in our work at $T=90$ K is
$P=0.1$ $\mu$W at the threshold and $\approx 3$ $\mu$W at four times the
threshold. This is two order of magnitude smaller than in Ref. 8 at the same
condition and $T=80$ K. As a result, one may consider that the main
contribution to the resistance drop in Ref. 8 may originate as a consequence
of heating (as we discuss below) while the contribution of the CDW sliding to
the total drop of resistance may be less than 10 %. This is simply one order
of magnitude smaller than the magnetoresistance effect and reported as an
indication of axion transport in (TaSe4)2I.

Transport properties in quasi 1D metals have been studied under the assumption
that the single particle gap originates from the nesting of a single band at
+$k_{F}$ and -$k_{F}$. This mechanism, the Peierls instability, stands for the
large part of the CDW compounds. However a large difference occurs when the
CDW order opens a gap between Weyl fermions of opposite chirality. Unlike the
single-band CDW ground state, the involvement of multibands in Weyl-CDWs
compounds results in a gapless CDW collective mode that does not contribute to
the conductivity unless the Weyl cones are tilted [32]. Considering the
simplest two-band Weyl model in the undistorted phase, it consists of two
small Fermi pockets around each Weyl node, separated by the wave vector $q$
(that determined by x-ray scattering to be consistent with previous results).
At the CDW phase transition there is condensation of particle-hole pairs
consisting of a hole in one Fermi pocket and an electron separated by momentum
$q$. These electron and hole states exhibit the same velocity, unlike for a 1D
CDW. Thus, for the sliding mode to contribute to the DC conductivity in a
Weyl-CDW, it must have an asymmetry between the velocities in different Weyl
pockets. This can be accomplished by shifting the nodes in energy or inducing
a tilt to the Weyl cones. Is it the case for (TaSe4)2I ? We note however that
the sliding properties of (TaSe4)2I are very similar to those of other
materials exhibiting CDW sliding.

An important point from the experimental point of view is how Joule power
dissipated in the sample can lead to a strong anisotropic magnetoresistance
effect. Strong heating creates strong thermal inhomogeneities because the
thermal exchange can be different at different parts of the sample. Logically,
one assume that such exchange will be better from the bottom surface of the
sample which is in contact with the high thermal conductivity substrate
compared with the top surface of the sample. Such difference leads to a strong
thermal gradient from top to the bottom of the crystal which creates a
corresponding thermoelectric current. Being oriented transverse to the applied
electric field this current does not give a contribution to the total drop of
voltage. It will be no effect under the application of a magnetic field
parallel to this thermal current ($B\perp I$ configuration) because the
Lorentz force is zero for thermo-electrons. Another picture should be observed
when $B\parallel I$. In this case the thermoelectric current which
continuously increases with the increase of transport current is affected by
the Lorentz force that leads to an additional scattering and additional
heating correspondingly.

As said before, the CDW ground state has a very long range order. However the
CDW cannot solely be assumed to behave elastically when the CDW velocity
passes through strong and isolated sharp discontinuities. The conflict between
different winding rates at the interface separating regions with different
velocities is released by phase slippage. In a path surrounding these vortices
the phase changes by $2\pi$. The topological defects of ICDW are dislocations
very similar to vortices in superfluids and their properties are close of
those of dislocation lines in crystals. There are simple cases of
dislocations: screw dislocations parallel to the chains and edge dislocations
perpendicular to chains with a dislocation core with dimensions
$\xi_{p}\xi_{p}$ for an edge or $\xi_{p}\xi_{\parallel}$ for a screw where
$\xi_{p}$ and $\xi_{\parallel}$ are the BCS amplitude coherence lengths. The
CDW can be pinned at the surface, near strong impurities in the volume, at the
electrodes. It was also shown that the coherence of the CDW motion is broken
when a thermal gradient is applied [33, 34, 35] between electrodes with the
detection of two different velocities. As the CDW order parameter is
suppressed in the dislocation core, the properties of the Weyl semi-metal,
which are characterized by negative MR, could partially be restored. However,
in this scenario the amplitude of negative MR should be very small because
this effect takes place in a relatively small part of sample. However if the
properties of Weyl semi-metal are to be truly restored it should also be
accompanied by a significant (at least tens of per cent) positive transverse
MR that was not observed in Ref. 8.

In the context of the axion insulator Wang and Zhang [7] have identified the
CDW dislocations as axion strings. An axion string is a one-dimensional
dislocation of the axion field around which the axion field $\theta$ changes
by $2\pi$. Such chiral modes would carry a dissipationless current. It appears
from Ref. 8 that the magnetoconductivity is only observed when the Joule power
dissipated surpasses some threshold: 60mW at $T=80$ K, 200 mW at $T=105$ K and
no effect at higher temperature. Then one may think that at this so huge power
dissipated, in this system out of equilibrium, amplitude modes are excited and
“hot” filaments in which the CDW order parameter is suppressed can cross at
high velocity from one electrode to the other. Interaction between
dislocations in the lattice and dynamical axion strings have been also studied
[36].

In conclusion, the result of magneto-conductivity in the sliding state of
(TaSe4)2I has risen a very large interest since it provides the experimental
proof, which lacked, of the axion dynamics in condensed matter [8]. Here we
have raised some points which questions this statement. The symmetry breaking
in (TaSe4)2I results from electron-phonon interactions and not from electron-
electron ones. The distorted phase of (TaSe4)2I consists of the condensation
of a mixed acoustic/optic ionic modulation from the interaction of a Ta-
tetramerization with a transverse acoustic wave. The Peierls transition,
probably of the first order, is rather an order-disorder transition than a
displacive one. The new approach on (TaSe4)2I originates from the chirality of
the I$422$ space group in which it crystallizes. The 24 Weyl points derived
from first principle calculations are connected with $q$ vectors that are
irrelevant with the CDW $q$-vector determined by x-rays three decades ago. Our
results do not show any extra conductivity in the sliding state of (TaSe4)2I
when magnetic and electric fields are collinear. A detail analysis of the Ref.
8 magneto-conductivity data shows that it has only been detected abruptly
below 100 K and specifically at a single temperature, $T=80$ K (while the
phase transition is at $T=263$ K). And under a very large voltage which
induces a large Joule power dissipation. As a possible explanation of the
discrepancy between our results as well as those from Ref. 11 we have
suggested possible hot filaments with cores having the properties of the
normal state.

We do not disregard all the new fascinating theoretical and experimental
discoveries around these Weyl semimetals. We however think that, on the basis
of present magneto-conductivity results, the assertion that (TaSe4)2I would be
an axionic charge density wave is not justified and this conclusion should be
considered as, at least, premature. New transport experiments are necessary
for the temperature dependence of the axionic magnetoresistance in a large
temperature scale and under different thermal gradients. Measurements of
Shapiro steps in the sliding state will also allow to determine if the
application of a collinear magnetic field increases the CDW velocity and
therefore if the axion field acts on the CDW long range condensate.

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

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