# Incommensurate antiferromagnetic order in CePtAl3

M. Stekiel<EMAIL_ADDRESS><EMAIL_ADDRESS>Jülich Centre
for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ),
Forschungszentrum Jülich GmbH, Lichtenbergstr. 1, D-85747 Garching  Physik-
Department, Technische Universität München (TUM), D-85748 Garching, Germany
P. Čermák  Jülich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz
Zentrum (MLZ), Forschungszentrum Jülich GmbH, Lichtenbergstr. 1, D-85747
Garching  Charles University, Faculty of Mathematics and Physics, Department
of Condensed Matter Physics, Ke Karlovu 5, 121 16, Praha, Czech Republic  C.
Franz  Jülich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz
Zentrum (MLZ), Forschungszentrum Jülich GmbH, Lichtenbergstr. 1, D-85747
Garching  MLZ, TUM, D-85748 Garching, Germany  W. Simeth  Paul Scherrer
Institut, Villigen, Switzerland  S. Weber  Physik-Department, Technische
Universität München (TUM), D-85748 Garching, Germany  E. Ressouche  University
Grenoble Alpes, CEA, IRIG, MEM, MDN, 38000 Grenoble, France  W. Schmidt  JCNS
at Institute Laue-Langevin, Forschungszentrum Jülich GmbH, 38042 Grenoble,
France  K. Nemkovski  Jülich Centre for Neutron Science (JCNS) at Heinz Maier-
Leibnitz Zentrum (MLZ), Forschungszentrum Jülich GmbH, Lichtenbergstr. 1,
D-85747 Garching  ISIS Neutron and Muon Source, STFC Rutherford Appleton
Laboratory, Harwell Campus, Didcot, Oxon OX11 0QX, United Kingdom  H. Deng
Jülich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum
(MLZ), Forschungszentrum Jülich GmbH, Lichtenbergstr. 1, D-85747 Garching
Institut für Kristallographie at MLZ, RWTH Aachen, Lichtenbergstrasse 1,
D-85747 Garching  A. Bauer  Physik-Department, Technische Universität München
(TUM), D-85748 Garching, Germany  Center for QuantumEngineering (ZQE), TUM,
85748 Garching, Germany  C. Pfleiderer  Physik-Department, Technische
Universität München (TUM), D-85748 Garching, Germany  Center for
QuantumEngineering (ZQE), TUM, 85748 Garching, Germany  Munich Center for
Quantum Science and Technology (MCQST), TUM, 85748 Garching, Germany  A.
Schneidewind  Jülich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz
Zentrum (MLZ), Forschungszentrum Jülich GmbH, Lichtenbergstr. 1, D-85747
Garching

###### Abstract

We report a neutron diffraction study of single-crystal CePtAl3 complemented
by measurements of the specific heat under applied magnetic field. Below
$T_{\mathrm{N}}\approx 3$ K CePtAl3 develops incommensurate antiferromagnetic
order with a single modulation vector $\boldsymbol{k}$=$(0.676\,0\,0)$.
Residual magnetic scattering intensity above $T_{\mathrm{N}}$ and a broadening
of the specific heat anomaly at $T_{\mathrm{N}}$ may be consistently described
in terms of a Gaussian distribution of transition temperatures with a standard
deviation $\sigma\approx 0.5$ K. The distribution of $T_{\mathrm{N}}$ may be
attributed to the observation of occupational and positional disorder between
the Pt and Al sites consistent with structural information inferred from
neutron diffraction. Measurements under magnetic field reveal changes of the
magnetic domain populations when the field is applied along the $[0\,1\,0]$
direction consistent with a transition from cycloidal to amplitude-modulated
magnetic order around 2.5 T.

##  I Introduction

Cerium based intermetallics comprise a wide variety of physical phenomena
owing to the coupling between electronic and magnetic degrees of freedom and
their relationship to the crystal structure and its symmetry. Perhaps most
extensively studied are materials with the ThCr2Si2 structure, featuring heavy
fermion superconductivity, quantum phase transitions, as well as large
magneto-elastic coupling phenomena Shatruk (2019); Pfleiderer (2009). Born out
of this long history, a structurally related materials class that has
attracted increasing interest in recent years comprises Ce$TX_{3}$ systems,
where $T$ is a transition metal and $X=$ Al, Ga, Si, or Ge. For instance, in
CeCuAl3 and CeAuAl3 anomalous thermal transport properties and inelastic
neutron and x-ray scattering experiments have revealed magnetoelastic hybrid
excitations even for nominally weak magnetoelastic coupling Paschen et al.
(1998); Adroja et al. (2015); Čermák et al. (2019). Moreover,
antiferromagnetic (AFM) order with low transition temperatures have been
reported, e.g. , for $T=$ Cu and Au, where CeCuAl3 exhibits an amplitude
modulated, collinear, AFM structure, with a modulation vector
$\boldsymbol{k}=(\frac{2}{5}\,\frac{3}{5}\,0)$ below $T_{\mathrm{N}}$ = 2.7 K
Klicpera et al. (2015), while CeAuAl3 exhibits a helical AFM structure with
$\boldsymbol{k}=(0\,0\,0.52)$ below $T_{\mathrm{N}}$ = 1.2 K Adroja et al.
(2015).

A key aspect of the $I4mm$ space group, in which most members of the
Ce$TX_{3}$ class crystallize, is the presence of a four-fold symmetry axis
along the $[0\,0\,1]$ direction. For $\boldsymbol{k}$ vectors in the basal
plane as in CeCuAl3 this permits, in principle, the formation of multi-$k$
magnetic structures. This possibility was considered unlikely for CeCuAl3
Klicpera et al. (2015); however, multi-$k$ structures were observed in various
other cerium intermetallics Schweizer et al. (2008); Puphal et al. (2020);
Donni et al. (1993). Thus, to unambiguously determine the magnetic state, the
number of modulation vectors must be determined, and measurements under
symmetry-breaking conditions must be performed to distinguish a potential
multi-$k$ structure from the formation of magnetic domains. On this note, the
putative presence of multi-$k$ order in the Ce$T$Al3 family would identify
another exciting facet of correlated electron systems. Further, the low Néel
temperatures in these systems suggest a possible vicinity to a quantum phase
transition. This raises the question if other members of the Ce$T$Al3 family
exist, that are either nonmagnetic in the zero temperature limit or even
coincidentally located at a quantum critical point.

In this paper we report the observation of long-range AFM order in CePtAl3.
This system crystallizes in the BaNiSn3-type structure, with a non-
centrosymmetric, tetragonal $I4mm$ space groupFranz et al. (2016). Previous
reports on the possible existence of long-range magnetic order in CePtAl3 were
inconclusive Mock et al. (1999); Franz et al. (2016). The electrical
resistivity, magnetization, and specific heat of polycrystalline samples were
found to be characteristic of spin-glass behavior $\leq$0.8 K Mock et al.
(1999). However, the authors of that study noted Mock et al. (1999) that their
CePtAl3 sample showed a high degree of crystalline disorder, speculating on an
intrinsic lack of long-range magnetic order. Likewise, based on the electrical
resistivity, Franz et al. (2016) suggested that CePtAl3 does not exhibit
magnetic order down to 0.1 K. Moreover, the magnetization of single-crystal
CePtAl3 shows an almost linear magnetic field dependence up to 9 T Franz
(2014), without signs of saturation. The magnetic moments at 9 T and 2 K are 1
and 0.3 $\mu_{\mathrm{B}}$ for field applied along the $a$ and $c$-axis,
respectively, which is far from the value expected for an isolated Ce3+ atom
of 2.14 $\mu_{\mathrm{B}}$. This suggests the presence of crystal field
effects that influence the magnetism in CePtAl3 and establishes an easy
magnetization plane $(0\,0\,1)$ and a hard magnetic axis along the $[0\,0\,1]$
direction. Taken together, these data suggested a possible vicinity to a
quantum phase transition, motivating the detailed microscopic study reported
in this paper.

##  II Experimental methods

Centimeter-sized single crystals of CePtAl3 were grown from polycrystalline
rods using an optical floating-zone furnace, described in more detail in Refs.
Franz, 2014; Neubauer et al., 2011; Bauer et al., 2016. Both the stability of
the floating-zone growth, which was well-behaved, and the metallurgical
appearance of the ingot reject the possibility of macroscopic phase
segregation. Rather, behavior characteristic of homogeneous sample was
observed. The crystals measured in this study were prepared from the same
batch studied by Franz et al. (2016). Samples were cut from the single-
crystalline rod as required for the different experimental techniques.
Demagnetization effects were negligible for all samples studied. The specific
heat was measured in a Quantum Design physical properties measurement system
at temperatures down to 2 K under applied magnetic fields up to 14 T. Constant
field scans as a function of temperature were performed on a
1$\times$1$\times$1 mm3 single-crystal cube of CePtAl3 using a large heat-
pulse method. Constant temperature scans as a function of an applied magnetic
field were carried out on a 2$\times$1$\times$6 mm3 single crystal using a
small heat-pulse method. In all specific heat measurements, the magnetic field
was applied along the $[1\,0\,0]$ direction. Neutron diffraction was carried
out at several different beamlines using a large cylinder-shaped crystal with
a length and diameter of 12 and 6 mm, respectively (volume=340 mm3).

Initial single-crystal neutron diffraction measurements were performed on the
triple-axes-spectrometer PANDA Heinz Maier-Leibnitz Zentrum (2015a) at the
MLZ. The crystal was mounted in a 3He cryostat and cooled to 0.5 K. The
intensities of selected reflections were measured as a function of temperature
to investigate the magnetic transition. The modulation vector of the magnetic
order was determined from measurements at the Diffuse Neutron Scattering
spectrometer (DNS) Heinz Maier-Leibnitz Zentrum (2015b); Schweika and Böni
(2001) at MLZ using polarization analysis. The incoming wavelength of 4.2 Å
and the one-dimensional detector bank at the DNS allowed us to simultaneously
collect data for a range of momentum transfers 0.25$<$$Q$$<$2.7 Å-1. The
sample was rotated by 1∘ steps in a 140∘ range to map out the neutron
scattering intensity on a plane in reciprocal space. Here, $xyz$-polarization
analysis was employed with the neutron polarization set parallel to the
momentum transfer, which allowed us to distinguish between magnetic and
nuclear scattering contributions on the spin-flip (SF) and non spin-flip (NSF)
channels, respectively Schweika (2010); Boothroyd (2020). The crystal was
cooled to 0.45 K using a 3He cryostat. Data were analyzed and reduced with the
mantid package Arnold et al. (2014) allowing us to project the measured
intensities on selected crystal planes.

Further single-crystal neutron diffraction measurements were performed on POLI
Heinz Maier-Leibnitz Zentrum (2015c) at MLZ and on D23 Ressouche et al. (1999)
at the ILL. Both diffractometers operate a sample stage with an $\omega$
rotation axis and a lifting detector stage allowing us to measure out-of-plane
scattering intensity. Measurements at POLI aimed at an accurate determination
of the crystal structure.

To minimize extinction effects, a small half-disc-shaped crystal with
dimensions 6 $\times$ 1 mm (diameter $\times$ thickness) and a volume of 14
mm3 was used at an incoming wavelength of 3.355 Å. Data were first recorded at
ambient conditions. Following this, the sample was mounted in the 3He cryostat
and cooled to 0.45 K. Measurements at D23 aimed at the determination of the
magnetic structure and were performed with a 6 T vertical field magnet at 1.5
K at an incoming wavelength of 1.275 Å. The same crystal that was used at the
DNS and PANDA ($V$=340 mm3) was mounted in two configurations, first with the
$[0\,1\,0]$ direction vertical and second with the $[0\,0\,1]$ direction
vertical.

The experimental protocol for measurements under magnetic field at D23 was as
follows. The sample was cooled to 1.5 K in zero field, and the magnetic field
was applied along the $[0\,1\,0]$ direction, _i.e._ along the $b$ axis. The
field was ramped from $\mu_{0}H=0$ to 5.5 T and then down to $\mu_{0}H=0$ in
steps. At each field, a series of $Q$ and $\omega$-scans was measured to
determine the position and intensity of selected reflections. Following this,
the sample was heated to ambient temperature, reoriented, and cooled down to
1.5 K again, so that the magnetic field was applied along the $[0\,0\,1]$
direction, _i.e._ , along the $c$ axis. For this orientation we measured
magnetic and nuclear reflections up to 4 T under increasing magnetic field.
Integration of the reflections from the diffraction measurements was done with
the davinci program Sazonov (2018). For the determination of magnetic
structures, refinements were carried out using jana2006 Petříček et al. (2014)
with the symmetry analysis input from isodistort Stokes et al. (2022).

##  III Results and Interpretation

The presentation of our experimental results is organized as follows. The
specific heat data are presented in Sec. III.1. Following this, the results of
four independent neutron diffraction measurements are reported. Measurements
focusing on the accurate determination of the crystal structure are presented
in Sec. III.2. The results of the search for the magnetic modulation vector
and the characterization of the magnetic order parameter are presented in Sec.
III.3. Finally, determination of the AFM order and its behavior under applied
magnetic fields are reported in Sec. III.4.

###  III.1 Specific heat

The specific heat of CePtAl3 was measured as a function of temperature at
various applied magnetic fields up to 14 T, as shown in Fig. 1, where we
focused on the low-temperature part in the regime of the magnetic transition.
At zero magnetic field, the specific heat shows a broad maximum centered at
$T_{\textrm{N}}$=3 K with a full-width-at-half-maximum (FWHM) of $\approx$1.5
K. Under increasing applied field, there is no visible change in the specific
heat curve up to 3 T. At 6 and 9 T, the amplitude of the maximum is decreased
and slightly shifted toward lower temperatures. The maximum has vanished at 14
T.

The field dependence was explored further in measurements as a function of
magnetic field at fixed temperature, as shown in Fig. 1(b). At 3 K, the
specific heat exhibits a maximum at 2.5 T marked as $H_{\mathrm{c}}$ and a
monotonic decrease under increasing field above $H_{\mathrm{c}}$.

As shown below, neutron diffraction allows us to associate the broad maxima in
the specific heat at 3 K and zero field with the transition to AFM order,
while the local maximum at $\mu_{0}H_{\mathrm{c}}=$ 2.5 T coincides with a
field-driven magnetic transition.

Figure 1: Specific heat of CePtAl3. (a) Specific heat as a function of
temperature at various magnetic fields up to 14 T. Data are shifted vertically
by 2 J/(mol K) for clarity. The dashed line marks characteristic temperature
$T_{\mathrm{N}}$ at which the field sweep was measured as shown in (b). In
(a), the maximum in specific heat at $T_{\mathrm{N}}$=3 K marks the transition
to the magnetically ordered state, consistent with neutron scattering. (b)
Specific heat as a function of magnetic field at $T$=3 K. In (b), the local
maximum of the 3 K field sweep is marked with $H_{\mathrm{c}}$ at 2.5 T and
coincides with the magnetic transition observed with neutron scattering.

###  III.2 Crystal structure

Single-crystal neutron diffraction at ambient conditions confirmed that
CePtAl3 crystallizes in the BaNiSn3 structure with a non-centrosymmetric
$I4mm$ space group, as previously reported by Franz et al. (2016) and shown in
Fig. 2. Lattice parameters and atomic positions are summarized in Table 1. A
large number of high-$Q$ reflections allowed for a reliable refinement of the
displacement parameters.

The refinement for the BaNiSn3 structure showed negative displacement
parameters on the Al 2$a$ site and a rather high refinement quality factor
$R_{\mathrm{obs}}$=12.24. The refinement was found to be improved,
$R_{\mathrm{obs}}$=5.67, when assuming antisite disorder between the Al and Pt
2$a$ sites. Further, the displacement ellipsoid of the Al 2$a$ site was found
to be strongly elongated along the $c$ axis, while other sites were found to
show a fairly isotropic displacement, as illustrated in Fig. 2(a). This was
taken as an indication of positional disorder. Splitting, in turn, the Al 2$a$
site into separate sites occupied by Pt and Al atoms gave a final figure of
merit $R_{\mathrm{obs}}$=4.98, where the occupancies were constrained to
retain the stoichiometric composition of CePtAl3. Refinements without such a
constraint did not improve the quality of the fit. The final model is
summarized in Table 1 and illustrated in Fig. 2(b).

Table 1: Details of the CePtAl3 crystal structure refinement based on neutron diffraction at ambient temperature. Space group $I4mm$, $a$=$b$=4.326 Å, $c$=10.655 Å. The number of measured:independent:observed reflections is 343:175:156, where $R_{\mathrm{int}}$=8.21, and observed reflections are with $I>3\sigma(I)$. The refinement was performed on $F^{2}$ with $R_{\mathrm{obs}}$=4.98 and $wR_{\mathrm{obs}}$=11.11. The columns contain the symbol of the Wyckoff site, atoms occupying the site, occupancy, $z$ coordinate of the Wyckoff site (2$a$: [0, 0, $z$] and 4$b$:[0, 0.5, $z$]), and associated, refined displacement parameters, respectively. Site | Atom | Occupancy | $z$ | U (Å2)
---|---|---|---|---
2$a$ | Ce | 1 | 0 (fixed) | U11=0.0168 (17)
U33=0.0158 (20)
2$a$ | Pt | 0.76 | 0.636 (1) | U11=0.0126 (12)
Al | 0.24 | U33=0.0157 (12)
2$a$ | Al | 0.76 | 0.410 (1) | Uiso=0.0136 (23)
Pt | 0.24 | 0.376 (2) | Uiso=0.0383 (65)
4$b$ | Al | 1 | 0.251 (1) | U11=0.0173 (28)
U22=0.0172 (28)
U33=0.0151 (16)
Figure 2: Crystal structure of CePtAl3. (a) Archetypal BaNiSn3 structure. Red
arrow in (a) points at the Al atom on the 2$a$ site which exhibits strong
elongation of displacement ellipsoid and is modeled by splitting the
equivalent sites (b) BaNiSn3 structure modified with occupational and
positional disorder on the Al 2$a$ and Pt 2$a$ sites. Occupations of the
disordered sites and their positions are indicated by the labels.

Cooling the crystal to 0.45 K, the intensities and positions of selected
reflections were monitored to verify whether CePtAl3 undergoes any structural
modifications. No significant changes were seen, suggesting that the peak
observed in the specific heat at 3 K is due to a magnetic transition that does
not coincide with a strong response of the lattice. The intensities of the
Bragg reflections also showed that the magnetic structure is characterized by
a modulation vector $\boldsymbol{k}\neq 0$. The refinement of the 147 nuclear
reflections measured at 0.45 K strongly favors the disorder model established
at ambient temperature. The refined parameters are close to those listed in
Table 1, _i.e._ , the crystal structure is identical at 0.45 K and ambient
temperature, except for the displacement parameters, which at 0.45 K are
reduced by a factor of two, as expected for low temperatures.

###  III.3 Spontaneous magnetic order

To determine the modulation vectors of the spontaneous magnetic order in
CePtAl3, _i.e._ at zero magnetic field and low temperatures, we mapped the
$(H0L)$ plane in reciprocal space using neutron diffraction at 0.45 K. In
addition to the nuclear reflections observed in the NSF channel, shown in Fig.
3(a), 10 weak magnetic reflections were detected in the SF channel, as shown
in Fig. 3(b). The intensities of these magnetic reflections are roughly two
orders of magnitude smaller than the nuclear reflections. They may be indexed
by a modulation vector $\boldsymbol{k}$=$(0.676(4)\,0\,0)$. The value
$k_{x}$=0.676 (4) is close to commensurate order with
$k_{x}$=$\frac{2}{3}$=$0.\overline{6}$. However, further measurements
confirmed that the modulation vector is, in fact, incommensurate.

Figure 3: Neutron diffraction intensity in the ($H0L$) plane of CePtAl3 at
0.45 K. (a) Non-spin-flip (NSF) channel, where nuclear reflections are
observed. (b) Spin-flip (SF) channel, where magnetic reflections with a
modulation vector $\boldsymbol{k}$=$(0.676\,0\,0)$ are observed.

The magnetic transition temperature was determined by means of neutron
diffraction at PANDA. Upon cooling, the intensities of the magnetic and
nuclear reflections at $(0.324\,0\,1)$ and $(0\,0\,2)$, respectively, were
measured as shown in Fig. 4. Below 3.5 K, the intensity of the magnetic
reflection increased, while the nuclear reflection did not change, indicating
lack of an accompanying structural transition or sizable magnetostriction
above the detection limit of our measurements, as mentioned in Sec. III.2.

The nuclear reflection consisted of a sharp, resolution-limited diffraction
peak with a FWHM of 0.91∘. The additional presence of diffuse shoulders is
well described by a broad Gaussian with FWHM=3.88∘. We attribute the diffuse
component to Huang scattering Huang (2000); Peisl (1975). It arises from the
atomic disorder, which causes strain fields responsible for broad, diffuse
signal contributions around some Bragg reflections. In comparison, the shape
of the magnetic reflection is well described by a single Gaussian, without any
qualitative change in line shape across the magnetic transition.

Table 2: Parameters obtained from fitting the intensity of the $(0.324\,0\,1)$ magnetic reflection with different models shown in Fig. 4(c). Model | Parameters
---|---
$1-(T/T_{\mathrm{N}})^{\alpha}$ |  $T_{\mathrm{N}}$=3.26 (3) K |  $\alpha$=1.89 (11)
$(1-T/T_{\mathrm{N}})^{\alpha}$ |  $T_{\mathrm{N}}$=3.15 (6) K |  $\alpha$=0.66 (5)
$[1-(T/T_{\mathrm{N}})^{\alpha}]\ast$ |  $\overline{T}_{\mathrm{N}}$=3.02 (15) K |  $\alpha$=2.7 (8)
$\ast\mathrm{exp}(-(T_{\mathrm{N}}-\overline{T}_{\mathrm{N}})^{2}/2\sigma_{T_{\mathrm{N}}}^{2})$ |  $\sigma_{T_{\mathrm{N}}}$=0.48 (7) K |
Figure 4: Intensities of nuclear and magnetic reflections across the magnetic
transition. (a) Nuclear $(0\,0\,2)$ reflection. It can be fitted with two
Gaussian components (green and red lines) of different full width at half
maximum (FWHMs), as indicated by arrows and labels. (b) Magnetic
$(0.324\,0\,1)$ reflection. In (a) and (b), data are shown with points, and
solid lines are guides to the eye. (c) Intensity in the vicinity of the
magnetic reflection (gray triangles) at the position marked with an arrow in
(b) and the FWHM of the magnetic reflection (green triangles) as a function of
temperature. (d) Integrated intensity of reflections in (a) and (b) as a
function of temperature. The intensity of the magnetic reflection is not
accurately reproduced with a power-law behavior (blue and green lines). A
power-law dependence convoluted with a Gaussian distribution (solid, purple
line), that may be attributed to the structural inhomogeneities in the sample
provides the best fit.

The temperature dependence of the intensity of the magnetic reflection may be
equally well described by a power-law behavior $I\propto(1-T/T_{N})^{\alpha}$
and a modified, heuristic expression $I\propto 1-(T/T_{N})^{\alpha}$. Both
functions are depicted in Fig. 4(c) together with the measured intensities.
Neither fit includes a point of inflection that accounts for the residual
intensity above $T_{\mathrm{N}}$. Such residual intensity is commonly
attributed to critical scattering that originates from slow, long-range
correlations in the vicinity of the transition and gives rise to magnetic
diffuse scattering. As a function of temperature, these correlations are most
pronounced at the transition temperature Wilkinson and Shull (1956); Krivoglaz
(1996). In Fig. 4(c) the width of the magnetic reflection and the intensity in
the vicinity of the magnetic reflection as a function of temperature are
shown. The width of the reflection increased with increasing temperature,
consistent with critical fluctuations. However, the magnetic diffuse intensity
measured at the position marked with an arrow in Fig. 4(b), which corresponds
to $(hkl)=(0.3\,0\,1.045)$, and shown in Fig. 4(c) did not exhibit a maximum
around $T_{\mathrm{N}}$. The diffuse scattering scales as the intensity of the
magnetic reflection, suggesting static disorder of the long-range magnetic
order, rather than fluctuations with a specific $Q$-dependence for the
resolution of our setup. Considering the crystalline disorder characterized in
Sec. III.2, which reveals antisite disorder, the metallurgical appearance of
our sample without hints for macroscopic phase segregation, and the broad peak
in the specific heat, see Sec. III.1, we suggest therefore an alternative
explanation of the residual scattering intensity above $T_{\mathrm{N}}$.

Namely, we propose that the atomic disorder and the structural inhomogeneities
cause a spread of transition temperatures $T_{\mathrm{N}}$ that accounts for
the residual scattering intensity above the transition, in addition to
conventional critical scattering. The notion that structural inhomogeneities
on the atomic scale may give rise to such a scattering contribution above the
critical temperature was previously found to describe the magnetic transitions
in polycrystalline samples of other materials Regulski et al. (2004); Manchón-
Gordón et al. (2020). Considering all information available, it represents
perhaps the simplest scenario.

Incorporating the mathematical formalism reported by Regulski et al. (2004),
we describe the spread of the transition temperatures seen in neutron
scattering by a Gaussian distribution, with an average transition temperature
$\overline{T}^{n}_{\mathrm{N}}$ and a standard deviation $\sigma^{n}_{T_{N}}$.
The measured intensity is then a convolution of the power-law behavior with
the Gaussian distribution as depicted in Fig. 4(c). Using the modified power-
law function $1-(T/T_{\mathrm{N}})^{\alpha}$, which proved to be insensitive
to the temperature range fitted, the model is in excellent agreement with
experiment across the entire temperature range, where
$\overline{T}^{n}_{\mathrm{N}}$=3.03 (15) K and
$\sigma^{n}_{T_{\mathrm{N}}}$=0.48 (7) K. It is worthwhile to point out that
the convoluted model assumes only one additional parameter than a power-law,
while significantly improving the quality of the fit.

The main conclusion suggested by this analysis also accounts for the broad
signature in the specific heat shown in Fig. 1, using an analogous model where
an underlying specific heat curve is convoluted by the distribution of
transition temperatures determined in neutron scattering. Although the exact
parameterization of the specific heat is not straightforward, the width of the
resulting convolution is dominated by the contribution from the distribution
of transition temperatures 111Convolution of two Gaussians with variance
$\sigma^{2}$ and $\tau^{2}$, results in a Gaussian with variance
$\sigma^{2}+\tau^{2}$. A broad contribution from a distribution of transition
temperatures will hence dominate the convoluted Gaussian., _i.e._ , the
resulting curve depends little on the precise form of the parameterization in
the disorder-free compound. Notably, the transition temperature
$\overline{T}^{c_{p}}_{\mathrm{N}}$=3 K and the FWHM of $\approx$1.5 K, which
corresponds to a standard deviation $\sigma_{T_{\mathrm{N}}}^{c_{p}}$=0.64 K
observed in the specific heat, are very close to the values determined in
neutron scattering, $\overline{T}^{n}_{\mathrm{N}}$=3.03 (15) K and
$\sigma^{n}_{T_{\mathrm{N}}}$=0.48 (7) K.

###  III.4 Magnetic order under magnetic field

Various magnetic reflections were measured at 1.5 K as a function of magnetic
field up to 5.5 T. The main observation inferred from these measurements is
that the $(hkl)\pm(0.676\,0\,0)$ and $(hkl)\pm(0\,0.676\,0)$ families of
magnetic reflections behave differently under applied magnetic field. The
general trend may be illustrated by means of the $(0.324\,0\,1)$ and
$(0\,0.324\,1)$ reflections shown in Fig. 5.

Figure 5: Diffraction intensity as a function of magnetic field. Reflections
were measured in increasing (blue) and decreasing (orange) field applied along
the $[0\,1\,0]$ axis and in increasing field applied along the $[0\,0\,1]$
axis (black). (a) Intensity of the $(0\,0.32\,1)$ reflection (open symbols).
(b) Intensity of the $(0.32\,0\,1)$ reflection (closed symbols). (c) Domain
populations as determined by scaling the intensities of reflections in (a) and
(b) under increasing field applied along the $[0\,1\,0]$ direction (blue
squares) and determined from the refinement (green diamonds). Solid lines are
guides to the eye. The vertical line marks the field of 2.5 T, the mid point
of domain repopulation.

Under increasing magnetic field applied along the $[0\,1\,0]$ direction (blue
symbols in Fig. 5), the intensity of the $(0\,0.324\,1)$ reflection increases
and reaches a maximum with a plateau between 1 and 1.5 T. Further increasing
the field reduces the intensity of the $(0\,0.324\,1)$ reflection, which
vanishes $>$3 T up to 5.5 T, the highest field studied. Subsequently
decreasing the field to 1 T (orange symbols in Fig. 5), the intensity displays
some hysteresis as highlighted by a curve guiding the eye. Below 1 T, the
intensity of the $(0\,0.324\,1)$ reflection does not drop to the value
observed prior to increasing the field. Instead, it remains high, with a small
drop just before the magnetic field reaches zero.

The behavior observed so far is contrasted by the intensity of the
$(0.324\,0\,1)$ reflection, which is shown in Fig. 5. Under increasing
magnetic field applied along $[0\,1\,0]$ its intensity decreases almost to
zero at 1 T, followed by an increase, reaching a maximum at 3 T, before slowly
decreasing again when increasing the field further. Decreasing the field next,
reveals the reverse evolution of the intensity. Analogous to the
$(0\,0.324\,1)$ reflection, the intensity does not increase to reach the value
before changing the field, but remains low with a small upturn just before the
magnetic field reaches zero.

Turning now to the behavior for magnetic field along the $[0\,0\,1]$ direction
(black symbols in Fig. 5) the evolution of both reflections $(0.324\,0\,1)$
and $(0\,0.324\,1)$ is the same. Both exhibit essentially the same intensity
and a monotonic decrease up to 4 T.

It is important to note that we did not observe any changes of the magnetic
modulation vector as such or widths of the magnetic reflections for all
applied fields studied under both increasing and decreasing field. The
magnetic modulation vector determined under applied fields is
$\boldsymbol{k}$=$(0.677\,(1)\;0\;0)$, consistent with the value presented in
Sec. III.3. In addition, we did not observe significant intensity of the
second-order mixed satellites, such as $(0.324\,0.676\,1)$ and others, that
would be indicative of a multi-$k$ magnetic structure. Instead, the field
dependence of the magnetic reflections is clearly characteristic of the
formation of a multi domain, single-$k$ AFM structure.

Namely, for a single-domain, multi-$k$ structure, the intensities of the
$(0\,0.324\,1)$ and $(0.324\,0\,1)$ reflections would have to be the same in
some field range, whereas a small field of order 0.25 T already changes the
intensities of these reflections, as shown in Fig. 5. Further, up to 1 T, the
magnetic field changes the domain population, almost saturating the volume of
the domain with the magnetic modulation vector along the field direction
($\boldsymbol{k}\parallel\boldsymbol{H}$), related to the intensity of the
$(0\,0.324\,1)$ reflection.

Above 1.5 T, an unexpected exchange of intensity may be observed between the
two reflections and a re-population of the magnetic domains. Decreasing the
field causes a hysteresis in the behavior of the intensities of the magnetic
reflections. The difference of intensity after removing the applied magnetic
field is, finally, characteristic of the presence of magnetic domains. Here,
the relative intensities of the reflections allow us to determine the field
dependence of the domain populations, as shown in Fig. 5(c).

###  III.5 Real-space depiction of magnetic order

Establishing that CePtAl3 exhibits a single-$k$ magnetic structure allows us
to infer the orientation of the magnetic moments in real space. The single-$k$
character of the magnetic order lowers the symmetry of the tetragonal $I4mm$
space group to orthorhombic $Imm2$, where two irreducible representations
$\sigma_{1}$ and $\sigma_{2}$ are allowed. For
$\boldsymbol{k}$=$(0.676\,0\,0)$=$(k\,0\,0)$, labeled $k_{\perp}$ since
$\boldsymbol{k}\perp\boldsymbol{H}\parallel[0\,1\,0]$, the $\sigma_{1}$
representation describes an amplitude modulated antiferromagnet with the
magnetic moment associated with the Ce atom at Wyckoff position $[x,y,z]$

$\boldsymbol{m}^{k_{\perp}}_{\sigma_{1}}(\boldsymbol{r})=[0,m_{1}\sin(2\pi
kx),0].$ (1)

In comparison, the $\sigma_{2}$ representation describes a cycloidal
modulation of magnetic moments in the $ac$-plane, with a moment

$\boldsymbol{m}^{k_{\perp}}_{\sigma_{2}}(\boldsymbol{r})=[m_{2}\sin(2\pi
kx),0,m_{3}\cos(2\pi kx)].$ (2)

The parameters $m_{i}$ describe the amplitude modulation along different axes.
They may be refined using the experimental data. Note that, for the other
domain with $\boldsymbol{k}$=$(0\,0.676\,0)$ labeled $k_{\parallel}$, the
modulation vector direction is different, and the magnetic moments are given
by

$\boldsymbol{m}^{k_{\parallel}}_{\sigma_{1}}(\boldsymbol{r})=[m_{1}\sin(2\pi
ky),0,0],$ (3)

and

$\boldsymbol{m}^{k_{\parallel}}_{\sigma_{2}}(\boldsymbol{r})=[0,m_{2}\sin(2\pi
ky),m_{3}\cos(2\pi ky)].$ (4)

Table 3: Parameters of the CePtAl3 magnetic structure determined from diffraction measurements at high magnetic field applied along the $[0\,1\,0]$ direction. Consecutive columns correspond to the magnetic field, domain population ratio, goodness of fit (GOF), $R$-factor for magnetic satellites, chosen representation, and magnetic moment components. $\mu_{0}H\,{\rm(T)}$ |  $k_{\perp}$:$k_{\parallel}$ | GOF | $R_{sat}$ | Repr. |  $m_{i}$ ($\mu_{\mathrm{B}}$)
---|---|---|---|---|---
0 | 52:48 | 4.89 | 12.15 | $\sigma_{2}$ |  $m_{2}$=0.852 (25)
|  |  |  |  |  $m_{3}$=0.484 (37)
0.75 | 19:81 | 9.25 | 14.75 | $\sigma_{2}$ |  $m_{2}$=0.836 (82)
|  |  |  |  |  $m_{3}$=0.494 (130)
1.5 | 13:87 | 8.44 | 16.71 | $\sigma_{2}$ |  $m_{2}$=0.846 (77)
|  |  |  |  |  $m_{3}$=0.52 (12)
5 | 95:5 | 13.5 | 8.99 | $\sigma_{1}$ |  $m_{1}$=0.64 (15)
5 | 95:5 | 13.62 | 8.12 | $\sigma_{2}$ |  $m_{2}$=0.65 (16)
|  |  |  |  |  $m_{3}$=0 (0.5)
Figure 6: Depiction of the incommensurate component of the magnetic order of
CePtAl3 under applied magnetic field. (a) and (b) Cycloidal phase with
$\boldsymbol{k}\parallel\boldsymbol{H}$ observed $\leq$2.5 T, viewed along (a)
the $[1\,0\,0]$ direction and (b) the $[0\,0\,1]$ direction. (c) and (d)
Amplitude-modulated phase with $\boldsymbol{k}\perp\boldsymbol{H}$ observed
between 2.5 and 5.5 T with the moments oriented along (c) the $[0\,1\,0]$
direction or (d) the $[1\,0\,0]$ direction; view along the $[0\,0\,1]$
direction ($c$ axis). On the left-hand side of each panel, a coordinate system
is shown which corresponds to the size of a nuclear unit cell. Next to each
panel, the directions of the modulation vector $\boldsymbol{k}$ and the
applied magnetic field $\boldsymbol{H}$ are shown. For the cycloidal
structure, colors mark the azimuthal angle between the magnetic moment and the
$[0\,0\,1]$ direction. For the amplitude-modulated structure, colors mark the
modulus of the magnetic moment.

The refinement of the dataset collected at $\mu_{0}H$=0, 0.75 and 1.5 T
strongly supports a cycloidal structure, where the refined magnetic moment
components are listed in Table 3, and the arrangement of the moments in real
space is shown in Fig. 6. For the data collected at 5 T the refinement is not
conclusive, and both cycloidal and amplitude-modulated magnetic structures
account for the data equally well.

One of the causes of this uncertainty may be the relatively small number of
reflections measured. For the cycloidal model, the $m_{3}$ parameter, which is
the modulation amplitude along the $c$ axis, is poorly constrained, and the
best refinement is obtained when setting it to zero. As a result, at 5 T, both
models correspond to the amplitude-modulated magnetic structure with moments
lying either along the $b$ or $a$ crystal axis, for the $\sigma_{1}$ and
$\sigma_{2}$ models, respectively, as shown in Fig. 6. Since high magnetic
fields tend to align the microscopic magnetic moments along the field
direction, the model with the $\sigma_{1}$ representation seems more likely.
However, further investigations will be necessary to resolve these
uncertainties.

###  III.6 Further discussion

The evolution of the magnetic structure in CePtAl3 under magnetic field
applied along the $[0\,1\,0]$ direction may be summarized as follows. At
$\mu_{0}H$=0, two equally populated magnetic domains coexist, supporting
cycloidal magnetic order described by Eq. (2) and Eq. (4). One domain exhibits
a modulation vector $\boldsymbol{k_{\perp}}=(0.676\,0\,0)$, while the other
exhibits $\boldsymbol{k_{\parallel}}=(0\,0.676\,0)$.

Application of the magnetic field along the $[0\,1\,0]$ direction changes the
domain populations, initially favoring the domain with
$\boldsymbol{k_{\parallel}}$=$(0\,0.676\,0)$. This domain population becomes
saturated between 1 and 1.5 T, where the magnetic moments are described by Eq.
(4) as depicted in Fig. 6(a) and Fig. 6(b). Further, between 1.5 and 3 T, an
exchange of the intensity is observed between the magnetic reflections with
$\boldsymbol{k_{\parallel}}$ and $\boldsymbol{k_{\perp}}$, as shown in Fig.
5(c), corresponding to a domain repopulation. This repopulation of the domains
is associated with a transition to an amplitude modulated structure, where the
moments are either oriented along $[1\,0\,0]$ or $[0\,1\,0]$ with a modulation
amplitude $m\approx$ 0.65 $\mu_{\mathrm{B}}$. In contrast, when the magnetic
field was applied along $[0\,0\,1]$, we did not see any evidence suggesting a
reorientation of the magnetic moments, and the domains remained equally
populated at all applied magnetic fields.

For magnetic field exceeding 3 T, the intensities of the magnetic reflections
decreased monotonically with increasing field for both field directions. This
is consistent with a reduction of the modulation amplitude and the increase of
the uniform magnetization detected in the magnetization Franz (2014). Indeed,
we observed an increase of intensity at specific $\boldsymbol{k}=0$
reflections, reflecting the uniform magnetization in the scattering process.
Moreover, for field applied along the $[0\,1\,0]$ direction, the magnetization
measured at 2 K and 5 T of 0.75 $\mu_{\mathrm{B}}$ per cerium atom Franz
(2014) was of the same order of magnitude as the amplitude of the modulation,
$m_{3}$=0.65 $\mu_{\mathrm{B}}$, inferred from our neutron diffraction
measurements. Overall, the magnetic structure is consistent with the magnetic
anisotropies determined from previous measurements Franz et al. (2016); Franz
(2014), which establish an easy magnetization plane $(0\,0\,1)$ and a hard
magnetic axis along the $[0\,0\,1]$ direction, since the main components of
the modulated part of the magnetization lie in the $(0\,0\,1)$ plane, as
summarized in Table 3.

Most likely, the tilting of the magnetic moment away from the easy $(0\,0\,1)$
plane is the result of Dzyaloshinskii–Moriya interactions allowed for the non-
centrosymmetric $I4mm$ space group, see Tab. II in Ref. Bauer et al., 2022.
Alternatively, the field dependence may suggest coexisting local and itinerant
magnetism in CePtAl3, as reported for other cerium compounds Nakano et al.
(2019); Hoshino and Kuramoto (2013), where the itinerant electrons might carry
the magnetic moment component along the $c$ axis, driving the
incommensurability of the magnetic order.

Within the experimental resolution of our study, we did not observe any
changes in the length scales of the magnetic modulations reflected in the
constant component of the modulation vector $k$=$0.677\,(1)$ under applied
field. Our experimental setup allowed us to clearly distinguish that the
modulation is not commensurate with $k$=$\frac{2}{3}$. The mechanism causing
the formation of the incommensurability and the pinning of the magnetic
modulation length cannot be explored further in our study. Since it is
unchanged for both field directions and up to relatively high magnetic fields,
it seems possible that the pinning involves structural disorder and follows
the underlying atomic length scales of the disorder.

##  IV Conclusions

In conclusion, we reported on a detailed neutron diffraction study of the
crystal and magnetic structure of CePtAl3 complemented by specific heat
measurements. For our study, we have grown a large single crystal of CePtAl3
by means of optical floating-zone method, which exhibits occupational and
positional disorder on atomic length scales in neutron diffraction
measurements. In the simplest scenario, the disorder affects the long-range
magnetic order, which is described well in terms of a spread of the transition
temperatures with an average transition temperature
$\overline{T}^{n}_{\mathrm{N}}$=3.03 (15) K and standard deviation
$\sigma^{n}_{T_{\mathrm{N}}}$=0.48 (7) K. This model is quantitatively
consistent between the temperature dependence of the magnetic order parameter
observed in neutron diffraction and the distribution of anomalies in the
specific heat at the magnetic transition where
$\overline{T}^{c_{p}}_{\mathrm{N}}$=3 K and
$\sigma^{c_{p}}_{T_{\mathrm{N}}}$=0.64 K.

Measurements under applied magnetic field provided further information on the
magnetic structure of CePtAl3. For fields up to 1 T applied along $[0\,1\,0]$,
CePtAl3 exhibits the characteristics of a multi domain state with single-$k$,
incommensurate, cycloidal magnetic order. Between 1 and 1.5 T, a single domain
state is stable. Further increasing the field causes a magnetic transition
between 1.5 and 3 T, suggesting a change from cycloidal to amplitude-modulated
magnetic order. Above 3 T, the magnetic order may be described by amplitude-
modulated moments in the tetragonal plane. For fields applied along
$[0\,0\,1]$, no transitions were observed, and CePtAl3 remains in a multi
domain state up to 4 T, the highest field studied. Putative mechanisms at the
heart of the AFM order may be Dzyaloshinskii–Moriya spin-orbit interactions
or, more speculatively, a mixed itinerant and local-moment magnetism carried
by the conduction and 4$f$-electrons, respectively.

Taken together, the evidence for structural disorder and smearing of the
magnetic phase transition in CePtAl3 at zero magnetic field and the magnetic
field dependence of the AFM structure from cycloidal to amplitude modulated
order identify a highly unusual interplay of magnetic order, magnetic
textures, and structural properties in the presence of disorder in an
$f$-electron compound.

##  V Acknowledgements

We kindly acknowledge Thomas Müller for help with the analysis of the DNS
data, Alistair Cameron for support at D23, as well as Marc Wilde and Vivek
Kumar for helpful discussions. We also thank the staff at the MLZ and ILL for
support. Parts of the neutron measurements were performed at POLI, jointly
operated by the RWTH Aachen University and JCNS within the JARA-FIT
collaboration.

This study was funded by the Deutsche Forschungsgemeinschaft under Project No.
WI33203-1, TRR80 (From Electronic Correlations to Functionality, Project No.
107745057, Project E1), SPP2137 (Skyrmionics, Project No. 403191981, Grant No.
PF393/19), and the excellence cluster MCQST under Germany’s Excellence
Strategy EXC-2111 (Project No. 390814868). Financial support by the European
Research Council through Advanced Grants No. 291079 (TOPFIT) and No. 788031
(ExQuiSid) and through the European Union’s Horizon 2020 research and
innovation program as well as the Czech Science Foundation GAČR under the
Junior Star grant No. 21-24965M (MaMBA) is gratefully acknowledged.

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# Towards Dialogues for Joint Human-AI Reasoning and Value Alignment

Elfia Bezou-Vrakatseli  Oana Cocarascu  Sanjay Modgil King’s College London
{elfia.bezou_vrakatseli, oana.cocarascu<EMAIL_ADDRESS>
###### Abstract

We argue that enabling human-AI dialogue, purposed to support joint reasoning
(i.e., ‘inquiry’), is important for ensuring that AI decision making is
aligned with human values and preferences. In particular, we point to logic-
based models of argumentation and dialogue, and suggest that the traditional
focus on persuasion dialogues be replaced by a focus on inquiry dialogues, and
the distinct challenges that joint inquiry raises. Given recent dramatic
advances in the performance of large language models (LLMs), and the
anticipated increase in their use for decision making, we provide a roadmap
for research into inquiry dialogues for supporting joint human-LLM reasoning
tasks that are ethically salient, and that thereby require that decisions are
value aligned.

##  1 Introduction

As AI becomes more integrated into our lives, one needs to ensure both
explainability of AI decisions, and that AI decisions accommodate inputs from
human users. This requirement is especially salient when decisions have
ethical impact, and decisions need to be aligned with the preferences and
values of human stakeholders. Indeed, addressing the so called ‘value
alignment’ problem (?; ?) – ensuring that AI (in particular machine learning)
systems achieve their goals in ways that are not anti-ethical to human
preferences/values – has acquired renewed urgency in light of recent dramatic
advances in large language models (LLMs), and anticipated uses of LLMs in
supporting decision making (?; ?; ?; ?; ?).

In this paper we make the following two claims so as to then advocate for a
programme of research that: 1) advances work on argumentation-based dialogues;
2) enhances the reasoning and conversational capabilities of LLMs; 3) enables
joint human-LLM reasoning. Firstly, we claim that human-AI dialogue is
required to support joint reasoning, enabling human input into decision making
processes. In particular, these joint reasoning or ‘inquiry’ dialogues should
accommodate input and reasoning about human values and preferences. This will
not only ensure that individual decisions are value-aligned, but that humans
can be supported in refining their ethical positions, and the
values/preferences informing decisions can then be learnt by AI systems so as
to inform their future value-aligned decisions. Secondly, we claim that
dialogues generalising argumentative formalisations of non-monotonic reasoning
(?; ?) present the most suitable candidates for enabling joint human-LLM
inquiry.

In Section 2 we further develop these claims by reference to motivating
examples, and then enumerate a non-exhaustive list of research challenges if
one is to realise the long term goal of joint human-LLM inquiry for value
alignment. In Section 3 we review current progress towards realising these
research challenges, and key next steps in the research programme. We conclude
in Section 4.

##  2 Motivating Research into Joint Inquiry for Value Alignment

The last two decades have witnessed significant advances in machine learning,
and accompanying concerns that, as AI systems become more autonomous and
sophisticated on a trajectory to AGI and beyond (?), the overriding imperative
of maximising reward functions may result in unforeseen decisions and actions
that are harmful to humans (?; ?). One of the most prominent emerging
solutions to this problem of ‘value alignment’ is Russell et.al’s Cooperative
Inverse Reinforcement Learning (CIRL) framework (?; ?). CIRL proposes that AI
systems remain in a constant state of uncertainty about what the human value
function is, and so are perpetually engaged in the primary goal of learning
the value function through observation, query, and instruction. The challenge
is to then ensure that the AI simultaneously acts so as to realise other goals
for which it has been tasked111CIRL can be interpreted as advocating
‘enculturation of’ AI, analogous to how young children acquire an
understanding of values through observation, questioning and instruction. Of
course, the analogy breaks down when one considers that AI systems are
expected to perform complex tasks, often surpassing human capabilities, while
simultaneously being ‘ethically enculturated’.. As emphasised in (?), this
entails that delegation of decision making tasks to AI systems requires
simultaneously eliciting human input, and in particular ethically salient
information that reveals stakeholder preferences and values; information that
can then be learnt in order to inform future decisions. Consider the following
simple example of an LLM specialised to serve as a personal assistant (PA) for
a human user (H):

> PA: “Given your budget and preferences for landscape and weather, I propose
> two options: (a) 7 days at the Dubai Beach Resort; (b) 7 days at the Miami
> Beach Resort.”
>
> H: “I prefer (b).”
>
> PA: “Why?”
>
> H: “Because according to Amnesty International, Dubai has unethical labour
> practices.”
>
> PA: “Ok. I will prepare an itinerary for Miami.”

This short dialogue illustrates how, based on H’s provision of factual
information (i.e., budget constraints) and aesthetic preferences (weather and
landscape), PA proposes two options and then elicits an ethically salient
preference from H, effectively leading to a joint value aligned decision in
favour of option (b), and ideally, PA leveraging said ethical preference in
later decisions. Any such reasoning process should ideally be prescriptively
governed by rational principles for non-monotonic reasoning, wherein, as new
information comes to light or there are conflicting reasons for alternative
claims (inferences), preferences and values can be used to arbitrate in favour
of a claim over its contradictory.

We thus turn to argumentative formalisations of non-monotonic (nm) reasoning,
wherein arguments constructed (via chaining premises to a claim through
application of deductive and/or default inference rules) from a given
individual’s belief base $\mathcal{B}$, are related by a defeat relation. For
example, if argument $A$’s claim contradicts argument $B$’s claim, then $A$
and $B$ attack each other. $A$ successfully attacks (i.e., ‘defeats’) $B$,
only if $A\nprec B$ (where $\prec$ is a preference relation over the
constructed arguments). The argument framework ($AF$) of arguments related by
defeats can then be evaluated under various Dung semantics (?), so as to
identify the justified (winning) arguments. The claims of these arguments can
then be shown to equate with various nm logical consequence relations defined
directly over $\mathcal{B}$ (e.g., see (?; ?; ?; ?)). Argument game proof
theories (e.g., (?)) can then be deployed to establish whether a given
argument in an $AF$ is justified under a given semantics, and hence whether
its claim is an nm consequence from the underlying $\mathcal{B}$. These
argument game proof theories can be adapted (?) to yield dialogical
formalisations of nm reasoning (e.g., (?; ?)) in which agents can exchange
locutions – not just arguments but claims, questions etc. – that conform to
protocols generalising rules in the aforementioned argument games. The
protocols and evaluation of the resultant _graph of locutions_ are defined so
that under certain conditions:

> a communicated claim $\alpha$ is established by dialogue $d$ iff $\alpha$ is
> the claim of a justified argument in $AF_{\mathcal{B}}$

where instead of $AF_{\mathcal{B}}$ being defined by a static given belief
base $\mathcal{B}$, as in the paradigmatic case of single agent reasoning,
$AF_{\mathcal{B}}$ is incrementally defined either by the publicly asserted
beliefs $\mathcal{B}$ asserted in the dialogue $d$, or the union of the belief
bases of the agents participating in $d$. One thus obtains (given the
aforementioned soundness and completeness results for argumentative
characterisations of nm consequence relations) communicative accounts of
distributed, i.e., ‘joint’, _nm_ reasoning ($\,\mid\\!\sim\,$ below denotes
the consequence relation of some nm logic):

> a communicated claim $\alpha$ is established by dialogue $d$ iff
> $\mathcal{B}\,\mid\\!\sim\,\alpha$

Consider our example dialogue. The budget and aesthetic information provided
by H, combined with other travel related information, is used as premises in
two mutually attacking arguments $A$ and $B$, respectively claiming the Dubai
and Miami options. PA claims a preference for $B$, and upon questioning,
argues against (i.e., defeats) $B$ with an argument $C$ claiming unethical
labour practices in Dubai. Hence $A$ is defended by $C$’s defeat on $B$ and
the dialogue establishes option (b).

Key to enabling such dialogues is the use of argument schemes (AS) and
critical questions (CQs) (?) that have been developed within the field of
informal logic. AS are templates capturing stereotypical patterns of intra-
argument reasoning, and can be instantiated by natural and/or formal logic
languages. For example, one can suppose both $A$ and $B$ to be instances of
the Argument for Action (AA) scheme (?): ‘In circumstances $S$, doing action
$Act$ will achieve goal $G$ and so promote value $V$’ (where $S$ includes,
amongst other things, budget and aesthetic information, $G$ is the goal of
‘going on holiday’, $V$ the value of ‘pleasure’ and $Act$ the respective
decision options). Associated with AA are sixteen critical questions which
identify possible criteria for attacking, or indeed supporting, any concrete
instance of AA. For example, the CQ ‘Is it the case that $S$ ?’ can be
addressed as a question by agent $Ag$ to an agent $Ag^{\prime}$ who has
submitted an argument $X$ instantiating AA. In response, $Ag^{\prime}$ is
dialectically obligated to provide an argument claiming $S$.222E.g., an
argument claiming $Act$ = ‘lockdown’ may cite $S$ = ‘there is a pandemic’, and
is not justified until such point that in response to questioning whether it
is the case that there is a pandemic, relevant evidence is cited to support
this claim. CQs can also be used to challenge arguments. In our running
example, H uses the CQ ‘Does action $A$ demote some other value’ to submit the
argument claiming unethical labour practices. $C$ is itself an instance of the
argument from expert opinion (EO) scheme which cites an expert having asserted
a claim as support for the claim. EO has its own CQ which can similarly be
used to respond to $C$. Thus, AS and CQs help map out the space of possible
reasoning, and a key requirement for realising the use of AS and CQs in
dialogues is the following:

> Strategic considerations should inform the choice of AS or CQs at any given
> point in a dialogue ($R_{strat}$)

Suppose that instead of $C$, H submitted the locution $C^{\prime}$: “Dubai has
unethical labour practices”. Now note that $C^{\prime}$ is an incomplete
argument – an ‘enthymeme’ (?) – consisting only of a claim. Use of enthymemes
are characteristic of real world dialogues, and ideally, the dialogue would
evolve so that either the PA or H is prompted to supply evidence in support of
the claim, to, as it were, ‘flesh out’ the claim $C^{\prime}$ into a fully
fledged argument $C$. Thus, PA might question ‘why $C^{\prime}$’, so that H is
prompted to submit the expert opinion as a premise. Indeed, H may herself
submit $C^{\prime}$ as a query; she believes $C^{\prime}$ to be the case, but
is inviting PA (or perhaps some other agent) to provide the missing premise
and so complete $C^{\prime}$ into an argument. In either case, until the
dialectical obligation is met, $C^{\prime}$ does not constitute a valid defeat
on $A$, and $B$’s claim is not established. Hence the following requirement
for realising inquiry dialogues:

> Dialogues should be guided so that declarative locutions can be combined to
> define complete arguments ($R_{enth}$)

Consider the following example dialogue333Admittedly, this does not represent
a typical decision making scenario; rather it is used to illustrate
requirements for ethical reasoning. between a human H and a ‘Socratic LLM’
SocrAPP:

> 1\. H: “I propose that we should make abortion illegal.”
>
> 2\. SocrAPP: “Why?”
>
> 3\. H: “Because it amounts to the early termination of life.”
>
> 4\. SocrAPP: “So?”
>
> 5\. H: “All humans have a right to life.”
>
> 6\. SocrAPP: “What about the right of a woman to choose?”
>
> 7\. H: “What do you mean?”
>
> 8\. SocrAPP: “Your argument appeals to the right to life, but making
> abortion illegal violates the right of a woman to choose.”
>
> 9\. H: “But the right to life takes precedence over the right to choose.”
>
> 10\. SocrAPP: “Why?”
>
> :
>
> n ($>$ 11). SocrAPP: “Here are links to studies showing that in developing
> countries, giving women the right to choose reduces poverty . Here are some
> articles that discuss and analyse these findings.”
>

In 1–5, SocrAPP elicits (in 2 and 4) information from H. This information
completes the initial claim (the enthymeme in 1) into an argument $A$,
instantiating a variant of scheme AA – which we call AA(R+) – that appeals to
a right (to life) rather than a value. In 6, SocrAPP uses a CQ for AA(R+) –
Does the action violate some other right? – to submit an enthymeme for an
argument $B$ instantiating a scheme (AA(R-)) prohibiting an action if it
violates an established right. We note that schemes such as AA(R+) and AA(R-)
and the CQ we posit for the former, are novel, and specialised for ethical
reasoning. Indeed, experience with the use of AS $\&$ CQs for guiding real-
world deliberation in the medical domain (?) highlights the limiting
generality of AS $\&$ CQs (?), and the need for AS $\&$ CQs specialised for
medical reasoning. Hence we posit the following requirement for joint inquiry
dialogues that facilitate value alignment:

> Schemes and Critical Questions specialised for ethical reasoning
> ($R_{ScCQ_{Ethical}}$)

In response to H’s query in 7, SocrAPP fulfills an explanatory remit (in 8),
completing 6 into the argument $B$, and additionally making explicit that $B$
and $A$ attack each other. In line 9, H declares a preference for the right to
life over the right to choose (i.e., the preference $B\prec A$) so that only
$A$ defeats $B$ (the dialogue has therefore thus far established H’s initial
claim). In line 10, SocrAPP queries H’s rationale for this preference. H’s
initial claim is now ‘dis-established’, and H is now required to provide a
‘metalevel’ argument for why one argument is preferred to another in order to
re-stablish her initial claim.

Now, recall that we have thus far proposed sound and complete inquiry
dialogues, in the sense that evaluation of a graph of locutions establishes a
claim $\alpha$ iff $\alpha$ is the claim of a justified argument in the $AF$
constructed from $\mathcal{B}$ (the contents of declarative locutions
exchanged, or of the participating agents’ belief bases), which in turn
equates with $\mathcal{B}\,\mid\\!\sim\,\alpha$ in some nm logic. However, in
real-world (especially ethically salient) dialogues, one does not always
expect consensus as to the preferences used to determine the success of
attacks and defeats; rather one expects reasoning about possibly contradictory
preferences, and argumentation based resolution of these conflicts. Hence, one
requires:

> Metalevel Schemes and Critical Questions specialised for metalevel reasoning
> ($R_{ScCQ_{Meta}}$)444 (?) also posit meta-schemes tailored to reasoning
> about other object-level features of $AF$, including e.g., the rationale for
> why arguments attack each other, when said rationale is not based on
> negation, e.g., when arguments for two actions conflict because they have
> mutually exclusive pre-conditions.

and

> Soundness and Completeness of dialogues that accommodate argumentation based
> metalevel reasoning about preferences and values ($R_{SC-Pref}$)

Let us suppose the dialogue continues, but (as is typically the case when
appealing to rights), there is no way to resolve the conflict between
arguments expressing contradictory preferences over rights. The claim is then
subject to consequentialist evaluation, and in line $n$, SocrAPP is responding
to a prompt requesting evidence for the positive and negative consequences of
legalising/prohibiting abortion (said evidence instantiating further variants
of the $AA$ scheme that justify actions on consequentialist grounds).

An important feature of our proposed inquiry dialogues for value alignment is
that they facilitate rational exploration of the space of possible ethical
evaluations one might take with respect to a decision. As illustrated above,
the implication is that although a user might be committed to a rights-based
ethical perspective, they would ideally be exposed to alternative viewpoints,
such as consequentialist criteria, when evaluating decisions. Indeed, recent
philosophical approaches eschew a ‘one size fits all’ approach, and instead
advocate judgements that aggregate evaluations of decision options under
different ethical theories (?; ?). Hence, the requirement for object-level
domain specific ethical AS and CQs, and metalevel AS and CQs that may be
deployed to aggregate the evaluations yielded by object-level arguments.
Indeed, the implication is that a user may be encouraged to refine or even
revise their ethical preferences, with the proviso that the AI interlocutor’s
role is not to persuade the user to commit to a particular ethical judgement,
but rather to explore the space of ethical reasoning. This then implies use of
other dialectical moves, such as the well known Socratic move demonstrating
that an interlocutor has contradicted herself (?). For example, suppose that
after line 5, SocrAPP queries whether H is for or against the death penalty,
and H replies to say she is in favour. SocrAPP can then reveal that H has
contradicted herself (given her appeal to the right to life), prompting H to
refine her ethical stance: “All humans, except those that take the life of
another, have the right to life”. Hence the requirement that:

> Dialogues accommodate dialectical moves other than the moving of
> attacking/supporting arguments, and results in changes to the beliefs of
> interlocutors ($R_{Dia}$)

In this section we have, through the use of illustrative examples, enumerated
a non-exhaustive list of requirements for inquiry dialogues that formalise
joint non-monotonic reasoning; dialogues purposed to accommodate human ethical
preferences and values that may be refined and subsequently learnt by AI
interlocutors that are simultaneously tasked with supporting human users in
their decision making. In the next section we briefly review progress towards
realising these requirements, and then focus on challenges that need to be
addressed if LLMs are to successfully engage in such dialogues.

##  3 Progress Towards Realising Joint Inquiry for Value Alignment

###  3.1 Towards Argumentation-based Dialogues for Value Aligned Inquiry

There is a sparsity of work on formalising Inquiry Dialogues such that the
aforementioned soundness and completeness results hold. A notable exception is
the work of (?) in which agents can challenge arguments and collaboratively
construct arguments in the language of Defeasible Logic Programming (?).
Rather, the majority of research focuses on argumentation-based ‘zero sum’
persuasion dialogues (?) that are inherently adversarial, and thus clearly
unsuitable given the collaborative goal of joint reasoning that we are
interested in. There are then key requirements specific to inquiry that do not
arise in the case of persuasion. Notably, arguments are constructed by
individual agents in the latter, whereas inquiry deploys the epistemic
resources of multiple agents in constructing individual arguments. Moreover,
in persuasion dialogues, an agent cannot target their own argument with
defeats, whereas inquiry should licence targeting one’s own argument, given
that the aim is to get to ‘truth of the matter’ in respect of factual
information, and to support decision options that are rationally justified as
being optimal for users. Indeed, given these collaborative aims, the
assumptions under which one expects the previous section’s soundness and
completeness results for dialogues to hold – namely that interlocutors are
honest and are exhaustive in submitting locutions licensed by the protocol –
are reasonable. In light of what we perceive to be a dearth of formal models
for sound and complete inquiry dialogues, we briefly point to progress made
towards realising the requirements enumerated in Section 2.

Firstly, the ASPIC+ framework (?) for formalising argumentative
characterisation of non-monotonic reasoning, has been extended to accommodate
metalevel reasoning about preferences and values (?; ?). Dialogical
formulations of this extended framework have been developed (?), but it
remains to accommodate moves distinctive of inquiry rather than persuasion
(e.g. allowing multiple interlocutors to collaboratively construct individual
arguments), and to show soundness and completeness results ($R_{SC-Pref}$).
Moreover, enthymemes are not accommodated, but sound and complete inquiry
dialogues that accommodate enthymemes and are based on the standard ASPIC+
framework have been developed (?; ?) (addressing $R_{enth}$), and integration
of this work with (?) represents a significant next research objective.
Finally, recent argumentative formalisations of non-monotonic reasoning have
been developed to accommodate attacks that dialectically demonstrate the
inconsistency of premises of challenged arguments (?; ?), partly addressing
$R_{Dia}$ (it remains to formalise changes to beliefs in response to such
moves).

Advances have also been made in addressing $R_{ScCQ_{Ethical}}$, as detailed
in (?), in which prevalent AS used in ethical reasoning on the debate platform
Kialo (https://www.kialo.com) are identified and have been used to guide
formulation of a new, specialized taxonomy for ethical reasoning. The
(ongoing) creation of this taxonomy categorises common AS used in ethical
debates and introduces new AS specifically tailored for such discussions.
Furthermore, this taxonomy proposes meta-schemes ($R_{ScCQ_{Meta}}$) that
facilitate reasoning about and comparing ethical evaluations yielded by object
level arguments instantiating ethical schemes.

###  3.2 Towards Large Language Model Interlocutors

Large Language Models (LLMs) (?; ?; ?; ?) have significantly advanced the
state-of-the-art on a wide range of language understanding tasks, by
harnessing vast amounts of data, and excel in generating coherent and relevant
text based on minimal prompts. This technological leap has ushered in a new
era of AI, where machines can engage in more human-like conversations and
provide insights with unprecedented accuracy (?; ?; ?; ?; ?; ?; ?; ?; ?), with
multiple studies investigating the limits of their abilities, including the
ability to learn (?), reason (?; ?; ?; ?), and comprehend intricate language
structures such as metaphors (?). The remarkable capabilities of LLMs have led
to their widespread adoption across various fields, including finance (?; ?),
robotics (?; ?), medicine (?; ?), and even crisis counselling (?).

However, despite their success, LLMs face challenges when faced with complex
reasoning tasks, especially involving knowledge and symbolic reasoning (?),
and exhibit deficiencies in both planning and reasoning abilities (?).
Dialogue, specifically, remains a challenging task for LLMs (?; ?; ?; ?; ?).
In order for LLMs to engage in joint inquiry as described in the previous
sections, LLMs must be able to instantiate AS and CQs effectively. We discuss
below a non-exhaustive selection of challenges for realising this goal.

Firstly, LLMs need to be able to classify arguments using AS and to identify
how one argument relates to another, through the use of their CQs. This is an
open challenge but, given their impressive capabilities, there is now the
potential to overcome previous hurdles in AI natural language processing,
particularly regarding the instantiation of AS and CQs. Indeed, our
preliminary investigations suggest that not only are LLMs capable of
classifying arguments according to the AS they instantiate, but are also able
to articulate why said AS have been identified. However, as with human
annotators of argumentative corpus (?; ?; ?), LLMs struggle to differentiate
between similar AS.

Building on these investigations suggests a number of open challenges.
Firstly, harnessing the abilities of LLMs to identify arguments that support
or challenge a given claim, the schemes these arguments instantiate, and hence
the ability to flesh out enthymemes (recall $R_{enth}$). Secondly, while LLMs
can be promted to generate example dialogues on a given topic, the ability to
identify the critical questions that arguments are responding to is still
limited.

Moreover, effective instantiation of AS and CQs requires LLMs to not only
recognise, but also apply these reasoning structures, in a contextually
appropriate manner. The challenges discussed above go hand in hand with the
LLMs’ need to retain conversational histories over prolonged interactions.
Maintaining conversational histories in a short-term memory module is
currently an active research area (?; ?), which could be leveraged to maintain
coherence across extended dialogues and multiple interactions.

Finally, one would want LLMs to develop reasoning capacities enabling more
complex dialectical moves ($R_{Dia}$), and to strategise ($R_{strat}$).
Besides identifying points of support and attack, there is a need for
subsequent instantiation of AS to construct arguments to support/attack,
supported by a dialogue manager, with evaluation supported by a protocol
manager. To effectively mimic human-like discourse management, and address
$R_{strat}$, LLMs should additionally be able to navigate the flow of
conversation, deciding when to probe deeper or when to guide towards
conclusions. This can be viewed as a search problem in a game tree, where the
choice of moves amounts to the choice of AS and CQs.

One technique that has been developed to improve the performance and maximize
the utility of LLMs is prompt engineering (e.g., (?; ?; ?)), which is
increasingly essential for proficient interaction with LLMs (?; ?). Prompts
are instructions given to an LLM by customizing it and/or enhancing its
capabilities to enforce rules, automate processes, and ensure desired
attributes and volumes of generated content (?; ?). Effectively, prompt
engineering is the means by which LLMs are ‘programmed’ via prompts, dictating
how the models should process information.

Chain of thought (CoT) prompting (?), which consists in breaking down an
initial task/question into intermediate reasoning steps, holds promise for
addressing the challenges we have identified. Overall, intermediate reasoning
has been shown to improve LLMs’ performance (?) and CoT is currently used for
improving the reasoning abilities of LLMs (?; ?; ?; ?). Essentially, CoT
represents a series of intermediate natural language reasoning steps; for
instance, step-by-step (?), which guides the model through a series of
reasoning steps, or decomposition (?), where complex problems are broken down
into simpler sub-problems. Especially promising for joint inquiry is maieutic
prompting (?; ?) which, building on the two aforementioned types, utilises a
recursive dialogue-based prompting, emphasising logically consistent reasoning
through recursive explanations at each step. This may be helpful for
addressing $R_{Dia}$ and supporting LLMs in developing the reasoning
capacities for addressing different dialectical moves.

##  4 Conclusions

This paper advocates for a focus on argumentation-based inquiry dialogues so
as to integrate human ethical preferences in joint human-LLM reasoning and
decision making. We have argued that such dialogues are warranted by state-of-
the-art proposals for value alignment, and moreover support humans in refining
their ethical positions. We have enumerated a number of research challenges
for realising said dialogues, in the areas of computational argument and
dialogue, and LLMs, and pointed to progress made. While recent advancements in
LLMs offer promising capabilities, significant challenges in dialogue
management persist. Addressing these challenges will be key to developing AI
systems that can engage effectively in joint inquiry for value alignment.

## Acknowledgments

This work was supported by the UK Research and Innovation Centre for Doctoral
Training in Safe and Trusted Artificial Intelligence [grant number
EP/S023356/1].

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