# On the connection between magnetic interactions and the spin-wave gap of the
insulating phase of NaOsO3

Nikolaos Ntallis  Department of Physics and Astronomy, Uppsala University,
Uppsala 751 20, Sweden  Vladislav Borisov  Department of Physics and
Astronomy, Uppsala University, Uppsala 751 20, Sweden  Yaroslav O. Kvashnin
Department of Physics and Astronomy, Uppsala University, Uppsala 751 20,
Sweden  Danny Thonig  School of Science and Technology, Örebro University,
SE-701 82, Örebro, Sweden  Erik Sjöqvist  Department of Physics and Astronomy,
Uppsala University, Uppsala 751 20, Sweden  Anders Bergman  Department of
Physics and Astronomy, Uppsala University, Uppsala 751 20, Sweden  Anna Delin
Department of Applied Physics, School of Engineering Sciences, KTH Royal
Institute of Technology, AlbaNova University Center, SE-10691 Stockholm,
Sweden  Swedish e-Science Research Center (SeRC), KTH Royal Institute of
Technology, SE-10044 Stockholm, Sweden  Olle Eriksson  Department of Physics
and Astronomy, Uppsala University, Uppsala 751 20, Sweden  School of Science
and Technology, Örebro University, SE-701 82, Örebro, Sweden  Manuel Pereiro
<EMAIL_ADDRESS>Department of Physics and Astronomy, Uppsala
University, Uppsala 751 20, Sweden

###### Abstract

The scenario of a metal-insulator transition driven by the onset of
antiferromagnetic order in NaOsO3 calls for a trustworthy derivation of the
underlying effective spin Hamiltonian. To determine the latter we rely on ab
initio electronic-structure calculations, linear spin-wave theory, and
comparison to experimental data of the corresponding magnon spectrum. We
arrive this way to Heisenberg couplings that are $\lesssim$45% to$\lesssim$63%
smaller than values presently proposed in the literature and Dzyaloshinskii-
Moriya interactions in the region of 15% of the Heisenberg exchange $J$. These
couplings together with the symmetric anisotropic exchange interaction and
single-ion magnetocrystalline anisotropy successfully reproduce the magnon
dispersion obtained by resonant inelastic X-ray scattering measurements. In
particular, the spin-wave gap fully agrees with the measured one. We find that
the spin-wave gap is defined from a subtle interplay between the single-ion
anisotropy, the Dzyaloshinskii-Moriya exchange and the symmetric anisotropic
exchange interactions. The results reported here underpin the local-moment
description of NaOsO3, when it comes to analyzing the magnetic excitation
spectra. Interestingly, this comes about from a microscopic theory that
describes the electron system as Bloch states, adjusted to a mean-field
solution to Hubbard-like interactions.

In the context of correlated electronic systems, 5$d$ oxides hold a distinct
position, realizing the peculiar regime where Coulomb interactions (the
Hubbard $U$ in particular), the $d$–$d$ hopping matrix elements (referred to
as $t$’s), and the spin-orbit (SO) coupling ($\lambda$) have similar
magnitudes ($\lambda\\!\sim\\!t\\!\sim\\!U$). In contrast, in 3$d$ and 4$f$
based magnets, it is possible to identify a smallest energy scale, since the
SO coupling defines a much smaller energy scale in 3$d$ transition-metal
compounds ($\lambda\\!\ll\\!t\\!<\\!U$), and the 4$f$–4$f$ hopping is minute
in rare-earth systems ($t\\!\ll\\!\lambda\\!<\\!U$).

The SO coupling in iridates and osmates, for example, is of order
$0.5-0.6\,\mathrm{eV}$. In addition, the 5$d$ orbitals, being much more
extended than, e.g., the rare-earth 4$f$ states, causes the 5$d$ electrons on
neighboring ions to interact much more effectively. In other words, reasonably
well localized magnetic moments can still be in place but ligand-mediated
superexchange is greatly enhanced as compared to in, e.g., 4$f$ insulators.
Combined with the strong SO coupling, it may generate highly anisotropic
intersite magnetic interactions, with the remarkable situation encountered in
$t_{2g}^{5}$ iridates of having either antisymmetric Dzyaloshinskii-Moriya
(DM) Jackeli_2009 ; yadav  or symmetric Kitaev Jackeli_2009 ; Yamali_2014 ;
Vamshi_NJP  effective coupling parameters that are even larger than the
isotropic Heisenberg constant.

Since $t\\!\sim\\!U$, metal-insulator transitions (MIT’s) may also occur.
Extensively discussed in this respect is the osmium oxide perovskite compound
NaOsO3. However, rather than a Mott MIT, the scenario of a Slater MIT has been
proposed for NaOsO3 Shi_2009 ; Calder_2012 ; Du_2012 ; Jung_2013 ;
Vecchio_2013 , where the formation of antiferromagnetic (AF) order is opening
the insulating gap.

In this frame of reference, detailed knowledge of the underlying magnetic
interactions is crucial. Here we shed light on this matter by means of ab
initio calculations and subsequent atomistic spin-dynamics simulations
employing the derived effective magnetic interactions. The computed magnon
spectra are compared with experimental data as obtained by resonant inelastic
X-ray spectroscopy (RIXS) Calder_2017 . We calculate from ab initio density
functional theory (DFT) a Heisenberg exchange of $\sim 4-6\,\mathrm{meV}$,
that can be easily reconciled with the bandwidth of the magnon spectrum;
$\sim$ 80 meV, reported on the basis of RIXS measurementsCalder_2017 .
However, our analysis shows that a single-ion anisotropy (SIA), the DM
interaction, Heisenberg exchange as well as the symmetric anisotropic exchange
interaction, are large, and necessary for theory to reproduce the magnon
spectrum of NaOsO3.

From the structural point of view, NaOsO3 has the octahedral environment of
Os5+O6, so that the electronic configuration is 5d3, suggesting that the
$t_{2g}$-band is half filled. Moreover, this compound shows Curie-Weiss
metallic nature and abruptly but continuously turns to an AF insulator at
$410\,\mathrm{K}$ Shi_2009 . The structural, electronic and magnetic
properties of bulk NaOsO3 are studied theoretically using density functional
theory (DFT)Hohenberg1964 ; Kohn1965  within the generalized-gradient
approximation in the PBE parameterizationPBE1996 . Electronic properties are
calculated within the all-electron full-potential fully relativistic approach,
with linear muffin-tin orbitals as basis functions in both LDA+U and LSDA+U
approximations, as implemented in the RSPt electronic structure code Wills1987
; Wills2000 ; Wills2010 , with the U values ranged from 0 to $4\,\mathrm{eV}$.

Figure 1(a) shows the experimental crystal structure of NaOsO3. The cell
dimensions are a=$5.3842$ Å, b=$5.3282$ Å and c=$7.5804$ Å. The magnetic
ground state for all used values of U has a G-type antiferromagnetic (AF)
ordering, as shown in Fig. 1(a). In Fig. 2, the spin-, orbital- and total
magnetic moment per Os atom is plotted with respect to the U value for the
different approximations, i.e. LDA+U and LSDA+U. The two approaches produce
different values of the magnetic moments as function of U. Notably, the LDA+U
approximation would produce a zero magnetic moment for Os atoms by decreasing
U down to 2 eV, according to Fig. 2. This result is natural, since all
exchange splitting in this level of approximation is in the static Hubbard
type interaction, while the spin-density functional by construction provides
no exchange splitting. In the limit of finite U, where both of the
approximations produce almost the same orbital moment, the difference on the
spin part is close to $0.35\,\mathrm{\mu_{B}}$ per atom. Also, one can
reproduce the experimental values of the Os moment ($\sim$ 1 $\mu_{B}$) for
both approximations, albeit for different values of U.

a)

b)

Figure 1:  a) $Pnma$ crystal structure of NaOsO3. Red and yellow spheres
indicate O and Na ions, respectively. b) $G$-type AF order within the Os
sublattice. The NN exchange paths (within the $ac$ plane and along the $b$
axis) are highlighted. Figure 2:  Calculated total (solid symbols) and spin
(unfilled symbols) magnetic moments in a) together with orbital moments
(opposite to the spin moments) per atom of NaOsO3 in b). The moments are
plotted for different values of U and for LSDA+U and LDA+U implementations
with fixed J${}_{H}=0.6$ eV. The dotted line represents the magnetic moment
measured in Ref. Calder_2012 .

Different types of magnetic interactions, such as the Heisenberg and DM
interaction, were evaluated for the G-type antiferromagnetic reference state
based on the fully-relativistic generalization Kvashnin2020  of the
Lichtenstein-Katsnelson-Antropov-Gubanov (LKAG) formula LKAG1987 , which we
have recently applied to different correlated systems Borisov2020 . The
purpose of this is to map the electronic system onto a generalized classical
Heisenberg model:

$H=-\sum\limits_{i\neq
j}e_{i}^{\alpha}\hat{J}_{ij}^{\alpha\beta}e_{j}^{\beta},\hskip
10.0pt\alpha,\beta=x,y,z,$ (1)

where the unit vectors $\mathbf{e}_{i}$ indicate the direction of local spins
and the fully-relativistic exchange tensor $J_{ij}^{\alpha\beta}$ contains
contributions from the Heisenberg exchange as well as the DM interaction
$\hat{D}$ and the symmetric anisotropic exchange $\hat{\Gamma}$ defined by

$\displaystyle D_{ij}^{z}$
$\displaystyle=\frac{1}{2}(J_{ij}^{xy}-J_{ij}^{yx})$ (2)
$\displaystyle\Gamma_{ij}^{z}$
$\displaystyle=\frac{1}{2}(J_{ij}^{xy}+J_{ij}^{yx})$ (3)

and similar expressions for the $x$\- and $y$-components.

According to Fig. 2, the values of U which produce total magnetic moments
closest to the experimental ones are U $\approx 0.8\,\mathrm{eV}$ for LSDA+U,
similarly to Middey2014 ; Mohapatra2018 ; Liu2020 , and U $\approx
3.5\,\mathrm{eV}$ for LDA+U. Figure 3 shows the calculated magnetic
interaction strengths with respect to the atomic distance. All interaction
strengths decay quite fast with respect to distance indicating that the major
interactions to be considered are the first and second order ones. The LDA+U
approximation (with U = $3.5\,\mathrm{eV}$) produces Heisenberg exchange
interaction that is roughly twice the magnitude compared to the LSDA+U data
(with U = $0.8\,\mathrm{eV}$). Similarly the LDA+U calculation results in a
value of $D$ that is approximately three or four times larger than the one for
LSDA+U approximation. In order to assess the quality of the exchange
interactions we used them to evaluate adiabatic magnon spectra, that can be
compared to experimental results (see discussion below). We note, however,
already here that such an analysis reproduces experimental results quite
poorly from a theory based on LDA+U (see supplementary information (SM) for
further details on the LDA+U results)supmat , while LSDA+U calculations give
reasonable results. For this reason we focus in the rest of this paper on
results from LSDA+U calculations.

Figure 3:  Calculated interaction strengths $J+D+\Gamma$ with respect to the
atomic distance for U=0.8 eV, J${}_{H}=0.6$ eV (top panel) and U=1.0 eV,
J${}_{H}=0.6$ eV (bottom panel). All interaction are marked following the
prescription $1\rightarrow 3$ given in Fig. 1.  Figure 4:  a) Calculated DM
vectors $\mathbf{D}$ for the nearest neighbor bonds in NaOsO3 shown by the
orange ($\mathbf{D}_{1}$), green ($\mathbf{D}^{a}_{3}$) and magenta
($\mathbf{D}^{b}_{3}$) arrows. The $\mathbf{D}^{a,b}_{3}$ vectors are
magnified by the factor of 2.5 for clarity. The G-type antiferromagnetic
ordering of Os magnetic moments $\mathbf{m}$ oriented along the $y$-direction
is depicted by the red and blue arrows. The magnetic interactions were
obtained using LSDA+$U$ with $U=0.8\,\mathrm{eV}$ and
$J_{H}=0.6\,\mathrm{eV}$. b) The nearest-neighbor Heisenberg exchange $J_{1}$
along the [100] and [001] directions and the $y$-component of the DM
interaction $\mathbf{D}_{1}$ in the ab plane as functions of the correlation
strength $U$.

Despite the same magnetic pattern along the $x$ and $z$ directions and very
similar distances ($d_{x}=3.787\,\mathrm{A}$ and $d_{z}=3.790\,\mathrm{A}$),
the Heisenberg exchange interactions $J_{1x}$ and $J_{1z}$ are markedly
different. This could be related to the large effect of magnetic anisotropy
which aligns the magnetic moments and the Néel vector preferably along the $y$
direction. It is interesting that the $J_{1z}/J_{1x}$ ratio increases as a
function of the correlation strength U from 0.62 at
$\mathrm{U}=0.8\,\mathrm{eV}$ to 0.85 at $\mathrm{U}=4\,\mathrm{eV}$. This may
suggest that the system becomes magnetically more isotropic for stronger
correlations where the system is more insulating, based on the calculated
value of the electronic band gap. We also notice that the scaled Heisenberg
parameters $J/m_{s}^{2}$ are far from the $\nicefrac{{1}}{{U}}$ scaling
expected for insulators. This can be explained by the fact that correlations
in the narrow-gap insulator NaOsO3 are not strong compared to the electron
hopping amplitude $t$, so that $\nicefrac{{t}}{{U}}$ is not a small parameter
and the perturbation theory, which could lead to the $\nicefrac{{1}}{{U}}$
scaling, is not applicable. Interestingly, the nearest-neighbor parameters
$J_{1}/m_{s}^{2}$ increase as functions of U up to
$\mathrm{U}=3.0-3.5\,\mathrm{eV}$ which deserves a more detailed study in the
future. On the other hand, the DM interaction shows a non-monotonous variation
of its $y$-component, which is dominating for the bonds along the
$xy$-directions, with a maximum around $\mathrm{U}=1\,\mathrm{eV}$ followed by
a decreasing trend for larger U values.

The numerical values of the exchange interactions are collected in Table 1 for
LSDA+U calculations, while the DM vectors for the nearest-neighbor bonds are
shown in Fig. 4. These interactions were used to calculate the magnon spectra,
which was evaluated by rewriting Eq. (1), in the form of a bilinear, effective
Hamiltonian $\mathcal{H}_{i,j}$ containing the isotropic Heisenberg, the DM as
well as symmetric anisotropic exchange. Adding single-site magnetic anisotropy
we end up with the following expression for the full Hamiltonian of the spin
system:

$\displaystyle\mathcal{H}^{i,j}_{\mathrm{mod}}$ $\displaystyle=$
$\displaystyle-
J_{ij}\mathbf{{S}}_{i}\cdot\mathbf{{S}}_{j}-\mathbf{D}_{ij}\cdot\mathbf{{S}}_{i}\times\mathbf{{S}}_{j}-\mathbf{{S}}_{i}\Gamma_{ij}\mathbf{{S}}_{j}$
$\displaystyle-$ $\displaystyle
K\sum_{k=i,j}\left(\mathbf{S}_{k}\cdot\mathbf{e}^{\mathrm{Y}}_{k}\right)^{2},$

where $\mathbf{{S}}_{i}$, $\mathbf{{S}}_{j}$ are normalized atomic spin
moments. The isotropic Heisenberg coupling is given by $J_{ij}$ and
$\mathbf{D}_{ij}$ stands for the DM vector, while the tensor $\Gamma_{ij}$
represents the symmetric anisotropic exchange interaction between atomic sites
$i$ and $j$. The Heisenberg exchange interaction $J_{ij}$, the norm of
$\mathbf{D}_{ij}$ and mean value of $\Gamma_{ij}$ tensor take the values
$J_{1}$, $\left|\mathbf{D}_{1}\right|$, $\Gamma_{1}$ and $J_{2}$,
$\left|\mathbf{D}_{2}\right|$, $\Gamma_{2}$ for bonds 1 and 2 (cf. Fig. 1(b)),
respectively. The orientation of the DM vectors depend on the position in the
unit cell, see Fig. 4. The DM vectors $\mathbf{D}^{a}_{3}$ and
$\mathbf{D}^{b}_{3}$ for the third-neighbours atoms are slightly different
(cf. Table 1 and Fig. 4) because the cell dimensions in the ab plane are also
slightly different (a$\neq$b). As regards the magnetic anisotropy, each site
possesses an easy-axis orientation directed along the local magnetic y-axis
$\mathbf{e}^{\mathrm{Y}}_{k}$. The magnetic anisotropy parameter ($K$ in Eq.
(On the connection between magnetic interactions and the spin-wave gap of the
insulating phase of NaOsO3)) was calculated based on the sum of eigenvalues in
the one-shot relativistic calculations using the RSPt code Wills1987 ;
Wills2000 ; Wills2010  where the magnetization axis is oriented along the
$x$-, $y$\- and $z$-directions. The obtained estimates are
$K=2.05\,\mathrm{meV}$ for U=$0.8\,\mathrm{eV}$ and $J_{H}=0.6\,\mathrm{eV}$
while the calculation with U=$1.0\,\mathrm{eV}$ and $J_{H}=0.6\,\mathrm{eV}$
resulted in $K=2.44\,\mathrm{meV}$.

Table 1: Intersite interaction vectors, effective couplings, and DM vectors for different Os-O-Os bond angles. The units of the coupling parameters are given in meV. The reference system is specified in Fig. 1(b). $\theta_{1}=155.2^{\circ}$ | $U=0.8\,\mathrm{eV}$ | $U=1.0\,\mathrm{eV}$
---|---|---
$J_{1}$ | -6.64 | -7.58
$\Gamma_{1}$ | 0.62 | 0.57
$\mathbf{r}_{ij}$ | $\left(\pm 0.5,\pm 0.5,0\right)$ | $\left(\pm 0.5,\pm 0.5,0\right)$
$\mathbf{D}_{1}$ | $\pm\left(0.04,-0.92,-0.37\right)$ | $\pm\left(0.08,-0.92,-0.40\right)$
$\theta_{2}=153.9^{\circ}$ |  |
$J_{2}$ | -4.12 | -5.11
$\Gamma_{2}$ | 0.07 | 0.05
$\mathbf{r}_{ij}$ | $\left(0.0,0.0,\pm 0.5\right)$ | $\left(0.0,0.0,\pm 0.5\right)$
$\mathbf{D}_{2}$ | $\left(-0.50,0.15,0.00\right)$ | $\left(-0.41,0.07,0.00\right)$
$\theta_{3}$=$\emptyset$ |  |
$J_{3}$ | 0.67 | 0.68
$\Gamma_{3}$ | 0.18 | 0.18
$\mathbf{r}_{ij}$ | $\left(\pm 0.5,\pm 0.5,\pm 0.5\right)$ | $\left(\pm 0.5,\pm 0.5,\pm 0.5\right)$
$\mathbf{D}^{a}_{3}$ | $\pm\left(0.20,-0.01,-0.18\right)$ | $\pm\left(0.20,-0.03,-0.18\right)$
$\mathbf{D}^{b}_{3}$ | $\pm\left(-0.20,0.01,-0.11\right)$ | $\pm\left(0.20,0.00,-0.10\right)$
Figure 5:  Adiabatic magnon spectrum of NaOsO3 by progressively adding energy
contributions to the effective spin Hamiltonian. We illustrate the cases with
contributions given by $J$, $J\\!+\\!\mathbf{D}+\\!\mathbf{\Gamma}$,
$J\\!+\\!\mathbf{\Gamma}+\\!K$, $J\\!+\\!\mathbf{D}+\\!\mathbf{\Gamma}+\\!K$,
for U=0.8 eV, J${}_{H}=0.6$ eV from LSDA+U theory (top panel) and U=1.0 eV,
J${}_{H}=0.6$ eV from LSDA+U theory (bottom panel). Experimental data provided
by RIXS measurements are shown in black dots Calder_2017 .

In order to validate the parameters calculated by ab initio electronic
structure theory, we performed calculations of magnon dispersions, based on a
collinear AF ground state, in the framework of linear spin-wave theory (LSWT)
yadav  by using the UppASD software asd . Effective coupling parameters as
collected in Table 1 were utilized, along with a moment as illustrated in Fig.
2a), from LSDA+U approximation using U=0.8-1.0 eV and JH= 0.6 eV.

In order to understand which interactions dominate the spin-wave gap of
NaOsO3, we calculated adiabatic magnon spectra by progressively adding
different energy terms as present in Eq. (On the connection between magnetic
interactions and the spin-wave gap of the insulating phase of NaOsO3), see
Fig. 7. It is evident from the figure that the uniaxial single ion anisotropy
is the term responsible for the spin-wave gap and the value of U that best fit
the experimental spin-wave gap is achieved by U=1.0 eV while for U=0.8 eV, the
gap is slightly too low (the magnon excitations are evaluated for a wider
range of values of U in the SMsupmat ). Furthermore, the contribution of the
DM interaction together with the symmetric anisotropic exchange interaction
even though they are small, they are necessary to correctly describe the full
experimental spin-wave dispersion. Interestingly, the calculated profile of
the magnon spectra for U=1 eV agrees quite well with the experimental spectra,
even though the exchange coupling parameters together with the anisotropy
constant differ substantially with respect to the ones suggested from the
experimental data in Ref. Calder_2017 . A direct fitting procedure of the
magnon spectra, with a relative large number of parameters in a spin-
Hamiltonian, can agree with experimental data but the parameters of the
Hamiltonian obtained from such fittings, might be completely off compared to
interactions calculated from ab initio electronic structure theory. This
illustrates that ab initio theory is crucial for extracting proper interaction
types, strengths and ranges.

Figure 6:  Dynamical structure factor of NaOsO3 for a) U=0.8 eV, J${}_{H}=0.6$
eV and at T=5K, b) for U=0.8 eV, J${}_{H}=0.6$ eV and at T=300K, c) for U=1.0
eV, J${}_{H}=0.6$ eV and at T=5K and d) U=1.0 eV, J${}_{H}=0.6$ eV and at
T=300K. All the aforementioned parameters are calculated at the level of
LSDA+U theory. Energy is convoluted with a 0.7 meV full width at half maximum
Gaussian. Experimental data provided by RIXS measurements are shown by green
dots Calder_2017 .

The calculated adiabatic magnon excitation energies are in good agreement with
the experimental data reported in Ref. Calder_2017 , as shown in Fig. 7.
However, AMS is formally calculated at T$=0$K, whereas the experimental
measurements are done at room temperature, i.e. T$=300$K. Therefore theory and
experiments refer to data obtained at different temperatures. In order to shed
more light on this point, we show in Fig. 6 the dynamical structure factor
calculated in the framework of LSWT, which is proportional to the differential
cross section measured in scattering experiments. It depends on the crystal
momentum ($\bf q$) and energy transfer ($\omega$) and is defined as a
3$\times$3 matrix:

$S({\bf q},\omega)=\frac{1}{2\pi N}\sum_{i,j}e^{i{\bf q}({\bf r}_{i}-{\bf
r}_{j})}\int_{-\infty}^{\infty}e^{-i\omega t}\langle{\bf S}_{i}(t){\bf
S}^{T}_{j}(t)\rangle dt\,,$ (5)

where indices $i,j$ label the atomic positions and ${\bf r}_{i}$ is the
position vector of atom $i$. $N$ is the number of magnetic atoms in the unit
cell and ${\bf S}_{i}$ represents a column vector of the $x$, $y$, and $z$
spin operator components. The symbol $\langle...\rangle$ denotes the ensemble
average in thermal equilibrium with the environment at temperature $T$. Since
we are dealing with magnons, i. e., bosons, $T$ is introduced in the
thermodynamic average by using Bose-Einstein statistics, with a Bose factor

$n(\omega)=\frac{1}{e^{\hbar\omega/k_{B}T}-1}\,.$ (6)

In Fig. 6(a),(c), we show $S({\bf q},\omega)$ for NaOsO3 using magnetic
interactions according to Eq. (On the connection between magnetic interactions
and the spin-wave gap of the insulating phase of NaOsO3) and Table 1 at
$T\\!=\\!5$ K and in Fig. 6(b),(d) the dynamical structure factor is
calculated for $T\\!=\\!300$ K, the value of the temperature at which the
experimental measurements have been reported in Ref. Calder_2017 . The
dispersion of the curve in Fig. 6 is naturally the same as in Fig. 7 but
additional information is obtained from the intensity, which reveals how
likely a scattering event is. Notably, for $T\\!=\\!5$ K only in the vicinity
of the $\Gamma$ point does the cross section show a significant intensity
indicating that at very low temperatures only these specific modes would be
captured. On the other hand, at a temperature of $T\\!=\\!300$ K, the two
lowest energy branches show a considerable intensity along the $\bf q$-path in
agreement with experimental data. Note here that according to Fig. 7, the
first acoustic and first optical branches are very close in energy and thus
both of them are activated by increasing temperature. Figure 6 also reveals
that at 300 K, the experimental conditions of the RIXS measurements, the
experimental apparatus only can capture the acoustic and first optical
branches while the additional third and fourth optical branches remain hidden
to the experiment.

In summary, the very rare situation in which ab initio calculations of
Heisenberg exchange results in $\lesssim$45% to $\lesssim$63% smaller values
compared to those obtained through fitting of experimental magnon dispersions
by LSWT, has led us to a careful analysis of the spin excitation spectrum of
the osmate perovskite NaOsO3. To this end, we carried out atomistic spin-
dynamics simulations using as input the effective magnetic interactions
determined from ab initio calculations. These interactions involved not only
Heisenberg but also DM and symmetric anisotropic intersite couplings, plus
single-ion spin anisotropies. The effective DM interaction parameters turn out
to be sizable, $\sim$14% of the Heisenberg $J_{1}$’s for U=$1\,\mathrm{eV}$,
while the single-ion anisotropies are also robust. Our theoretical results
reproduce with good accuracy the observations, albeit with a distinct
difference in that the spectra along the X $\rightarrow\Gamma\rightarrow$ Z
path are slightly steeper than the measured spectra. Moreover, the third and
fourth optical branches demonstrated in the theory, are not found in the
experimental measurements. We propose a mechanism based on thermal
fluctuations for why these two modes are hidden from experimental detection.
The analysis put forth here differs from that of Ref. Calder_2017  in that our
calculations provide somewhat smaller magnetic anisotropy, but notably, a
sizable DM exchange which was not suggested in Ref. Calder_2017 . We note that
the DM interaction can on general grounds not be neglected in NaOsO3 since the
O ion mediating superexchange between two Os NN’s does not sit at an inversion
center and SO interactions are large for the 5$d$ shell. Hence, one should
anticipate that the DM interaction is large, which our theoretical
calculations confirm. We have demonstrated that the single ion anisotropy
along with DM interaction and to a less extent the symmetric anisotropic
exchange interaction, clearly has a role in the spin-wave gap of NaOsO3. The
contribution of the DM and $\Gamma$ interaction as the microscopic mechanism
behind a spin-wave gap is seldom discussed, and NaOsO3 is a unique material in
this sense.

While our results for the magnon modes faithfully reproduce the dispersion
reported on the basis of RIXS experiments Calder_2017 , we also point out that
in the regime of observed magnons, the optical and acoustic modes are
sometimes close in energy and in order to identify which mode is observed, it
is important to theoretically analyse the scattering amplitudes. Furthermore,
we find that a full account of all interaction parameters are needed in order
to reproduce experimental observations of this complex compound.

Finally, we note that it was argued in Ref. Calder_2017  that NaOsO3 is a
system on the boundary between local-moment and itinerant magnetism, based on
the fact that the SOC is very strong with a spin gap of $58\,\mathrm{meV}$
around the $\Gamma$ point. Our results can explain experimental magnon spectra
by assuming an effective on-site Coulomb repulsion of $U_{\rm
eff}=U-J_{H}=0.4\,\mathrm{eV}$ which drives the system out of the purely
itinerant regime.

Acknowledgements. The authors acknowledge financial support from Knut and
Alice Wallenberg Foundation through Grant No. 2018.0060. A.D. acknowledges
financial support from the Swedish Research Council (VR) through Grants No.
2015-04608, No. 2016-05980, and No. 2019-05304. O.E. also acknowledges support
from eSSENCE, SNIC, the Swedish Research Council (VR) and the ERC (synergy
grant FASTCORR, project 854843). D.T. acknowledges support from the Swedish
Research Council (VR) through Grant No. 2019-03666. E.S. acknowledges
financial support from the Swedish Research Council (VR) through Grant No.
2017-03832. The work of Y.O.K. is supported by VR under the project No.
2019-03569. Some of the computations were performed on resources provided by
the Swedish National Infrastructure for Computing (SNIC) at the National
Supercomputer Center (NSC), Linköping University, the PDC Centre for High
Performance Computing (PDC-HPC), KTH, partially funded by the Swedish Research
Council through grant agreement no. 2016-07213 and the High Performance
Computing Center North (HPC2N), Umeå University.

##  I Supplementary Information

Table 2: Calculated values of $J_{1}$, $D_{1}$ and $\Gamma_{1}$, for LDA+U approximation. Only values of $U$ and $J_{H}$ (in eV) are shown that produce a magnetic moment per Os atom close to the experimental value. All values of couplings are in meV. $U$ | $J_{H}$ | $J_{1}$ | $D_{1}$ | $\Gamma_{1}$
---|---|---|---|---
$3.5$ | $0.6$ | $-14.42$ | $-1.67$ | $0.54$
$4.0$ | $0.6$ | $-16.87$ | $-1.67$ | $0.44$
Figure 7:  Adiabatic magnon spectrum of NaOsO3 for LDA+U approximation. Only
U=$3.5$ eV, $J_{H}=0.6$ eV and U=$4.0$ eV, $J_{H}=0.6$ eV are shown as they
produce a magnetic moment per atom close to the experimental one. Experimental
data provided by RIXS measurements are shown in black dots Calder_2017 .

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