SuppSupplementary References

# Signatures of a liquid-crystal transition in spin-wave excitations of
skyrmions

Narayan Mohanta∗ Materials Science and Technology Division, Oak Ridge National
Laboratory, Oak Ridge, TN 37831, USA  Andrew D. Christianson  Materials
Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN
37831, USA  Satoshi Okamoto  Materials Science and Technology Division, Oak
Ridge National Laboratory, Oak Ridge, TN 37831, USA  Elbio Dagotto  Materials
Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN
37831, USA  Department of Physics and Astronomy, The University of Tennessee,
Knoxville, TN 37996, USA
∗Email<EMAIL_ADDRESS>
###### Abstract

Abstract: Understanding the spin-wave excitations of chiral magnetic order,
such as the skyrmion crystal (SkX), is of fundamental interest to confirm such
exotic magnetic order. The SkX is realized by competing Dzyaloshinskii-Moriya
and ferromagnetic-exchange interactions with a magnetic field or anisotropy.
Here we compute the dynamical spin structure factor, using Monte Carlo and
spin dynamics simulations, extracting the spin-wave spectrum in the SkX, in
the vicinity of the paramagnet to SkX transition. Inside the SkX, we find six
spin-wave modes, which are supplemented by another mode originating from the
ferromagnetic background. Above the critical temperature $T_{s}$ for the
skyrmion crystallization, we find a diffusive regime, reminiscent of the
liquid-to-crystal transition, revealing that topological spin texture of
skyrmionic character starts to develop above $T_{s}$ as the precursor of the
SkX. We discuss the opportunities for the detection of the spin waves of the
SkX using inelastic-neutron-scattering experiments in manganite-iridate
heterostructures.

Introduction
Chiral magnetic order occurring in several condensed-matter systems has added
a new dimension to the realization of novel functionalities and exotic
fundamental physics Barron (2008); Fert  _et al._ (2017). Magnetic skyrmions,
the vortex-like spin swirls, belong to the class of chiral magnetic textures
having a quantized topological charge Nagaosa and Tokura (2013). In the
presence of competing spin exchange interactions, the skyrmions often arrange
themselves naturally in a triangular array to form a skyrmion crystal (SkX) in
real lattices Ezawa (2011); Buhrandt and Fritz (2013); Rohart and Thiaville
(2013); Ozawa  _et al._ (2017). The existence of the SkX phase has been
established in a surge of recent experiments by Lorentz transmission electron
microscopy, spin-resolved scanning tunneling microscopy, and elastic small-
angle neutron scattering (SANS) Rößler  _et al._ (2006); Mühlbauer  _et al._
(2009); Yu  _et al._ (2010, 2011); Heinze  _et al._ (2011); Yu  _et al._
(2014); Park  _et al._ (2014); Woo  _et al._ (2016); Hervé  _et al._ (2018);
Mühlbauer  _et al._ (2019). A magnetic skyrmion generates an emergent electro-
magnetic field Schulz  _et al._ (2012); Nagaosa  _et al._ (2012), giving rise
to topological Hall effect in metals. This effect also serves as an indirect
yet effective probe for the chiral magnetic order in the SkX phase Neubauer
_et al._ (2009); Kanazawa  _et al._ (2011); Taguchi  _et al._ (2001); Huang
and Chien (2012); Rana  _et al._ (2016); Wang  _et al._ (2018a); Hamamoto  _et
al._ (2015); Yi  _et al._ (2009); Nakamura  _et al._ (2018); Swekis  _et al._
(2019); Vir  _et al._ (2019); Mohanta  _et al._ (2020).

The SkX phase exhibits spin-wave excitations which are Goldstone modes
associated with the spontaneously-broken translational symmetry of the chiral
spin arrangement. The dispersion of these spin-wave modes is determined by the
properties of the SkX. Three SkX spin-wave modes – viz. clockwise,
counterclockwise circulation and breathing modes – have been predicted in
previous theoretical analysis Mochizuki (2012); Petrova and Tchernyshyov
(2011); Garst  _et al._ (2017) and subsequently reported in experiments using
broadband microwave spectroscopy Schwarze  _et al._ (2015); Onose  _et al._
(2012); Ehlers  _et al._ (2016). Some of the spin-wave modes exhibit non-
trivial topological properties, revealing in the form of chiral edge states in
the SkX Roldán-Molina  _et al._ (2016). The transition to the SkX phase takes
place within a range of external magnetic field strengths, when the
temperature is slowly lowered down Mohanta  _et al._ (2019).

An important puzzle which remained unresolved for years is the role of spin
fluctuations in the transition to the SkX phase. This is particularly
interesting since specific heat measurements report a first-order phase
transition Bauer  _et al._ (2013) to the helimagnetic phase at zero magnetic
field while neutron-scattering experiments reveal a Landau soft-mode mechanism
of weak crystallization to the SkX phase at a finite field Wilhelm  _et al._
(2011); Samatham and Ganesan (2013); Kindervater  _et al._ (2019). These
experiments suggest that the emergence of the SkX phase occurs via a precursor
regime in which non-trivial topological spin textures of skyrmionic character
are developed within a paramagnetic background with abundance of fluctuations,
similar to the precursor phenomenon proposed in the context of liquid-crystal
transitions Brazovskii  _et al._ (1987). At low finite temperatures, the
positional correlations of the skyrmions decay with distance as power laws
while the orientational correlations remain finite Nishikawa  _et al._ (2019).
With increasing temperature, a transition takes place from the solid to a
liquid phase of skyrmions with asymptotically vanishing long-range
correlations.

Here, we theoretically investigate the spin-wave excitations of a two-
dimensional SkX phase and its evolution at higher temperatures, in the
vicinity of the paramagnetic-to-SkX phase transition, by computing the
dynamical spin structure factor $S(\mathbf{q},\omega)$, expecting its future
direct detection in inelastic SANS experiments. We use a spin Hamiltonian to
first find out the stable spin configurations at low temperatures using the
Metropolis Monte Carlo annealing method and then numerically solve the Landau-
Lifshitz equation of motion to capture the spin dynamics. The computed
$S(\mathbf{q},\omega)$ reveals an exotic array of six gapless spin-wave modes
in the SkX phase, stabilized within a range of the external perpendicular
magnetic fields, accompanied by another gapless mode originating from the
ferromagnetic background. In the diffusive regime for the skyrmion
crystallization, $S(\mathbf{q},\omega=0)$ profile reveals a clear transition
from a ring-shaped profile to a six-peak profile, characteristic of the SkX
phase, confirming the previous experimental findings that the transition to
the SkX phase upon cooling slowly from a higher temperature occurs via a
precursor regime. A gradual decrease in the $S(\mathbf{q=0},\omega)$ intensity
with temperature, at the SkX spin-wave modes, also indicates a phase
transition similar to the liquid-crystal transition.

The spin spiral (SS) phase, which is the natural solution at zero magnetic
field, exhibits two gapless spin-wave modes from the spiral order and a soft
ferromagnetic mode that appears at finite energies and goes away with
increasing the DMI strength. The field-polarized ferromagnetic (FM) phase
exhibits a spin-wave mode which appears with an excitation gap due to the
Larmor precession in a magnetic field.

A particularly interesting platform that hosts a two-dimensional SkX is oxide
interfaces, such as the interface between the ferromagnetic metal La1-xSrxMnO3
($0.2\\!\lessapprox\\!x\\!\lessapprox\\!0.5$) and the non-magnetic semimetal
SrIrO3. The Dzyaloshinskii-Moriya interaction (DMI), which arises at this
interface due to the strong spin-orbit coupling in SrIrO3 and the broken
inversion symmetry, stabilizes a Néel-type SkX Mohanta  _et al._ (2019). We
discuss the feasibility of the experimental detection of our results in this
manganite/iridate interfaces based on the state-of-the-art SANS spectrometers
and the developments required for a successful future confirmation of our
predictions.

Figure 1: Spin textures in the spin spiral (SS) and the skyrmion crystal (SkX)
phases. The spin textures were spontaneously generated in the Monte Carlo
annealing process in a 100$\times$100 square lattice with periodic boundaries:
(a) SS texture realized at zero magnetic field ($H_{z}\\!=\\!0$), (b) nearly-
triangular crystal of skyrmions realized at a magnetic field
$H_{z}\\!=\\!0.18J$ ($\approx 3.74$ T). Parameters used are ferromagnetic
exchange interaction strength $J\\!=\\!1$, spin amplitude $S\\!=\\!3/2$,
Dzyaloshinkii-Moriya interaction strength $D\\!=\\!0.5J$, easy-plane
anisotropy amplitude $A\\!=\\!0.01J$ and temperature $T\\!=\\!0.001J$
($\approx 14$ mK). The white arrows and the colorbar respectively show the in-
plane ($x$ and $y$) components and the perpendicular ($z$) component of the
spins. The insets show an expanded view of the spin configuration inside the
region denoted by the green square on the top-left corner.

Results
Model Hamiltonian. We consider a model Hamiltonian, relevant to two-
dimensional material systems such as the La1-xSrxMnO3/SrIrO3, Pt/Co and Fe/Ir
interfaces. We consider the manganite/iridate interface as our example system
and choose parameters that are relevant to this interface. The manganite is
considered to be doped suitably to be in the ferromagnetic phase in which the
localized spins can be described effectively by a Heisenberg exchange
interaction. The influence of the strong spin-orbit coupling of the iridate,
in the absence of explicit structural inversion symmetry, leads to the DMI at
the interface. The total Hamiltonian, in the presence of an external magnetic
field, is given by

$\displaystyle\mathcal{H}\\!=\\!$ $\displaystyle-J\sum_{\langle
ij\rangle}\mathbf{S}_{i}\cdot\mathbf{S}_{j}-D\sum_{\langle
ij\rangle}(\hat{z}\times\hat{r}_{ij})\cdot(\mathbf{S}_{i}\times\mathbf{S}_{j})$
$\displaystyle-H_{z}\sum_{i}S_{zi}-A\sum_{i}|S_{zi}|^{2},$ (1)

Figure 2: Spin waves in the skyrmion crystal (SkX) phase. (a)-(f) Constant-
energy contours of the dynamical spin structure factor $S(\mathbf{q},\omega)$
calculated in the momentum plane $\mathbf{q}=(q_{x},q_{y})$ in the SkX phase,
realized at a magnetic field $H_{z}\\!=\\!0.18J$ ($\approx 3.74$ T), at
energies (a) $\omega\\!=\\!0$, (b) $\omega\\!=\\!J$, (c) $\omega\\!=\\!3J$,
(d) $\omega\\!=\\!6J$, (e) $\omega\\!=\\!8J$, and (f) $\omega\\!=\\!11J$. Plot
(a) shows the three characteristic momenta $\mathbf{Q}_{1}$, $\mathbf{Q}_{2}$
and $\mathbf{Q}_{3}$ of the SkX. (g)-(h) Plots of $S(\mathbf{q},\omega)$ along
the momentum paths (g)
$\Gamma-\mathbf{Q}_{1}-\mathbf{Q}_{2}-\mathbf{Q}_{3}-\Gamma$ and (h)
$\Gamma-\rm X-\rm M-\Gamma$. The spin-wave mode originating from the $\Gamma$
(0,0) point is due to the ferromagnetic background in the SkX. The rest six
branches arise from the triangular SkX spin configuration.

where $J$ is the nearest-neighbor ferromagnetic exchange parameter, $D$ is the
DMI strength, $H_{z}$ is the magnetic field applied perpendicular to the
interfacial plane, and $A$ is the easy-plane magnetic anisotropy originating
from the combined interfacial strain and Rashba spin-orbit coupling Banerjee
_et al._ (2014). We used $A\\!=\\!0.01J$ throughout this paper.
$\mathbf{S}_{i}$ is the localized spin of amplitude $S\\!=\\!3/2$ on the Mn
$t_{2g}$ orbitals at site $i$ on the manganite side of the interface. By
comparing the critical temperature for the zero-field FM phase (at
$D\\!=\\!0$) extracted from our Hamiltonian with the experimental findings
Nichols  _et al._ (2016), it was found that the Heisenberg exchange energy
constant at the manganite/iridate films varies within the range $1.2~{}{\rm
meV}\lessapprox J\lessapprox 3.9~{}{\rm meV}$, depending upon the thickness of
the film Mohanta  _et al._ (2019). In the description below, we present
energies in units of $J$ and reveal magnetic field and temperature scales, at
selected places, using $J=1.2~{}{\rm meV}$. The dipole-dipole interaction
usually has an important contribution in ferromagnetic materials. However,
recent experiments suggest that the short-range exchange interaction dominates
over the long-range dipole-dipole interaction in Sr-doped manganite compounds
Tiwari  _et al._ (2020). While the long-range dipole-dipole interaction may be
sizeable and help in stabilizing the magnetic phases in bulk manganite
systems, here we focus on the short-range exchange interactions which are more
relevant to the two-dimensional geometry.

Low-temperature spin configurations. We present the spin-wave excitations,
obtained using a $100\\!\times\\!100$ lattice at different magnetic fields,
first at a low temperature $T\\!=\\!0.001J$ and then discuss the high-
temperature diffusive regime in which the liquid-crystal transition takes
place. The SS phase, which is stabilized at zero magnetic field, evolves into
a triangular SkX within a range of magnetic fields and finally becomes a
fully-polarized FM phase at higher fields. A detailed analysis of the phase
diagram, spanned by the magnetic field and the temperature, was done in our
previous work Mohanta  _et al._ (2019). In Fig. 1(a) and (b), we show the spin
configurations of the SS phase and the SkX phase, obtained in a typical
annealing session of the Monte Carlo simulations, respectively at zero field
($H_{z}\\!=\\!0$) and at a finite field ($H_{z}\\!=\\!0.18J$). Since the DM
vectors lie predominantly at the interface plane, the SS and the SkX textures
are Néel-type in nature, unlike in three-dimensional chiral magnets such as
MnSi in which out-of-plane DM vectors lead to Bloch-type textures. A change in
the sign of $D$ reverses the sense of rotation of the spins in going from the
skyrmion center towards the radially outward direction. At the considered DMI
strength $D\\!=\\!0.5J$, which is close to the value realizable at oxide
interfaces Matsuno  _et al._ (2016); Wang  _et al._ (2018b); Büttner  _et al._
(2018), the period of the SS and the skyrmion diameter are nearly 12 lattice
spacings. At smaller values of $D$, these length scales increase almost
exponentially, $D\\!=\\!0$ ($H_{z}\\!=\\!0$) being the rotationally-invariant
Heisenberg FM phase.

Figure 3: Spin waves in the diffusive regime for skyrmion crystallization. (a)
Temperature variation of the skyrmion number $N_{sk}$ and the average
magnetization component $\langle M_{z}\rangle$, showing the gradual transition
to the triangular SkX below $T\\!\lessapprox\\!2.5J$. Panels (b) and (c) are
the spin configuration and the zero-energy contour of the dynamical spin
structure factor $S(\mathbf{q},\omega)$ at temperature $T\\!=\\!J$ and
magnetic field $H_{z}\\!=\\!0.18J$. Panels (d) and (e) are the spin
configuration and the zero-energy contour of $S(\mathbf{q},\omega)$ at
temperature $T\\!=\\!2J$ and magnetic field $H_{z}\\!=\\!0.18J$. Panels (f)
and (g) show the plots of $S(\mathbf{q},\omega)$ along the momentum path
$\Gamma-\mathbf{Q}_{1}-\mathbf{Q}_{2}-\mathbf{Q}_{3}-\Gamma$ at the two
temperatures, representatives of the diffusive regime for the SkX
crystallization.

Spin waves in the skyrmion crystal. The SkX phase is formed within an
intermediate magnetic-field range in which, upon cooling, the skyrmions
establish a long-range order to form a triangular crystal. At the DMI strength
$D\\!=\\!0.5J$, the SkX phase is realized within the field range
$0.13J\\!\lessapprox\\!H_{z}\\!\lessapprox\\!0.20J$. We focus on the SkX phase
obtained at $H_{z}\\!=\\!0.18J$, shown in Fig. 1(b). In Fig. 2(a)-(f), we show
the constant-energy contours of the dynamical spin structure factor
$S(\mathbf{q},\omega)$ at different energies. The plot at $\omega\\!=\\!0$ in
Fig. 2(a) depicts the six-peak structure of the SkX phase, as revealed in
elastic neutron-scattering experiments (see e.g. Fig.2E in Ref. Mühlbauer  _et
al._ , 2009). The three characteristic momenta of the triangular SkX phase are
denoted by $\mathbf{Q}_{1}$, $\mathbf{Q}_{2}$ and $\mathbf{Q}_{3}$. The peak
at the $\Gamma$ point originates from the zero-momentum magnetic ordering in
the ferromagnetic background of the SkX. The constant-energy contours of
$S(\mathbf{q},\omega)$ at different finite energies are shown in Fig.
2(b)-(f). At a finite energy $\omega\\!=\\!J$, the seven elastic peaks evolve
into circles as shown in Fig. 2(b). With the increase in energy, these circles
grow in size and deviate their shape to rounded squares near energy
$\omega\\!=\\!6J$ (Fig. 2(d)). At much higher energies, the topology of the
energy contour changes (characteristic of the Lifshitz transition), seven
pockets appear at the corners of the Brillouin zone (Fig. 2(e)) and these
modes disappear at energies above $\omega\\!\approx\\!13J$.

In Fig. 2(g), we present the momentum$-$energy dependence of
$S(\mathbf{q},\omega)$ of the SkX phase, the main result of the paper, along
the high-symmetry path
$\Gamma-\mathbf{Q}_{1}-\mathbf{Q}_{2}-\mathbf{Q}_{3}-\Gamma$. The plot unveils
a complex structure for the spin-wave modes, six of which originate from the
SkX and another from the ferromagnetic background. The spin-wave modes from
the SkX appear to follow a parabolic dispersion relation ($\omega\propto
q^{2}$) in the low-energy regime $\omega\lessapprox 0.2J$, around each elastic
peak. As shown in Fig. 2(h), the spin-wave modes of the SkX along the
$\Gamma-{\rm X}$ direction are two-fold degenerate. The rich features shown in
Fig. 2(g) are a remarkable result and its possible experimental confirmation
is addressed below.

In the presence of an additional AC magnetic field, three spin-wave modes of
the SkX phase were numerically identified in Ref. [Mochizuki, 2012]. Two of
these three modes are rotational modes (clockwise and counterclockwise) that
appear with in-plane AC magnetic field. The third one which was found with
out-of-plane AC magnetic field is called the “breathing” mode, where the
skyrmion core expands and shrinks alternatively. These modes were subsequently
reported in experiments using the broadband microwave spectroscopy in the
skyrmion-host compounds MnSi, Fe1-xCoxSi, Cu2OSeO3 and GaV4S8 Schwarze  _et
al._ (2015); Onose  _et al._ (2012); Ehlers  _et al._ (2016). We speculate
that these three modes correspond to the excitations at energy
$\omega\\!\approx\\!0.5J$ at the $\Gamma$ point in Fig. 2(g). Previous efforts
Petrova and Tchernyshyov (2011); Garst  _et al._ (2017); dos Santos  _et al._
(2018) analytically obtained the spin-wave modes of the SkX that we explore
here. Below, we discuss the feasibility of the detection of the spin-wave
modes, obtained in our calculations at constant magnetic field, in inelastic
neutron-scattering experiments.

Figure 4:  Spin-wave mode at zero wave vector across the paramagnet-skyrmion
crystal transition. The energy dependence of the dynamical spin structure
factor $S(\mathbf{q\\!=\\!0},\omega)$ at different temperatures across the
phase transition to the skyrmion crystal is shown. The linear decay (inset) of
the $S(\mathbf{q\\!=\\!0},\omega)$ amplitude at energy $\omega/J\\!=\\!0.51$
suggests the absence of a sharp transition. Figure 5: Spin waves in the spin
spiral (SS) phase. (a)-(f) Constant-energy contours of the dynamical spin
structure factor $S(\mathbf{q},\omega)$ calculated in the SS phase, realized
at zero magnetic field, at energies (a) $\omega\\!=\\!0$, (b)
$\omega\\!=\\!J$, (c) $\omega\\!=\\!3J$, (d) $\omega\\!=\\!6J$, (e)
$\omega\\!=\\!8J$, and (f) $\omega\\!=\\!11J$. $\mathbf{Q}_{1}$
($=\\!-\mathbf{Q}_{2}$) is the characteristic momentum of the SS phase, as
shown in plot (a). (g)-(h) Plot of $S(\mathbf{q},\omega)$ along the momentum
paths (g) $\mathbf{Q}_{1}-\Gamma-\mathbf{Q}_{2}$ and (h) $\Gamma-\rm X-\rm
M-\Gamma$. Panel (g) is a symmetrized expansion of the lower right corner of
panel (h).The ‘soft’ spin-wave mode, which arises above a finite energy around
the $\Gamma$ point (0,0), is due to the ferromagnetic correlation within the
stripes in the SS phase. The two other spin-wave modes arise from the spiral
order in the spin configuration.

Diffusive regime for skyrmion crystallization. We now focus on the high-
temperature regime in which the crystallization of the skyrmions takes place
as the temperature is lowered down. In our model, the triangular SkX phase is
stabilized below a temperature $T_{s}\\!\lessapprox\\!0.8J$ (which, in Kelvin
unit, lies in the range 11 K $\lessapprox$ $T_{s}$ $\lessapprox$ 35 K,
depending on the thickness of the manganite layer), as revealed by the
temperature dependence of the skyrmion number
$N_{sk}=\frac{1}{4\pi}\int\mathbf{S}\cdot(\frac{\partial\mathbf{S}}{\partial
x}\times\frac{\partial\mathbf{S}}{\partial y})dxdy$, plotted in Fig. 3(a). The
average magnetization component $\langle M_{z}\rangle$ also becomes nearly
constant below $T_{s}$. However, the development of the skyrmionic correlation
begins at temperatures below $T\\!\approx\\!2.5J$, much above $T_{s}$, in the
annealing process when the temperature is reduced gradually. The temperature
range $0.8J$ $\lessapprox$ $T$ $\lessapprox$ $2.5J$ is the weak
crystallization regime at the considered values of $D$ and $H_{z}$. The
snapshots of the spin configuration at two temperatures $T\\!=\\!J$ and
$T\\!=\\!2J$ are shown, respectively in Fig. 3(b) and (d). This diffusive
regime may also involve an extended phase of nucleated skyrmions, which we
explored in our previous study Mohanta  _et al._ (2019). The constant-energy
contour of the dynamical spin structure factor $S(\mathbf{q},\omega)$ at
$\omega\\!=\\!0$ and at temperature $T\\!=\\!2J$, in Fig. 3(e), shows a ring-
shaped profile. At the smaller temperature $T\\!=\\!J$, in Fig. 3(c), the six-
peak structure, which is a reminiscent of the triangular crystal of skyrmions,
appears with the ring in the background. Such a paramagnetic-to-skyrmion
lattice transition has been observed in the transition metal helimagnet MnSi
using small-angle neutron scattering, neutron-resonance spin-echo
spectroscopy, and microwave spectroscopy experiments Kindervater  _et al._
(2019).

Figures 3(f) and (g) show the energy-momentum dependence of
$S(\mathbf{q},\omega)$ along the momentum path
$\Gamma-\mathbf{Q}_{1}-\mathbf{Q}_{2}-\mathbf{Q}_{3}-\Gamma$ at the two
temperatures $T\\!=\\!J$ and $T\\!=\\!2J$, respectively. At $T\\!=\\!J$, the
spin-wave modes of the SkX can be resolved well from the merged background of
the diffusively-scattered intensities. On the contrary, at $T\\!=\\!2J$, the
ring-shaped elastic profile do not reveal any clear mode of the spin-wave
branches. Interestingly, however, the modes near $\omega\\!=\\!0.51J$ at
$\mathbf{q=0}$ persist, despite a smearing in the intensities within a narrow
energy range. In Fig. 4, we show the evolution of this $\omega\\!=\\!0.51J$
mode at different temperatures across the paramagnet-SkX transition and find
that the intensity of $S(\mathbf{q},\omega)$ decreases linearly below the
temperature $T_{s}\leq 3J$. The absence of a sharp transition in the intensity
of $S(\mathbf{q},\omega)$ further indicates the occurrence of a liquid to
crystal transition leading to the SkX phase. The measurement of these smeared
modes of $S(\mathbf{q},\omega)$ at zero wave vector can be useful to
distinguish the elastic neutron-scattering pattern of a SkX from the nearly-
similar one of the hexagonal ferrofluids Gollwitzer  _et al._ (2006), iron
oxide nanoparticles Fu  _et al._ (2016), hexagonal magnets Takagi  _et al._
(2018) and possibly ferromagnetic domains.

The transition from the ring shape to the six-peak structure of elastic
$S(\mathbf{q},\omega)$ profile upon reducing temperature in the diffusive
regime establishes the fact that fluctuating skyrmionic correlation starts to
develop at temperatures much above the SkX-ordering temperature $T_{s}$ as the
precursor of the long-range triangular crystal of skyrmions. Such a precursor
phenomenon has been reported in the cubic helimagnets FeGe and MnSi using ac-
magnetic susceptibility and magneto-heat capacity measurements Wilhelm  _et
al._ (2011); Samatham and Ganesan (2013).

Spin waves in the spin spiral. We now turn our attention to the SS phase that
appears spontaneously in the absence of any external magnetic field at low
temperatures in our model. The SS phase has a single characteristic momentum
$\mathbf{Q}_{1}$ that is governed by the period of the spiral texture. It is
shown in the constant-energy contour of $S(\mathbf{q},\omega)$ at
$\omega\\!=\\!0$ in Fig. 5(a). With an increase in energy, the two elastic
peaks at $\mathbf{Q}_{1}$ and $\mathbf{Q}_{2}=-\mathbf{Q}_{1}$, similarly as
in the SkX phase, evolve into circles around these characteristic momenta, as
shown by the constant-energy contour of $S(\mathbf{q},\omega)$ at
$\omega\\!=\\!J$ in Fig. 5(b). In addition to the two spin-wave modes of the
spiral texture, a soft ferromagnetic mode appears at finite energies, as shown
by the circle around the $\Gamma$ point in Fig. 5(b). The intensity of this
ferromagnetic mode decreases with a decrease in the period of the SS texture,
i.e. with an increase in the DMI strength $D$. The constant-energy contours of
$S(\mathbf{q},\omega)$ at different energies in Fig. 5(c)-(f) show the spin-
wave modes behaving in a similar manner as in the FM or in the SkX phase.

We show the momentum$-$energy dependence of $S(\mathbf{q},\omega)$ of the SS
phase along the momentum path $\mathbf{Q}_{1}-\Gamma-\mathbf{Q}_{2}$ in Fig.
5(g), and along $\Gamma-\rm X-\rm M-\Gamma$ in Fig. 5(h). The plots show two
spin-wave modes from the spiral order and one mode from the ferromagnetic
order with a bandwidth $\sim$$13J$. The spin-wave modes from the SS phase
follow nearly a parabolic dispersion relation ($\omega\propto q^{2}$) in the
low-energy regime $\omega\lessapprox J$. The two-fold degenerate spin-wave
excitation at $\omega\\!\approx\\!0.8J$ at the $\Gamma$ point (Fig. 5(g)) is
detectable in microwave spectroscopy experiments.

Theoretical calculations Garst  _et al._ (2017) and inelastic neutron-
scattering experiments in the context of chiral magnets Kugler  _et al._
(2015); Schwarze  _et al._ (2015); Grigoriev  _et al._ (2015); Dussaux  _et
al._ (2016); Tucker  _et al._ (2016); Portnichenko  _et al._ (2016); Bauer
_et al._ (2017); Turgut  _et al._ (2017); Siegfried  _et al._ (2017); Weiler
_et al._ (2017) analyzed the “helimagnon” excitation modes of the SS phase.

A ferromagnetic spin-wave mode, corresponding to the zero wave-vector magnetic
order, is expected to robustly emerge from $\omega\\!=\\!0$ in the case of the
conical spin spiral phase, that appears in three-dimensional chiral magnets at
zero magnetic field and in two-dimensional oxide interfaces with an in-plane
magnetic field Garst  _et al._ (2017).

Spin waves in the field-polarized ferromagnet. Having explored all other
magnetic phases in our model, we now discuss, for concreteness, the spin-wave
mode of the field-polarized FM phase. We realized this FM phase at a magnetic
field $H_{z}\\!=\\!0.5J$ in the MC simulations. Figure 6(a) shows the
dynamical spin structure factor $S(\mathbf{q},\omega)$ computed along a
momentum path $\Gamma~{}(0,0)-\rm X~{}(\pi,0)-\rm M~{}(\pi,\pi)-\Gamma$ in the
Brillouin zone.

Figure 6: Spin waves in the field-polarized ferromagnetic (FM) phase. (a) The
dynamical spin structure factor $S(\mathbf{q},\omega)$ calculated in the FM
phase, realized at a magnetic field $H_{z}\\!=\\!0.5J$ ($\approx 10.37$ T),
along the momentum path $\Gamma~{}(0,0)-\rm X~{}(\pi,0)-\rm
M~{}(\pi,\pi)-\Gamma$. Plots (b) to (e) show the constant-energy contours of
$S(\mathbf{q},\omega)$ at energies (b) $\omega\\!=\\!0.5J$, (c)
$\omega\\!=\\!2J$, (d) $\omega\\!=\\!6J$, and (e) $\omega\\!=\\!11J$.

The plot describes the well-known spin-wave dispersion relation
$\omega\\!=\\!H_{z}+4JS(\cos{q_{x}}+\cos{q_{y}})$ with a bandwidth $\sim$$12J$
and an energy gap $\Delta\omega\\!=\\!0.5J$. Figures 6(b)-(e) show the
constant-energy contours of the spin-wave dispersion at different energies.
The point-like peak in $S(\mathbf{q},\omega)$ at the $\Gamma$ point at energy
$\omega\\!=\\!0.5J$ evolves into a circle with increasing $\omega$ and the
circle takes the shape of a rounded square at $\omega\\!\approx\\!6J$. With
further increase in $\omega$ and four pockets appear at the corners of the
Brillouin zone, like in the cases of the SkX and the SS phases. These hole
pockets disappear above the upper limit, $\omega\\!\approx\\!12.5J$, of the
energy spectrum.

The sharpness of the $S(\mathbf{q},\omega)$ dispersion in the
frequency$-$momentum space is controlled primarily by the lattice size and the
time step. At smaller lattice sizes, the $S(\mathbf{q},\omega)$ peaks appear
discontinuously along the dispersion curve. With larger lattice sizes, the
increase in the number of peaks form a continuous sharply-defined curve. An
analysis of finite-size effects on the spin-wave dispersion was done in Ref.
Chen and Landau, 1994. On the other hand, a larger time step introduces tails
along the $\omega$ axis surrounding the peak positions with an amplitude that
decays in an oscillatory manner with distance from the $\omega$ of the peak. A
sufficiently small time step reduces these tails significantly from the
dispersion curve. In three-dimensional materials with a finite DMI, the fully-
polarized FM phase at a finite magnetic field exhibits an anisotropy in the
spin-wave dispersion, the anisotropy being proportional to the DMI strength
$D$ and the perpendicular momentum $k_{z}$ Garst  _et al._ (2017); Grigoriev
_et al._ (2015). In two dimensions, the anisotropy goes away and the spin-wave
dispersion is similar to a two-dimensional ferromagnet.

Discussion
The presented dynamical spin structure factor of the skyrmion crystal reveals
a rich structure of the spin-wave excitations that can be achieved at the
maganite/iridate or similar interfaces where a skyrmion crystal is expected to
be formed. Its detection in inelastic neutron-scattering experiments can offer
an opportunity to effectively probe the topological magnetic textures which is
often difficult to deduce conclusively from only the topological Hall effect
measurements. The challenge for the successful detection of the SkX spin-wave
modes in inelastic scattering experiments lies in obtaining a sufficiently
high signal-to-noise ratio from a single interface. For this purpose,
multilayer heterostructures, composed of alternate manganite and iridate
layers can provide a workable platform. We note that with the mass of a
typical La1-xSrxMnO3/SrIrO3/SrTiO3 superlattice (of thickness 1-10 $\mu$m
deposited on a 1$\times$1 cm2 substrate) is $\sim$0.00064 g (with average
density $\sim$6.4 g/cm3), implying that a sample mass of 0.1 g can be achieved
by stacking $\sim$15-150 of these superlattices. Considering that LaMnO3 has a
molar mass 239.3 g/mol, a 0.1 g sample contains 0.4 millimoles, while a
generic criterion for inelastic neutron scattering experiments is to have
samples of about 6.2 (3.5) millimoles for S=3/2 (S=2) Mn spins. Thus,
measurements with these small samples are likely a factor 10 beyond current
capabilities, such as the Spallation Neutron Source in the United States, the
Institut Laue-Langevin in France, the ISIS neutron source in the United
Kingdom, and J-PARC in Japan. There are additional considerations such as
neutron absorption, background from the substrate scattering, bandwidth of the
excitations, achieving appropriate coverage in reciprocal space, etc for a
successful neutron-scattering experiment. However, such experiments are likely
feasible at planned or under construction next generation neutron sources,
such as the European Spallation Source or the Second Target Station at the
Spallation Neutron Source, Oak Ridge National Laboratory, where improvements
in instrumentation and neutron flux are anticipated to result in performance
gains in excess of an order of magnitude SNS ; Ins ; CSP .

Although the challenges related to the thin-film geometry of oxide interfaces
are considerable with regards to sample mass, if robust results could be
experimentally gathered they will enrich the understanding of the dynamics of
the skyrmions which are of high technological interests owing to their
potential for wide-spread applications in the high-density data storage
Everschor-Sitte  _et al._ (2018) and magnonic devices Ma  _et al._ (2015).
Another merit of the presented results and the proposed inelastic measurements
would be to distinguish the elastic neutron-scattering pattern of a skyrmion
crystal from the nearly-similar one of the hexagonal ferrofluids Gollwitzer
_et al._ (2006), iron oxide nanoparticles Fu  _et al._ (2016), and hexagonal
magnets Takagi  _et al._ (2018).

To conclude, we theoretically investigated the spin-wave excitations of a two-
dimensional Dzyaloshinskii-Moriya magnet, in the context of the two-
dimensional interface between lanthanum manganite and strontium iridate, by
computing the dynamical spin structure factor. We explore the excitation modes
appearing in the SkX, SS, and FM phases obtained using Monte Carlo simulations
and Landau-Lifshitz spin dynamics at different magnetic fields. In particular,
we focus on the SkX phase which shows a complex interesting structure in the
spin dynamical spin structure factor at low temperatures. We also present the
spin-wave excitations in the diffusive regime above the SkX-ordering
temperature $T_{s}$ and show that fluctuating skyrmionic correlation develops
at temperatures much above $T_{s}$ as the precursor of the long-range SkX
phase, indicative of a liquid-crystal transition. We finally discussed the
possible challenges in the experimental detection and propose a multilayer
heterostructure geometry that might overcome some of the obstacles to
successfully observe these spin-wave modes from the SkX phase at oxide
interfaces.

Methods
Monte Carlo annealing. We obtain the spin configurations on a square lattice
of size 100$\times$100 with periodic boundary conditions using the Monte Carlo
(MC) annealing procedure. For all considered magnetic fields, the annealing
process was started at a high temperature $T\\!=\\!30J$ with a completely
random spin configuration and the temperature was lowered slowly down to a low
value $T\\!=\\!0.001J$ in $1000$ temperature steps. At each temperature,
$10^{9}$ MC spin updates were performed. In each spin update step, a new spin
direction was chosen randomly within a small cone spanned around the initial
spin direction. The MC sampling method provides a full and uniform coverage of
the phase space spanned by the spin angles, as illustrated in Fig. 7. The new
spin configuration was accepted or rejected according to the standard
Metropolis algorithm by comparing the total energies of the previous and the
new trial spin configurations, calculated using the Hamiltonian in Eq. (1).

Figure 7: Monte Carlo sampling in the phase space. Typical Monte Carlo
sampling of $10^{4}$ spin angles ($\theta$, $\phi$) in the phase space,
showing a uniform coverage of the unit sphere by the sampled spin angles
(black dots).

Landau-Lifshitz spin dynamics. Starting with the initial conditions obtained
from MC simulations at a given temperature, the spins are integrated
numerically and averaged over the initial values using the Landau-Lifshitz
equations of motion

$\displaystyle\frac{d\mathbf{S_{r}}(t)}{dt}=\frac{\partial\mathcal{H}}{\partial\mathbf{S_{r}}(t)}\times\mathbf{S_{r}}(t),$
(2)

where $\mathcal{H}$ is the Hamiltonian described in Eq. (1) and
$\mathbf{S_{r}}(t)$ is the spin vector at position $\mathbf{r}$ and time $t$.

The central object of investigation is the dynamical spin structure factor
$S(\mathbf{q},\omega)$, expressed as

$\displaystyle S(\mathbf{q},\omega)\\!$
$\displaystyle=\\!\sum_{\alpha,\beta}\Big{(}\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}+\eta^{2}}\Big{)}$
$\displaystyle\times\frac{1}{2\pi
N}\sum_{\mathbf{r},\mathbf{r}^{\prime}}e^{-i\mathbf{q}\cdot(\mathbf{r}-\mathbf{r}^{\prime})}\int_{-\infty}^{\infty}\\!dt~{}e^{-i\omega
t}C^{\alpha\beta}(\mathbf{r}-\mathbf{r^{\prime}},t)$ (3)

which is the space-time Fourier transform of the dynamical spin correlation
function $C^{\alpha\beta}(\mathbf{r}-\mathbf{r^{\prime}},t)$

$\displaystyle
C^{\alpha\beta}(\mathbf{r}-\mathbf{r^{\prime}},t)=\Big{[}\langle
S^{\alpha}_{\mathbf{r}}(t)S^{\beta}_{\mathbf{r^{\prime}}}(0)\rangle\\!-\\!\langle
S^{\alpha}_{\mathbf{r}}(t)\rangle\langle
S^{\beta}_{\mathbf{r^{\prime}}}(0)\rangle\Big{]},$ (4)

where $\alpha$ and $\beta$ represent the $x,y,$ and $z$ components. $N$ is the
total number of lattice sites and the parameter $\eta\\!=\\!0.01$ was used to
obtain $S(\mathbf{q},\omega)$ at $q\\!=\\!0$. The factor
$(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}+\eta^{2}})$ was used
in Eq. (3) to include the off-diagonal components of the correlation function
$C^{\alpha\beta}(\mathbf{r}-\mathbf{r^{\prime}},t)$ ($\alpha\\!\neq\beta$).

While solving Eq. (2), we express the spins using spherical polar coordinates
and evaluate the evolution of the polar and the azimuthal angles $\theta$ and
$\phi$. The time integration in Eq. (2) was carried out using the fourth-
order Runge-Kutta method to a maximum time $t_{max}\\!=\\!1000J^{-1}$ with a
time step $\Delta t\\!=\\!0.001J^{-1}$. The finite time cutoff may introduce
small oscillations in $S(\mathbf{q},\omega)$ along the $\omega$ axis,
resulting from the time Fourier transform. These oscillations were minimized
using the convolution of the spin-spin correlation function with a resolution
function in frequency, equivalent to the energy resolution of the SANS
spectrometer. A Gaussian broadening function with a broadening parameter
$\sigma_{\omega}$ yields the modulated dynamical spin structure factor Chen
and Landau (1994)

$\displaystyle S(\mathbf{q},\omega)\\!$
$\displaystyle=\frac{1}{\sqrt{2\pi}\sigma_{\omega}}\int_{-\infty}^{\infty}S^{\prime}(\mathbf{q},\omega^{\prime})exp\Big{[}-\frac{(\omega-\omega^{\prime})^{2}}{2\sigma_{\omega}^{2}}\Big{]}d\omega^{\prime},$
(5)

where $S^{\prime}(\mathbf{q},\omega^{\prime})$ is the dynamical spin structure
factor, calculated using a time cutoff. In the results presented in this
paper, $\sigma_{\omega}\\!=\\!0.001J$ was sufficient to resolve different
spin-wave modes.

The Gilbert damping may also introduce a broadening of the spin-wave bands.
However, the Gilbert damping coefficient ($\alpha$) for manganite systems is
very low ( $\sim 10^{-4}$) [see, for example, Ref. Qin  _et al._ , 2017].
These small values of $\alpha$ will introduce only a tiny amount of broadening
to the higher-energy spin-wave excitations, and therefore, it can be safely
ignored in the context of manganite thin films.

The computed $S(\mathbf{q},\omega)$ was averaged using 50 independent
thermalized realizations of the MC spin configurations to reduce the noise
arising from small statistical imperfections in those configurations.
Increasing this number did not produce noticeable modifications in the
results. One advantage of the current approach over linear spin-wave theory
techniques is the scope to examine spin dynamics at finite temperatures, where
the spin system does not have a well-established long-range order.

Acknowledgments
All members of this collaboration were supported by the U.S. Department of
Energy (DOE), Office of Science, Basic Energy Sciences (BES), Materials
Sciences and Engineering Division. The authors acknowledge discussion with
Elizabeth M. Skoropata and Ho Nyung Lee on the specification of their
manganite-iridate thin films.

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