# Texture fluctuations and emergent dynamics in coupled nanomagnets

Samuel D. Slöetjes  Björgvin Hjörvarsson  Vassilios Kapaklis  Department of
Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden

###### Abstract

We analyse the thermal fluctuations of magnetization textures in two stray
field coupled elements, forming mesospins. To this end, the energy landscape
associated with the thermal dynamics of the textures is mapped out and
asymmetric energy barriers are identified. These barriers are modified by
changing the gap that separates the mesospins. Moreover, the coupling between
the edges leads to an anisotropy in the curvature of the energy surface at the
metastable minima. This yields a dynamic mode splitting of the edge modes and
affects the attempt switching frequencies. Thus, we elucidate the mechanism
with which the magnons in the thermal bath generate the stochastic
fluctuations of the magnetization at the edges.

††preprint: AIP/123-QED

Single domain magnetic nanoislands with binary magnetization states -
mesospins \- are used as building blocks for magnetic metamaterials.
Contemporary examples are e.g. artificial spin ices (ASI) [1, 2, 3, 4, 5],
Ising chains and lattices [6, 7], and reconfigurable magnonic crystals [8, 9,
10]. These magnetic metamaterials can be viewed as having thermal
fluctuations, associated with switching of the magnetic states of the
mesospins [11, 12, 13, 14, 15], in an activated process [16, 12, 13, 17].
However, the magnetic fluctuations are not solely restricted to the switching
of rigid mesospins, but can also occur in the interior textures of the
mesospin, which necessitates a more nuanced view of their thermal excitations.
Furthermore, a finite temperature leads to the excitation of magnons, which
take place on timescales of the magnetization dynamics, much smaller than
those of the fluctuations of individual mesospins [18].

Resonances can occur in mesospins when the wavelength of magnons matches their
extension. At frequencies below the uniform (Kittel) mode, there is a band
gap, characterized by the absence of magnon modes in the interior of the
mesospins. However, dynamic edge modes may occur at frequencies inside to the
band gap, which can be used for spectral detection of topological defects in
ASI [19]. We have previously confirmed the existence of additional modes [20,
17], associated with a binary stochastic fluctuation of the magnetization at
the edges, from hereon called edge fluctuations. These edge fluctuations cause
switching between S- or C-states [21], and their presence results in a
residual entropy in magnetic metamaterials [22, 17]. They have recently also
been confirmed experimentally, and were found to occur at shorter time- and
length-scales than conventional mesospin switching [17]. Thus so far, the
intra-nanomagnet excitations are well understood, however an understanding of
the coupling between intra- and inter-mesospin degrees of freedom has not been
developed for magnetic metamaterials.

Figure 1: (a) Definition of the the geometrical parameters and the edge
magnetization states ($A$ and $B$). (b) The energy barriers associated with
switching the edge magnetization of one of the edges, with indication of the
points $A$, $S$ and $B$, for different gap values $g$. (c) Morphology of the
energy landscape as a function of the perpendicular edge components for a
$g=30$ nm gap between the elements.

For Ising-like mesospins, the edge fluctuations of the texture occur in the
transverse direction with respect to the switching of the net magnetization
(e.g. in the example of Fig. 1a, $m_{y}$ is the relevant reaction coordinate
in phase space). The height of the barrier associated with these fluctuations
depends on the size of the nanomagnets, and the corresponding balance between
exchange and demagnetization energies. In the case of a single mesospin, the
relevant phase space for the edge fluctuations can be captured in a
1-dimensional energy landscape. When adding one more mesospin, the texture
excitations are influences by inter-mesospin interactions, with the phase
space extending over a 2-dimensional energy landscape. The two nanomagnet
system is in the ground state when the two mesospins are magnetized in the
same direction, in which case the flux exiting one mesospin will be absorbed
by the other, which results in a reduction of the transverse magnetization at
the edges. In terms of the energy landscape, this situation leads to a single,
global energy minimum, which is associated with to the two edges being
aligned. In this case, the coupling causes the splitting of dynamic edge modes
into one in-phase and one out-of-phase mode. On the other hand, when the two
mesospins are oppositely magnetized (as illustrated in Fig.1a), there will be
two degenerate metastable states for the edge magnetization, leading to non-
trivial behaviour upon thermal excitation.

In this work, we discuss the fluctuations of two oppositely magnetized
mesospins. To this end, we start by exploring and quantifying the energy
landscape associated with the inter-mesospin excitations. We then perform
micromagnetic simulations with a temperature implementation, in order to
inspect the behaviour of the system under thermal excitation, for a range of
different temperatures and spacings between the mesospins. The observed
switching events are used to infer the energy barriers and attempt frequencies
utilizing Arrhenius analysis [23]. Finally, the Arrhenius prefactors are
discussed, relating them to the dynamic modes of the system and the curvatures
in the energy landscape.

For the numerical investigation, we performed micromagnetic simulations in
Mumax3 [24], which is based on the use of a finite difference method to solve
the Landau-Lifshitz-Gilbert (LLG) equation of motion. The size of the stadium
shaped mesospin used in this work is $l\times w\times h$ = $360\times
120\times 4$ nm3, using rectangular cells of size $2.5\times 2.5\times 4$ nm3.
We use parameters that are relevant for permalloy, i.e. saturation
magnetization $M_{s}=1\cdot 10^{6}$ A/m, an exchange stiffness of $A_{ex}$ =
$1\cdot 10^{-12}$ J/m, and a Gilbert damping of $\alpha=0.001$. More
information on the details of the simulations can be found in Slöetjes  _et
al._ [20].

Two mesospins are placed in line, separated by a gap, $g$, that defines the
edge-to-edge distance, as illustrated in Fig. 1a. We define the magnetization
in the edges as the averaged transverse magnetization over the half circle
that constitutes an edge, i.e.
$m_{y,1}=\frac{1}{V}\int_{\mathrm{edge_{1}}}m_{y}(\mathbf{r},t)d^{3}\mathbf{r}$
for the edge of mesospin 1, and similarly for mesospin 2, where $V$ is the
volume of the mesospin edge. The lowest energy configuration, state $A$, has
opposing transverse magnetization components, $m_{y,1}=-m_{y,2}$, and state
$B$ has aligned transverse components, $m_{y,1}=m_{y,2}$, see Fig. 1a.

We represent these states in an energy landscape, parameterized by $m_{y,1}$
and $m_{y,2}$, see Fig. 1c. This is an approximate representation of the
energy landscape, since the Zeeman energy, $E_{Z}$, is included in the total
energy, therefore it only serves to indicate the morphology. Aside from the
states $A$ and $B$ (and the degenerate states $A$´ and $B$´) one can observe
the saddle points at $S$ and the top of the energy barrier at $C$. As the gap
$g$ between the mesospins increases, the points $A$ and $B$ shift towards the
same energy, while the points $S$ and $C$ merge having the same height.
Bringing the mesospins closer together causes the energy hill to emerge at
point $C$, pushing the states $A$ and $B$ outwards, and breaks the two-fold
symmetry. Thus, the gap between the mesospins provides a handle to tailor the
energy landscape.

When switching from State $A$ to the opposite State $A$´, it is not
energetically favorable to take the direct route $A-C-A$´, corresponding to
both moments switching simultaneously. Instead, it is more favorable to take
the route $A-B-A$´ via the saddlepoints, which corresponds to consecutive
switchings of the nanomagnet edges. Due to the symmetry of the energy
landscape, it is only necessary to know the minimum energy path $A-S-B$ for an
understanding of the fluctuations. We map out the energy barrier along the
path that is highlighted in Fig. 1c (see Appendix A for the method), the
resulting morphologies are shown in Fig. 1b. As previously noted, the energy
landscape becomes increasingly asymmetric as the gap between the mesospins is
reduced. The states $A$ and $B$ are simultaneously pushed away from the
center. Furthermore, the saddle point moves away from the center position,
towards state $B$. From hereon, we refer to the energy difference between A
and S as the high energy barrier ($E_{H}$), and the energy difference between
B and S as the low energy barrier ($E_{L}$).

Next, we will discuss the thermally induced dynamics of the two oppositely
magnetized mesospins. To this end, we use micromagnetic simulations at a
finite temperature, as described in Leliaert  _et al._ [25]. The thermal bath
is imitated by transforming the LLG equation to a Langevin equation using a
stochastic thermal field, given by:

$\mu_{0}\mathbf{H}^{\mathrm{therm}}=\bm{\eta}\sqrt{\frac{2\alpha
k_{B}T}{M_{s}\gamma V\Delta t}}$ (1)

where $k_{B}$, $T$, $\gamma$, $V$, and $\Delta t$ are the Boltzmann constant,
temperature, gyromagnetic ratio, volume of the cell, and the time step,
respectively. In addition, $\bm{\eta}$ is a random vector which changes
direction and size at every timestep. We run the simulations for temperatures
from $T=25$ to 450 K in steps of 50 K, and eight different gaps, ranging from
$g=$ 25 to 200 nm. We analyze the timetraces of the perpendicular component of
the magnetization in the two edges $m_{y,j}(t)$, where $j$ denotes the
particular edge. Examples of these time traces are shown in Fig. 2a, for a
large ($g=200$ nm) and a small ($g=25$ nm) spacing. Each edge can be seen to
switch between the two different states, and the absolute value of the
perpendicular magnetization component increases as the spacing becomes
smaller. The thermal energy $k_{B}T$ required to switch the edge magnetization
needs to be increased by more than an order of magnitude for $g=25$ nm
compared to $g=200$ nm, in order to obtain a similar switching rate.
Furthermore, the magnetization at the edges is weakly anti-correlated when the
mesospins are placed far apart, and strongly anti-correlated when they are
close together. This correlation can be quantified using the Pearson
correlation coefficient, $\rho(m_{y,1},m_{y,2})$, which is shown in the
Appendix C for different spacings and temperatures.

Figure 2: (a) Timetraces of the edge magnetization for two different gaps and
temperatures. The shaded regions mark the system being in State $B$, the
regions without shading mark state $A$. (b) Switching events for a gap of 25
nm and a temperature of 400 K. Upper panel: a true switching event, lower
panel: a false switching event. (c) Arrhenius plots for the relaxation times
associated with the high barrier, $\tau_{A}$. (d) The energy barriers EH and
EL against the inverse gap values obtained via Arrhenius law, and direct
mapping of the energy barrier. The dashed lines are linear fits in the case of
$E_{L}$ and the low gap regime of $E_{H}$, and a parabolic fit to the high gap
regime of $E_{H}$.

When the two mesospins are strongly coupled, the edges seemingly switch
simultaneously, but zooming in on the switching process (upper panel of Fig.
2b) reveals that the two edges switch in a consecutive manner, as was
predicted on the basis of the energy landscape. Another possibility for a
fluctuation is that the edge magnetization switches back to its initial state,
as seen in the lower panel of Fig. 2b. We have observed (see Appendix D) that
the process $A-B-A^{\prime}$ occurs more often that the process $A-B-A$, while
in principle these two processes are the same in terms of energy barrier. This
asymmetry becomes more pronounced as the gap between the nanomagnets is
reduced.

We will now set out to determine the thermal energetics of the textures in the
mesospins. The average time in each state is determined, and we use Arrhenius
law to extract the energy barriers and attempt frequency:

$\tau_{i}=\tau_{0,i}e^{\frac{E_{i}}{k_{B}T}}$ (2)

where $i$ indicates $A$ or $B$ for $\tau$, and $H$ and $L$ for $E$
respectively, with $\tau_{0,i}$ being the inverse attempt frequency.
Generally, for inverse gaps larger than $1/g=0.027$ nm-1, the errors for
$E_{L}$ diverge, due to the high temperatures used for the relatively low
barriers, resulting in erroneous estimations of the switching time. For the
largest gap, 200 nm, there is no notable difference between EL and EH, and the
energy barrier ($E$ $=$ 8 meV) is close to that of a single mesospin [20] ($E$
$=$ 5 meV). Bringing the mesospins closer together leads to an expected
increase in $E_{H}$ and $E_{L}$. Whereas $E_{L}$ scales with the inverse gap,
$E_{H}$ shows two different scaling regimes, as shown in Fig. 2d. It can be
seen that up to an inverse gap of $1/g<0.02$ nm-1, $E_{H}$ shows a parabolic
scaling with $1/g$, whereas for values $1/g\geq 0.02$, there is a linear
scaling. The transition between the two regimes occurs between $1/g=1/65$ nm-1
and $1/g=1/50$ nm-1, which means that the radius of the half circle at the
edge (60 nm) defines the scale of the transition length. Overall, good
agreement can be found between the energies obtained via the stochastic
simulations and the energy landscape method, confirming the Arrhenius law.

Next, we shall focus on the attempt frequency, $f_{0}$, calculated as the
inverse of $\tau_{0,i}$. Similar to the energy barriers, the attempt
frequencies for State $A$ and $B$ also show a diverging behaviour upon a
decrease of the gap (see Fig. 3a). At the largest spacing, the attempt
frequency at $A$ and $B$ become equal, although they are not equal to that of
a single mesospin [20]. Interestingly, it can be observed that $f_{0,A}$
increases when the nanomagnets are being brought closer together, while
$f_{0,B}$ shows a slight decrease. In order to understand the scaling of the
attempt frequencies, we shall inspect the mode frequencies at the edge of the
mesospins, as well as the curvature in the energy landscape at $A$ and $B$.

The spectrum of magnon modes at the $A$ state is different from those at $B$,
see Fig. 3b. The edge modes reside in the frequency band below the Kittel mode
($\sim$ 6 GHz) due to the demagnetizing field counteracting the effective
field at the edges, thus lowering the eigenfrequencies with respect to the
modes inside the mesospin [20]. We can observe, for both cases, that a mode
splitting occurs as the mesospins are brought closer together. The mode
splitting in $B$ happens at a larger spacing than the splitting in $A$, which
can be understood as being due to the difference in strength of the dipole
coupling: in $B$, the sources of the stray field are closer together than in
$A$. In both cases, the two resulting branches after the splitting have
opposite phases in the in- and out-of-plane directions of the precession. By
using phase maps of the mesospins with the full inner texture taken into
account (see Appendix E), we determine the in-plane
($\Delta\varphi^{\parallel}$) and out-of-plane ($\Delta\varphi^{\perp}$) phase
differences of the modes in the two edges. The out-of-plane components are
very small compared to the in-plane ones, i.e. the precessional motion is
highly elliptical, therefore we will only consider the in-plane component,
where in- and out-of-phase oscillations are indicated by $++$ and $+-$,
respectively. From Fig. 3b, it can be seen that the phases of the high and low
frequency branches in $B$ are inverted with respect to the branches in $A$.
Moreover, at State $A$, both modes increase in frequency as the gap is
decreased, whereas in $B$, the $-+$ mode decreases in frequency, which
suggests that the $-+$ mode is responsible for switching from $B$ to $A$.

Figure 3: (a) Attempt frequency dependence on the gap for the states $A$ and
$B$. (b) Dynamic mode spectra for the states $A$ and $B$, generated by a small
thermal field. (c) The minimum energy path emerging from $A$ and $B$ (blue)
with the corresponding direction of the in-phase and out-of-phase modes
(black). The contours in the background are the same as in Fig. 1c (d)
Curvatures in the direction of the minimum energy path.

We turn to the energy landscape to investigate this scenario, where we map the
trajectories of the modes at $A$ and $B$, shown in Fig. 3c, for a gap of
$g=25$ nm. The $-+$ and $++$ modes represent straight paths in the diagonal
direction in the $(m_{y,1},m_{y,2})$ phase space. The mode splitting occurs
because the curvatures of the $(m_{y,1},m_{y,2})$-energy landscape at the
metastable minima become anisotropic, which is a result of the dipolar
coupling between the edges. We plot the minimum energy path from $A$ to $B$ in
the vicinity of the minima, and inspect the tangent with the directions of the
dynamic modes. At point $A$, the path is mostly tangential to the $++$ mode.
In contrast, the path emerging from point $B$ is mostly tangential to the $-+$
mode, which indeed shows that it is the $-+$ mode is the dominant contribution
for switching from $B$ to $A$. These findings suggests that the transitions
from $A$ to be $B$ can be stimulated using an applied microwave field with a
frequency in the $++$ branch, and vice versa with a frequency in the $-+$
branch, although the latter will be problematic in a ferromagnetic resonance
experiment, because of the symmetry of the mode.

Since the damping is low, we can assume that $f_{0,i}$ $\approx$ $f_{i}$, i.e.
the attempt frequency is approximately equal to the eigenmode in the direction
of the minimum energy path [26]. The frequency is related to the curvatures in
the energy landscape via $f\propto\sqrt{(}K)$ [27], where $K$ is the total
curvature, given in terms of the principle curvatures by
$K=\kappa_{\parallel}\kappa_{\perp}$, here $\kappa_{j}=d^{2}E/dm_{j}^{2}$,
where $j$ runs over the in- and out-of-plane components (see Appendix B for
more details). Thus, upon decreasing $g$, the curvature in the $-+$ direction
at point $B$ should decrease, while at point $A$, both curvatures in the $-+$
and $++$ directions increase. We found that the curvature associated with
bringing the magnetization out of the plane, $\kappa_{\perp}$, is more or less
constant as a function of $g$, which is due to the strong out-of-plane
demagnetizing field overwhelming any effects of proximity to the other edge.
The in-plane component, $\kappa_{\parallel}$ varies strongly, as seen in Fig.
3d. At $A$, $\kappa$ shows an increase with $1/g$, in qualitative agreement
with $f_{0,A}$, whereas the curvature at $B$ remains constant, which is not in
line with the decrease of $f_{0,B}$. A possible explanation for this
discrepancy could be attributed to the path $B-A^{\prime}$ being favored over
$B-A$ as the gap is reduced (see Appendix D), thus suppressing one of the
channels for switching. The suppression of this channel effectively lowers the
entropic contribution, by reducing the multiplicity. It has been seen before
in works by Desplat  _et al._ [28], Desplat and Kim [27], that entropic
contributions can have large effects on the attempt frequency, which occur
when the positive curvatures at the saddle point are small, allowing for an
increased number of paths from $A$ to $B$. The positive curvature at the
saddle point increases with the height of the energy barrier, thus restricting
the saddle point and lowering the amount of paths. However, this effect is
compensated by the increase in the curvature at the minima.

In conclusion, we have mapped out the energetics associated with the coupling
of intra- and inter-mesospin degrees of freedom in a system of two interacting
mesospins. It was found that, upon thermal excitation, a stochastic switching
occurs between two degenerate states, $A$ and $B$, and that the reciprocal gap
between the nanomagnets provides a handle on the symmetry and barrier height
in the energy landscape, modifying in turn the correlations between the edges.
These correlations feature a strong temperature dependency, that could
potentially be harvested in nanomagnetic neuromorphic-like systems, having the
opportunity to operate at sub-ps timescales controlled by light, as
demonstrated by Pancaldi  _et al._ [29], and Mishra  _et al._ [30]. Moreover,
by associating the dynamic modes of the system with directions in the energy
landscape, we were able to explain the trends in the attempt frequency upon a
variation of the gap value. As such, we shed light on how the thermal bath,
i.e. the magnonic excitations, couples to the stochastic fluctuations on the
next length scale. The stochastic fluctuations of the edges may in turn affect
fluctuations on the next length scale, i.e. the full switching of the
mesospin. For example, a certain state $A$ can induce a preferred helicity for
switching the magnetization in the $m_{x}$ direction, similar to the symmetry
breaking pointed out by Leo  _et al._ [31]. The double degeneracy of the
metastable states of the edge magnetization can in this case cause an increase
of a factor two in the switching rate of the mesospin island. Tunable
magnetization texture states have also been recently shown to be useful
building block of arrays, where their magnonic properties can be harnessed for
hardware implementations of reservoir computing schemes [32, 33].

##  I Acknowledgments

We wish to thank Louise Desplat and Matías Pablo Grassi for helpful
discussions. The authors acknowledge support STINT (Project No. KO2016-6889)
and the Swedish Research Council (Project No. 2019-03581). SDS gratefully
acknowledges financial support from the Carl Tryggers Foundation (Project No.
19:175 and 21:1219).

## Author Declarations

### Data availability

The data that support the findings are available upon request.

### Conflict of Interest

The authors have no conflicts of interest to disclose.

##  II Appendix A: Energy barrier calculations

Figure 4: (a) The two mesospins, prepared in state $B$. The field is applied
to region 1, which is defined by the half circle that constitutes the edge on
mesospin 1. This field pushes the state towards the saddle point $S$. (b) The
paths from a point in the vicinity of $S$ to $A$ and to $B$, mapping out the
majority of the minimum energy path from $A$ to $B$. (c) The projections of
the paths shown in (b) on the $(m_{y,1},E)$, $(m_{y,2},E)$, and the
$(m_{y,1},m_{y,2})$-planes. (d) The simulated path with the cusps removed,
after fitting to a 10th order polynomial.

The energy barrier morphology is mapped out by “pushing" the system to a point
close to the saddle point using an external field, and then letting it relax
according to the Landau Lifshitz Gilbert (LLG) equation, which is given by:

$\frac{d\mathbf{m}}{dt}=-\frac{\gamma}{1+\alpha^{2}}[\mathbf{m}\times\mathbf{H}+\alpha\mathbf{m}\times(\mathbf{m}\times\mathbf{H})]$
(3)

where $\gamma$, $\alpha$, $\mathbf{m}$, and $\mathbf{H}$ are the gyromagnetic
ratio, Gilbert damping constant, magnetization vector, and the effective
field, respectively. The effective field acts as a proxy for including the
different energy contributions in the equation of motion via

$\mathbf{H}=-\frac{1}{\mu_{0}M_{s}}\frac{\partial E}{\partial\mathbf{m}}$ (4)

where $\mu_{0}$ is the permeability of free space, and $M_{s}$ is the
saturation magnetization. The total energy is a sum of the separate energies:
$E=E_{demag}+E_{exchange}+E_{Zeeman}$. The first term on the right hand side
of the LLG equation describes the precession of the moment around the
effective field, and the second term describes the movement of the
magnetization towards the effective field. For the LLG equation to correctly
predict the minimum energy path, the precession needs to be suppressed,
therefore we set the damping to the high value of $\alpha=10$.

The system is first configured in either the $A$ or $B$ state (see Fig. 4a).
Then, the field is applied to region 1 to push one of the edges to a point
close to the saddle point $S$, after which the field is taken away, and the
system evolves for 60 ns to either local minimum, $A$ or $B$. During this
process, the energy is recorded every $1\times 10^{-12}$ s, giving a highly
resolved trajectory in phase space. The resulting curves can be seen in Fig.
4b,c. The minimum energy path indicates that the majority of the change occurs
in $m_{y,1}$, with a small response of $m_{y,2}$.

A cusp is visible near the saddle point, which is a result of the system
having relaxed in an external field, thus not being precisely positioned on
the minimum energy path. Thus, the cusp signifies the system moving from the
position in the field to the minimum energy path. This cusp is relatively
small and can be removed, after which we fit a 10th order polynomial to the
curve projected on the $(m_{y,1},E)$-plane. This fitting procedure mostly
serves to bridge the gap around the saddle point (see Fig. 4d). Strictly, the
fitting should take place in three dimensions ($m_{y,1}$ $m_{y,2}$, and $E$),
but since $m_{y,2}$ varies very little, we are only interested in the height
of the energy barrier and later on the curvature at $S$, fitting the curve
only to the projection is justified. With this method we recover the minimum
energy path from $A$ to $B$, yielding the energy barriers, $E_{L}$ and
$E_{H}$.

This method is relatively simple and can be entirely carried out using Mumax3
in the LLG framework. The only drawback is that it demands prior knowledge of
the minimum energy path through phase space, in contrast to more sophisticated
methods such as the Nudged Elastic Band (NUB) method, and variations thereof
[34].

##  III Appendix B: Curvature calculations

As stated in the main text, we evaluate the curvatures by taking the second
derivative of the energy to the magnetization, $\kappa=\partial^{2}E/\partial
m^{2}$, at certain points in the energy landscape.

###  III.1 Curvature at S

From the minimum energy path, we obtain the in-plane curvature at the saddle
point, $\kappa_{\parallel}^{S}$, for which $\partial E/\partial m_{y,1}=0$ at
the top of the energy barrier. We evaluate the curvature only in the
$(m_{y,1},E)$-plane, as the change in the $m_{y,2}$-component is negligible at
$S$.

Figure 5: (a) In-plane curvatures of the minimum energy path at $A$, $B$, and
$S$. (b) Out of plane curvatures at A and B. (c) The “wiggling" of the
magnetization at $A$, for different applied field strengths, and (d) the
curvatures for different guiding fields. The dashed line denotes the
extrapolation to 0.

###  III.2 Curvatures at A and B

Next we calculate the curvature at $A$ and $B$ via a different method, as the
morphology that we previously mapped out only provides one half of the
metastable minimum. Instead, we use a “guiding" field parallel to the edge
magnetization to wiggle the edge around $A$ and $B$. The guiding field method
is similar to the one used for mapping out the energy landscape. However, to
obtain the precise value of the curvature, we should eliminate the Zeeman
energy, $E_{Z}$, contribution. In order to eliminate this contribution, we
extrapolate to the zero field value of the curvature after calculating the
curvature for different sizes of guiding fields, from 15 mT to 1 mT in steps
of 1 mT (see Fig. 5c). Mathematically, this can be expressed as:

$\kappa^{A,B}_{\parallel}=\lim_{\mathbf{B}_{\mathbf{ext}}\to
0}\frac{\partial^{2}E}{\partial m_{y,1}^{2}}(\mathbf{B}_{\mathbf{ext}})$ (5)

The extrapolation method is based on a spline method, performed in Matlab. An
example of such an extrapolation for a gap value of $g=25$ nm, is shown in
Fig. 5d.

Here, it is necessary to take into account both perpendicular components,
$m_{y,1}$ and $m_{y,2}$, since the angle of the path with the $m_{y,1}$ axis
can be quite substantial at $A$ and $B$, and varies as a function of 1/g,
especially at State $B$ (see Fig. 5d). We can determine the curvature of the
parabola projected on the $m_{y,1}$-axis, given by $\partial^{2}E/\partial
m_{y,1}^{2}$, which in turn provides the total curvature of the parabola
through:

$E^{\prime\prime}=\frac{\partial^{2}E}{\partial m_{y,1}^{2}}cos^{2}(\theta)$
(6)

where $\theta$ is the angle between the $m_{y,1}$ and $m_{y,2}$ components of
the path.

The square root of the in-plane curvatures for all the simulated gap values
are shown in Fig. 5a.

###  III.3 Out of plane curvatures at A and B

The out-of-plane curvatures were determined by simply relaxing the
magnetization to either $A$ or $B$, by slightly pushing the magnetization out
of the plane using an external field and measuring the energy versus the
magnetization response, $m_{z,1}$. Here, the contribution from the Zeeman
interaction can be neglected, since the field is perpendicular to the
magnetization, and $E_{Zeeman}=-\mu_{0}\mathbf{m}\cdot\mathbf{H}$. A parabola
can the be fitted to find the out-of plane curvature, $\kappa^{A,B}_{\perp}$.
The result is shown in Fig. 5b.

##  IV Appendix C: Correlations

The edge magnetization correlations were quantified by the Pearson correlation
coefficient, given by:

$\rho(m_{y,1},m_{y,2})=\frac{1}{N}\frac{\int(m_{y,1}(t)-\mu_{y,1})(m_{y,2}(t)-\mu_{y,2})dt}{\sigma_{y,1}\sigma_{y,2}}$
(7)

The results of this calculation for some values of the gap are shown in Fig.
6. It can be seen that the correlations decrease with increasing temperature,
as well as an increasing gap. We fitted these data points using the following
function 111This function can be derived using
$\rho(m_{y,1},m_{y,2})=\frac{1}{T}\int m_{y,1}m_{y,2}dt$, and considering just
two intervals, $\tau_{A}$, and $\tau_{B}$, in which case
$T=\tau_{A}+\tau_{B}$. Now, $\tau_{A}$ runs from $t_{0}$ to $t_{1}$, and
$\tau_{B}$ from $t_{1}$ to $t_{2}$, thus the integral can be split into the
two contributions. Assuming that $m_{y,1}m_{y,2}=-1$ in State A, and
$m_{y,1}m_{y,2}=1$ in State B (the normalized case), we obtain
$\rho(\tau_{A},\tau_{B})=\frac{\tau_{B}-\tau_{A}}{\tau_{A}+\tau_{B}}$, which
can be seen to converge to -1 as $\tau_{B}\rightarrow 0$, and 1 if
$\tau_{A}\rightarrow 0$. Substituting the Arrhenius law, and assuming equal
attempt frequencies for A and B gives
$\rho=\frac{1-e^{\frac{E_{H}-E_{L}}{k_{B}T}}}{1+e^{\frac{E_{H}-E_{L}}{k_{B}T}}}$,
which is equal to Eq. 8.:

$\rho(m_{y,1},m_{y,2})=-\tanh\left(\frac{E_{H}-E_{L}}{2k_{B}T}\right)$ (8)

This implies that the correlations are only dependent on the difference
between the energy barriers, as well as the temperature.

Figure 6: Correlation values: The symbols are calculated from the time traces
for the spacings indicated in the legend and in nm. The dashed lines are fits
using equation 8.

##  V Appendix D: Ratio of A-B-A to A-B-A’ switching events

Figure 7 shows the ratio of switching processes in which the system starts at
state A, propagates to state B, and then goes to either state A or A’. It can
be seen that this ratio scales mostly with the inverse gap, the temperature
not playing a large role.

Figure 7: The asymmetry between A-B-A’ (‘true switches’) versus A-B-A (‘back
switches’) against the inverse gap (upper panel) and temperature (lower
panel). The two outliers at are likely the result of low statistics.

##  VI Appendix E: Dynamic modes and phase determination

The dynamic modes are produced by applying a thermal field at low temperatures
in order to excite all the possible modes. This is done for a given time
interval, during which the integrated magnetization at the edges is recorded,
and subsequently taking the Fourier transform
$\mathbf{m}^{\mathrm{edge}}(f)~{}=~{}\mathcal{F}\\{\mathbf{m}^{\mathrm{edge}}(t)\\}$.
The amplitudes are then given by
$n^{\mathrm{edge}}(f)~{}=~{}|m_{y}^{\mathrm{edge}}(f)|^{2}+|m_{z}^{\mathrm{edge}}(f)|^{2}$.

The phases of the oscillations were determined by plotting the sine of the
phase, multiplied by the magnon intensity. To this end, we first prepare the
system in either state A or B using an external magnetic field. We then run
the simulation for a certain amount of time, exposed to a low enough
temperature such that the system does not switch from A to B or vice versa.
During the simulation, we save the spatial magnetization every $5\cdot
10^{-12}$ s. We then stack these states in a 3D matrix $(x,y,t)$ and take a
Fourier transform to the frequency space:

$m_{y,z}(x,y,f)=\mathcal{F}\\{m_{y,z}(x,y,t)\\},$ (9)

after which the magnon density can be obtained through:

$n(x,y,z,f)=|m_{y}(x,y,f)|^{2}+|m_{z}(x,y,f)|^{2}.$ (10)

From the Fourier transformation, we can also the obtain the phase information
of the magnons by means of the angle of the Fourier components in the complex
plane:

$\phi_{y,z}(x,y,f)=\angle\mathcal{F}\\{m_{y,z}(x,y,t)\\}$ (11)

We then generate a 3D intensity plot with the phase included as
$sin(\phi_{y})$, and the amplitude as $n$, giving a total amplitude of
$n\times sin(\phi_{y})$. The 3D intensity plot is generated using the
‘vol3d.m’ matlab code, which uses voxels to represent intensities in 3D space.
The code for vol3d.m is written by Woodford [36], and is available from Matlab
File Exchange. The resulting plots are shown in Fig. 8.

Figure 8: The intensity and phase of the magnonic excitations for (a) State A,
in which case the edge excitations lie below 5 GHz, and the inner excitations
which lie above this value. The relevant edge excitations are highlighted by a
pink square. (b) Side view of the modes, where the two modes representing the
in- and out-of-phase modes, can be clearly distinguished. The thermal noise
causes some spreading of the mode intensity. (c) The edge mode cross section
for State B.

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# LLMs as Academic Reading Companions: Extending HCI Through Synthetic
Personae

Celia Chen  College of Information Studies
University of Maryland<EMAIL_ADDRESS>and  Alex Leitch  College of
Information Studies
University of Maryland<EMAIL_ADDRESS>
###### Abstract.

This position paper argues that large language models (LLMs) constitute
promising yet underutilized academic reading companions capable of enhancing
learning. We detail an exploratory study examining Anthropic’s Claude.ai, an
LLM-based interactive assistant that helps students comprehend complex
qualitative literature content. The study compares quantitative survey data
and qualitative interviews assessing outcomes between a control group and an
experimental group leveraging Anthropic’s Claude.ai over a semester across two
graduate courses. Initial findings demonstrate tangible improvements in
reading comprehension and engagement among participants using the AI agent
versus unsupported independent study. However, there is potential for
overreliance and ethical considerations that warrant continued investigation.
By documenting an early integration of an LLM reading companion into an
educational context, this work contributes pragmatic insights to guide
development of synthetic personae supporting learning. Broader impacts compel
policy and industry actions to uphold responsible design in order to maximize
benefits of AI integration while prioritizing student wellbeing.

LLMs, education, AI ethics, synthetic personae

††ccs: Applied computing Computers in other domains Publishing††ccs:
Information systems Information systems applications Document management and
text processing Optical character recognition††ccs: Information systems
Information systems applications Document management and text processing
Markup languages††ccs: Human-centered computing HCI design and evaluation
methods Usability testing

##  1\. Introduction

The recent, explosive popularity of large language models (LLMs) like ChatGPT
has sent shockwaves through higher education, evoking both optimism and fear.
Alarmist voices dominate as universities scramble to update academic policies
aiming to deter cheating, citing concerns these emerging “artificially
intelligent peers” erode original human scholarship [8].

These reactive stances overlook many previous waves of disruptive classroom
innovation, such as SMARTboards [13], incorporation of laptops and personal
computing devices, and smartphones in the classroom. These previous technical
interventions focused on changing the material conditions of education to
incorporate dynamic information manipulation tools, which typically had a
level of “charisma” in the sense used by Morgan G. Ames in their 2019 work
“The Charisma Machine” [1]. Ames [1] argues that technologies become
charismatic by not just doing something, but also by displaying a dream for
the world through that doing. This construction means that SMARTboards not
only allow an educator to wipe clean a board or share a document, but allow a
school that has such technology to show they are aligned with a specific
vision of what the future finds valuable: having sufficient resources to
deploy a screen that can connect to the internet - itself a well of expense
and dreaming - rather than a slab of slate. The actual function of the
technology, or the quality of the education, becomes secondary to the image of
innovation presented by the screen itself.

Past research examining the impact of these technical-material deployments on
actual students provide good reason to support the current wave of caution
about LLMs [1], [15], [16]. The problem shared in the research is that
typically, these charismatic - visionary - machines are deployed in a way that
expects users to “pick it up” by deploying sufficient curiosity to lead deep,
independent explorations, often without documentation support or structured
tutorial assistance. This has been explored elsewhere, particularly in
Warschauer and Ames [15] via the construction of the “technologically
precocious boy-child,” a legacy of Seymour Papert’s [10] work on childhood
programming/problem-solving skills development. The problem of unsupported
technical intervention is presented again by Wood and Ashfield [16], who
showcase dilemmas around effectively leveraging interactive whiteboards’
affordances to enhance in-place pedagogical practices.

Taken as a whole, these present a clear case for caution as to how any
particular “innovative system” can quickly move the focus of work from the
educational outcomes to other priorities. They compel policymakers,
technologists, and educators to humility. Whether or not LLMs yet fit the
definition of a charismatic machine is an open question: we do not know if
they offer a strong vision for a future, since they tend not to have a
specific physical form as yet. They nonetheless fit the definition of a
charismatic technology: they are being freely deployed at a breakneck pace
into every text-based user productivity system to support a visionary type of
technical future, once again without training support. Users are encouraged to
treat these engines as simple search tools, which they are not [6], [11].
Rather than reactive policies or utopian solutions, integrating emergent tools
like generative AI requires contextual wisdom and ethical foresight to
maximize benefits while avoiding potential pitfalls.

Following Hayles and Burdick’s work in Writing Machines [7], the LLM as a
conversational companion seems to realize a long-held dream of simulation:
someone to talk to, who can talk back, who will not get bored, and will
flatter their conversational partner sufficiently that such partner will treat
them as a person. Some models - notably Claude.ai - are better than others at
the key role of support, which is emotional validation during the learning
process. This type of emotional validation is care work [5] that is both
draining and difficult to train into the typical working population of the
university, other students or graduate students. As with most care work,
emotional validation, low-level question answering, and repetitive patience is
not well-compensated, and to date cannot be generalized. It is impossible in
current educational models to provide every student with their own always-on
tutorial assistant who can conversationally answer technically complex
questions with a relatively high degree of accuracy, in a timely manner. The
risk, though, is that the LLM hallucinates with such fluency that it not only
flatters its audience, but like any more human charm engine, also lies to them
effectively enough to preclude learning.

The flattery and validation of the simulation, coupled to a deep archive,
offers this charismatic vision of education: approached earnestly, it is now
possible to get clear, reasonably specific answers with an inhuman degree of
patience. The promise is that with an LLM to guide them, anyone can once again
learn anything, as before with internet search. The question then becomes: how
good are the answers, and how will this resource be used in practice.
Answering this question requires that LLMs should be examined as a tool, and
their integration or rejection in the classroom space should be explored
directly to discover their actual pedagogical potential before they are
generally celebrated or disdained.

Our position contends reflective scholarship playing out such integration in
context offers urgent guidance. Specifically, exploratory efforts highlighting
available benefits and pitfalls of generative AI assistants provides
actionable intelligence steering our best path forward. The following study
documents one such investigation, assessing performance of an AI reading
companion named Claude designed to scaffold comprehension of complex texts.

##  2\. Proposed Study Methodology

This position paper argues that LLMs have promising yet underutilized
capability as academic reading companions that can enhance learning.
Specifically, we detail an exploratory study examining an LLM-based tool from
Anthropic called Claude.ai which serves as an interactive assistant helping
students comprehend complex textbook content.

Among available LLMs, this study elected to utilize Claude.ai for its
commitment to safety and transparency, aiming to uphold ethical standards
vital for educational deployment. Research shows risks of potential harms from
overly trusting LLMs without critical evaluation [3]. Unlike alternatives such
as ChatGPT, Claude.ai refrains from posing as an expert on topics like
assignments to avoid enabling cheating. Its capabilities for long-form
summarization facilitate explaining complex texts while its affective
responses help smooth dialogues around difficult content [12]. There is
presently limited third-party peer-reviewed validation of some claimed
capabilities. Overall though, Anthropic’s alignment of Claude.ai with ethical
values made its exploratory integration the best choice for this educational
study, while highlighting needs for continued research assessing LLMs impacts
on vulnerable populations.

The study incorporates a between-subjects experimental design with
participants randomly assigned to either a control group or an experimental
group aided by the Claude.ai reading companion LLM. The target population
comprises 60 students (30 per group) enrolled across two interaction design
courses at the University of Maryland who volunteer to participate.
Quantitative data collection involves pre, mid, and post-study surveys
assessing self-reported comprehension and engagement. The experimental group
will also submit textual dialogues with the LLM assistant. Qualitative methods
further encompass post-study interviews with 5 students from the aided group
regarding their experience.

Recruitment will occur through an in-class announcement emphasizing the
voluntary nature of participation without impact on grades. Interested
eligible students can review details and provide consent. Inclusion criteria
entail adults age 18+ able to read English at a college level with regular
computer/internet access. Power calculations determined a total sample of 52
(26 per group) has 80% power to detect a moderate effect size (the minimum
target). By recruiting 60 participants, the study exceeds thresholds for
sufficient statistical conclusion validity.

We predict initial findings will demonstrate tangible improvements in reading
comprehension and interest among those using Claude.ai as a reading companion.
This would provide empirical evidence of benefits for thoughtfully
incorporating LLMs as academic aids. Findings may inform design guidelines
maximizing performance of LLMs while upholding inclusive, equitable ideals.

##  3\. Addressing Skepticism Towards LLMs in Academia

Integrating emergent technologies like LLMs into educational contexts
inevitably evokes skepticism and reasonable counter-perspectives that should
be discussed.

Some may argue LLMs enable laziness or cheating among students rather than
properly enhancing learning. Critics also express concern that these AI tools
take agency away from human readers and writers; deskilling students in ways
reminiscent of other classroom technologies such as calculators, spell-
checking and grammar-checking software [2]. Additionally, technical flaws in
LLMs regarding potential bias, inaccuracy, or harm must be weighed given
prominent examples like racist or nonsensical outputs [3].

A common question asks whether reliance on AI assistance diminishes students’
own critical thinking and metacognitive skills over time [4]. Structured
scaffolding that later fades under an appropriate framework constitutes
established educational practice [14], avoiding perpetual dependence akin to
losing skills. Claude.ai’s constraints against posing as an expert also reduce
this risk. Empirical evidence would confirm or refute such an argument through
measures of student outcomes and self-efficacy.

Another frequent criticism contends that LLM-based agents could propagate
biases or misinformation among impressionable learners who lack evaluation
abilities. While concerning, any experimental pilot would establish ethical
safeguards and debrief participants on the tool’s actual capabilities to
mitigate such dangers. Responsible informed consent and moderated testing
limits vulnerabilities, producing better risk-benefit assessments.

Ultimately the core argument of this paper stands - that thoughtful
integration of reading companions warrants exploration [9]. With careful
design and oversight, LLMs introduction could constitute a net benefit over
potential costs. Our position contends not ignorance or reactionary policies,
but responsible investigatory efforts assessing evidence-based applications
that will best guide appropriate adoption of policies to meet students’ needs.

##  4\. Conclusion

This position paper’s central stance affirms that LLMs can serve as promising
academic reading companions for students and deserve constructive exploration.
Despite potential counter-arguments regarding risks of plagiarism, harm,
deskilling, or technical flaws, we argue that integration with careful
oversight could constitute a net benefit over downsides. Our exploratory study
examined one such LLM tool, Anthropic’s Claude.ai finding initial empirical
evidence of improved reading comprehension and engagement versus typical
independent study. While emergent tensions around human-AI collaboration in
learning contexts will require vigilance, updating reactive policies to more
responsibly harness these technologies through evidence-based research offers
the most hope moving forward. The next steps for the field entail larger
randomized control trials building on these preliminary insights to recommend
design guidelines for maximizing the powerful performance of LLMs in clearly
augmenting mental abilities while maintaining student well-being and ethics at
the core. Furthermore, multi-stakeholder involvement including educators,
students, AI developers and policymakers would strengthen developing large
language models that enhance rather than endanger vulnerable populations
through negligent practices, mindful of historically recurring sociotechnical
dilemmas during waves of classroom computerization. Nevertheless, the immense
promise of leveraging LLMs productivity justifies continued progress
empowering students through overseen academic partnerships, rather than
abandoning innovation out of fallacious fears or complacency with the status
quo amidst accelerating demands for human capital development. This research
constitutes an initial yet actionable step in that direction - thoughtfully
pioneering integration of assistive reading LLMs while contributing actionable
intelligence about their prospects for consequently updating institutional
strategies through participatory paradigms upholding inclusive, equitable
ideals in increasingly digital environments.

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