# Magnetic and electronic properties of Eu5In2Sb6

M. Victoria Ale Crivillero  Max Planck Institute for Chemical Physics of
Solids, Nöthnitzer Strasse 40, 01187 Dresden, Germany  Sahana Rößler  Max
Planck Institute for Chemical Physics of Solids, Nöthnitzer Strasse 40, 01187
Dresden, Germany  S. Granovsky  Institut für Festkörper- und Materialphysik,
Technische Universität Dresden, D-01062 Dresden, Germany  M. Doerr  Institut
für Festkörper- und Materialphysik, Technische Universität Dresden, D-01062
Dresden, Germany  M. S. Cook  Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA  Priscila F. S. Rosa  Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA  J. Müller  Institute of Physics, Goethe-
University Frankfurt, 60438 Frankfurt(M), Germany  S. Wirth
<EMAIL_ADDRESS>Max Planck Institute for Chemical Physics of Solids,
Nöthnitzer Strasse 40, 01187 Dresden, Germany

###### Abstract

The intermetallic compound Eu5In2Sb6, an antiferromagnetic material with
nonsymmorphic crystalline structure, is investigated by magnetic, electronic
transport and specific heat measurements. Being a Zintl phase, insulating
behavior is expected. Our thermodynamic and magnetotransport measurements
along different crystallographic directions strongly indicate polaron
formation well above the magnetic ordering temperatures. Pronounced
anisotropies of the magnetic and transport properties even above the magnetic
ordering temperature are observed despite the Eu2+ configuration which testify
to complex and competing magnetic interactions between these ions and give
rise to intricate phase diagrams discussed in detail. Our results provide a
comprehensive framework for further detailed study of this multifaceted
compound with possible nontrivial topology.

## Introduction

Materials in which the electronic and magnetic properties are strongly coupled
hold great promise for applications in spin electronics [1]. From a
fundamental point of view, the underlying entanglement of magnetic, electronic
and structural degrees of freedom in correlated electron materials can give
rise to unexpected and often spectacular physical phenomena, e.g. high-
temperature superconductivity in cuprates [2] and colossal magnetoresistance
(CMR) in magnetic semiconductors and manganites [3, 4]. A characteristic of
these coupled degrees of freedom is the appearance of diverse electronic
phases, along with phase separation and pattern formation [5, 6, 7, 8]. In
this respect, Eu compounds are of particular interest because of their often
strong exchange interaction between the charge carriers and the localized spin
of Eu2+ ions [9, 10, 11], which can give rise to a localization of the charge
carriers (of low density) around the Eu2+ spins—so called magnetic polarons.
In addition, the ground state of Eu2+ is an isotropic ${}^{8}S_{7/2}$
configuration, limiting crystalline electric field contributions to anisotropy
to higher order. In general, Eu compounds can be viewed as model systems for
CMR effects and polaron formation [12, 13, 14].

Considering Eu-based compounds, research concentrated on EuO, Eu chalcogenides
and EuB6, mostly because of their relatively simple crystallographic
structures [11, 15, 16]. Recently, the topological properties of materials and
their relation to crystal symmetry has been highlighted [17, 18].
Specifically, materials crystallizing in nonsymmorphic space groups are prone
to topological states [19, 20]. Interestingly, space group $Pbam$ (No. 55)
fulfills these requirements [20] including the Zintl phase Ba5In2Sb6 [21].
This drew attention to the Eu-based counterpart Eu5In2Sb6 [22] which was known
as a narrow gap semiconductor [23]. Indeed, Eu5In2Sb6 exhibits an
extraordinarily large CMR effect and signatures of polaron formation, with the
possibility of axion insulating states within the antiferromagnetic regime
[22]. In addition, band structure calculations for Eu5In2Sb6 in its
paramagnetic state predicted nontrivial surface states [24] (similar to the
isostructural nonmagnetic compound Ba5In2Sb6 [20]), which have not been
observed [25].

However, the complex structure of Eu5In2Sb6 allows for three
crystallographically different Eu sites [23], and the Eu sublattices may
magnetically order independently [25]. In this case, the Dzyaloshinskii–Moriya
interaction needs to be taken into consideration. Independent of the
crystallographic site, Eu is divalent [26] and hence, fulfills the Zintl rule.
It is noteworthy that, even though the Eu2+ ions with orbital angular momentum
$L=0$ are isotropic, there appear to be multiple, anisotropic exchange
interactions [22, 25]. All this results in a complex magnetic structure which
is not yet resolved experimentally.

Here we report on a comprehensive study of Eu5In2Sb6 by magnetic, electronic
transport and specific heat measurements in an effort to establish the
magnetic field–temperature ($H$–$T$) phase diagrams with $H$ applied along
different crystallographic directions. The former two properties are highly
anisotropic despite the Eu2+ state indicating complex magnetic interactions.
The temperature evolution of the studied properties establish Eu5In2Sb6 as a
rare example of a material exhibiting polaron formation in an antiferromagnet,

Figure 1: (a) Magnetization in dependence on magnetic field for up and down
sweeps. Data at 2 K are shown for fields along all three crystallographic
directions, for $H\parallel c$ also data at 5 K and 8 K are presented. Lines
indicate measurements up to 14 T, markers up to 7 T. Inset: Three groups of
crystallographically different Eu positions (reddish color, marked Eu1, Eu2
and Eu3) within the crystal structure of Eu5In2Sb6 (Sb in green, In not shown)
viewed along the $c$-axis. (b), (c) Temperature evolution (in steps of 1 K) of
magnetization curves for $H\parallel a$ and $H\parallel b$ emphasizing the
metamagnetic transitions at $H_{\rm m}(T)$ (arrows).

likely also of anisotropic nature. Such electronically inhomogeneous
properties in an intermetallic compound with precise electron count, i.e. in
an insulating environment, provide a rich playground for possibly new quantum
states, specifically when time-reversal symmetry is broken by incorporating
magnetic (here rare earth) elements [27] and materials of nonsymmorphic
symmetry are investigated [19].

## Results and Discussions

### Magnetic properties

As already mentioned, the crystallographic structure of Eu5In2Sb6 gives rise
to three different Eu sites, cf. inset to Fig. 1(a): Eu(2) and Eu(3) are
surrounded by six Sb (with the octahedra around Eu(3) slightly larger than for
Eu(2)), while Eu(1) has two In and seven Sb close neighbours [23]. The nearest
Eu distances are between Eu(2)–Eu(3) (3.7987 Å) whereas the nearest Eu to an
Eu(1) is spaced 4.072 Å apart. Magnetic ac susceptibility $\chi^{\prime}(T)$
clearly shows two magnetic transitions at $T_{\rm N1}\approx$ 14.1 K and
$T_{\rm N2}\approx$ 7.2 K [28, 22], see Fig. 2(a) for $H\parallel a$, which
are both suppressed in applied magnetic fields of $\mu_{0}H=$ 2 T.

The magnetization curves $M(H)$ for fields applied along the main
crystallographic axes of Eu5In2Sb6 largely conform to the behavior expected
for an antiferromagnet, Fig. 1; there is an almost linear increase of $M(H)$,
specifically for $H\parallel c$, and a saturation at high fields. The
saturation value of $M_{\rm sat}\approx 7.0\;\mu_{\rm B}$/Eu for $H\parallel
c$ is consistent with the expected saturation magnetic moment of Eu2+,
$g\mu_{\rm B}J=7\mu_{\rm B}$. For $H\parallel a$ and $H\parallel b$ slightly
higher values of $M_{\rm sat}\approx 7.5\;\mu_{\rm B}$/Eu were observed.

It is important to emphasize again the Eu2+ state with $L=0$. In this case,
magnetic anisotropy can be attributed to anisotropic exchange, dipolar
interactions or even crystalline electric fields of higher order. Hence, the
clear anisotropic behavior below $T_{\rm N1}$, as seen in the magnetization
curves $M(H)$, Fig. 1, for fields applied along the main crystallographic
axes, is likely due to anisotropic exchange between the Eu-sites. Notably,
there are magnetic transitions for $H\parallel a,b$, but not for $H\parallel
c$, which indicates that the magnetic moments are (primarily) aligned within
the $a$-$b$ plane [25]. Figures 1(b) and (c) exhibit the disappearance of the
metamagnetic transitions with increasing temperature for $H\parallel a$ and
$H\parallel b$, respectively. Interestingly, this transition is completely
vanished at $T\gtrsim T_{\rm N2}$ for $H\parallel a$, but can be followed up
to $T\approx T_{\rm N1}$ for $H\parallel b$. In the latter case, there is only
a weak shift of the transition field $H_{\rm m}$ up to $T_{\rm N1}$.
Furthermore, there is no notable remanent magnetization found after applying a
magnetic field along any crystallographic direction, as expected for an
antiferromagnetic spin configuration. However, it is essential to note that
the precise magnetic structure of Eu5In2Sb6 has not been reported yet. In the
absence of detailed experimental insight, we consider a predominantly A-type
antiferromagnetic order with antiparallel stacking along $c$, as suggested by
density functional theory (DFT) calculations [25], see upper inset to Fig.
2(a). In all likelihood, the actual spin configuration is

Figure 2: (a) Temperature dependent susceptibility $\chi^{\prime}(T)$ for
different magnetic fields $H$ applied along the crystallographic $a$-axis
illustrating the temperature shifts of the transitions with $H$. Lower inset:
Slow $T$-sweep (0.1 K/min) reveals a small offset for different ac-field (2
Oe) drive frequencies for the low-$T$ transition. Upper inset: suggested spin
configuration. (b) $\chi^{\prime}(T)$ for different $H$ along the $c$-axis.
Inset: Imaginary component of susceptibility, $\chi^{\prime\prime}(T)$ for the
same conditions. For comparison, $\chi^{\prime\prime}(T)$ at 317 Hz and 7 T is
shown. Different samples were measured in (a) and (b).

more complex and non-collinear, and may even support very weak ferromagnetism
by spin-canting resulting from Dzyaloshinskii-Moriya interactions allowed in
this low crystallographic symmetry [22, 25, 29].

At even lower temperature, $T\lesssim 3$ K, there is a third transition
observed only for $H\parallel a$. This transition is readily suppressed to
below 2 K at fields as small as 0.1 T, and exhibits a very weak frequency
dependence, see lower inset to Fig. 2(a). At present, we cannot rule out In
inclusions (resulting from the sample flux growth) causing this transition.

In line with the above-suggested spin configuration, $\chi^{\prime}(T)$
exhibits a different behavior for $H\parallel c$, Fig. 2(b): Here, the
transition at $T_{\rm N1}$ becomes ever more pronounced while shifting to
lower temperature with increasing dc-field $H$. Also, the transition at
$T_{\rm N2}$, albeit shifted down to about 3.3 K, can still be recognized at 5
T. The shallow maximum of $\chi^{\prime}(T\sim\rm{25\;K},\mu_{0}H=\rm{7\;T})$
and the peculiar $T$-dependence of the imaginary component of the
susceptibility, $\chi^{\prime\prime}(T)$, inset of Fig. 2(b), will be
discussed below.

The numerous transitions and different dependencies of $\chi^{\prime}$
certainly testify to an intricate response of Eu5In2Sb6 to magnetic fields. To
gain further insight, the isothermal magnetization was measured upon rotating
the sample in constant applied fields. The results of one exemplary
measurement is presented in Fig. 3 for fields of $\mu_{0}H=$ 0.1, 3 T and at
$T=2$ K, with the sample being rotated around its crystallographic $a$ axis
(for more results on sample rotation see Supplementary Information, Fig. S1).
At small fields (0.1 T), we find the expected two-fold symmetry: There is
little response of the local moments for $H\parallel b$, where $b$ is likely
the easy magnetization direction [22, 25] along which the moments are already
aligned in the ground state, while the moments can more easily be turned
towards the $c$ direction even by a small field. At $\mu_{0}H=$ 3 T, the curve
is just offset for a broad angular range around the $c$ direction. In
addition, however, there is a pronounced maximum for $H\parallel b$, which is
quickly suppressed within a range of $\pm 20^{\circ}$. This observation, along

Figure 3: Magnetization in dependence on sample orientation with respect to
$H$ for two applied fields ($\mu_{0}H=$ 0.1 T, 3 T) below and above
$\mu_{0}H_{\rm m}\approx 2.2$ T at $T=2$ K. The sample is rotated in the
$b$–$c$ plane, i.e. around the $a$-axis, with the alignment of the respective
axis with the direction of $H$ marked by arrows.

with the jump in magnetization at $H_{m}$ for $H\parallel b$ shown in Fig.
1(c), is a clear indication for a field-induced change in the magnetic
structure (possibly a spin-flop transition). This assignment is further
supported by the very weak temperature dependence of $H_{m}(T)$ and is
consistent with the $b$ axis being the magnetically easy direction.

There are several indications for Eu5In2Sb6 being a candidate material for the
formation of magnetic polarons near the magnetic ordering temperature, ranging
from colossal magnetoresistance [22] and piezoresistance [30] to
characteristic changes of the Eu2+ electron spin resonance (ESR) [31]. Here,
magnetic polarons refer to magnetically ordered clusters within which
conduction electrons are localized via strong exchange interaction with the
4$f$ moments, giving rise to spin-polarization [32]. The $\chi^{\prime}(T)$
data for small applied field exhibit a tiny deviation from

Figure 4: (a) Specific heat divided by temperature, $C_{p}/T$, in dependence
on $T$ for applied fields $H\parallel b$ up to 9 T. The insets show enlarged
views of $C_{p}(T)$ within certain temperature ranges (in units of J/mol K).
(b) $C_{\rm ph}$ contribution from a Debye model (green curve) fitted to the
experimental $C_{p}(T)$-data (black) within 60 K $\leq T\leq$ 200 K and used
to estimate $C_{\rm mag}/T$ (magenta). Inset: photograph of one representative
Eu5In2Sb6 sample with the long dimension corresponding to the $c$ axis.

Curie-Weiss behavior below about 180 K, and a more obvious one around 50–60 K
(see Supplementary Information, Fig. S2) which was argued to be caused by the
formation of magnetic polarons and correlations between them, respectively.
Very likely, the polarons grow in size with increasing applied field such that
correlations become more pronounced. Consequently, the deviations from Curie-
Weiss behavior are stronger for larger field, even resulting in a shallow
maximum of $\chi^{\prime}(T)$ at 7 T, see Fig. 2(b). This picture is
corroborated by the imaginary component of the susceptibility,
$\chi^{\prime\prime}(T)$, presented in the inset of Fig. 2(b).
$\chi^{\prime\prime}$ is related to dissipative magnetic processes and is
typically zero in antiferromagnetic phases. Interestingly, for large enough
applied fields, $\chi^{\prime\prime}(T\lesssim 50{\rm K})$ is finite,
indicating ferro- or ferrimagnetic contributions, and negative, consistent
with an inhomogeneous state of magnetization [33]. Moreover,
$\chi^{\prime\prime}$ is positive, if measured at 317 Hz and 7 T, emphasizing
a changed response to higher drive frequencies.

The behavior just described is also observed when the magnetic field is
applied along the crystallographic $b$ axis. In comparison to other compounds
for which a magnetic polaron scenario is considered, the phenomenology found
in Eu5In2Sb6 is consistent with findings for EuB6 (see, e.g., susceptibility
data in Ref. [34]), but differs from the one reported in Sr doped LaCoO3
samples [35]. In the latter case, the spin-glass behavior of ferromagnetic
clusters embedded in a non-ferromagnetic matrix is thought to be revealed by a
frequency-dependent cusp in the ac susceptibility data.

### Heat capacity

Results of the measurements of the specific heat $C_{p}(T,H)$ for different
magnetic fields $H\parallel b$ are presented in Fig. 4. Here, the main panel
shows $C_{p}/T$ as a function of $T$, while the insets zoom into $C_{p}$-data
within two temperature ranges of interest. The two transitions at $T_{\rm N1}$
and $T_{\rm N2}$ as well as their suppression in magnetic field are clearly
revealed. Importantly, the impact of applied magnetic fields can be recognized
up to temperatures of about 60 K whereas at higher temperatures there are no
obvious differences between the data obtained at different magnetic fields.
The experimentally observed zero-field $C_{p}/T$-data exhibit a pronounced
maximum at $T_{\rm max}\sim 45$ K which shifts to lower temperatures with
increasing magnetic field ($T_{\rm max}^{\rm 9T}\sim 37$ K). Consequently, we
will relate this maximum to the magnetic properties of Eu5In2Sb6.

To analyse $C_{p}$, magnetic, electronic and phonon contributions to the
specific heat are considered, $C_{p}(T,H)=C_{\rm mag}+C_{\rm el}+C_{\rm ph}$.
Here, $C_{\rm el}=\gamma\,T$ and $C_{\rm ph}$ are described by a linear-in-$T$
and a Debye model, respectively. Given the number of parameters, the above
description of $C_{p}$ cannot unambiguously be fitted to our experimental
data. Therefore, $(C_{\rm el}+C_{\rm ph})$ was fitted to the experimental data
only for 60 K$\,\leq T\leq\,$200 K, i.e. in a $T$-range above the temperature
below which differences for different magnetic fields were seen. The fit
yields a Debye temperature of $\theta_{\rm D}=$ 197 K [green curve in Fig.
4(b)] and a vanishing value for $\gamma$. Using these fit results to subtract
the corresponding contributions from the experimental data in the low-
temperature range, $C_{\rm mag}/T=C_{p}/T-C_{\rm ph}/T$ shown as magenta curve
in Fig. 4(b), reveals a significant magnetic contribution to the specific heat
even above $T_{\rm N1}$. A very similar behavior was observed for a second set
of samples measured up to 280 K to improve on the Debye fit, see Supplementary
Information Fig. S3. Integrating $C_{\rm mag}/T$ up to $T=15$ K, i.e. just
above $T_{\rm N1}$ but well below $T_{\rm max}$ yields about 80% of the
expected magnetic entropy $S_{\rm mag}=5R\ln(2S+1)\approx$ 86.4 J/mol K (where
$R$ and $S$ are the gas constant and the spin momentum of Eu2+, respectively),
i.e. the integration has to be carried out up to higher temperatures to
recover the full magnetic entropy. This inference holds despite the
uncertainty of our analysis as well as for

Figure 5: $H$–$T$ phase diagrams for magnetic fields aligned along different
crystallographic directions, left: $H\parallel a$, center: $H\parallel b$,
right: $H\parallel c$, as extracted from ac and dc susceptibility $\chi$,
magnetization $M$ and specific heat $C_{p}$ measurements. Violet arrows depict
a possible magnetization orientation with respect to magnetic field (gray
arrows). To enable comparison, sample demagnetization effects are taken into
consideration.

our data in magnetic field (see Supplementary Information Fig. S3) and
reinforces our conclusion above that the feature related to $T_{\rm max}$ is
magnetic in nature.

Near $T_{\rm max}\sim 45$ K a strong deviation of the susceptibility from a
Curie-Weiss law was observed (Fig. S2) which was previously ascribed to the
onset of interactions between polarons [22]. Following this interpretation, we
assume this maximum to result from a _field-dependent_ Schottky-like anomaly.
Indeed, there are examples of such Schottky-like anomalies, where the
temperature $T_{\rm max}$ decreases with increasing magnetic field [36]. Also,
a low-temperature Schottky anomaly in LaCoO3 has been previously linked to
magnetic polarons [37]. If an interpretation of $T_{\rm max}$ based on a
polaron-interaction scenario is correct, the _formation_ of polarons is
expected to take place at significantly higher temperatures. Here we note that
there is a small hump observed in $C_{p}(T)$ at $T_{\rm hump}\sim 130$ K,
lower inset to Fig. 4(a), a temperature which is somewhat lower compared to
the one expected for polaron formation from susceptibility, Fig. S2, and Refs.
[22, 31].

A Schottky contribution $C_{\rm Sch}$ to $C_{p}$ is associated with a two
level system [38]. We speculate that such a two-level scenario may result from
ferromagnetic (fm) correlations related to the magnetic polarons on the one
hand, and antiferromagnetic interactions between Eu ions on the other hand.
The latter are expected to prevail at lower temperature near $T_{\rm N1}$
while fm correlations are likely the dominant ones at higher temperatures,
where they are expected to be stabilized in magnetic fields. Indeed, assuming
a lattice background independent of magnetic field and fitting the specific
heat data well above $T_{\rm N1}$ with a Debye term and a magnetic field-
dependent Schottky contribution reveals a systematic increase of the level
splitting from $\delta\sim 2.2$ meV at $\mu_{0}H=0$ to about 4.5 meV at 9 T,
while at the same time the magnitude of the magnetic Schottky contribution
shows a $\sim 30$ % decrease with increasing field. This explains the apparent
shift of $T_{\rm max}$ in $C_{p}/T(T)$ to lower temperatures with increasing
field.

Albeit a Schottky anomaly linked to polaron interaction is consistent with the
observed temperature evolution of $C_{p}$, we cannot exclude alternative
explanations. For instance, a temperature well above $T_{\rm N1}$ required to
gain the full magnetic entropy might also result from some partial ordering
due to frustration between different Eu sites. We wish to stress, however,
that—independent of any fitting or interpretation—there is a considerable
amount of magnetic contribution to $C_{p}$ found at temperatures well above
$T_{\rm N1}$.

### Magnetic phase diagrams

In order to gain more insight into the magnetic behavior we construct $H$–$T$
phase diagrams for the different crystallographic directions making use of our
magnetization, susceptibility and specific heat data. The resulting phase
diagrams for $H\parallel a$ and $H\parallel c$ are qualitatively similar, with
$c$ clearly being the magnetically hard direction. We note that for
$H\parallel a$ the jump of $M(H)$ at 2 K and $\sim$1.6 T [Fig. 1(b)] is
indicative of a metamagnetic transition, similar to the observation for
$H\parallel b$ [Fig. 1(c)]. However, in contrast to the latter case, the
transition for $H\parallel a$ is readily suppressed near $T_{\rm N2}$. This
makes an assignment of this transition for $H\parallel a$ to $H_{\rm m}$ or
$H_{\rm N2}$ difficult. Moreover, only in case of $H\parallel a$ is there a
slight change of slope of $M(H)$ near 5 T, Fig. 1(a), and a third transition
seen in $\chi^{\prime}(T)$ at $\mu_{0}H\leq 0.1$ T, Fig. 2(a).

A more complex behavior is found for the $b$ axis, i.e. the magnetically easy
direction, where a field-induced change in the magnetic structure (likely a
spin-flop transition) at $H_{\rm m}$ is observed. This gives rise to a
bicritical point, where $H_{\rm m}(T)$ terminates at $H_{\rm N1}(T)$. Here,
$T_{\rm N1}(H)$ exhibits the characteristic kink at the bicritical point [39].
The non-monotonic $H$-dependence of $T_{\rm N2}(H)$ indicates a

Figure 6: Resistivity $\rho(T)$ in dependence on temperature for current $I$
along different directions: (a) $I\parallel a$ and $H\parallel c$ and (b)
$I\parallel c$ and $H\parallel b$. Inset in (a): Zoom into the low-$T$ range
on a logarithmic $T$-scale. At zero field, a kink is observed around 27 K
(arrow). Inset in (b): Ratio $r$ of resistivities for the different
configurations shown in (a) and (b). The zero-field maximum near 27 K is
suppressed with field. At $H=9$ T, $r$ is of order 0.3 (green curve). All
panels share the same color code for magnetic fields. (c) Ratio of
resistivities measured at fields of 9 T and 0 T, as shown in (a). (d)
Photographs of the sample, scale bar 400 $\mu$m.

stiffening of the ordering taking place at $T_{\rm N2}$ for fields below the
spin-flop transition, while above $H_{\rm m}$ the ordering is successively
suppressed with increasing field. One may speculate that a similar interplay
of exchange interactions is at play for $T_{\rm N1}(H)$ at very small fields
$H\parallel a$. All these observations point to a complex magnetic exchange
and ordering mechanism in Eu5In2Sb6 as can be expected from the
crystallographic structure.

### Resistivity measurements

Electronic transport in general, but specifically in a material as complex as
Eu5In2Sb6 can be influenced by a number of factors (this may also include In
inclusions originating from sample growth). Indeed, resistivity measurements
carried out on different samples exhibited somewhat different
$\rho(T)$-results, in particular for $T<T_{\rm N1}$ (compare results shown in
Fig. 6 and Supplementary Information Fig. S4). Therefore, resistivity data
were not included in constructing the phase diagrams of Fig. 5. Rather, we
make use of the insight gained from other measurements so far when discussing
the electronic transport results in the following.

In Fig. 6 results of the transversal resistivity $\rho(T,H)$ are compared for
currents $I$ along the $a$ and the $c$ direction, respectively. Clearly, the
strong anisotropy of Eu5In2Sb6 is also seen in $\rho(T)$, specifically in zero
field. For, $I\parallel a$ the resistivity $\rho_{I\parallel a}(T)$ rises much
more rapidly upon cooling than $\rho_{I\parallel c}(T)$. This behavior (also
seen in Fig. S4) may be related to the crystal structure of Eu5In2Sb6 which is
characterized by infinite [In2Sb6]10- ribbons oriented parallel to the
crystallographic $c$ axis [23] and hence, favors transport for
$\rho_{I\parallel c}(T)$. Near 27 K, $\rho_{I\parallel a}(T)$ exhibits a kink
into a much less-steep behavior [see arrows in insets to Fig. 6(a) and Fig.
S4] before dropping by several orders of magnitude below $T_{\rm N1}$. For
$I\parallel c$, a smaller hump is detected, in agreement with Ref. [22]. In
consequence, the ratio of resistivities, $r(T)=(\rho_{I\parallel
a})/(\rho_{I\parallel c})$ measured for the two different current directions
exhibits a pronounced maximum near 27 K, inset of Fig. 6(b). The second,
sharper maximum at 12.7 K is likely related to the antiferromagnetic order at
$T_{\rm N1}$. Larger spin fluctuations can be expected in the plane
perpendicular to the magnetically hard $c$ direction, increasing
$\rho_{I\parallel a}$, an effect that quickly subsides for decreasing $T$,
i.e. away from the magnetic transition.

As mentioned above, $\chi^{\prime}(T)$ exhibits a deviation from Curie-Weiss
behavior below about 50 K (Fig. S2). Therefore, the rise of $r(T)$ for
$T\lesssim 50$ K is very likely linked to the magnetic behavior of Eu5In2Sb6,
in line with the specific heat results discussed above. More specifically,
ellipsoidal polarons have been suggested for Eu5In2Sb6 which start to interact
below about 50 K [30]. We speculate that these ellipsoidal polarons are more
extended perpendicular to the magnetically hard $c$ direction and strongly
interact below $\sim$27 K, possibly causing the plateau-like behavior of
$\rho_{I\parallel a}(T)$. A further growth of the polarons upon decreasing $T$
also increases interaction along the $c$ direction resulting in the drop of
$r(T)$ below about 25 K. Applying magnetic fields is expected to level out
inhomogeneities in the magnetic state of Eu5In2Sb6. Our finding of a
significantly suppressed $r(T)$ already at 3 T and an almost flat behavior at
9 T, inset of Fig. 6(b), strongly supports such a polaron scenario.

As expected in a polaron scenario, large MR effects are observed [22],
especially in a temperature range $T_{\rm N1}\lesssim T\lesssim 27$ K. This
can be inferred from the ratio of resistivities $\rho(9\,{\rm T})/\rho(0\,{\rm
T})$, plotted in dependence on $T$ for $I\parallel a$ in Fig. 6(c). This ratio
reaches values well below 10-5, corresponding to a CMR $[$defined as
$(\rho(H)-\rho(0))/\rho(0)]$ of more than $-$99.999%. The anisotropy of $\rho$
in Eu5In2Sb6 is also reflected in the MR: For $I\parallel c$, the ratio
$\rho(9{\rm T})/\rho(0{\rm T})$ goes only down to $\sim\\!1.6\cdot 10^{-4}$
close to $T_{\rm N1}$.

One characteristic of materials exhibiting polaron formation is a low carrier
concentration $n$ such that energy gain by carrier localization is achieved
[32]. Here, Hall measurements are called for. However, the determination of
$n$ is straightforward only if inhomogeneous magnetic states and spurious
contributions to the Hall signal are avoided. Within this limitation (i.e. for
$\mu_{0}H\geq$ 7 T where a linear Hall response was observed) we obtain
$n\approx 1.5\cdot 10^{17}$ cm-3 at $T=100$ K assuming a single (hole-like)
band. Note that this corresponds to about 0.02 carriers per Eu site, a number
very similar [40] to the one found for EuB6 which is a prototypical material
for polaron formation [41, 13].

## Conclusions

The CMR effect observed in Eu5In2Sb6 is very likely related to the formation
of magnetic polarons, similar to EuB6 or the doped manganites. However, there
are also marked differences: i) As seen from Fig. 6(c) the MR in Eu5In2Sb6 is
always negative whereas the MR in EuB6 and doped manganites turns positive
below the magnetic ordering temperature (for discussions, see Refs. [42, 3]).
Likely, this is related to the ferromagnetic ordering in the latter two
materials while Eu5In2Sb6 orders antiferromagnetically. Therefore, Eu5In2Sb6
provides a rare opportunity to study magnetic polaron formation in an
antiferromagnetic environment. ii) The large MR effects in Eu5In2Sb6 extend to
comparatively high temperatures. The resistivity is still suppressed by an
_order of magnitude_ in $\mu_{0}H=9$ T at around 75 K, i.e. at more than six
times $T_{\rm N1}$. By comparison, the polaron formation in EuB6 only _sets
in_ at around 35 K [41, 43], which corresponds to about three times the
ferromagnetic ordering temperature. iii) The highly anisotropic magnetic and
transport properties of Eu5In2Sb6 are certainly related to its complex
crystallographic structure. This is in line with the observation of highly
anisotropic electronic transport properties in the manganites with layered
perovskite structure [44].

The latter, anisotropic properties stem from anisotropic exchange
interactions, given the magnetically isotropic Eu2+ ground state configuration
${}^{8}S_{7/2}$. Earlier spin-resolved band structure calculations suggested a
spin configuration with ferromagnetically coupled Eu moments in the $ab$
plane, which are alternatingly stacked in an antiparallel staggered order
[25]. Our angular dependent magnetization measurements support such a picture.
In particular, a spin-flop transition for $H\parallel b$ indicates a preferred
alignment of magnetic momentums along the $b$ direction.

The here established phase diagrams can guide future exploration of this
complex material. Clearly, the magnetic structure needs to be resolved
further, including a possible weakly ferromagnetic component resulting from
spin canting due to Dzyaloshinskii-Moriya exchange. The insights provided here
are essential for an evaluation of the surface topology in this nonsymmorphic
material. In particular, Eu5In2Sb6 may provide a test case for studying the
impact of electronic inhomogeneities on topology [45, 46].

## Methods

Single crystalline samples Eu5In2Sb6 were grown by a combined In-Sb self-flux
technique [22]. The crystallographic structure was confirmed by x-ray
diffractometry. For sample orientation a real-time Laue x-ray system was
employed (Laue-Camera GmbH). For magnetization measurements up to magnetic
fields of 14 T a vibrating sample magnetometer (VSM, Oxford Instruments Inc.)
was utilized. Magnetic measurements up to 7 T were also conducted using a
magnetic property measurement systems (MPMS3, Quantum Design Inc.). If not
noted otherwise, susceptibility was measured with an applied ac field of 10 Oe
after cooling in zero applied field (ZFC). The angle dependence of
magnetization was measured by means of a horizontal rotator inside an MPMS-XL,
Quantum Design Inc. Specific heat was measured using a physical property
measurement system (PPMS, Quantum Design Inc.) equipped with a calorimeter
that utilizes a quasi-adiabatic thermal relaxation technique. Electronic
transport was investigated using the same PPMS system. In an effort to
accurately measure the high sample resistances at low temperatures, an
external, lock-in-based circuitry hooked up to the PPMS was implemented. In
addition, to enable electronic transport measurements, a somewhat thicker
sample was used, Fig. 6(c).

## Data availability

Full sets of crystallographic data generated and/or analysed during the
current study have been deposited with the joint CCDC and FIZ Karlsruhe
deposition service. The data can be obtained free of charge from The Cambridge
Crystallographic Data Centre via www.ccdc.cam.ac.uk/structures citing
deposition number CSD 2214610. All other data generated and analysed during
this study are included in this published article (and its supplementary
information files) and/or are available from the corresponding author on
reasonable request.

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## Acknowledgements

The authors thank U. K. Rößler for insightful discussions, Tim Thyzel for help
with the analysis of the specific heat data, Ralf Koban for technical
assistance and Horst Borrmann for help with the data deposition. SR
acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) through SFB 1143. Work at the Max-Planck-Institute for
Chemical Physics of Solids in Dresden and at Goethe University Frankfurt was
funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation), Project No. 449866704. Work at Los Alamos was performed under the
auspices of the U.S. Department of Energy, Office of Basic Energy Sciences,
Division of Materials Science and Engineering. MSC acknowledges support from
the Laboratory Directed Research and Development program.

## Author contributions

S.R., P.F.S.R. and S.W. conceived the experiments, P.F.S.R. prepared the
samples, M.V.A.C., S.R., S.G., M.D. and M.S.C. conducted the experiments,
M.V.A.C., S.R., J.M. and S.W. analysed the results, M.V.A.C., J.M. and S.W.
wrote the manuscript. All authors reviewed the manuscript.

## Additional Information

Supplementary information accompanies this paper at http://www.nature.com/srep

Competing interests: The authors declare no potential conflict of interest.


w{\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}\in\mathcal{C}^{\approx}(\mathcal{L}(G(v,0))){\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}.$

We give a $\approx$-implementation for $\mathbf{G}_{\operatorname{MPCP}}$. It
is straightforward to construct FSMs for both
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
and
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$.
They are involved in the initial decision and $\approx$ does not affect their
projected languages. Thus, the projection by erasure can be applied to obtain
FSMs
$A_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
and
$A_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$.
We construct an FSM
$A_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
for
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
with control state $\texttt{i}\in\\{1,\ldots,n\\}$,
$\texttt{j}\in\\{1,\ldots,\max(\lvert u_{i}\rvert\mid
i\in\\{1,\ldots,n\\})\\}$, $\texttt{d}\in\\{0,1,2\\}$, and
$\texttt{x}\in\\{u,v\\}$, where $\lvert w\rvert$ denotes the length of a word.
The FSM is constructed in a way such that

$\displaystyle
w{\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}\in\mathcal{C}^{\approx}(\mathcal{L}(G(u,0))){\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}\quad\text{
if and only if }\quad\texttt{d}\text{ is }2\text{ and }\texttt{x}\text{ is
}u\qquad\text{ as well as }$ $\displaystyle
w{\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}\in\mathcal{C}^{\approx}(\mathcal{L}(G(v,0))){\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}\quad\text{
if and only if }\quad\texttt{d}\text{ is }2\text{ and }\texttt{x}\text{ is
}v.$

We first explain that this characterisation suffices to show that
$\\{\\!\\!\\{A_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\}\\!\\!\\}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}}$
$\approx$-implements $\mathbf{G}$. The control state d counts the number of
received $d$-messages. Thus, there will be no more messages to
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
in any channel once d is $2$ by construction of
$\mathbf{G}_{\operatorname{MPCP}}$. Once in a state for which d is $2$,
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
sends message $\textit{ack-}u$ to
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
if x is $u$ and message $\textit{ack-}v$ if x is $v$. With the
characterisation, this message $\textit{ack-}x$ matches the message
$\textit{c-}x$ sent from
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
to
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$
in the beginning and, thus,
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
will be able to receive it and conclude the execution.

Now, we will explain how to construct the FSM
$A_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$.
Intuitively,
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
keeps a tile number, which it tries to match against, and stores this in i. It
is initially set to $0$ to indicate no tile has been chosen yet. The index j
denotes the position of the letter it needs to match in tile $u_{\texttt{i}}$
next and, thus, is initialised to $1$. The variable d indicates the number of
$d$-messages received so far, so initially d is $0$. With this,
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
knows when it needs to send $\textit{ack-}x$. The FSM for
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
tries to match the received messages against the tiles of $u$, so x is
initialised to $u$. If this matching fails at some point, x is set to $v$ as
it learned that $v$ was chosen initially by
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$.

In any of the following cases: if a received message is a $d$-message, d is
solely increased by $1$:

  * •

If x is $u$ and i is $0$,
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
receives a message $z$ from
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
and sets i to $z$ (technically the integer represented by $z$).

  * •

If x is $u$ and i is not $0$,
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
receives a message $z$ from
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$.

    * –

If $z$ is the same as $u_{\texttt{i}}[\texttt{j}]$, we increment j by $1$ and
check if $\texttt{j}>\lvert u_{\texttt{i}}\rvert$ and, if so, set i to $0$ and
j to $1$

    * –

If not, we set x to $v$

  * •

Once x is $v$,
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
can simply receive all remaining messages in any order.

The described FSM can be used for
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
because it reliably checks whether a presented sequence of indices and words
belongs to tile set $u$ or $v$. It can do so because
$\mathcal{C}^{\approx}(\mathcal{L}(G(u,0))){\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}\cap\mathcal{C}^{\approx}(\mathcal{L}(G(v,0))){\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}=\emptyset$
by assumption.

We prove $\Rightarrow_{2}$ by contraposition. Suppose the MPCP instance has a
solution. Let $i_{1}$, $\ldots$, $i_{k}$ be a non-empty sequence of indices
such that $u_{i_{1}}u_{i_{2}}\cdots u_{i_{k}}=v_{i_{1}}v_{i_{2}}\cdots
v_{i_{k}}$ and $i_{1}=1$. It is easy to see that

$w_{x}\coloneq{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?i_{1}{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?[x_{i_{1}}].\,\cdots\,.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?i_{k}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?[x_{i_{k}}].\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?d.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?d\in\mathcal{L}(G(x,0)){\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}\text{
for }x\in\\{u,v\\}.$

By definition of $\approx$, we can re-arrange the previous sequences such that

${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?i_{1}.\,\cdots\,.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?i_{k}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?[x_{i_{1}}].\,\cdots\,.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?[x_{i_{k}}].\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?d.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?d\in\mathcal{C}^{\approx}(\mathcal{L}(G(x,0))){\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}\text{
for }x\in\\{u,v\\}.$

Because $i_{1},\ldots,i_{k}$ is a solution to the instance of MPCP, it holds
that

${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?[u_{i_{1}}].\,\cdots\,.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?[u_{i_{k}}]={\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?[v_{i_{1}}].\,\cdots\,.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?[v_{i_{k}}]$

and, thus,

${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?i_{1}.\,\cdots\,.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?i_{k}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?[u_{i_{1}}].\,\cdots\,.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?[u_{i_{k}}].\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?d.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?d\text{
is in
}\mathcal{C}^{\approx}(\mathcal{L}(G(v,0))){\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}.$

This shows that
$\mathcal{C}^{\approx}(\mathcal{L}(G(u,0))){\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}\cap\mathcal{C}^{\approx}(\mathcal{L}(G(v,0))){\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}\neq\emptyset$.

Lastly, we prove $\Leftarrow_{2}$. We know that the MPCP instance has no
solution. Thus, there cannot be a non-empty sequence of indices $i_{1}$,
$i_{2}$, …, $i_{k}$ such that $u_{i_{1}}u_{i_{2}}\cdots
u_{i_{k}}=v_{i_{1}}v_{i_{2}}\cdots v_{i_{k}}$ and $i_{1}=1$. For any possible
word
$w_{u}\in\mathcal{C}^{\approx}(\mathcal{L}(G(u,0))){\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}$
and word
$w_{v}\in\mathcal{C}^{\approx}(\mathcal{L}(G(v,0))){\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}$.

We consider the sequence of receive events
$w_{x}{\Downarrow}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?\\_}$
with sender
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
and the sequence of messages
$w_{x}{\Downarrow}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?\\_}$
from
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$
for $x\in\\{u,v\\}$. The intra-role indistinguishability relation $\approx$
allows to reorder events of both but for a non-empty intersection of both
sets, we would still need to find a word $w_{u}$ and $w_{v}$ such that

$w_{u}{\Downarrow}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?\\_}=w_{v}{\Downarrow}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?\\_}\quad\text{
and }\quad
w_{u}{\Downarrow}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?\\_}=w_{v}{\Downarrow}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\bm{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?\\_}.$

However, $G(x,0)$ for $x\in\\{u,v\\}$ is constructed in a way that this is
only possible if the MPCP instance has a solution. Therefore, the intersection
is empty which proves our claim.


Copyright for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).

PhysioCHI: Towards Best Practices for Integrating Physiological Signals in
HCI, May 11, 2024, Honolulu, HI, USA

[1] [1]You can use this document as the template for preparing your
publication. We recommend using the latest version of the ceurart style.

[email=poorvesh@vt.edu,] [1]

<EMAIL_ADDRESS>]

<EMAIL_ADDRESS>]

<EMAIL_ADDRESS>]

<EMAIL_ADDRESS>]

[1]Corresponding author. [1]These authors contributed equally.

# Integrating Physiological Data with Large Language Models for Empathic
Human-AI Interaction

Poorvesh Dongre  Department of Computer Science, Virginia Tech, Blacksburg,
VA, 24060  Empathic Computing Lab, Auckland Bioengineering Insitute,
University of Auckland, New Zealand  Majid Behravan  Kunal Gupta  Mark
Billinghurst  Denis Gračanin

(2024)

###### Abstract

This paper explores enhancing empathy in Large Language Models (LLMs) by
integrating them with physiological data. We propose a physiological computing
approach that includes developing deep learning models that use physiological
data for recognizing psychological states and integrating the predicted states
with LLMs for empathic interaction. We showcase the application of this
approach in an Empathic LLM (EmLLM) chatbot for stress monitoring and control.
We also discuss the results of a pilot study that evaluates this EmLLM chatbot
based on its ability to accurately predict user stress, provide human-like
responses, and assess the therapeutic alliance with the user.

###### keywords:

Physiological Computing Large Language Models (LLMs) Deep Learning

##  1 Introduction

In recent years, Large Language Models (LLMs) have made substantial strides in
showcasing their expertise across multiple disciplines, such as computer
science, medicine, and psychology [1]. They have demonstrated remarkable
proficiency in processing complex linguistic structures and producing text
that resonates with human-like coherence and relevance. While LLMs can process
emotional nuances from our text inputs, they currently lack the capability to
discern our mental and emotional states accurately. Enhancing LLMs’ ability to
interpret mental states could make them more adept in applications such as
psychotherapy, which would benefit from emotional understanding.

Physiological data can provide insights into a user’s mental and emotional
states and could enable LLMs to engage with users more empathically. This
research proposes a multifaceted approach to enhance empathic interaction with
LLMs. First, we aim to construct user profiles that go beyond static
preferences by incorporating users’ physiological data. Second, we use deep
learning to infer the user’s mental and emotional state with raw physiological
data. Finally, we integrate the prediction into a pre-trained LLM to develop
physiology-driven Empathic LLMs (EmLLM) that are aware of the user’s mental
and emotional states while interacting with them.

We present the application of our approach in an EmLLM chatbot for everyday
stress monitoring and control. Users’ physiological data, including
Electrodermal Activity (EDA), Blood Volume Pressure (BVP), and Skin
Temperature (ST), is collected using the Empatica EmbracePlus smartwatch. An
end-to-end multi-channeled 1D Convolution Neural Network (CNN) trained on the
Wearable Stress and Affect Detection (WESAD) dataset [2] is applied to the
collected raw data to predict if a user is feeling stressed. Finally, the
Falcon-7B LLM [3] is customized to include the predicted labels to develop an
EmLLM chatbot for stress monitoring and control. We discuss the findings from
a pilot study with 8 students to evaluate the chatbot’s ability to predict
user stress, provide human-like responses, and assess therapeutic alliance.

##  2 Background

Empathy is defined as the ability to feel the same way as others (affective
empathy) and understand the thoughts and perspectives of others (cognitive
empathy) [4]. Humans can have varying degrees of empathy based on their
backgrounds and experiences [5]. On the other hand, LLMs are designed to
understand and generate human-like text and can mimic empathy but don’t
actually feel emotions. However, LLMs can be trained, customized, and
integrated with multimodal human data to help them understand human thoughts
and emotions. Such LLMs could be useful in applications such as customer
service, education and training, virtual assistants, and psychotherapy.

Psychotherapy involves conversations between a trained therapist or counselor
and the individual or group seeking mental health support. With technological
progress, Artificial Intelligence (AI) chatbots have emerged to provide mental
health support to users. For example, Woebot, a fully automated chatbot,
follows principles of Cognitive Behavior Therapy (CBT) to deliver accessible
mental health support to college students [6]. BioBase, another AI chatbot,
combines features of wearable devices (tracking sleep, heart rate, and
activity) and CBT to provide mental health support to users [7]. However, a
limitation of such chatbots is that they have a fixed conversational style for
interacting with users, which can feel generic and robotic.

LLMs have emerged as transformative tools in the field of psychotherapy and
offer unique opportunities to provide users with personalized mental health
support. Existing research explores the potential applications and challenges
of using LLMs in psychotherapy. Demszky et al. [8] used LLMs for psychological
measurement, experimentation, and practice, highlighting their ability to
simulate realistic conversations for training mental health professionals. Lai
et al. [9] proposed the Psy-LLM framework that serves as an assisting tool for
mental healthcare professionals. However, there are concerns about the LLMs’
ability to produce reliable outputs consistently.

Another limitation of existing AI chatbots is that users must actively engage
with them to update their mental and emotional states. However, it’s not
always possible for users to engage with the chatbot or keep track of their
mental states throughout the day. In contrast, wearable devices could
automatically track the user’s psychological state and passively inform the
chatbot. Integrating wearable sensors with LLMs is becoming more common in
biomedical applications. Belyaeva et al. [10] developed a multimodal LLM for
healthcare that uses high-dimensional modalities, including wearable data, to
estimate underlying disease risk. Liu et al. [11] introduce “BioSignal
Copilot,” a system that utilizes LLMs to interpret biomedical signals and
draft clinical reports. These studies exemplify how LLMs can translate complex
physiological data into actionable insights, thereby paving the way to enhance
personalized psychotherapy approaches.

##  3 Physiology-Driven Empathic LLMs (EmLLMs)

To enhance empathy in LLMs and make them more suitable for applications such
as psychotherapy, we propose a physiological computing approach that primarily
comprises two components : (1) A deep learning model to predict user’s
psychological states with their physiological data; (2) Integrating the
predictions with a custom LLM that interacts with users based on their
psychological states. This section describes the conceptual approach of the
physiology-driven EmLLM (Figure 2).

###  3.1 Pyscho-Physiological Inference

Human physiology and psychology are intricately linked and profoundly
influence each other. The method for estimating psychological states from
peripheral physiological signals is called psycho-physiological inference
[12]. Researchers extract various time-domain, frequency-domain, and
statistical features to capture relevant information from the raw
physiological data [2, 13, 14, 15]. These features are then applied to various
Machine Learning (ML) algorithms, including Decision Tree (DT), Random Forest
(RF), and Support Vector Machine (SVM) to predict users’ psychological states
[2, 16]. However, there is no consensus on the best physiological features,
and researchers typically use the features that give them the best accuracy.

Deep Learning (DL) networks can incorporate the feature engineering process in
the modeling task and remove the dependence on hand-crafted features [17].
CNNs have the ability to automatically extract useful time-varying features
from raw physiological data and send it to a fully connected Multilayer
Perception (MLP) to perform classification [18]. CNNs can also be combined
with recurrent networks, such as Long Short-Term Memory (LSTM), to develop
hybrid networks for time-series classification [19, 20]. With CNNs, raw data
sampled at varying frequencies from different sensors can be fused by applying
convolutional layers to each univariate series individually and then
concatenating the learned features to an MLP for classification [21]. Finally,
deep networks can also predict multiple related tasks in one network [22],
thereby eliminating the need to develop different models for predicting
related psychological states.

###  3.2 Dynamic LLM Adaptation

Pre-trained LLMs such as GPT-3 can perform a wide range of natural language
processing tasks out of the box but require some customization for specialized
applications. The common ways of customizing LLMs include techniques such as
prompt engineering and model fine-tuning. Prompt engineering is a lightweight
yet powerful technique for customizing the behavior of LLMs by giving them a
set of instructions to perform better on particular tasks. The task
performance largely depends on how the prompts are crafted, and various
prompting techniques [23] could be used to instruct an LLM to get the desired
output. Prompt engineering is a promising technique for integrating psycho-
physiological inferences with LLMs to make them more empathetic. Another
approach for customizing LLMs is model fine-tuning, which refers to the
process of updating the parameters of a pretrained LLM on a task-specific
dataset. There is little scope for integrating psycho-physiological inferences
with LLM fine-tuning, but it is a powerful approach to control LLMs’ output.

To develop the EmLLMs, the predicted user psychological states from the deep
learning model can be used as part of the crafted prompts for the LLM. These
prompts can also instruct the LLM to adopt an empathetic personality [23] and
follow chain-of-empathy reasoning in conversations to understand what the user
is going through [24]. LLMs can also be instructed to follow the principles of
various psychotherapy techniques, such as CBT and Active Listening, to enhance
their empathy. Moreover, fine-tuning the LLMs on specialized datasets can
generate more context-specific responses, enhancing the user experience. LLMs
can also be trained from scratch on specialized datasets [9], but this
approach requires more time and resources. Lastly, LLMs can go beyond natural
language comprehension and can be trained on multimodal data to power
conversation around user health and wellness [10].

##  4 Physiology-Driven EmLLM Prototype

Based on the conceptual framework of EmLLM, we developed a prototype chatbot
that passively monitors user stress throughout the day using their
physiological data and actively interacts with them at the end of the day
based on how much stress they experienced. We use the Empatica EmbracePlus
smartwatch [25] to collect users’ raw physiological data, including BVP, EDA,
and ST, which is used as input to an end-to-end multi-channeled 1D CNN model.

The model inputs the raw physiological data individually through separate
channels and applies a sequence of three 1D convolution layers with a kernel
size of 3 and a stride length proportional to the sampling rate of the
physiological signals. The convoluted features from each channel are then
pooled (1D max-pooling), flattened, and concatenated to pass it over to fully
connected layers for classification. We use the “Adam” optimizer and “Binary
Cross Entropy” loss function to train our model on the WESAD dataset [2] to
predict if a participant is feeling “stressed” or “not stressed”. We followed
an approach similar to Schmidt et al. [2] for binary stress classification and
combined all participants’ data from baseline and amusement conditions to a
non-stress class. The authors used a variety of hand-crafted features to train
several ML models, and the maximum accuracy and F1 score with the wrist data
from all participants was 84.1% and 86.9 F1 score. However, our model achieved
a state-of-the-art accuracy of 85.1% and 89.0 F1 score.

To develop the EmLLM chatbot for stress-related psychotherapy, we start with
fine-tuning the Falcon-7B LLM [3] using the Quantized Low-Rank Adaption
(QLoRA) technique [26] on the mental health conversational dataset [27]. The
Falcon-7B LLM was selected because it is open source and has extensive
adaptability. The QLoRA technique involves normalizing and reducing the
model’s weights to a 4-bit quantized format, significantly decreasing the
memory requirement. The mental health dataset comprises a rich collection of
conversational pairs, including patient questions and corresponding answers
provided by mental health professionals. Next, the fine-tuned LLM is prompt-
engineered to take as input the predicted user stress levels, act like a
trained psychologist, follow the principles of CBT, and provide a reasonable
explanation whenever it cannot answer a question. Lastly, we develop a web
user interface for the EmLLM chatbot for psychotherapy.

##  5 Pilot Study

###  5.1 Study Protocol

The study aimed to evaluate the chatbot’s ability to accurately predict user
stress, provide human-like responses, and assess the therapeutic support
provided to the user. To achieve this, we conducted an in-the-wild study to
monitor the user’s physiology in their work environment throughout the day and
make them interact with the chatbot at the end of the day. The different parts
of the study protocol are detailed below-

Preparation: Participants arrive at the study location before they begin their
daily tasks. First, they complete the pre-task Short Stress State
Questionnaire (SSSQ) [28]. Then, the participants are equipped with the
Empatica EmbracePlus smartwatch.

Data Collection: After completing the preparation phase, the participants were
asked to resume their daily tasks. However, they were instructed to keep
wearing the smartwatch for the entire study duration. Participants were
advised not to loosen or change the position of the smartwatch. They are asked
to avoid any mood-altering substances, including coffee and tobacco, and not
to do any strenuous exercise during the study period.

Interaction: At the end of the day, participants returned to the study
location. First, the smartwatch was removed from the participants’ wrists, and
then they were asked to complete the post-task SSSQ. Finally, the participants
were asked to interact with the EmLLM chatbot for at least 15 minutes. The
chatbot greets the participant by their name and introduces itself. It then
informs the participant about their predicted stress and asks if they want to
discuss their day with it. After completing the session, the participants were
asked to complete the Godspeed questionnaire [29] and the Session Rating Scale
(SRS) [30]. More details about the survey instruments are given in the section
below. We also asked participants to comment or give feedback.

###  5.2 Survey Instruments

The survey instruments used in this study, SSSQ, Godspeed questionnaire, and
SRS, assess the chatbot’s ability to predict user stress, provide human-like
responses, and give therapeutic support to the user, respectively. The SSSQ is
an adaptation of the Dundee Stress State Questionnaire (DSSSQ), which measures
three higher-order dimensions of subjective stress states: engagement,
distress, and worry [28]. The pre-and-post-SSSQ provides a reflection of the
stress states that the user experienced during the day while performing their
daily tasks. We use the results of the pre-and-post-SSSQ to validate the
stress predictions.

The Godspeed questionnaire is a tool used in human-robot interaction research
to gather users’ feedback about their experiences interacting with robots
[29]. We use its answers to assess the user’s perception of the chatbot’s
friendliness, perceived intelligence, ease of use, and likability instantly
after the counseling session. The SRS is a tool commonly used in psychotherapy
to assess clients’ subjective experiences of individual therapy sessions [30].
We use its responses to gather user feedback immediately after each session to
assess the therapeutic alliance between the user and the chatbot.

###  5.3 Results

We used the study protocol to evaluate the performance of the EmLLM chatbot
for psychotherapy among 8 students. Among these, 5 identified themselves as
men and 4 as women, ages 23 to 37 (mean age = 30.5). For our pilot study, we
wanted to focus on a population that experiences high degrees of work-related
mental stress. Therefore, we focus on Ph.D. students as they experience high-
stress levels resulting from their academic pursuits [31, 32].

Figure 1: Proposed EmLLM approach.

Figure 2: Average stress states of all participants.

####  5.3.1 Quantitative Results

Figure 2 shows the average values of the three stress states for all
participants before and after they performed their daily work-related tasks.
As expected, almost all participants experienced high levels of worry and
engagement before and after performing their daily tasks. The post-task worry
was less than the pre-task, but participants felt more engaged later in the
day, suggesting an inability to detach from work psychologically, indicating
more stress [33]. Some participants also experienced higher degrees of
distress after their workday ended.

The results of the stress prediction model from physiological data show that 6
out of 8 participants experienced some stressful events in their workday. The
predictions are consistent with the findings of the SSSQ, indicating that the
stress prediction model is more likely to predict the stress state of
engagement. The two participants with no signs of stress in the prediction
model also showed high levels of engagement, with one scoring high in all
three stress states in both pre-task and post-task SSSQ. These participants,
however, belong to special groups, with one having medically diagnosed
Attention-Deficit/Hyperactivity Disorder (ADHD) and the other with a hearing
disability.

Chatbot’s mean competence score was 3.5, and its mean responsiveness score was
3.63. However, it appears that participants were not very impressed by its
ability to give human-like responses (2.88), conversational elegance (2.88),
and pleasantness (3). The chatbot also seemed to have developed a good
therapeutic alliance with participants, with a mean quality rating of 3.25 and
a mean empathy score of 3.63. The participants also felt that the
psychotherapy session with the chatbot was relevant (3.75), a good fit (3.38),
and right (3.25) for them.

####  5.3.2 Qualitative Results

Along with the survey responses, the participants also gave their comments and
feedback on their interaction with the chatbot. Most participants thought the
EmLLM chatbot was “therapist-like” and gave “excellent counsel.” They were
impressed that the chatbot “asked questions to make them think” and “gave
context” for asking these questions instead of giving generic information like
ChatGPT. However, a common concern raised by all participants was that
sometimes the chatbot would “start talking in Spanish”. This could be because
the mental health dataset used to fine-tune the LLM had some conversation
pairs in Spanish. Some participants were also surprised by the stress
prediction made by the chatbot about their day, as it was opposite to their
expectations. Participants also wanted the chatbot to remember their
conversation and give them more details about their day from the smartwatch
data. However, one participant was concerned about their chat history and
thought “someone would see” their conversation with the chatbot.

##  6 Discussion

Accuracy of Psycho-Physiological Inference The stress prediction model in this
study combines the physiological data of all participants subjected to
baseline, amusement, and stressful conditions. The data from baseline and
amusement conditions are further combined to make stress prediction a binary
classification task. Combining the data does give a stress prediction model of
higher accuracy, but it raises concerns about the model’s diagnosticity,
sensitivity, reliability, and validity. Diagnosticity and sensitivity refer to
the precision of inference, reliability captures the generality of inference
across people, and validity represents the consistency of inference across
conditions [34]. HCI researchers must investigate these topics when using
psycho-physiological data in their studies.

Does more data necessarily mean more accuracy? ML research for predicting
human psychological states such as stress, emotion, and thermal comfort with
wearable data uses both physiological and non-physiological data.
Physiological data such as BVP, EDA, and ST are scientifically proven to be
related to human psychology. Non-physiological data collected from wearable
devices, including ACC, is useful in predicting human activity but has little
relation to the mental states of humans. Data-hungry ML models will give
better accuracy with more data and features, but including non-physiological
data to infer psychological states is fundamentally questionable and can
result in models that overfit.

Are hand-crafted features indeed better? Another practice that potentially
overfits psychophysiological ML models is using several hand-crafted features.
Researchers might use techniques such as correlation analysis, principal
component analysis, and regularization to remove redundant features and avoid
overfitting. However, there is no consensus on the best features; researchers
typically use features that give them the best accuracy. A possible solution
can be using end-to-end deep learning models incorporating the feature
extraction process from raw data in the classification pipeline without
relying on hand-crafted features. In this study, we developed an end-to-end
multichannel 1D CNN that gave us state-of-the-art results, but this model can
be further improved by combining it with recurrent networks.

Biocybernetic Adaptations in LLMs Biocybernetic adaptation is a category of
physiological computing systems that use physiological data to infer users’
psychological states and adapt based on these states to improve user
experience [35]. This research integrates psycho-physiological inferences with
LLMs using prompt engineering to facilitate empathic human-AI interactions.
However, the behavior of LLMs is complex, and prompt engineering alone cannot
ensure the necessary adaptation in LLMs. Training and fine-tuning LLMs on
specialized multi-modal datasets can help control their output, but this
requires adapting the model’s input layers and learning algorithms to process
and correlate data across different sensory modalities effectively.

Privacy, Security, and Ethics Developing technological solutions for
applications such as mental health support introduces substantial privacy,
security, and ethical challenges for users [36]. Inadequate management of
psycho-physiological data and users’ intimate conversations with the chatbot
could lead to breaches of confidentiality and unintentional disclosures.
Implementing robust data protection measures and stringent ethical guidelines
is crucial to ensure the privacy and security of sensitive information. Such
precautions are essential for maintaining user trust and safeguarding the
well-being of individuals who rely on these digital interventions.

##  7 Conclusion

Our research advances empathic AI by integrating Large Language Models (LLMs)
with physiological data, creating the EmLLM chatbot for stress monitoring and
control. This integration enables the chatbot to interpret users’
psychological states more accurately, which could enhance the effectiveness of
digital psychotherapy tools. The pilot study validates our approach, showing
improved stress prediction and the ability to build a therapeutic rapport.
Future work should focus on developing AI that can provide personalized mental
health support by further understanding individual physiological responses,
aiming to make digital therapy more effective and responsive to users’ needs.

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