]

Current addresses: ]Faculty of Engineering and Physical Sciences, University
of Southampton, Southampton SO17 1BJ, United Kingdom

# Skyrmion and incommensurate spin dynamics in centrosymmetric Gd2PdSi3

M. Gomilšek<EMAIL_ADDRESS>Jožef Stefan Institute, Jamova c. 39,
SI-1000 Ljubljana, Slovenia  Faculty of Mathematics and Physics, University of
Ljubljana, Jadranska u. 19, SI-1000 Ljubljana, Slovenia  Department of
Physics, Durham University, South Road, Durham DH1 3LE, United Kingdom  T. J.
Hicken  Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institut, 5232
Villigen PSI, Switzerland  Department of Physics, Durham University, South
Road, Durham DH1 3LE, United Kingdom  M. N. Wilson  Department of Physics and
Physical Oceanography, Memorial University, A1B 3X7, Canada  Department of
Physics, Durham University, South Road, Durham DH1 3LE, United Kingdom  K. J.
A. Franke  School of Physics and Astronomy, University of Leeds, LS2 9JT,
United Kingdom  Department of Physics, Durham University, South Road, Durham
DH1 3LE, United Kingdom  B. M. Huddart  Clarendon Laboratory, University of
Oxford, Department of Physics, Oxford OX1 3PU, United Kingdom  Department of
Physics, Durham University, South Road, Durham DH1 3LE, United Kingdom  A.
Štefančič  University of Warwick, Department of Physics, Coventry CV4 7AL,
United Kingdom  S. J. R. Holt  [  Max Planck Institute for the Structure and
Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany  University
of Warwick, Department of Physics, Coventry CV4 7AL, United Kingdom  G.
Balakrishnan  University of Warwick, Department of Physics, Coventry CV4 7AL,
United Kingdom  D. A. Mayoh  University of Warwick, Department of Physics,
Coventry CV4 7AL, United Kingdom  M. T. Birch  Max Planck Institute for
Intelligent Systems, Heisenbergstrasse 3, D-70569 Stuttgart, Germany  RIKEN
Center for Emergent Matter Science, JP-351-0198 Wako, Japan  Department of
Physics, Durham University, South Road, Durham DH1 3LE, United Kingdom  S. H.
Moody  Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institut,
5232 Villigen PSI, Switzerland  Department of Physics, Durham University,
South Road, Durham DH1 3LE, United Kingdom  H. Luetkens  Laboratory for Muon
Spin Spectroscopy, Paul Scherrer Institut, 5232 Villigen PSI, Switzerland  Z.
Guguchia  Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institut, 5232
Villigen PSI, Switzerland  M. T. F. Telling  ISIS Facility, STFC Rutherford
Appleton Laboratory, Didcot, Oxfordshire OX11 0QX, United Kingdom  P. J. Baker
ISIS Facility, STFC Rutherford Appleton Laboratory, Didcot, Oxfordshire OX11
0QX, United Kingdom  S. J. Clark  Department of Physics, Durham University,
South Road, Durham DH1 3LE, United Kingdom  T. Lancaster  Department of
Physics, Durham University, South Road, Durham DH1 3LE, United Kingdom

(August 31, 2024)

###### Abstract

Skyrmions are particle-like vortices of magnetization with non-trivial
topology, which are usually stabilized by Dzyaloshinskii–Moriya interactions
(DMI) in noncentrosymmetric bulk materials. Exceptions are centrosymmetric Gd-
and Eu-based skyrmion-lattice (SkL) hosts with net zero DMI, where both the
dominant SkL stabilization mechanisms and ground-state magnetic properties
remain controversial. In this Letter we use muon spectroscopy ($\mu$SR) to
address these by investigating both static and dynamic spin properties of the
most-studied centrosymmetric SkL host, Gd2PdSi3. We find that spin
fluctuations in the non-coplanar SkL phase are highly anisotropic, implying
that spin anisotropy plays a prominent role in stabilizing this phase.
Intriguingly, we also observe strongly-anisotropic spin fluctuations in the
ground-state (IC-1) incommensurate magnetic phase of the material, indicating
that it is a complex multi-$q$ magnetic structure. On the other hand, the
higher-field coplanar IC-2 phase is found to be single-$q$ with mostly
isotropic spin dynamics.

††preprint: APS/123-QED

Topological spin textures can give rise to exotic spin dynamics with a range
of potential applications [1, 2]. Especially promising are materials hosting
skyrmions, which are topologically-protected non-coplanar magnetization
vortices that behave as extended particles [3, 1]. Usually found in bulk
noncentrosymmetric materials, where they are stabilized by
Dzyaloshinskii–Moriya interactions (DMI) that select a preferred skyrmion
helicity [4], they were also found to be stabilized by competing magnetic
interactions in bulk centrosymmetric compounds with no preferred helicity and
no net DMI [1, 2]. Examples are Gd2PdSi3, with a triangular spin lattice [5,
6, 7, 8, 9], Gd3Ru4Al12 with a breathing kagome spin lattice [10, 11, 12], and
GdRu2Si2 [13, 14, 15, 16, 17] and EuAl4 [18, 19, 20] with tetragonal spin
lattices. Common to these highly-symmetric rare-earth materials is that they
host incommensurate skyrmion-lattice (SkL) phases with ultra-small
($1.9$–$3.5\text{\,}\mathrm{nm}$) skyrmions that are stable under much higher
applied magnetic fields ($\sim 1\text{\,}\mathrm{T}$) and over a much wider
range of $T$ than their DMI-stabilized counterparts.

The dominant stabilization mechanism of centrosymmetric skyrmions is
controversial, with suggestions including: (i) short-range geometrical
frustration [21, 22, 23, 11, 24], (ii) long-range
Ruderman–Kittel–Kasuya–Yosida (RKKY) plus dipolar [25, 26, 17, 15] or
biquadratic exchange [27, 28, 29, 30], and (iii) competition of orbital-
dependent exchange [31, 32]. Most, but not all [27, 28], scenarios require
spin anisotropy. Furthermore, the zero-field (ZF) ground state is also
contentious [9], with early studies of Gd2PdSi3 suggesting a possible exotic,
triple-$q$ magnetic structure (e.g., a lattice of merons and antimerons, which
are half-skyrmion-like spin textures) [5, 6], while recent calculations under
scenario (ii) indicated a simpler, single-$q$ helical ground state [25], as in
DMI-stabilized SkL hosts [33]. Investigations of centrosymmetric SkL hosts
have mostly focused on static and topological transport properties of their
phases, with less attention [20] paid to their emergent spin dynamics [22],
which could help illuminate both their stabilization mechanism and their
single- or multi-$q$ nature. In contrast, characteristic dynamics of SkL and
related spin textures have been observed in a range of DMI-stablized SkL hosts
[2, 1, 34, 35, 36]. There, muon spectroscopy ($\mu$SR), a local-magnetic-probe
technique sensitive to spin fluctuations over a unique frequency range [37],
has proven valuable [38, 39, 40, 41, 42, 43]. However, the narrow $T$ range of
DMI-stabilized skyrmions [1, 2, 33] presented a limitation.

Fig. 1: (a) Real part of ZFC AC susceptibility of Gd2PdSi3 in fields applied
along the $c$ axis. Phase boundaries are highlighted by white and $\mu$SR
scans by green dashed lines. (b) Sample and detector arrangement for $\mu$SR
measurements. The initial muon spin ${\bm{\mathbf{S}}}_{\mu}(0)$ (violet)
points at an angle $\theta$ from the out-of-plane applied field
${\bm{\mathbf{H}}}\parallel c$ (black), which points along the longitudinal
detector direction ${\mathrm{L}}$ (blue). The transverse detector direction
${\mathrm{T}}$ (red) lies within the sample’s hexagonal $ab$ plane.

In this Letter we investigate the most-studied centrosymmetric SkL host
Gd2PdSi3 [5] using $\mu$SR and AC susceptibility over a range of applied
fields and $T$, complemented by density functional theory (DFT) calculations
of muon stopping sites [44]. We find clear signatures of spin reorientation
transitions between the incommensurate magnetic phases of the material and
reveal the highly anisotropic character of their spin fluctuations. By
exploiting the wide $T$ stability of these phases, we are also able to
characterize the low-energy dispersion relations of their spin-wave
excitations. In the higher-field coplanar incommensurate IC-2 phase these are
found to be consistent with a single-$q$ fan-like spin texture [5, 6] with
nearly isotropic spin fluctuations. However, in the ground-state ZF
incommensurate IC-1 phase in-plane ($ab$-plane), we find that low-energy spin
fluctuations dominate while out-of-plane ($c$-axis) spin fluctuations are
almost completely suppressed. This indicates that the IC-1 phase is indeed a
complex, triple-$q$ magnetic structure [5, 6], not the recently-predicted
single-$q$ helical structure [25], disfavouring SkL stabilization scenario
(ii). Finally, in the SkL phase we find that out-of-plane spin fluctuations
dominate instead, with in-plane fluctuations suppressed as a power-law in $T$.
Our results suggest that symmetric spin anisotropy is an important ingredient
in stabilizing the SkL phase in centrosymmetric rare-earth magnets.

Single crystals of Gd2PdSi3 were synthesized, with five high-quality,
$${\approx}9\text{\,}{\mathrm{mm}}^{2}$\times$0.6\text{\,}\mathrm{mm}$$ plate-
like single crystals extracted from a larger boule with the $c$ axis normal to
the plates [44]. AC magnetic susceptibility measurements reproduced the phase
diagram from previous studies [5, 6, 7], including the SkL phase [Fig. 1(a)].
For $\mu$SR measurements, the five single-crystal plates were assembled in a
mosaic with coaligned $c$ axes [44]. The initial muon spin orientation pointed
along the normal ($c$-axis) direction in longitudinal-field (LF) measurements
($\theta=0$), and at an angle of
$\theta\approx$50\text{\,}\mathrm{\SIUnitSymbolDegree}$$ away from the $c$
axis in ZF and transverse-field (TF) measurements [Fig. 1(b)]. In ZF and TF
both the longitudinal- (${\mathrm{L}}\parallel c$) and transverse-
(${\mathrm{T}}\perp c$) muon spin component could thus be measured separately,
while in LF the ${\mathrm{L}}$ muon spin component was measured. All
measurements were made after first zero field cooling (ZFC) the sample to base
$T$, while magnetic fields were always applied along the out-of-plane
($c$-axis) direction, which is an axis of threefold crystallographic symmetry
and thus a magnetic eigenaxis of the material.

Fig. 2: (a) Transverse-direction (${\mathrm{T}}$) ZFC muon asymmetry data at
early (left) and late times (right) in the IC-1 phase in ZF. Solid lines on
the left panel are guides to the eye (damped cosines), while dashed lines are
fits using the Gaussian model described in the text. (b) Characteristic in-
plane quasi-static magnetic fields (black) and dynamical muon relaxation rates
(violet) in the IC-1 phase extracted from these data. Inset shows the
contributions to muon relaxation due to out-of-plane ($c$-axis, blue) and in-
plane ($ab$-plane, black) internal field fluctuations. (c) ZFC muon relaxation
due to in-plane field fluctuations in the IC-2 phase under a
$2\text{\,}\mathrm{T}$ applied field. Vertical lines in (b) and (c) highlight
transition temperatures $T_{N}$. Solid lines over $\lambda_{\mathrm{T}}$ (b)
and $\lambda_{\mathrm{L}}$ (c) at $T<T_{N}$ are guides to the eye (sum of two
power laws), while dashed lines are their single-power-law low-$T$ limits.
Dashed lines at $T>T_{N}$ are SCR fits.

We first performed $\mu$SR measurements on warming from the IC-1 ground-state
of Gd2PdSi3 in ZF on the GPS instrument at the Paul Scherrer Institut (PSI),
Switzerland. At low $T$, a highly-damped oscillation in the ${\mathrm{T}}$
muon spin component was observed at early times
$t\ll$0.1\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ [Fig. 2(a)] due to a broad
distribution of quasi-static local fields at the muon site, which arise from
long-range incommensurate magnetic ordering of Gd^3+ spins. At late times, up
to $t\approx$10\text{\,}\mathrm{\SIUnitSymbolMicro s}$$, a further exponential
relaxation was observed due to dynamical fluctuations Gd^3+ spins [37] slower
than the quasi-static muon precession frequency (faster fluctuations would
wipe out early-time oscillations). A damped oscillation model proved
problematic to fit as the damping was too high [44]. The measured
${\mathrm{T}}$ muon asymmetry data were thus fit using
$A(t)=\left[a_{s}-a_{r}\right]e^{-\sigma^{2}t^{2}}+a_{r}e^{-\lambda
t}+a_{\mathrm{bgd}}$, where $a_{s}=\mathrm{const.}$ is the total sample
asymmetry, the early-time relaxation rate $\sigma$ is proportional to the
width of the quasi-static local-magnetic-field vector distribution (and thus
to the average magnitude of quasi-static internal fields), $a_{r}$ is the
late-time relaxing asymmetry due to the fraction of quasi-static local fields
that at early times point along the measured muon spin component, $\lambda$ is
the late-time dynamical relaxation rate due to slow fluctuations of local
fields orthogonally to the measured muon spin component [37, 44], and
$a_{\mathrm{bgd}}$ is a constant background contribution due to muons stopping
in the silver sample holder.

Fit results for the ${\mathrm{T}}$ direction are shown in Fig. 2(b) with the
extracted transition temperature $T_{N}\approx$22(1)\text{\,}\mathrm{K}$$
consistent with AC susceptibility [Fig. 1(a)]. In the itinerant paramagnetic
(PM) regime at $T>T_{N}$ there are no average quasi-static fields
($\sigma\approx 0$), while spin fluctuations are described reasonably well by
the 3D self-consistent renormalization (SCR) theory prediction
$\lambda_{\mathrm{T}}\propto T/|T-T_{N}|$ [37], as also observed in the
paramagnetic regime of the itinerant skyrmion host MnSi [45]. Around $T\approx
T_{N}$ spin fluctuations exhibit a stronger critical divergence characteristic
of a continuous phase transition as we enter the IC-1 phase. Finally, in the
IC-1 phase at $T<T_{N}$ the local field strength increases and saturates as a
typical order parameter $\sigma\propto[1-(T/T_{N})^{3/2}]^{\beta}$ [37, 20]
with $\beta=0.72(3)$, while slow spin fluctuations cross over into a low-$T$
power-law $\lambda_{\mathrm{T}}\propto T^{p}$ with $p=0.95(9)$. A low-$T$
power-law dependence of the dynamical relaxation rate could be understood
within the two-magnon (Raman) process spin-wave-theory prediction that in the
low-$T$ limit, with a single magnon band, we would have [46, 47]

$p=\frac{2D}{s}-1,$ (1)

where $D$ is the dimensionality of spin-wave excitations and $s$ is the
dominant power in their dispersion relation $\omega\propto|q-q_{0}|^{s}$
around the ordering wave vector $q_{0}$. Usually, $s=2$ for ferromagnetic
($q_{0}=0$) and $s=1$ for antiferromagnetic and incommensurate spin states
($q_{0}\neq 0$) [48]. From our ${\mathrm{T}}$-direction measurements of the
incommensurate IC-1 phase we have $p\approx 1$, which would, in this single-
magnon-band approximation, give $D=s=1$, i.e., that IC-1 would be a 1D
(single-$q$) magnetic structure.

To test this scenario, we also fitted the ${\mathrm{L}}$ direction ZF $\mu$SR
data using the same model to obtain the relaxation rate
$\lambda_{\mathrm{L}}$. Assuming bulk uniaxial symmetry with a magnetic
eigenaxis $c$ and eigenplane $ab$ we can further write [37, 44]:
$\lambda_{\mathrm{L}}=2\lambda_{ab}$ and
$\lambda_{\mathrm{T}}=\lambda_{ab}+\lambda_{c}$, where $\lambda_{ab}$ and
$\lambda_{c}$ are the contributions to the muon relaxation rate due to
dynamical fluctuations of in-plane and out-of-plane magnetic fields at the
muon site, respectively. At the nearly in-plane muon site we obtain from DFT
calculations [44] $\lambda_{ab}$ and $\lambda_{c}$, furthermore, correspond to
dynamical fluctuations of in-plane and out-of-plane Gd^3+ spin components,
respectively, if we assume predominantly ferromagnetic interlayer correlations
[44], as found in Refs. [25, 26]. Inset in Fig. 2(b) shows the extracted in-
plane $\lambda_{ab}$ and out-of-plane $\lambda_{c}$ [44]. Remarkably, in
contrast to the single-magnon-band approximation, where we should have
$\lambda_{ab}\propto\lambda_{c}\propto T^{p}$, we instead find
$\lambda_{ab}\propto T$ and $\lambda_{c}\approx 0\ll\lambda_{ab}$ in the IC-1
phase. Our first results is thus that the IC-1 phase is not, in fact, a simple
single-$q$ magnetic structure as was recently predicted [25]. Instead, there
appear to be multiple, highly-anisotropic spin fluctuation modes in the IC-1
phase, as would be expected for large, multi-$q$ spin textures [22, 34], like
the hypothesized meron–antimeron (M–AM) lattice state [5, 6]. Their
predominantly in-plane nature appears consistent with easy-plane anisotropy
found in this phase [9].

Fig. 3: (a) ZFC muon relaxation rates due to out-of-plane ($c$-axis, blue) and
in-plane ($ab$-plane, black) internal field fluctuations in the SkL and IC-2
phases in a $0.75\text{\,}\mathrm{T}$ applied field. (b) Relaxing asymmetry
due to quasi-static local fields within the $ab$ plane (blue) and along the
$c$ axis (black) under the same conditions. Vertical lines highlight
transition temperatures.

Next, we turn to the IC-2 and SkL phases. To study these we performed separate
$\mu$SR measurements under a TF of $0.75\text{\,}\mathrm{T}$ at the GPS
instrument at PSI, Switzerland, and a LF of $0.75\text{\,}\mathrm{T}$ on the
HiFi instrument at the ISIS Facility, Rutherford Appleton Laboratory (RAL),
UK. The advantage of the latter were much higher statistics at late times,
though with lower early-time temporal resolution. In the measurements we
warmed from the low-$T$ SkL to an intermediate IC-2, and finally to the PM
phase [Fig. 1(a)]. The late-time $\mu$SR data were well described by the same
model as ZF data [44], which simplifies to $A(t)=a_{r}e^{-\lambda
t}+a_{\mathrm{bgd}}$ at late times. From the ${\mathrm{T}}$ direction from TF
data and the ${\mathrm{L}}$ direction from LF data we extracted the in-plane
$\lambda_{ab}$ and out-of-plane $\lambda_{c}$ spin-fluctuation contributions
to muon relaxation [44], via relations described above.

The extracted relaxation rates are shown in Fig. 3(a) with the extracted
transition temperatures $T_{N1}=$11(1)\text{\,}\mathrm{K}$$ and
$T_{N2}\approx$20(1)\text{\,}\mathrm{K}$$ consistent with AC susceptibility
[Fig. 1(a)]. We see that the in-plane relaxation rate $\lambda_{ab}$ in the
itinerant PM regime at $T>T_{N2}$ is again well described by 3D SCR theory,
with a critical divergence at $T\approx T_{N2}$ due to a continuous phase
transition to the IC-2 phase. In the IC-2 regime at $T_{N1}<T<T_{N2}$ we find
$\lambda_{c}\approx\lambda_{ab}$, i.e., spin fluctuations in this phase are
approximately isotropic. They can even be empirically fitted to a gapped model
$\lambda\propto e^{-\Delta/T}$ with the same energy scales $\Delta=47(3)$ and
$49(3)\text{\,}\mathrm{K}$ for $c$-axis and $ab$-plane spin fluctuations,
respectively. However, any low-$T$ power-law spin-wave behaviour [e.g., Eq. 1]
is masked by near-critical fluctuations in this $T$ region, and is interrupted
by a transition to the SkL phase at $T_{N1}$. This phase transition does not
show any critical divergence in the muon relaxation rate, just a change of
slope, implying that it is first-order, which is consistent with the
topologically non-trivial nature of the SkL phase [22, 1]. Similarly to the
IC-1 phase, we find that highly anisotropic spin fluctuations appear with
lowering $T<T_{N1}$. Intriguingly, while in the IC-1 phase spin fluctuations
were predominantly in-plane, $\lambda_{c}\approx 0\ll\lambda_{ab}\propto T$,
in the SkL phase the dominant spin fluctuations are instead out-of-plane, with
$\lambda_{c}\approx\mathrm{const.}\gg\lambda_{ab}\propto T^{p}$ and
$p=2.0(2)$. We note that the $\lambda_{ab}$ power $p\approx 2$ would
correspond to $D=3$, $s=2$ (i.e., 3D ferromagnetic) spin excitations under the
single-band spin-wave approximation [Eq. 1], but this is inconsistent with a
$T$-independent $\lambda_{c}$. Instead, it seems that there is a large spin
density of states (DOS) due to multiple spin fluctuation modes, as expected
for SkL phases [49, 22, 34, 35, 36], and that these are, furthermore, highly
anisotropic in terms of their spin polarization that points predominantly out-
of-plane (as, for example, expected for skyrmion breathing modes [49, 34]).

We next turn to quasi-static SkL and IC-2 signatures under the same applied
field. Fig. 3(b) shows the fitted late-time relaxing asymmetries
$a_{r,{\mathrm{T}}}$ and $a_{r,{\mathrm{L}}}$ in the ${\mathrm{T}}$ and
${\mathrm{L}}$ directions [Fig. 1(b)], respectively. Both change rather
abruptly around $T_{N2}$ due to the onset of magnetic order. Assuming bulk
uniaxial symmetry with a magnetic eigenaxis $c$ and eigenplane $ab$ we have,
in ordered magnetic phases [37, 44]:
$a_{r,{\mathrm{T}}}=a_{s,{\mathrm{T}}}\braket{\cos^{2}\phi}=a_{s,{\mathrm{T}}}\braket{\hat{B}_{a}^{2}}$
and
$a_{r,{\mathrm{L}}}=a_{s,{\mathrm{L}}}\braket{\cos^{2}\phi^{\prime}}=a_{s,{\mathrm{L}}}\braket{\hat{B}_{c}^{2}}$,
where $\phi$ and $\phi^{\prime}$ are the angles between the normalized quasi-
static local magnetic field direction
${\bm{\mathbf{\hat{B}}}}={\bm{\mathbf{B}}}/|{\bm{\mathbf{B}}}|$ at early times
and the ${\mathrm{T}}$ (in-plane) and ${\mathrm{L}}$ (out-of-plane) detector
directions, respectively, $a_{s,{\mathrm{T}}}$ and $a_{s,{\mathrm{L}}}$ are
the total sample asymmetries along these directions, and $\braket{\cdots}$
denotes a thermal expectation value. In Fig. 3(b) we see that
$a_{r,{\mathrm{T}}}\propto\braket{\hat{B}_{a}^{2}}=\braket{\hat{B}_{b}^{2}}$
exhibits a broad peak in the IC-2 phase at $T_{N1}<T<T_{N2}$, while
$a_{r,{\mathrm{L}}}\propto\braket{\hat{B}_{c}^{2}}=1-2\braket{\hat{B}_{a}^{2}}$
exhibits a minimum, both indicating mainly coplanar quasi-static magnetism in
this phase. On the other hand,
$a_{r,{\mathrm{T}}}\propto\braket{\hat{B}_{a}^{2}}$ drops and
$a_{r,{\mathrm{L}}}\propto\braket{\hat{B}_{c}^{2}}$ rises as $T<T_{N1}$ is
lowered in the SkL phase, implying that quasi-static local magnetic field
directions become more isotropic in this phase, as expected for intrinsically
non-coplanar SkL spin textures [1, 2, 3]. The observed coplanar nature of the
IC-2 phase and non-coplanar nature of the SkL phase in terms of quasi-static
magnetism is consistent with conclusions from resonant X-ray scattering (RXS)
in Ref. [5], where this was argued based on the measured ellipticity
($m_{c}/m_{ab}$) of magnetic moments ${\bm{\mathbf{m}}}({\bm{\mathbf{q}}})$ of
individual ${\bm{\mathbf{q}}}$-space magnetic Bragg peaks. However, our
conclusions are based on a complementary, real-space [9] measurement of the
anisotropy local quasi-static magnetic field directions
($\braket{\hat{B}_{c}^{2}}/\braket{\hat{B}_{ab}^{2}}$), which is only
accessible to local magnetic probes like the muon.

Finally, to assess the low-$T$ dynamics of the IC-2 phase we performed
additional $\mu$SR measurements under a $2\text{\,}\mathrm{T}$ LF at the HiFi
instrument at ISIS, RAL, UK. The measured late-time ${\mathrm{L}}$ muon
asymmetry data were well described by the same model as
$0.75\text{\,}\mathrm{T}$ data [44]. The fitted muon relaxation rate
$\lambda_{\mathrm{L}}=2\lambda_{ab}$ is shown in Fig. 2(c) with the extracted
transition temperature $T_{N}\approx$15(1)\text{\,}\mathrm{K}$$ consistent
with AC susceptibility [Fig. 1(a)]. In the itinerant PM regime at $T>T_{N}$
spin fluctuations are again described reasonably well by 3D SCR theory, near
$T\approx T_{N}$ they exhibit a steep critical divergence characteristic of a
continuous phase transition as we enter the IC-2 phase, and at $T<T_{N}$ they
smoothly cross over into a low-$T$ power-law $\lambda\propto T^{p}$ with
$p=1.2(3)$. The observed isotropy of spin fluctuations in this phase [Fig.
3(a)] makes the single-band spin-wave theory applicable with $p\approx 1$
implying that $D=s=1$ [Eq. 1]. In other words, we confirm that the IC-2 phase
is indeed a 1D (single-$q$) incommensurate magnetic structure, as previously
suggested [5, 6], in contrast to the multi-$q$ IC-1 and SkL phases found at
lower applied fields.

Our finding of a fundamentally different low-$T$ dependence of in-plane and
out-of-plane spin fluctuations in both the non-coplanar centrosymmetric SkL
and the ground-state IC-1 phases of Gd2PdSi3 contrasts with isotropic spin
fluctuations found in the coplanar single-$q$ IC-2 phase. This should be a
clue as to the stabilization mechanism behind centrosymmetric skyrmions.
However, in contrast to DMI-stabilized skyrmions, where their dynamics were
studied [49, 34, 35, 36], clear theoretical predictions for SkL spin dynamics
under different stabilization-mechanism scenarios are mostly missing.
Nevertheless, it seems unlikely that a spin model without strong intrinsic
anisotropy could explain the observed, highly-anisotropic SkL spin dynamics.
This expectation agrees with suggestions that spin anisotropy [22, 9],
combined with long-range interactions, could be important for stabilizing the
SkL phase [26, 29]. Furthermore, our observation that the ground-state IC-1
phase is a triple-$q$ magnetic structure suggests it is likely the exotic
meron–antimeron (M–AM) lattice state, as hypothesized in Ref. [5] based on RXS
and field-dependent Hall resistivity measurements. This intriguing result
represents another challenge to theory, as recent calculations suggested that
this phase should be single-$q$ under the RKKY+dipolar stabilization scenario
(ii) for centrosymmetric skyrmions [26]. Our finding that the ground-state is
in fact multi-$q$ thus disfavours this scenario as the dominant skyrmion
stabilization mechanism. Nevertheless, RKKY interactions were found to be
important in the related centrosymmetric SkL host GdRu2Si2 via ARPES [17] and
quantum-oscillation measurements [15], complemented by ab initio calculations,
so we do expect them to play a role in Gd2PdSi3 as well, though, perhaps, a
subdominant one.

In conclusion, in our local-probe study of Gd2PdSi3 we have found large
anisotropy of spin dynamics, with qualitatively different behaviour of the
dominant out-of-plane and subdominant in-plane spin fluctuations in its
centrosymmetric SkL phase. We have also established the exotic triple-$q$
nature of its incommensurate IC-1 ground state, confirming the hypothesis of
Ref. [5], and found that in it in-plane spin fluctuations dominate instead,
with out-of-plane fluctuations nearly fully suppressed at low $T$. The higher-
field IC-2 phase was found to be coplanar and single-$q$. Our results suggest
that the enigmatic stabilization mechanism behind SkL phases in
centrosymmetric Gd- and Eu-based materials is likely to be intimately related
to strong spin anisotropy. Further local-probe studies of spin dynamics in
this and related centrosymmetric compounds (e.g., other Gd-based SkL hosts
like GdRu2Si2 [13], or the wider $R$2PdSi3 family, where $R$ is a rare earth
[50, 51], which has similar itinerant electronic but different magnetic
properties) should be highly informative in exploring this connection.
Finally, the $\mu$SR methods used in this study to establish strong spin-
fluctuation anisotropy of centrosymmetric skyrmions, could be extended to
low-$T$ metastable DMI-stabilized skyrmions [1] to characterize their spin DOS
and the anisotropy of their low-$T$ spin dynamics.

Data presented in this paper will be made available on figshare.

We thank Martin Klanjšek and Matej Pregelj at the Jožef Stefan Institute,
Slovenia for helpful discussions. Parts of the work were carried out at the
STFC ISIS Muon Source, United Kingdom and at the Swiss Muon Source, Paul
Scherrer Institute, Switzerland and we are grateful for the provision of
beamtime and experimental support. We thank Raymond Fan and Paul Steadman for
enabling us to use the Quantum Design MPMS3 at the I10 support laboratory,
Diamond Light Source for AC susceptibility measurements. Computing resources
were provided by the Durham HPC Hamilton cluster. M.G. acknowledges the
financial support of the Slovenian Research and Innovation Agency through
Program No. P1-0125 and Projects No. Z1-1852, N1-0148, J1-2461, J1-50008, and
J1-50012. We acknowledge the financial support of the Engineering and Physical
Sciences Research Council (EPSRC, UK) through Grants No. EP/N032128/1 and
EP/N024028/1. The work at the University of Warwick was also funded by EPSRC,
UK Grant No. EP/T005963/1.

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# Thermal Geo-axions

Hooman Davoudiasl 111email<EMAIL_ADDRESS>Department of Physics, Brookhaven
National Laboratory, Upton, NY 11973, USA  Patrick Huber 222email:
<EMAIL_ADDRESS>Department of Physics, IPNAS, Virginia Tech, Blacksburg, VA
24061, USA

###### Abstract

We estimate the production rate of axion-type particles in the core of the
Earth, at a temperature $T\approx 5000$ K. We constrain thermal geo-axion
emission by demanding a core-cooling rate less than $\mathcal{O}{(100)}$
K/Gyr, as suggested by geophysics. This yields a “quasi-vacuum” (unaffected by
extreme stellar conditions) bound on the axion-electron fine structure
constant $\alpha_{ae}^{QV}\lesssim 10^{-18}$, stronger than the existing
accelerator (vacuum) bound by 4 orders of magnitude. We consider the prospects
for measuring the geo-axion flux through conversion into photons in a
geoscope; such measurements can further constrain $\alpha_{ae}^{QV}$.

A variety of scenarios for physics beyond the Standard Model (SM) give rise to
light pseudo-scalar particles, generically referred to as axions. The Peccei-
Quinn (PQ) solution to the SM strong CP problem provided the initial context
for axions Peccei:1977hh . Axion-type particles are ubiquitous in string
theory constructs and have also been considered in cosmological model building
Preskill:1982cy . There are stringent astrophysical and cosmological
constraints on the couplings of axions, as a result of which they are largely
assumed to be very weakly interacting. Some of the strongest bounds on axion-
SM couplings come from astrophysics, where stellar evolution and cooling
arguments imply that the axion (PQ) scale $f\gtrsim 10^{9}$ GeV. Such analyzes
are based on the requirement that new exotic processes should not
significantly perturb a standard picture of the energetics that govern the
evolution of various astrophysical objects. Since axions (or other light
weakly interacting particles) can directly drain energy out of such objects,
one can obtain bounds on the coupling of axions to matter. For a concise
summary of various astrophysical bounds, see Ref. Amsler:2008zzb . More recent
astrophysical bounds on axion-type particles have been presented in Ref.
Gondolo:2008dd .

In this work, we consider the possibility that the hot core of the Earth can
convert some of its thermal energy into a flux of axions of $\mathcal{O}{({\rm
eV})}$ energy Sivaram 333Non-thermal geo-axions, produced in radioactive
decays within the Earth, have been examined in Ref. Liolios:2007gu . This work
does not find a currently detectable signal even in the most favorable case
considered therein.. Then, it would be interesting to find out what bounds can
be obtained from geological considerations and also to determine the prospects
for discovering the geo-axions emanating from the terrestrial core. The
Earth’s core is at a temperature of around $5000\,\mathrm{K}$ corresponding to
$0.4\,\mathrm{eV}$. Although this is a much lower temperature than those of
stellar interiors, which have temperatures of order $\mathrm{keV}$, there are
a number of considerations that motivate our analysis.

First of all, the core of the Earth is only a short distance away, compared to
any astronomical object. This greatly enhances the prospects for measuring a
geo-axion flux and can potentially compensate for the low core temperature.
Secondly, the Earth’s core is quite different from other axion emitting
environments, being mainly made up of hot molten or crystallized iron. Hence,
in principle, the intuition and calculations that apply to stellar plasmas may
not be adequate to estimate geo-axion emission and new effects may need to be
considered. Finally, the Earth’s center is a far less extreme medium compared
to stellar media. The possibility of the dependence of axion properties on the
environment has been proposed medium1  in the context of reports of large
vacuum birefringence by PVLAS Zavattini:2005tm . This result if it were
confirmed would have implied an axion like particle with an axion-photon
coupling in stark violation of astrophysical bounds. As a consequence, a
number of models were developed to reconcile the laboratory result with the
astrophysical bounds medium2 . Although the initial PVLAS result could not be
reproduced Cantatore:2008zz , it highlighted the necessity for obtaining
complementary bounds on axions in a wide variety of production environments.
If axion couplings are temperature and/or density dependent, the geo-axion
bounds could be viewed as independent new data on axion physics in “quasi-
vacuum” conditions. Thus, in this letter we do not attempt to supersede
existing, stringent astrophysical bounds, but to supplement them by examining
axion production in a novel environment.

Motivated by the above discussion, we will next derive an estimate for the
thermal geo-axion flux. We will use geodynamical considerations to constrain
this flux and hence the axion-electron coupling $\alpha_{ae}$ in the core.
This bound is not competitive with its astrophysical counterparts, but, as
mentioned before, is derived in a very different regime. Note that collider
bounds on $\alpha_{ae}$ that are derived in a similar regime are much weaker
than our geo-axion bound. We will then consider detection of the geo-axion
flux, via magnetic conversion into photons, using a “geoscope,” in analogy
with the helioscope concept Sikivie:1983ip ; vanBibber:1988ge . A discussion
and a summary of our results are presented at the end of this work.

The core of the Earth is mainly made of iron (Fe). The inner core, which
extends to a radius of $R_{ic}\approx 1200$ km, is thought to be in solid
crystalline form at a temperature $T\sim 6000$ K. The outer core, which
extends to $R_{c}\approx 3500$ km, is made up of molten iron at $T\sim 4000$ K
core . Since Fe is a transition metal, with the electronic configuration
$[{\rm Ar}]\;3d^{6}\;4s^{2}$, both $3d$ and $4s$ electrons are important in
determining its properties. However, for a simplified treatment, we only
consider the $4s$ electrons as nearly free. The effective nuclear charge seen
by the $4s$ electrons is $Z_{\rm eff}\simeq 5.4$ Zeff .

Given that the solid iron core makes up a negligible mass of the total core,
we will ignore its contributions to our estimate. This is partly done to avoid
a complicated treatment of the interactions of electrons and phonons inside a
hot crystal, far from the plasma regime. However, we note that a more complete
analysis should take these effects into account. We adopt $T_{c}\approx 5000$
K$\approx 0.4$ eV core  as the mean temperature of the molten iron core. We
are also ignoring the contribution of other trace elements, such as nickel,
which have more or less the same properties as iron, for our purposes. Given
the metallic nature of the core, we will treat it as a plasma composed of a
degenerate gas of free electrons, with a Fermi energy $E_{F}\approx 10.3$ eV
FerroFe . The resulting Fermi momentum is given by
$p_{F}=\sqrt{2m_{e}\,E_{F}}\approx 3.3$ keV, where $m_{e}\simeq 0.5$ MeV is
the mass of the electron. These free electrons move in the background of Fe
ions with effective charge $Z_{\rm eff}\simeq 5.4$. The free electron density
in the core is given by GR

$n_{e}=\frac{p_{F}^{3}}{3\pi^{2}}$ (1)

and hence we get $n_{e}\approx 2\times 10^{23}$ cm-3. Let us define the radius

$a_{e}=\left(\frac{3}{4\pi n_{e}}\right)^{1/3},$ (2)

for the mobile charged particles in the plasma, which we take to be electrons
here. The quantity

$\Gamma\equiv\frac{Z_{\rm eff}^{2}\alpha}{a_{e}T_{c}},$ (3)

with $\alpha=1/137$, is a measure of the relative strength of Coulomb
interactions and the kinetic energy of the electrons. For the core parameters,
we get $a_{e}\approx 10^{-8}$ cm and $\Gamma\sim 10^{3}$. We take $\Gamma\gg
1$ as indicative of a strongly coupled plasma GR . Since the iron core of the
Earth is in a molten state and not yet a crystal, this interpretation is
reasonable, despite the large value of $\Gamma$. The effect of the geomagnetic
field in the core on the density of states close to the Fermi surface can be
neglected since the thermal energy is large compared to the energy difference
between successive Landau levels. In any case, we note that a more detailed
numerical treatment may reveal important corrections to the estimates that
follow.

Interestingly enough, there is an astrophysical environment that is described
by the above key features. This is the interior of White Dwarfs (WD’s) which
is a strongly coupled plasma of Carbon and Oxygen, supported by a degenerate
gas of electrons, similar to the iron core of the Earth. Hence, we adopt the
formalism used for WD cooling by axion emission in the bremsstrahlung process
$e\,N(Z,A)\to e\,N(Z,A)\,a$ Raffelt:1985nj , in order to estimate the geo-
axion flux; $Z$ is the ionic charge and $A$ is the atomic mass. We will ignore
Primakoff Primakoff  contributions to this flux, resulting from the
interactions of thermal photons in the plasma. This is justified, since the
density of such photons is roughly given by $({\rm eV})^{3}\sim 10^{15}$ cm-3,
which is much smaller than $n_{e}$ in the core.

For a plasma with only one species of nuclei, the energy emission rate, in
axions, per unit mass is given by Raffelt:1985nj

$\varepsilon_{a}=(Z^{2}\alpha^{2}\alpha_{ae})/(Am_{e}^{2}m_{u})\,T^{4}\xi(p_{F}),$
(4)

where $m_{u}\simeq 1.7\times 10^{-24}$ g is the atomic mass unit and
$\xi(p_{F})$ is a numerical factor which only depends on $p_{F}$. Numerical
calculations relevant for WD’s indicate that $\xi\simeq 1$ to a good
approximation, over a wide range of parameters in the strongly coupled regime
GR . We thus take $\xi\sim 1$ in our calculations. For geo-axion emission, we
then obtain

$\varepsilon_{a}\sim 10^{7}\alpha_{ae}\,T_{3}^{4}\;\;{\rm erg}\,{\rm
g}^{-1}{\rm s}^{-1},$ (5)

where we have set $Z=Z_{\rm eff}\simeq 5.4$, $A=56$, and $T_{3}\equiv
T/10^{3}$ K. Given a core mass density of $\rho_{c}\simeq 10$ g cm-3 and
$T_{c}\approx 5T_{3}$, we get

$\Phi_{a}\sim 10^{37}\alpha_{ae}\;\;{\rm erg}\,{\rm s}^{-1},$ (6)

for the flux of geo-axions.

It is interesting to inquire how geological considerations can constrain the
estimate in Eq. (6). As a simple criterion, and in the spirit of analogous
considerations for stellar objects, we will demand that the rate of core-
cooling $\Phi_{a}$ be less than that inferred from geodynamical
considerations. This rate has been estimated to be in the range of 100
K/Gyr$=10^{-7}$ K/yr CB2006 . Given that the heat capacity of the Earth’s core
is estimated to be $C_{\oplus}\sim 10^{34}$ erg/K CB2006 , we get for the
geological rate of core-cooling

$\Phi_{\oplus}\sim 10^{27}{\rm erg/yr}\sim 10^{19}\;\;{\rm erg}\,{\rm
s}^{-1},$ (7)

in agreement with Ref. core . Requiring $\Phi_{a}<\Phi_{\oplus}$ yields

$\alpha_{ae}^{\oplus}\lesssim 10^{-18}\quad({\rm core}{\rm-}{\rm cooling}).$
(8)

This bound is not strong compared to those from astrophysics. For example, the
bound from solar age is $\alpha_{ae}\lesssim 10^{-22}$ and the one from red
giant constraints is $\alpha_{ae}\lesssim 10^{-26}$ GR . However, the bound in
(8) is within a few orders of magnitude of the solar. Again, we note that the
bound in (8) is valid for a quasi-vacuum regime and not the extreme stellar
environments. The closest such bounds for quasi-vacuum environments are from
$e^{+}e^{-}$ collider experiments, and correspond to $\alpha_{ae}\lesssim
10^{-14}$ ($f\gtrsim 1000$ GeV) Amsler:2008zzb , weaker than our geo-axion
bound (8) by 4 orders of magnitude. Next, we will examine the prospects for
detecting a geo-axion flux consistent with this bound.

The geo-axion flux $F_{a}$, corresponding to $\Phi_{a}$ in Eq. (6), at
$R_{\oplus}\simeq 6.4\times 10^{3}$ km (surface of the Earth) is given by

$F_{a}=\Phi_{a}/(4\pi R_{\oplus}^{2})\sim\alpha_{ae}10^{30}\;\;{\rm eV}\,{\rm
cm}^{-2}{\rm s}^{-1}.$ (9)

Assuming average axion energy $\langle E_{a}\rangle\simeq 1$ eV, we get

$\frac{dN_{a}}{d{\cal A}\,dt}\sim\alpha_{ae}10^{30}\;\;{\rm cm}^{-2}{\rm
s}^{-1}$ (10)

for the flux of eV-axions at the surface of the Earth444For axions, the non-
radial flux also contributes, thus in principle Gauß’ law can not be used. In
this case, the difference compared to an exact treatment is about 5%..

In principle, there are two ways to detect these axions: The obvious choice is
to exploit their coupling to electrons $\alpha_{ea}$ via the axio-electric
effect, in analogy with the photo-electric effect. The cross section for the
axio-electric effect is reduced by a factor of Avignone:2008uk

$\frac{\alpha_{ea}}{2\alpha}\left(\frac{E_{a}}{m_{e}}\right)^{2}\sim
10^{-10}\alpha_{ea}$ (11)

compared to the ordinary photo-electric cross section. Using our bound derived
in Eq. (8) and the typical size of photo-electric cross sections of
$\mathcal{O}{(10^{-20})}\,\mathrm{cm}^{2}$, the resulting axio-electric cross
section is $\mathcal{O}{(10^{-48})}\,\mathrm{cm}^{2}$, which is quite small.
For comparison typical neutrino cross sections are
$\mathcal{O}{(10^{-42})}\,\mathrm{cm}^{2}$. Therefore, we next consider the
possibility to use magnetic axion-photon mixing, to convert axions into
photons. We use the simple formula GR

$P_{a\gamma}\simeq\left(\frac{Bg_{a\gamma}L}{2}\right)^{2},$ (12)

where $B$ is the strength of the transverse magnetic field along the axion
path, $g_{a\gamma}$ is the axion-photon coupling, and $L$ is the length of the
magnetic region. The above equation is valid when the $q\ll 1/L$ with

$q=|m_{a}^{2}-m_{\gamma}^{2}|/(2E_{a}),$ (13)

where $m_{a}$ is the axion mass and $m_{\gamma}$ is the photon effective mass.

Note that for $m_{a}\lesssim 10^{-3}$ eV, the geo-axion oscillation length
$q^{-1}\gtrsim 0.6$ m in vacuum with $m_{\gamma}=0$. We will assume this mass
range for the purposes of our discussion. An obvious choice would be to
consider an LHC-class magnet, such as the one used by the CAST experiment
Andriamonje:2004hi , with $B=10$ T and $L=10$ m. However, this magnet has a
cross sectional area of order 14 cm2. Given that the core of the Earth
subtends an angle of order $30^{\circ}$ as viewed from its surface, we see
that the CAST magnet will capture a very small portion of the relevant “field
of view.” Thus, we have to consider other magnets of similar strength, but
larger field of view. Fortunately, such magnets are used in Magnetic Resonance
Imaging (MRI), to carry out medical research. For example, the MRI machine at
the University of Illinois at Chicago has a magnetic field of 9.4 T, over
length scales of order 1 m MRI . Hence, we will use $B=10$ T and $L=1$ m, as
presently accessible values, for our estimates.

The current best laboratory bound on $g_{a\gamma}$ for nearly massless axions,
derived under vacuum conditions, was recently obtained by the GammeV
collaboration Chou:2007zzc : $g_{a\gamma}<3.5\times
10^{-7}\,\mathrm{GeV}^{-1}$. With this value the upper bound for the axion
photon conversion probability is

$P_{a\gamma}\simeq 3\times 10^{-12}\,.$ (14)

Thus, we conclude that magnetic detection is more promising than axio-electric
detection.

Using Eq. (10), we then get

$\frac{dN_{\gamma}}{d{\cal A}\,dt}\sim\alpha_{ae}10^{18}\;\;{\rm cm}^{-2}{\rm
s}^{-1}$ (15)

for the flux of converted photons in the signal. The bound in (8) then
suggests that a sensitivity to a photon flux of order $1\;{\rm cm}^{-2}{\rm
s}^{-1}$, using our reference geoscope parameters, is required to go beyond
the geodynamical constraint and look for a signal. Modern superconducting
transition edge bolometers have demonstrated single photon counting in the
near infrared with background rates as low as $10^{-3}\,\mathrm{Hz}$ noise
and quantum efficiencies close to unity qe . With such a detector photon
fluxes as small as $10^{-5}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$ can be
detected with an integration time of $10^{7}\,\mathrm{s}$. Our results on the
prospects of direct search for geo-axions are summarized in figure 1.

Figure 1:  The black lines show the resulting photon fluxes $\phi$ in units of
$\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$ for a geoscope with $L=1\,\mathrm{m}$ and
$B=10\,\mathrm{T}$ as a function of $\alpha_{ae}$ and $g_{a\gamma}$. The gray
lines show the current astro-physical bounds on $\alpha_{ae}$ from the cooling
of white dwarfs Amsler:2008zzb  and on $g_{a\gamma}$ from the non-observation
of solar axions by CAST Andriamonje:2004hi . The colored/shaded regions
indicate the parameter space excluded by photon regeneration Chou:2007zzc  and
the cooling of the Earth (this work).

Before closing, we would like to point out a few directions for improving our
estimates. First of all, our picture of the iron core is quite simplified. A
more detailed treatment of electron-ion interactions in the molten core, as
well as the inner core contribution, which was ignored here, could reveal
extra enhancements or suppressions that were left out in our analysis. This
could, in principle, require a numerical simulation of the strongly coupled
plasma (molten Fe) and the crystalline solid core. Another issue is the
possible role that the Fe $3d$-orbital electrons play, given that they are
delocalized over a few nuclei and may contribute to pseudo-scattering
processes inside the hot Fe medium. Also, there could be important bound-bound
and free-bound processes that result in the emission of axions from Fe atoms
at high temperatures. These processes have been ignored here, but could
provide contributions comparable to those we have estimated. In principle,
more detailed geodynamical analyzes may yield stronger bounds on non-
convective energy transfer out of the Earth’s core. This can result in tighter
bounds on axion-electron coupling $\alpha_{ae}$, in the regime we considered
here. Finally, our estimate of a geoscope signal assumed an axion flux
transverse to the magnetic field. Given the angular size of the core, as
viewed through the geoscope, we expect that effective transverse field is, on
average, suppressed by roughly $\cos^{2}30^{\circ}$, which does not affect our
conclusions, given the approximate nature of our estimates.

In summary, we have derived estimates on possible emission of axions from the
hot core of the Earth. Our analysis allows for possible dependence of axion
properties on non-vacuum production media, such as astrophysical environments.
We approximated the molten core as a strongly coupled plasma of free
degenerate electrons in the background of Fe nuclei. We adapted the existing
axion emission estimate from a White Dwarf interior, which is a strongly
coupled plasma supported by a degenerate electron gas. We obtained the bound
$\alpha_{ae}^{\oplus}\lesssim 10^{-18}$ on the axion-electron coupling, by
considering geodynamical constraints on core-cooling rates. Given that geo-
axions would originate from a far less extreme environment than stellar cores,
our bound is nearly a vacuum bound. Hence, our result improves existing
accelerator constraints on $\alpha_{ae}$, in vacuo, by 4 orders of magnitude.
We also estimated the signal strength to be expected in a dedicated search for
geo-axions using a geoscope, based on magnetic axion-photon conversion.

###### Acknowledgements.

We would like to thank G. Khodaparast, S. King, and Y. Semertzidis for useful
discussions. The work of H.D. is supported in part by the United States
Department of Energy under Grant Contract DE-AC02-98CH10886.

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