2022 1]Department of Mechanical Engineering, University of Delaware, 130
Academy St, Newark, 19717, DE, USA

# ModMag: A Modular Magnetic Micro-robotic Manipulation Device

Max Sokolich<EMAIL_ADDRESS>David Rivas<EMAIL_ADDRESS>Markos Duey
<EMAIL_ADDRESS>Daniel Borsykowsky<EMAIL_ADDRESS>Sambeeta Das
<EMAIL_ADDRESS>[

###### Abstract

Electromagnetic systems have been used extensively for the control
magnetically actuated objects, such as in microrheology and microrobotics
research. Therefore, optimizing the design of such systems is highly desired.
Some of the features that are lacking in most current designs are compactness,
portability, and versatility. Portability is especially relevant for
biomedical applications in which in vivo or in vitro testing may be conducted
in locations away from the laboratory microscope. This document describes the
design, fabrication and implementation of a compact, low cost, versatile, and
user friendly device (the ModMag) capable of controlling multiple
electromagnetic setups, including a two-dimensional 4-coil traditional
configuration, a 3-dimensional Helmholtz configuration, and a 3-dimensional
magnetic tweezer configuration. All electronics and circuitry for powering the
systems is contained in a compact 10”x6”x3” system which includes a 10”
touchscreen. A graphical user interface provides additional ease of use. The
system can also be controlled remotely, allowing for more flexibility and the
ability to interface with other software running on the remote computer such
as propriety camera software. Aside from the software and circuitry, we also
describe the design of the electromagnetic coil setups and provide examples of
the use of the ModMag in experiments.

Keywords. micro-robotics, cellular manipulation, magnetic actuation,
electromagnets, magnetic tweezers

Table 1: Specifications table Subject Area | Mechanical Engineering: Micro-Robotics
---|---
More Specific Subject Area | Magnetic Manipulation Systems
Name of Your Method | Modular Magnetic Manipulation System
Name and Reference of Original Method | Not Applicable
Resource Availability | CAD Files and Software Available Upon Request.
|

##  1 Introduction

###  1.1 Background

Micro-robots have drawn a lot of attention in the past few years due to their
potential to radically improve the efficacy of numerous tasks, such as cell
manipulation, targeted drug delivery, microsurgery, and mixing of particles
peyer2013bio . Despite this, conventional methods for powering macro-scale
robotic actuation methods due not apply at the micro-scale,since they cannot
be scaled down to size. Hence, autonomous and reliable control of these micro-
robots is an on going challenge. Different methodologies have been proposed in
recent years in a bid to efficiently control micro robots. Researchers have
validated the actuation of the micro machines employing bubble creation using
chemical reactions solovev2010catalytic  or by attaching the synthetic micro-
robots to swimming bacteria sakar2011modeling . External magnetic, acoustic,
or electric fields have also been employed to control and steer the micro-
robots sakar2011modeling ; aghakhani2020acoustically ; das2018experiments .

One of the most common methods of controlling micro-robots is magnetic
actuation. Magnetic actuation has multiple advantages over other actuation
methods; for example, the micro-robots can be controlled independent of line
of sight making them capable of deployment in variety of environments. In
addition, magnetic fields are regarded as a safe choice to use at the cellular
and tissue level for a large number of biomedical applications
fischer2011magnetically .

The examples listed above all use expensive equipment such as data
acquisitions devices and large bench top power supplies which limit their
ability to be used on other microscopes or transported to other locations.
Furthermore, the software is generally integrated onto a lab computer which
also limits portability.

In this paper, we demonstrate the design and testing of a handheld, portable,
low cost, modular electromagnetic actuation system, ModMag, for control of
micro-robots. We also show experimental examples of micro-robot control using
different types of magnetically driven micro-robots for applications in
extracellular research.

###  1.2 Electromagnetism

When an electric current passes through a wire, the movement of charge
generates a magnetic field. A solenoid, or electromagnet, uses many of these
wires wrapped concentrically around a cylinder to create a magnetic field in a
direction along its central axis. The strength of the magnetic field is
proportional to the current and increases with the number of times the wire is
wrapped around the cylinder. It is also proportional to the magnetic
permeability which can be increased by introducing certain materials into the
core of the solenoid. Generally, ferromagnetic materials with low coercivity,
such as soft iron, are the material of choice for their ability to enhance the
magnetic field generated by the solenoid while retaining low magnetization
when the current is turned off.

Ferromagnetic materials contain microscopic magnetic domains where groups of
magnetic moments naturally align in the same direction. When the domains are
all aligned randomly the material is demagnetized. However, the presence of an
external magnetic field can align the domains in the direction of the applied
magnetic field, causing the material to be magnetized. It may be helpful to
think of each of the magnetic domains as tiny magnets; when current passes
through the surrounding coil of wire, it forces each of these tiny magnets in
the same direction, thereby amplifying the field.

In a similar manner, when a ferromagnetic object, such as a magnetic micro-
robot, is placed in a magnetic field, its magnetic moment aligns with the
external field. This is the fundamental way in which magnetic microrobots are
controlled. Consider a self propelled, spherical, iron coated micro-sphere
suspended in a liquid solution. Typically, the micro-sphere moves in an
arbitrary direction and is subject to Brownian rotational fluctuations. An
external torque can be applied in order to steer the microsphere in a
particular direction. The torque, $\boldsymbol{\Gamma}$, is given by

$\boldsymbol{\Gamma}=\textbf{m}\times\textbf{B}$ (1)

where m is the magnetic moment of the microrobot and B is the magnetic field.
This torque therefore allows one to magnetically orient the microrobots.
Magnetic forces can also be applied to the microrobots by applying magnetic
gradients. The magnetic force, F, is given by

$\textbf{F}=\left(\textbf{m}\cdot\nabla\right)\textbf{B}$ (2)

Applying sufficiently strong magnetic gradients generally requires the use of
high permeability poles which are used to concentrate the magnetic flux
generated from an electromagnetic. This allows for strong enough magnetic
gradient forces to overcome the highly viscous drag forces that a microrobot
experiences. Magnetic devices that are used to magnetically manipulate objects
are known as magnetic tweezers. Figure 1 illustrates the process by which an
object with a magnetic moment can be oriented or translated in a uniform or
spatially varying field, respectively.

Figure 1: The left figure shows an object rotating due to a magnetic torque.
The white arrow indicates the object’s magnetic moment and the gray lines
represent magnetic field lines. The figure on the right shows an object moving
due to a magnetic force created by a magnetic field gradient vries_high_2004 .
The region to the right of the object has a greater density of field lines
which corresponds to a higher magnetic field strength.

##  2 Method Details

###  2.1 Mechanical

In determining the design of the portable electromagnetic system we focused on
three aspects. The first was that the software and circuitry should be
versatile in its ability to be used with multiple electromagnetic setups and,
therefore, microrobots with different actuation mechanisms. That is, it is
desirable for a system to be versatile and not require a complete system
redesign when using different microrobots or during different applications or
experiments. The electromagnetic setups we focused on were a 2D solenoid
array, a 3D Helmholtz-based, and a 3D magnetic tweezer system. Each of these
systems allows for immense versatility when it comes to micro-robotic
experiments. We also sought to create a portable and inexpensive system that
could also be used easily. Making the system portable and using a standalone
design allows for the system to be transferable to different microscopes and
imaging systems and also to settings outside the lab which could be useful in
medical applications.

As described above, three designs were used: a traditional 2D coil system, a
3D Helmholtz-based coil system, and a 3D magnetic tweezer system. The designs
of each are discussed in the following sections.

####  2.1.1 2D Traditional Coils

The traditional 2D electromagnetic setup consists of a 3D printed rectangular
exchangeable stand that fits inside the microscope slide holder, and 4
perpendicular electromagnets (see Fig. 2). The stand was designed such that it
fit onto the mechanical stage of a Zeiss Axioplan 2 upright microscope,
however it can be placed on any microscope stage surface. The Axioplan 2’s
limited stage space of approximately 150 x 100 x 25 cm in volume made it
difficult to design a stand with sufficient magnetic strength and negligible
disruption of the microscope’s functionality. As a result both the stand and
the electromagnetic coils were designed to be as compact as possible.

The 2D electromagnetic stand was designed in Solidworks and consists of only
one part (see Figure 2). The height of the stand measures roughly 23 mm to
accommodate for the rotation of all microscope objectives in the objective
turret. The region of interest was designed such that a standard 22 x 22 mm
cover-slip or 25 x 25 mm square glass slide could be loaded with the desired
samples roughly half way up the height of the stand. In addition, a 30 mm
petri dish fits perfect in the region of interest. This allows for the
electromagnets to point directly towards the samples. In addition, a custom
square-shaped holder avoids the need to use a microscope slide which would
require the electromagnets to rest at an angle. A slit in the stand was
introduced to allow for easy insertion and removal of the cover slip using
laboratory tweezers. The electromagnetic stand was printed on a Comgrow
Creality Ender 3 Pro 3D printer using 1.75 mm, white PLA filament. An STL file
of the stand was uploaded to Creality’s Cura software with the following
settings: printing temperature: 200 C, bed temperature 60 C, speed: 50 mm/s,
layer height: 0.12 mm, and infill: 20. The rest of the settings were left as
default. These settings provide a strong and durable stand for the
electromagnets to rest upon. 4 equally spaced holes 5 mm in diameter were
created along the same plane as the cover slip holder allowing for the
insertion of each electromagnet core. The holes allow for the electromagnetic
core to be as close to the cover slip as possible.

Figure 2: 2D electromagnetic stand

The electromagnetic core material is AISI 1008 carbon steel made of
99.31-99.7$\%$ iron, cold drawn and annealed at 925°C. The cores had raw
dimensions of 5 mm in diameter by 200 mm long. The relative permeability of
the core was listed as (86 $\pm$ 3.61) H/m. The cores were then cut to 50 mm
long. 24 AWG magnet wire was used for the electromagnetic coils. In order to
prevent the coil from unraveling, plastic end caps of 30 mm in diameter were
cut from 0.7 mm in thickness plastic. 5 mm holes were then drilled into the
center of these circles to allow for the electromagnetic core to be inserted.
To create an electromagnet, one end cap is positioned at the end of the core
and glued in place using Gorilla 2 part epoxy. The second end cap is then
glued 50 mm from the first end cap, or 5 mm from the opposite side. As a
result, the electromagnetic coil will be 50 mm long, leaving 5 mm at one side
to be inserted into the stand. Coils were then wound carefully using an
electric drill to exactly 980 turns of the 24 AWG magnet wire. Using a Tesla
meter, the electromagnetic strength at the face of the coil was measured to be
(201 $\pm$ 3) mT at a current of 2 Amps, and around 15 mT at the center of the
work space. The six electromagnet’s were then inserted and glued into the six
holes in the 3D printed stand.

####  2.1.2 3D Helmholtz-Based System

A 3D electromagnetic manipulation system was designed for micro-robotic
applications involving rotating homogeneous magnetic fields, which is well
suited for 3D magnetic steering or magnetic rolling. The custom coil system is
inspired by a Helmholtz design and is designed for mounting on the Zeiss
Axiovert 200 M inverted microscope (see Fig. 3). Measurements of the
microscope bed dimensions and distances between lenses were recorded. These
size constraints then defined the maximum radii for the coil rings. Calipers
were used to determine the microscope measurements. The Helmholtz coil system
consists of three pairs of rings each sharing their respective common axis.
The rings are wrapped with copper wire to form coils. As mentioned above, to
attain an optimally uniform field the distance between pairs of rings should
be equal to their radius. However, this is difficult to achieve in practice
and is also generally unnecessary for microrobotic applications, therefore we
loosened this constraint in order to allow the system to better fit the
workspace. As we will discuss later, we find that this system performs very
well for magnetic steering and rolling applications for which it is designed.
We do not notice any effects of non-uniformity in any of our experiments.

To optimally fit in the workspace, the large and medium sized rings were
mounted on the stage vertically while the small rings were mounted
horizontally, as shown in Fig. 3. Microscope dimensions allow for a maximum
diameter of 106 mm for the large rings without interfering with the objective
turret, and a minimum diameter of 36 mm for the small rings without
interfering with the condenser lens. The smallest pair of rings are spaced
such that their distance, $H_{s}$, is equal to their radius, $H_{s}=R=36mm$.
The medium sized pair is spaced slightly farther apart ($H_{m}=66mm$). For the
largest pair of rings, $H_{l}=83mm$. As a result, the ratios of distance over
radius of the small, medium, and large pairs of rings are 1.3, 1.8, and 1.5,
respectively. The platform unit is roughly 230 mm long and 140 mm wide with
integrated slots for mounting on the microscope. On either side of the
platform is an area for mounting a motherboard with electrical connections,
and a square divot for mounting a 2-axis micrometer for slide arm micro-
translation. The slide arm fits both a conventional glass microscope slide as
well as square cover-slips. Four slots are outfitted within the platform to
hold the large and medium coils with a transition fit. The small coil pair
sits on four stilts, while four small pegs separate the coil pair.

The platform, slide arm, and rings were designed within Solidworks CAD
software and printed using 3D Printing methods. All components are printed
with PLA using an Ender 3 Pro. Structural properties of PLA allow for adequate
rigidity within the frame to hold the weight of the primary components. The
coils are wrapped with 24 AWG copper wire using a standard hand drill to
achieve more uniform wire layers than what is achieved by hand. In order to
connect the ring to the drill, a drill adapter unit consisting of three
extending limbs was fabricated. Small pegs on each limb allowed for a
transition fit into receiving holes on the rings. The resulting number of
turns is approximately 368 for the small and medium coils and 260 for the
large coils. Applying 1 A to each of the coils yields a uniform magnetic field
over the sample area of 4 mT, 2 mT and 2 mT for the small, medium and large
rings respectively. This is more than adequate for actuating non-gradient
based magnetic micro-robots as shown in the experimental validation section.

Figure 3: Design schematics and image of the 3D Helmholtz-based System

####  2.1.3 3D Magnetic Tweezers

The magnetic tweezer system is inspired by the design presented in zhang20193d
. It consists of 6 magnetic poles made out of high permeability MuMetal. A
high magnetic permeability material is necessary for the construction of the
poles in order to attain efficient transport of the magnetic flux to the tips
of the poles. The system has an easy to use substrate holder in which a
microscope slide can be inserted into the workspace.

Figure 4: 3D magnetic tweezer system based on zhang20193d

This 3D design of magnetic tweezers allows for a concentrated and uniform
magnetic field gradient surrounding the microrobots. The 3 poles coming from
the top and bottom allow for magnetic symmetry as well as 3D maneuverability
of magnetic particles, something not achievable with in-plane tweezers. With 6
sources of magnetic flux, actuation can be made more precise, and movement
more robust.

The design aims to maximize field strength with large pole to pole tweezer
distances as well as minimal air gaps for better conduction. The bottom
hexagon orients the 3 bottom tweezers such that they point toward the bottom
center of the slide (1.5mm separated), and also supports the middle hexagon,
the slide holder in a transition fit. The top hexagon orients the top 3
tweezers in the same manner and fits around the slide holder. The simplicity
of the case design allows for convenient assembly/disassembly, as well as no
usage of hardware or adhesive. The case, including the coil/tweezers holders
was 3D printed entirely out of PLA and the tweezers were cut from 1.5mm
Mumetal, a nickel-iron soft ferromagnetic alloy with very high permeability.

The limiting design constraint of these systems was space availability due to
the microscope itself, and therefore the design may be altered or enhanced
based on one’s specific space constraints.

###  2.2 Electronics

####  2.2.1 Components

Figure 5: Electrical Schematic

The electrical system was designed to be functional with any and all of the
above mentioned electromagnetic setups while also operating as a standalone
device that is not constrained to a single lab computer. It is also fully
portable and does not require a bench-top external power supply. The
computational capability is provided by a Raspberry Pi Model 3 B+. The
Raspberry Pi is a small computer and capable of running a complete operating
system. The image file of Raspberry PI OS was burned to a SD card allowing a
full Desktop experience when hooked up to an HDMI monitor or display. We
employ a HMTECH 10.1 Inch Touchscreen Monitor with a resolution of 1024x600 to
control the electromagnets. The touchscreen displays a custom graphical user
interface which was written in Python. The computer is also equipped with 40
GPIO (general purpose input/output) pins which can be controlled through
software. However, electromagnets require significantly high currents and are
not able to be powered directly from the GPIO pins. Thus, six SEEU. AGAIN
BTS7960B 43A Double DC Stepper H-Bridge PWM drivers are used to vary the
current and polarity from an external power supply. The H-Bridge convert the
user defined currents generated from the GPIO pins and scale them accordingly
to each electromagnet. The external power supply consists of 3 Lithium-ion
18650 batteries yielding 12V with a peak current draw of 3 A. A charging
circuit board with a DC barrel jack allows the batteries to be recharged when
empty. Power can also be supplied by an external power source if desired. To
connect the controller to each of the 3 coil setups, a 20 pin IDC cable with
associated female adapter terminal was used. The electrical schematic is shown
in figure 5.

###  2.3 Software

####  2.3.1 Description

In order to accurately control current to each coil in real time, a custom
graphical user interface was designed in Python using the Tkinter library and
Gpiozero libraries. The software enables application of constant magnetic
fields in a user-defined direction as well as rolling, spinning, or vibrating
functionalities. It also is capable of being controlled using a game
controller for more responsiveness and precise control.

To apply a constant magnetic field, for example for microrobot steering, the
user-defined magnetic field direction and strength can be inputted and this is
used to modulate the current going to each coil. The associated vector is
broken into its x and y components and, since the magnetic field is
proportional to the applied current, the current sent through the x and y
coils is scaled proportionally. To enhance the field strength and uniformity,
an opposite polarity signal is sent to the oppositely facing coil. For
example, if a positive current is applied to the -x electromagnet coil, a
simultaneous negative current will be applied to the +x coil.

Figure 6: GUI python control panel displayed on 10 inch touchscreen. All
circuitry mounted on backside of device.

For the use of magnetic tweezers, the user is able to toggle between ”tweezer
on” and ”tweezer off” modes depending on whether the magnetic tweezers are
being used or one of the other electromagnetic systems. Magnetic tweezers
concentrate the magnetic flux generated from traditional coils towards the
tips of poles thereby generating adequate magnetic gradients capable of moving
a magnetic micro-robot without any additional propulsion mechanism. To
optimally move a microrobot, its magnetic moment should point in the same
direction as the magnetic gradient. Therefore, when the magnetic tweezer mode
is enabled in the software, current is only sent through one of the pairs of
coils in order to align the magnetic moment with the field gradient. Note that
the polarity of the magnetic field does not matter in this case since a single
pole always acts to attract the microrobots. This is because changing the
polarity changes the sign of B and this also causes the microrobot to rotate
such that m aligns with B, hence changing the sign of m as well.

All functions can also be controlled via a gaming controller. When a button is
pressed on the graphical user interface, it will connect to a wireless USB
gaming controller. The left joystick controls the 360∘ orientations or tweezer
functionalities while the right joystick controls the rotating magnetic field
direction, that is, the direction of a rolling magnetic microrobot. The device
is also capable generating a rotating magnetic field in any user defined
direction which allows for responsive and precise control of a rolling micro-
robot or an artificial bacteria flagella micro-robot/helical micro-robot. This
is done by applying a time varying sine wave to each of the X,Y,Z axis coils
using the following equations:

$B_{x}=A\cos(\gamma)\cos(\alpha)\cos(\omega t)+\sin(\alpha)\sin(\omega t)$ (3)
$B_{y}=-A\cos(\gamma)\sin(\alpha)\cos(\omega t)+(\cos(\alpha)\sin(\omega t)$
(4) $B_{z}=A\sin(\gamma)\cos(\omega t$ (5)

Where $\gamma$ is the azimuthal angle from the Z axis, $\alpha$ is the polar
angle from the Y axis, A is magnetic field magnitude and $\omega$ is the
frequency which controls the speed of the rolling microrobot. When controlling
magnetic rolling micro-robots, an aziumthal angle of 90∘ is set by default and
the polar angle can be changed to steer the direction of the rolling micro-
robot. See Figure 7 for an illustration of the rotation. For helical micro-
robot swimmers, changing both the azimuthal and polar angle results in full 3D
maneuverability.

In addition to the functions described above, there are also buttons on the
graphical user interface to magnetically vibrate a micro-robot. This is done
by turning on and off opposite facing electromagnets at user defined
frequencies. There are also quick direction buttons which allow for the
application of magnetic fields in the $\pm$x or $\pm$y directions without the
need to use the slider to select specific angles.

All of these functionalities can be controlled via the touchscreen display, or
wirelessly via the PiGPIO daemon. This allows not only the GUI python program
to be run and controlled on any other computer on the same Wi-Fi, but also
opens up the possibility for integrating the magnetic control system with
other software on the computer. For example, we have used this functionality
for tracking microrobots using Python’s OpenCV library and TrackPy while
controlling the magnetic software wirelessly for real-time feedback control..

Figure 7: Left: Schematic illustrating the effect of $\alpha$ and $\gamma$ on
rotation axis of microrobot. Right: A 1 Hz rotating magnetic field plotted
using equations (3),(4), and (5). 1 Hz sine waves with $\gamma$ = 45∘ and
$\alpha$ = 90∘ over a 1 second interval.

##  3 Experimental Validation

The 2D traditional coil setup is used to orient a variety of self-propelled
microrobots including electrophoretic or diffusiophoretic magnetic Janus
microspheres or bubble propelled microrobots. The 2D system was used in
rivas2022dynamics  to study the dynamics and control of Platinum and Nickel
coated bubble propelled micro-robots.

The 3D Helmholtz system is used to control a variety of magnetic rolling
micro-robots i.e. micro-spheres coated with a hemisphere of Nickel via
electron beam deposition. For example, the 3D Helmholtz system was used in
rivas_cellular_nodate  to demonstrate cellular manipulation using rolling
magnetic micro-robots. Furthermore, the system has also been used for the
actuation of magnetic helical micro-robotic swimmers.

The magnetic tweezers can be used to direct magnetic micro-beads using high
magnetic field gradients and is useful in a variety of applications where
self-propulsion is not possible, or high forces are required. The 3D magnetic
tweezers system was used in mallick_doxorubicin-loaded_nodate  where 4.5 um
paramagnetic micro-beads were coated with Doxorubicin and delivered via
magnetic gradients to kill cancer cells.

##  4 Conclusion

This document described the design and construction of a compact, versatile,
low cost, portable, and easy to use electromagnetic control system which can
be universally paired with any of three electromagnetic systems: traditional
planar coils, 3D magnetic tweezers, and 3D Helmholtz-based coils. Employing a
graphical user interface allowed for straightforward control. We intend to
continue making the system as robust and versatile as possible by
incorporating other micro-robotic off board actuation methods such as acoustic
and light actuation. We also envision that the portability of the system will
make it useful for controlling microrobots in biomedical applications such as
in medical imaging systems for in vitro or in vivo experiments.

##  5 Acknowledgements

The authors gratefully acknowledge the late Richard West for his help with the
cell lines. This project was supported by the Delaware INBRE program, with a
grant from the National Institute of General Medical Sciences – NIGMS (P20
GM103446) from the National Institutes of Health and the State of Delaware.
This work was also supported by NSF grant OIA2020973. This content is solely
the responsibility of the authors and does not necessarily represent the
official views of NIH.

## Declaration of interests

The authors declare that they have no known competing financial interests
personal relationships that could have appeared to influence the work reported
in this paper.

## References

  *   * (1) Peyer, K.E., Zhang, L., Nelson, B.J.: Bio-inspired magnetic swimming microrobots for biomedical applications. Nanoscale 5(4), 1259–1272 (2013)
  * (2) Solovev, A.A., Mei, Y., Schmidt, O.G.: Catalytic microstrider at the air–liquid interface. Advanced Materials 22(39), 4340–4344 (2010)
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Association schemes on triples over few vertices
Jose Maria P. Balmaceda

Institute of Mathematics
University of the Philippines Diliman, 1101 Diliman, Quezon City, Philippines

<EMAIL_ADDRESS>

Dom Vito A. Briones

Institute of Mathematics
University of the Philippines Diliman, 1101 Diliman, Quezon City, Philippines
<EMAIL_ADDRESS>

###### Abstract

In this paper, we obtain classification results for higher-dimensional
analogues of classical association schemes called association schemes on
triples (ASTs). We present an algorithm that enumerates all ASTs on a fixed
number of vertices whose nontrivial relations are invariant under the action
of some group. Applying this algorithm to three, four, and five vertices along
with appropriate group actions yields the unique AST over three vertices, the
unique symmetric ASTs over four or five vertices, the unique AST over four
vertices with two nontrivial relations, and the unique nontrivial circulant
AST over five vertices.

Keywords: association scheme on triples, algebraic combinatorics,
classification

MSC 2020: 05E30

##  1 Introduction

Initially introduced by Bose and Shimamoto for designing statistical
experiments [6, 3], classical association schemes have since found widespread
applications in various algebraic-combinatorial structures. By unifying
seemingly different mathematical objects under a single framework, such
applications include permutation groups [34, 25, 20, 11], discrete geometries,
graphs and directed graphs [20, 4, 3, 12, 24, 8, 10, 2, 13, 33, 9, 4], coding
theory and designs [9, 1], certain types of algebras [23, 14, 5, 35, 28], and
generalizations of groups and their representations [38, 36, 7, 37, 31].

In its relation form, a classical association scheme of order $m$ on a
nonempty set $\Omega$ is a partition $X=\\{R_{i}\\}_{i=0}^{m}$ of
$\Omega\times\Omega$ that satisfies certain symmetry requirements. These
requirements are robust and flexible enough to equip classical association
schemes with algebraic and combinatorial properties capable of describing the
variety of mathematical structures mentioned prior. For instance, the linear
span of the adjacency matrices of these relations is a Jacobson semisimple
associative binary algebra that satisfies certain duality properties [3].
Restrictions on the parameters of the adjacency algebra and its dual algebra
can then be used for the classification theorems of certain graphs [3]. With
these applications of classical association schemes in mind, Mesner and
Bhattacharya created a higher-dimensional analogue of association schemes
called association schemes on triples or ASTs [26]. In ASTs, the relations and
resulting adjacency algebras are ternary instead of binary. In the same paper,
some basic properties of ASTs were observed; in particular, close
relationships with ternary algebras, 2-designs, two-graphs, and two-transitive
permutation groups were discerned. However, the study of ASTs is still in its
infancy, as many structural properties of ASTs are still unknown. For example,
it is not yet known if there is an analogue for the desirable semisimplicity
and duality properties of classical association schemes in ASTs. However, [27]
introduced “identity pairs” and “inverse pairs” as analogues of the usual
multiplicative identity and multiplicative inverses in binary algebras in
hopes of providing tools that may someday illuminate the ternary algebra
structure of ASTs. In addition, [30] introduced another family of ASTs called
circulant ASTs whose nontrivial relations are invariant under a transitive
cyclic subgroup of the symmetric group.

Motivated to aid in the study of ASTs by classifying examples on a small
number of vertices, we mirror the classification of classical association
schemes with few vertices up to isomorphism. Such classifications of classical
association schemes are found in works such as [29, 21, 22, 15, 16, 18, 17,
32, 19]. We approach the classification problem for ASTs by providing an
algorithm which outputs the ASTs $X=\\{R_{i}\\}_{i=0}^{m}$ of order $m$ over
$n$ vertices such that the nontrivial relations of the ASTs are invariant
under the action of some group. Applying this algorithm to $n\in\\{3,4,5\\}$
and the appropriate groups, we obtain the unique AST over three vertices, the
unique symmetric ASTs over $n\in\\{4,5\\}$ vertices, the unique AST over four
vertices with two nontrivial relations, and the unique nontrivial circulant
AST over five vertices.

##  2 Preliminaries

In this section, we introduce association schemes on triples which are higher-
dimensional analogues of classical association schemes. Due to their relevance
in our examples, we also define symmetric ASTs, circulant ASTs, and
isomorphism of ASTs. Many of the definitions, notations, and theorems we use
are from [26] and [30].

We begin by defining order $m$ association schemes on triples over $n$
vertices.

######  Definition 1.

Let $\Omega$ be a finite set of cardinality $n\geq 3$. An association scheme
on triples (AST) of order $m\geq 4$ on $\Omega$ is a partition
$X=\\{R_{i}\\}_{i=0}^{m}$ of $\Omega\times\Omega\times\Omega$ that satisfies
the following properties.

  1. 1.

For any $i\in\\{0,\ldots,m\\}$ there exists a constant
$n_{i}^{(3)}\in\mathbb{N}\cup\\{0\\}$ such that

$|\\{z\in\Omega:(x,y,z)\in R_{i}\\}|=n_{i}^{(3)},$

for any pair of distinct elements $x,y\in\Omega$.

  2. 2.

For any $i,j,k,l\in\\{0,\ldots,m\\}$, there exists a constant
$p_{ijk}^{l}\in\mathbb{N}\cup\\{0\\}$ such that

$|\\{w:(w,y,z)\in R_{i},\;(x,w,z)\in R_{j},\;\text{and}\;(x,y,w)\in
R_{k}\\}|=p_{ijk}^{l},$

for any $(x,y,z)\in R_{l}$.

  3. 3.

For any $i\in\\{0,\ldots,m\\}$ and $\sigma\in S_{3}$, there exists a
$j\in\\{0,\ldots,m\\}$ such that

$R_{i}^{\sigma}:=\\{(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)}):(x_{1},x_{2},x_{3})\in
R_{i}\\}=R_{j}.$

  4. 4.

The first four relations are given by the following.

$\displaystyle R_{0}$ $\displaystyle=\\{(x,x,x):x\in\Omega\\},$ $\displaystyle
R_{1}$ $\displaystyle=\\{(y,x,x):x,y\in\Omega,\;x\neq y\\},$ $\displaystyle
R_{2}$ $\displaystyle=\\{(x,y,x):x,y\in\Omega,\;x\neq y\\},$ $\displaystyle
R_{3}$ $\displaystyle=\\{(x,x,y):x,y\in\Omega,\;x\neq y\\}.$

The relations $R_{0},\ldots,R_{3}$ are the trivial relations and the remaining
relations are the nontrivial relations. For example, analogous to how
transitive groups yield classical association schemes, two-transitive groups
yield ASTs [26]. Indeed, if $G$ is a two-transitive group acting on a set
$\Omega$ of $n\geq 4$ elements, the orbits of $G$ in its natural action on
$\Omega\times\Omega\times\Omega$ form an AST.

Paralleling the definition of symmetric classical association schemes, an AST
$X=\\{R_{i}\\}_{i=0}^{m}$ is symmetric when $R_{i}^{\sigma}=R_{i}$ for all
nontrivial relations $R_{i}\in X$ and all $\sigma\in S_{3}$. In other words,
each nontrivial relation is invariant under the action of $S_{3}$ by
coordinate permutation. The intersection numbers $p_{ijk}^{l}$ of these ASTs
were studied in [26], along with some relationships of symmetric ASTs with
partial 3-designs. It was also shown in the same paper that any nontrivial
relation of an AST that is invariant under coordinate permutation yields a
family of 2-designs. Moreover, they showed that 2-designs yield order $5$ ASTs
and that a converse exists for this construction.

If we instead consider ASTs on $\Omega$ where all nontrivial relations are
invariant under a common transitive cyclic subgroup of $S_{\Omega}$, we obtain
circulant ASTs [30]; that is, an AST $X=\\{R_{i}\\}_{i=0}^{m}$ is circulant if
there exists a transitive cyclic subgroup $G\leq S_{\Omega}$ such that for any
$i\in\\{0,\ldots,m\\}$, we have

$\\{(gx,gy,gz):g\in G,\;(x,y,z)\in R_{i}\\}\subseteq R_{i}.$

Such ASTs were introduced and studied in [30], where it was shown that
circulant ASTs arise in correspondence with so-called AST-regular partitions
of a certain subset of $\Omega\times\Omega$. In fact, they showed that in
circulant ASTs, each nontrivial relation is the disjoint union of so-called
“thin” circulant relations. For example, the ASTs obtained from the two-
transitive action of $AGL(1,p)$ on the Galois Field $GF(p)$, where $p$ is
prime, is circulant. This is due to the subgroup of translations in
$AGL(1,p)$, which is cyclic and which acts transitively on $GF(p)$.

To close this section, we provide an analogue of combinatorial isomorphisms
for classical association schemes [29] to association schemes on triples. To
do so, let

$\sigma(R)=\\{(\sigma(x),\sigma(y),\sigma(z)):(x,y,z)\in R\\}$

for each $R\subseteq\Omega\times\Omega\times\Omega$ and $\sigma\in
S_{\Omega}$.

######  Definition 2.

Let $X=\\{R_{i}\\}_{i=0}^{m}$ and $Y=\\{S_{i}\\}_{i=0}^{m}$ be two order $m$
ASTs over a set $\Omega$ of $n\geq 3$ vertices. We say that $X$ and $Y$ are
isomorphic ASTs if there exists $\sigma\in S_{\Omega}$ such that

$\\{\sigma(R_{i})\\}_{i=0}^{m}=\\{S_{i}\\}_{i=0}^{m},$

where the equality is on unordered sets.

In other words, two ASTs $X$ and $Y$ on $\Omega$ are isomorphic if relabelling
the elements of $\Omega$ using $\sigma$ and then rearranging the
$\sigma(R_{i})$ yield the relations $\\{S_{i}\\}_{i=0}^{m}$ of $Y$. For
computations in the algorithm later on, we restate this isomorphism condition
in terms of a group action by the symmetric group $S_{\Omega}$ on the set of
partitions of $\Omega\times\Omega\times\Omega$. Indeed, $S_{\Omega}$ acts on
the set of all unordered partitions of $\Omega\times\Omega\times\Omega$ by

$\sigma(\\{R_{i}\\}_{i=0}^{m})=\\{\sigma(R_{i})\\}_{i=0}^{m}.$ ($\dagger$)

It follows that two ASTs $X$ and $Y$ are isomorphic if and only if $X$ and $Y$
belong to the same orbit under this action of $S_{\Omega}$.

##  3 Algorithm for determining the number of isomorphism classes of ASTs

In this section, we present an algorithm for determining the number of
isomorphism classes of ASTs on a fixed number of vertices such that each
relation is invariant under some group action on
$\Omega\times\Omega\times\Omega$. We provide the algorithm below and prove its
validity.

######  Theorem 1.

Let $m\geq 4$, $\Omega$ be a set of size $n\geq 3$, and $G$ be a group acting
on $\Omega\times\Omega\times\Omega$.

  1. 1.

Let $\\{R_{i}\\}_{i=0}^{3}$ be the trivial relations, $\mathbb{O}$ be the set
of orbits of $G$ on
$\Omega\times\Omega\times\Omega\setminus\bigcup_{i=0}^{3}{R_{i}}$, and
$\mathbb{P}$ be the set of size $m-3$ (unordered) partitions of $\mathbb{O}$.

  2. 2.

Let $P\in\mathbb{P}$ and $U$ be an element of $P$. Define
$A_{U}=\\{\bigcup_{O\in U}O\\}$. Define $\hat{P}$ to be the partition of
$\Omega\times\Omega\times\Omega$ given by

$\hat{P}=\\{R_{0},R_{1},R_{2},R_{3}\\}\cup\\{A_{U}:U\in P\\}.$

Further, let $\hat{\mathbb{P}}=\\{\hat{P}:P\in\mathbb{P}\\}.$

  3. 3.

By checking conditions 1, 2, and 3 of Definition 1, let
$\mathbb{A}\subseteq\hat{\mathbb{P}}$ be the members of $\hat{\mathbb{P}}$
that are ASTs.

  4. 4.

Using the action of $S_{\Omega}$ described in ($\dagger$ ‣ 2), determine the
ASTs in $\mathbb{A}$ up to isomorphism by computing which ones belong to the
same orbit.

  5. 5.

Let $\mathbb{T}\subseteq\mathbb{A}$ be a transversal of the orbits in the
previous step.

Then $\mathbb{T}$ gives all ASTs of order $m$ on $\Omega$ (up to isomorphism)
such that each nontrivial relation is invariant under the action of $G$.

###### Proof.

Any AST of order $m$ is a size $m+1$ partition of
$\Omega\times\Omega\times\Omega$ whose first four relations are the trivial
relations $R_{0},R_{1},R_{2}$, and $R_{3}$. Since the trivial relations are
always included in any AST, the problem reduces to determining the size $m-3$
partitions of
$\Omega\times\Omega\times\Omega\setminus\bigcup_{i=0}^{3}{R_{i}}$ whose
members are invariant under $G$ and which serve as the nontrivial relations of
some AST on $\Omega$. In order to find such partitions, we note that a ternary
relation $R\subseteq\Omega\times\Omega\times\Omega$ is invariant under $G$ if
and only if $R$ is a union of orbits of $G$.

The first step in the algorithm is to obtain the set $\mathbb{O}$ of orbits of
$G$ and find the set $\mathbb{P}$ of all its size $m-3$ partitions. The second
step in the algorithm is to establish a search space for partitions of
$\Omega\times\Omega\times\Omega$ that contains all ASTs of size $m$ where each
nontrivial relation is invariant under $G$. The set $\hat{\mathbb{P}}$ is our
search space, as we have observed that any AST with the desired properties
will have each of its nontrivial relations as unions of members of
$\mathbb{O}$. The third step determines the subset $\mathbb{A}$ of
${\hat{\mathbb{P}}}$ that consists of the partitions of
$\Omega\times\Omega\times\Omega$ that are ASTs with the desired properties.
The fourth step then provides how many of the desired ASTs exist, up to
isomorphism. Finally, by taking a representative from each isomorphism class,
the fifth step yields all ASTs (up to isomorphism) of order $m$ on $\Omega$
such that each nontrivial relation is invariant under the action of $G$. ∎

##  4 Applications to ${n=3,4,5}$

In this section, we apply the results of Theorem 1 on $n\in\\{3,4,5\\}$ and
suitable choices of $G$. This yields the unique AST over three vertices, the
unique symmetric ASTs over $n\in\\{4,5\\}$ vertices, the unique AST over four
vertices with two nontrivial relations, and the unique nontrivial circulant
AST over five vertices. The data was obtained through source code written for
and executed by the computer algebra system GAP 4.11.1. The source code is
located in the appendix.

The first example uses the action of the trivial group to yield the unique AST
over three vertices. Necessarily, this AST is symmetric and circulant.

######  Example 1.

Let $\Omega=\\{1,2,3\\}$, $G$ be the trivial group, and $m-3$ range over
$\\{1,\ldots,6=|\mathbb{O}|\\}$. Any ternary relation is invariant under the
trivial group; hence, ranging over all possible partition sizes yields all
ASTs over three vertices. There is only one such AST and its sole nontrivial
relation is $\Omega\times\Omega\times\Omega\setminus\bigcup_{i=0}^{3}{R_{i}}$.

The second example uses the orbits of $S_{3}$ acting on
$\Omega\times\Omega\times\Omega$ by coordinate permutation as building blocks
for the nontrivial relations of ASTs. This yields the unique symmetric ASTs
over $n\in\\{4,5\\}$ vertices, each of which is also necessarily circulant.

######  Example 2.

Let $\Omega=\\{1,\ldots,n\\}$ with $n\in\\{4,5\\}$, $G=S_{3}$, and $m-3$ range
over $\\{1,\ldots,|\mathbb{O}|\\}$. The ternary relations invariant under
coordinate permutation are those who are unions of orbits of $S_{3}$; hence,
ranging over all possible partition sizes yields all symmetric ASTs over four
or five vertices. There is only one such AST for $n=4$ and only one such AST
for $n=5$. In each case, the AST has a lone nontrivial relation given by
$\Omega\times\Omega\times\Omega\setminus\bigcup_{i=0}^{3}{R_{i}}$.

The following example shows that the symmetric AST over four vertices obtained
from the prior example is also the only circulant AST over four vertices.

######  Example 3.

Let $\Omega=\\{1,2,3,4\\}$, $G$ be the cyclic group generated by $(1,2,3,4)\in
S_{4}$, and $m-3$ range over $\\{1,\ldots,|\mathbb{O}|\\}$. Any ternary
relation invariant under $G$ will be unions of orbits of $G$ in
$\Omega\times\Omega\times\Omega$; hence, ranging over all possible partition
sizes yields all circulant ASTs (with respect to $G$) over four vertices. In
fact, if $H$ were any transitive cyclic subgroup of $S_{4}$ and $Y$ were any
circulant AST with respect to $H$, then we may relabel the elements of
$\Omega$ to obtain an isomorphic AST that is circulant with respect to $G$.
Hence, our algorithm actually produces all circulant ASTs over four vertices,
up to isomorphism. There is only one such AST and it has a lone nontrivial
relation $\Omega\times\Omega\times\Omega\setminus\bigcup_{i=0}^{3}{R_{i}}$.

The next example again uses the trivial group but fixes $m$ to be 5. This
yields the unique size 6 AST on four vertices.

######  Example 4.

Let $\Omega=\\{1,2,3,4\\}$, $G$ be the trivial group, and $m=5$. Reasoning as
in Example 1, the algorithm produces all order 5 ASTs over four vertices. We
find that there is only one such AST. Its nontrivial relations $R_{4}$ and
$R_{5}$ are given by the following.

$\displaystyle R_{4}=$
$\displaystyle\Big{\\{}(1,2,3),(1,3,4),(1,4,2),(2,1,4),(2,3,1),(2,4,3),$
$\displaystyle(3,1,2),(3,2,4),(3,4,1),(4,1,3),(4,2,1),(4,3,2)\Big{\\}},$
$\displaystyle R_{5}=$
$\displaystyle\Big{\\{}(1,2,4),(1,3,2),(1,4,3),(2,1,3),(2,3,4),(2,4,1),$
$\displaystyle(3,1,4),(3,2,1),(3,4,2),(4,1,2),(4,2,3),(4,3,1)\Big{\\}}.$

In fact, [26] says that the two-transitive group PSL(2,3) acting on the
projective line $PG(1,3)$ yields an AST of size 6. In the same paper, an AST
of size 6 is also obtained from the action of the two-transitive group
$AGL(1,4)$ on the Galois field $GF(4)$. Hence, the unique AST of order 5 over
four vertices is the AST induced by the action of $PSL(2,3)$ on $PG(1,3)$ and
is isomorphic to the AST obtained from the action of $AGL(1,4)$ on $GF(4)$.

The last example uses the transitive action of the cyclic subgroup generated
by the permutation $(1,2,3,4,5)$ to obtain two circulant ASTs over five
vertices.

######  Example 5.

Let $\Omega=\\{1,2,3,4,5\\}$, $G$ be the cyclic group generated by
$(1,2,3,4,5)\in S_{5}$, and $m-3$ range over $\\{1,\ldots,|\mathbb{O}|\\}$.
Reasoning as in Example 3, our algorithm produces all circulant ASTs over five
vertices. There are two such ASTs. The first is the AST whose sole nontrivial
relation is $\Omega\times\Omega\times\Omega\setminus\bigcup_{i=0}^{3}{R_{i}}$,
as obtained before. The remaining AST has size 7, with nontrivial relations
$R_{4},R_{5}$, and $R_{6}$ as given below.

$\displaystyle R_{4}=$
$\displaystyle\Big{\\{}(1,2,3),(1,3,5),(1,4,2),(1,5,4),(2,1,5),(2,3,4),(2,4,1),(2,5,3),(3,1,4),(3,2,1),$
$\displaystyle(3,4,5),(3,5,2),(4,1,3),(4,2,5),(4,3,2),(4,5,1),(5,1,2),(5,2,4),(5,3,1),(5,4,3)\Big{\\}},$
$\displaystyle R_{5}=$
$\displaystyle\Big{\\{}(1,2,4),(1,3,2),(1,4,5),(1,5,3),(2,1,4),(2,3,5),(2,4,3),(2,5,1),(3,1,2),(3,2,5),$
$\displaystyle(3,4,1),(3,5,4),(4,1,5),(4,2,3),(4,3,1),(4,5,2),(5,1,3),(5,2,1),(5,3,4),(5,4,2)\Big{\\}},$
$\displaystyle R_{6}=$
$\displaystyle\Big{\\{}(1,2,5),(1,3,4),(1,4,3),(1,5,2),(2,1,3),(2,3,1),(2,4,5),(2,5,4),(3,1,5),(3,2,4),$
$\displaystyle(3,4,2),(3,5,1),(4,1,2),(4,2,1),(4,3,5),(4,5,3),(5,1,4),(5,2,3),(5,3,2),(5,4,1)\Big{\\}}.$

In fact, [26] says that the two-transitive action of the group $AGL(1,5)$ on
the Galois field $GF(5)$ yields an AST of size 7. Since this AST is circulant,
it follows that this must be the size 7 AST obtained above.

##  Appendix A Source codes

This appendix provides the source codes for GAP 4.11.1 utilized for obtaining
the classification results in the article.

  1. 1.

This is the program that executes the algorithm provided in Theorem 1. The
size of $\Omega$, the desired group action by $G$, and the order $m$ of the
desired ASTs are set manually, as guided by the comments therein. The program
is dependent upon the auxiliary programs in items 2, 3, 4, and 5 of this list.

        n:= #set the desired set size here#;
    set:=[1..n];
    cart:=Cartesian(set,set,set); #yields cartesian product

    #setting the trivial relations
    R:=[];
    R[1]:=[];
        for c in cart do
            if c[1]=c[2] and c[1]=c[3] then Append(R[1],[c]); fi;
        od;
    R[1]:=Set(R[1]);
    R[2]:=[];
        for c in cart do
            if not(c[1]=c[2]) and c[2]=c[3] then Append(R[2],[c]); fi;
        od;Ψ
    R[2]:=Set(R[2]);Ψ
    R[3]:=[];
        for c in cart do
            if not(c[1]=c[2]) and c[1]=c[3] then Append(R[3],[c]); fi;
        od;
    R[3]:=Set(R[3]);
    R[4]:=[];
        for c in cart do
            if not(c[1]=c[3]) and c[1]=c[2] then Append(R[4],[c]); fi;
        od;
    R[4]:=Set(R[4]);

    for i in[1..4] do
        SubtractSet(cart,R[i]);
    od;Ψ

    #determining the orbits of the desired group action
    symblocks:=#set desired action here#;
    #e.g. Orbits(SymmetricGroup(3),cart,Permuted);
    symblocks2:=[];
        for i in [1..Size(symblocks)] do
            symblocks2[i]:=Set(symblocks[i]);
        od;
    symblocks:=symblocks2;

    #determining partitions
    part:=PartitionsSet(symblocks,#set partition size here#);;

    #determining orbits of partitions
    distinct:=Orbits(SymmetricGroup(n),part,partact);;

    #obtain a representative for each partition orbit
    transversal:=[];
        for i in [1..Size(distinct)] do
            Append(transversal,[distinct[i][1]]);
        od;

    #obtain candidates for ASTs
    unioned:=[];
        for i in [1..Size(transversal)] do
            unioned[i]:=[];
            for j in [1..Size(transversal[i])] do
                unioned[i][j]:=[];
                unioned[i][j][1]:=[];
                for k in [1..Size(transversal[i][j])] do
                    unioned[i][j][1]:=UnionSet(unioned[i][j][1],
                    transversal[i][j][k]);
                od;
        od; od;

    S:=[];
        for i in [1..Size(unioned)] do
            S[i]:=ShallowCopy(R);
            for j in [1..Size(unioned[i])] do
                Append(S[i],[unioned[i][j][1]]);
            od;
        od;

    #Check for AST axioms
    for i in [1..Size(S)] do
        if
            (valencycheck(S[i],[1..n]))=true and
            (regularitycheck(S[i],[1..n]))=true and
            (permclosedcheck(S[i],[1..n]))=true then
            Print(i); Print("\n"); fi;
    od;

  2. 2.

This is an auxiliary program for determining the orbit of a partition of
$\Omega\times\Omega\times\Omega$ under the action given by ($\dagger$ ‣ 2).

        partact:= function(P,g) #P for a partition g for Sym elt

    local size, image, i,j;
    size:=Size(P);
    image:=[];

    for i in [1..size] do
        image[i]:=[];
            for j in [1..Size(P[i])] do
                image[i][j]:=Set(OnSetsTuples(P[i][j],g));
            od;
    image[i]:=Set(image[i]); od;

    return Set(image);

    end;;

  3. 3.

This is an auxiliary program for determining whether or not the first
condition of Definition 1 is satisfied.

        valencycheck:=function(R,S)
    Ψ
    local truth, rel,countinit,s,x,y,z,countproper;
    truth:=true;
    Ψ
    for rel in R do
        countinit:=0;
    Ψ
        for s in S do
            if ([S[1],S[2],s] in rel) then countinit:=countinit+1; fi;
        od;
        for x in S do
        for y in S do
            countproper:=0;
            if not(x=y) then
                for z in S do
                    if [x,y,z] in rel then countproper:=countproper+1; fi;
                od;
                if not(countproper=countinit) then return false; fi;
            fi;
        od; od;
    od;
    Ψ
    return truth;
    Ψ
    end;

  4. 4.

This is an auxiliary program for determining whether or not the second
condition of Definition 1 is satisfied.

        regularitycheck := function(R,S)
    Ψ
    local truth, index, i,j,k,l, a, b, countinit, countproper, z;
    truth:=true;
    index:=[1..Size(R)];
    Ψ
    for i in index do
    for j in index do
    for k in index do
    for l in index do
        countinit:=0;
        a:=R[l][1];
            for z in S do
                if ([z,a[2],a[3]] in R[i]) and ([a[1],z,a[3]] in R[j]) and
                ([a[1],a[2],z] in R[k]) then countinit:=countinit+1; fi;
            od;
        for b in R[l] do
        countproper:=0;
            for z in S do
               if ([z,b[2],b[3]] in R[i]) and ([b[1],z,b[3]] in R[j]) and
               ([b[1],b[2],z] in R[k]) then countproper:=countproper+1;fi;
            od;
        if not(countproper =countinit) then
        return false; fi;
        od;
    od;Ψod; od; od;
    Ψ
    return truth;
    Ψ
    end;;

  5. 5.

This is an auxiliary program for determining whether or not the third
condition of Definition 1 is satisfied.

        permclosedcheck:= function(R,S)
    Ψ
    local sym, truth, s, rel, relcopy, i;
    sym:=SymmetricGroup(3);
    truth:=true;

    for s in sym do
        for rel in R do
            relcopy:=[];
            for i in [1..Size(rel)] do
                relcopy[i]:=Permuted(rel[i],s);
            od;Ψ
            if not(Set(relcopy) in R) then return false; fi;
        od;
    od;
    Ψ
    return truth;
    Ψ
    end;;

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††thanks: Present Address: Institute of Radiation Physics, Helmholtz-Zentrum
Dresden-Rossendorf, 01328 Dresden, Germany.

# Antiferromagnetic resonance in the cubic iridium hexahalides (NH4)2IrCl6 and
K2IrCl6

L. Bhaskaran<EMAIL_ADDRESS>Dresden High Magnetic Field Laboratory (HLD-
EMFL) and Würzburg-Dresden Cluster of Excellence ct.qmat, Helmholtz-Zentrum
Dresden-Rossendorf, 01328 Dresden, Germany  A. N. Ponomaryov  Dresden High
Magnetic Field Laboratory (HLD-EMFL) and Würzburg-Dresden Cluster of
Excellence ct.qmat, Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden,
Germany  J. Wosnitza  Dresden High Magnetic Field Laboratory (HLD-EMFL) and
Würzburg-Dresden Cluster of Excellence ct.qmat, Helmholtz-Zentrum Dresden-
Rossendorf, 01328 Dresden, Germany  Institut für Festkörper- und
Materialphysik, TU Dresden, 01062 Dresden, Germany  N. Khan  A. A. Tsirlin
Experimental Physics VI, Center for Electronics Correlations and Magnetism,
Institute of Physics, University of Augsburg, 86135 Augsburg, Germany  M. E.
Zhitomirsky  University Grenoble Alpes, Grenoble INP, CEA, IRIG, PHELIQS,
38000 Grenoble, France  S. A. Zvyagin<EMAIL_ADDRESS>Dresden High Magnetic
Field Laboratory (HLD-EMFL) and Würzburg-Dresden Cluster of Excellence
ct.qmat, Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany

(october 16, 2021)

###### Abstract

We report on high-field electron spin resonance studies of two iridium
hexahalide compounds, (NH4)2IrCl6 and K2IrCl6. In the paramagnetic state, our
measurements reveal isotropic $g$-factors $g=1.79$(1) for the Ir4+ ions, in
agreement with their cubic symmetries. Most importantly, in the magnetically
ordered state, we observe two magnon modes with zero-field gaps of 11.3 and
14.2 K for (NH4)2IrCl6 and K2IrCl6, respectively. Based on that and using
linear spin-wave theory, we estimate the nearest-neighbor exchange couplings
and anisotropic Kitaev interactions, $J_{1}/k_{B}=10.3$ K, $K/k_{B}=0.7$ K for
(NH4)2IrCl6, and $J_{1}/k_{B}=13.8$ K, $K/k_{B}=0.9$ K for K2IrCl6, revealing
the nearest-neighbor Heisenberg coupling as the leading interaction term, with
only a weak Kitaev anisotropy.

Transition-metal compounds with partly filled $4d$ and $5d$ orbitals have
recently attracted enormous attention as promising candidates for realizing
novel magnetic phenomena, emerging from the interplay between crystal-field
effects and spin-orbit interactions [1, 2, 3]. In particular, honeycomb
magnets with $j\rm_{eff}=\frac{1}{2}$ moments and bond-anisotropic Kitaev
interactions have come under extensive scrutiny due to the possible
realization of quantum spin liquid states, characterized by unusual spin
dynamics with fractionalized Majorana fermions and flux excitations [4, 4, 5,
6, 7, 8].

While the majority of experimental and theoretical studies remains focused on
honeycomb materials as the most promising platform for the Kitaev interaction,
[9, 10, 11], there is also a significant interest in detecting anisotropic
exchange terms (including the Kitaev interaction) in Ir4+ compounds with
structures distinct from the honeycomb geometry [12, 13, 14, 15, 16, 17].
However, most of the nonhoneycomb iridates studied over the last decade still
evade detailed spectroscopic characterization with an unambiguous experimental
determination of their interaction parameters. In particular, the absence of
large single crystals as well as the notoriously strong neutron absorption by
Ir atoms (which is of crucial importance for inelastic neutron scattering
studies) and the insufficient energy resolution (in case of resonant inelastic
x-ray scattering, RIXS) have often hindered the experimental access to
relevant magnetic excitations.

Figure 1: (a) Schematic view of the type-III antiferromagnetic spin structure
of the IrCl6 fcc lattice with four sublattices. Spins belonging to different
sublattices are shown in red, green, blue, and orange colors. (b) Schematic
representation of nearest-neighbor Cl--mediated superexchange pathways across
the parallel edges of the octahedra formed by chlorine atoms. The Ir ions are
shown in gray while the Cl ions are in green color.

Here, we focus on two compounds from the family of Ir4+ antifluorites [18, 19,
20, 21], (NH4)2IrCl6 and K2IrCl6, that have been recently revisited by
crystallographic and thermodynamic studies [22, 23, 24]. These compounds
feature Ir4+ ions arranged on a face-centered cubic (fcc) lattice with four
sublattices [Fig. 1(a)] and distinct long-range superexchange pathways
mediated by the covalently bonded halogen ions [Fig. 1(b)]. The presence of
possible frustration in fcc lattices is interesting in its own right as it
shows an intricate competition of magnetic ground states, even in the absence
of Kitaev or any other exchange anisotropy [25, 26, 27]. In contrast to many
other iridates, these compounds are available as relatively large single
crystals amenable to lab spectroscopic probes, featuring a cubic symmetry of
the molecule that not only ensures the spin-orbit-coupled $j_{\rm
eff}=\frac{1}{2}$ state of the Ir4+ ions, but also restricts the number of
relevant interaction parameters.

In the following, we present electron spin resonance (ESR) studies of
(NH4)2IrCl6 and K2IrCl6, revealing the presence of two gapped magnon modes in
each of the compounds. Employing linear spin-wave-theory calculations, we
estimate the spin-Hamiltonian parameters and, consequently, establish the main
interaction terms.

Single crystals of (NH${}_{4})_{2}$IrCl6 and K2IrCl6 were grown from solutions
prepared by dissolving commercially available powders (Alfa Aesar) in
deionized water and keeping the solutions at 60 ∘C for several days. The
resulting crystals had well-defined hexagonal (111) faces and sizes of $\sim
2-3$ mm3. Magnetization measurements revealed magnetic ordering transitions at
2.15 K for (NH${}_{4})_{2}$IrCl6 and 3.1 K for K2IrCl6, in agreement with
previous studies [28, 20, 23, 22].

High-field electron spin resonance (ESR) measurements were performed employing
a transmission-type ESR spectrometer (similar to that described in Ref. [29]),
in magnetic fields up to 16 T. Measurements were done in the frequency range
of $60-400$ GHz, using a set of VDI microwave-chain radiation sources (product
of Virginia Diodes, Inc., USA). An InSb hot-electron bolometer (QMC
Instruments Ltd., UK) was used to record the spectra. We measured single
crystals in the Faraday configuration with magnetic field $H$ $\parallel$
[111] and in the Voigt configuration with $H$ $\perp$ [111]. For the powder
samples, we confined the specimens by a binding agent such as high vacuum
grease. The powders were obtained by crushing single crystals to ensure
consistency between single-crystal and powder measurements. The stable free-
radical molecule DPPH (2,2-diphenyl-1-picrylhydrazyl) was used as a frequency-
field marker.

Figure 2: Normalized ESR spectra measured in single-crystalline ($H$
$\parallel$ $[$111$]$ and $H$ $\perp$ $[$111$]$) and powder samples of (a)
(NH4)2IrCl6 and (b) K2IrCl6 at 50 K. The spectra are offset for clarity.
Lorentzian fits to the powder sample are shown by the dashed red lines, while
simulation of the single-crystal spectra are shown by the dashed blue lines.

ESR is traditionally recognized as a powerful tool to probe crystal-field
effects in solids. A very sensitive test of the presence of non-cubic
distortions in covalently bound [IrCl6]2- complexes is provided by a
$g$-factor anisotropy [18]. In Fig. 2, we show powder and single-crystal
spectra obtained for (NH4)2IrCl6 (a) and K2IrCl6 (b) at a temperature of 50 K.
Both powder spectra were fit using a Lorentzian function with $g=1.79(1)$, and
linewidths of 0.93 and 1.23 T for (NH4)2IrCl6 and K2IrCl6, respectively. These
parameters perfectly agree with the simulation results for the single-crystals
along both $H$ $\parallel$ [111] and $H$ $\perp$ [111], revealing isotropic
paramagnetic $g$-factors, and thus evidencing the cubic symmetry for both
compounds. The reduction of the $g$-factor from the free-spin value ($g=2$)
can be attributed to the covalent nature of the Ir and Cl bonds [30, 31].
Large covalency of the Ir4+ halides was indeed confirmed by recent RIXS
studies [24], detecting also minor deviations from the $j_{\rm
eff}=\frac{1}{2}$ state [23], despite the absence of any macroscopic symmetry
lowering [22].

Figure 3: (a) Temperature dependence of the resonance field (closed squares)
and linewidth (open squares) measured in (NH4)2IrCl6 at 270 GHz. (b) Examples
of normalized ESR spectra recorded at 270 GHz and different temperatures. The
spectra are offset for clarity. Magnetic field is applied along the [111]
direction. Figure 4: (a) Temperature dependence of the resonance field (solid
squares) and linewidth (open squares) measured in K2IrCl6 at 356 GHz. (b)
Examples of normalized ESR spectra recorded at 356 GHz and different
temperatures. The spectra are offset for clarity. Magnetic field is applied
along the [111] direction.

Upon cooling, the ESR spectra for $H$ $\parallel$ [111] exhibit significant
broadening, evidencing enhancement of magnetic correlations, with the maximum
in the vicinity of $T_{N}$ (Figs. 3 and 4). On further decreasing the
temperature, the ESR linewidth rapidly decreases, revealing the onset of long-
range ordering below $T_{N}$. This transition into the ordered state also
results in a pronounced low-temperature shift of the ESR field positions.

Two antiferromagnetic resonance (AFMR) modes, A1 and A2, were observed in the
magnetically ordered state. (Figs. 5 and 7). Selected AFMR spectra measured at
1.5 K with magnetic field applied along the [111] direction are shown in Fig.
6 for (NH${}_{4})_{2}$IrCl6 and in the inset of Fig. 7 for K2IrCl6.

The frequency-field dependences of the modes A1 and A2 can be described using
the equation

$h\nu=\Delta\pm g_{eff}\mu_{B}B,$ (1)

where $h$ is the Planck constant, $\nu$ represents the excitation frequency,
$g_{eff}$ is the effective $g$-factor, and $\mu_{B}$ the Bohr magneton. From
this linear fit we obtain the zero-field magnon gaps $\Delta$ = 235 $\pm$ 10
and 295 $\pm$ 10 GHz (which correspond to 11.3 and 14.2 K; $g_{eff}=0.8$) for
(NH4)2IrCl6 and K2IrCl6, respectively. For both materials, mode A1 shows a
linear increase in the resonance-field position, while some deviation from the
linear dependence was observed for mode A2 in high fields. Noticeably, the
mode A2 disappears above $\sim 6$ T. This field corresponds to a field-induced
transition reported for K2IrCl6 [32], whereas in (NH${}_{4})_{2}$IrCl6 the
changes in applied field appear to be limited to a domain repopulation wherein
the number of domains with magnetic moments oriented perpendicular to the
applied field increases [33].

Figure 5: Frequency-field diagram of ESR excitations in (NH4)2IrCl6 measured
at 1.5 K with magnetic field applied parallel to the $[$111$]$ direction. The
dashed lines represent linear fits using Eq. (1). Figure 6: Examples of
normalized ESR spectra for (NH4)2IrCl6 at different frequencies and $T=1.5$ K
with $H\parallel$ [$111$]. The spectra are offset for clarity. Figure 7:
Frequency-field diagram of ESR excitations in K2IrCl6 measured at 1.5 K with
magnetic field applied parallel to the $[$111$]$ direction. The dashed lines
represent linear fits using Eq. (1). In the inset, examples of normalized ESR
spectra for K2IrCl6 at different frequencies and $T=1.5$ K with $H$
$\parallel$ $[$111$]$ are shown. The spectra are offset for clarity.

A general form of spin-spin interactions in magnetic systems with an fcc
crystal structure was discussed by several authors [34, 13, 35, 22]. Apart
from the Heisenberg exchange it includes symmetry-allowed anisotropic terms
typical for materials with strong spin-orbit coupling. According to an ab-
initio calculation [22] two strongest spin-spin interactions in (NH4)2IrCl6
and K2IrCl6 are the Heisenberg exchange, $J_{1}>0$, and the Kitaev
interaction, $K>0$, for the nearest-neighbor (nn) pairs of Ir ions. In
addition, a weak next-nearest-neighbor (nnn) exchange $J_{2}>0$ is necessary
to explain the experimentally observed type-III antiferromagnetic order [36].
Accordingly, we consider the following Hamiltonian for effective $S=1/2$
moments of iridium ions:

$\hat{\cal H}=\sum_{\langle ij\rangle}^{\rm nn}\Bigl{(}J_{1}{\bf
S}_{i}\cdot{\bf
S}_{j}+KS_{i}^{\gamma}S_{j}^{\gamma}\Bigr{)}+J_{2}\sum_{\langle
ij\rangle}^{\rm nnn}{\bf S}_{i}\cdot{\bf S}_{j}\,,$ (2)

where ${\bf S}_{i}$ and ${\bf S}_{j}$ are spin-1/2 operators at site $i$ and
$j$, respectively. In the anisotropic Kitaev term, $\gamma$ is one of the
cubic axes ($x,y,z$) that is orthogonal to a given bond.

The Heisenberg fcc antiferromagnet with only the nearest-neighbor interactions
has highly degenerate classical ground states [37]. A small positive next-
nearest-neighbor exchange $J_{2}$ lifts this degeneracy in favor of the type-
III magnetic structure described by the propagation vector ${\bf
Q}=(2\pi/a,\pi/a,0)$ [38] (note, that ${\bf Q}^{\prime}=(0,\pi/a,2\pi/a)=-{\bf
Q}+{\bf G}$, where $\bf G$ is a reciprocal lattice vector). In addition, the
anisotropy $K>0$ locks magnetic moments in the $(0,1,0)$ direction:

${\bf S}_{i}=\sqrt{2}\,S\hat{\bf y}\cos\bigl{(}{\bf Q}\cdot{\bf
r}_{i}\pm\pi/4\bigr{)}\,.$ (3)

Together with two other equivalent ${\bf Q}$, obtained by permutation of
$Q_{x,y,z}$, there are in total six domains for the type-III collinear
structure.

Following a recent theoretical study of the excitation spectra in fcc
antiferromagnets [26], we have performed linear spin-wave calculations for the
Hamiltonian (2) with $J_{2},K>0$. The ESR response corresponds to long wave-
length excitations with ${\bf k}\to 0$. Accordingly, the ESR spectrum is
formed by two degenerate pairs of gapped magnons with the zero-field
excitation energies:

$\Delta_{l}=2\sqrt{KJ_{2}}\,,\quad\Delta_{u}=2\sqrt{4KJ_{1}+2K^{2}+KJ_{2}}\,.$
(4)

Note, that the size of the low-energy gap $\Delta_{l}$ is determined by both
$K$ and $J_{2}$, while the upper gap $\Delta_{u}$ depends mainly on the
strength of the Kitaev interaction. Moreover, in the absence of $K$ both gaps
vanish, while at $J_{2}\to 0$ they have substantially different magnitudes.

The next-nearest-neighbor isotropic exchange coupling was recently estimated
from the ab initio calculation as $J_{2}/k_{B}\simeq 0.2$ K for both
(NH4)2IrCl6 and K2IrCl6 [33, 22]. In addition, we use the expression for the
Curie-Weiss temperature [39]

$\theta_{\rm CW}=3J_{1}+K+\frac{3}{2}\,J_{2}\ .$ (5)

Experimental values for $\theta_{\rm CW}/k_{B}$ are 32 K for (NH4)2IrCl6 [33]
and 42.6 K for K2IrCl6 [22]. They are obtained by fits to the magnetic-
susceptibility data in the 100-400 K and 150-400 K range, respectively. The
corresponding effective moment of 1.73 $\mu_{B}$ confirms that mostly the
ground-state $j_{\rm eff}=\frac{1}{2}$ doublet is occupied in this temperature
range. Using Eq. (4) and Eq. (5) we estimate $J_{1}/k_{B}=10.3$ K and
$K/k_{B}=0.7$ K for (NH4)2IrCl6, and $J_{1}/k_{B}=13.8$ K and $K/k_{B}=0.9$ K
for K2IrCl6 ($K/J_{1}\approx 6-7\%$). The exchange coupling $J_{1}$ is
stronger for K2IrCl6 compared to (NH4)2IrCl6, due to the lattice parameter
difference ($a=9.77$ Å vs. 9.87 Å [22, 33], respectively), with the respective
negative pressure reducing magnetic interactions in (NH${}_{4})_{2}$IrCl6. It
is worth noting that our experimental value of $K$ is much smaller than the 5
K reported in Ref. [22]. This discrepancy may indicate the importance of
Hund’s coupling on the Cl ligand, an effect that has been neglected in the
superexchange theory used in Ref. [22] and generally leads to a ferromagnetic
contribution, which counteracts antiferromagnetic superexchange. Similar
effects were previously reported in Cu2+ halides [40].

Using the parameters, as obtained above, the size of low-energy gaps was also
estimated, yielding $\Delta_{l}/k_{B}\approx 1$ K. Our thorough search did not
reveal the low-energy AFMR modes (note also that our ESR probe has a cut-off
frequency limit of 60-70 GHz that is higher than the expected size of
$\Delta_{l}$). The absence of the low-energy ESR modes, which directly depends
on $J_{2}$, may indicate, on the other hand, that the next-nearest-neighbor
exchange in these compounds is very small. In such a case the quantum order-
by-disorder effect can play a significant role in the Kitaev fcc
antiferromagnets [41].

Summarizing, we presented systematic ESR studies of the two iridium hexahalide
compounds (NH4)2IrCl6 and K2IrCl6. In the paramagnetic state, our measurements
reveal isotropic $g$-factors $g=1.79$(1) for Ir4+ ions, confirming their cubic
symmetries. By measuring the ESR spectra in the ordered state, we were able to
determine the nearest-neighbor Heisenberg exchange $J_{1}$ and the Kitaev
interaction $K$ for both materials, revealing $K\ll J_{1}$. The intriguing
possibility of the realization of the quantum order-by-disorder effect as well
as details of the high-field behavior of magnetic excitations in these
frustrated compounds deserves further experimental and theoretical studies.

This work was supported by the Deutsche Forschungsgemeinschaft (DFG), through
ZV 6/2-2, the Würzburg-Dresden Cluster of Excellence on Complexity and
Topology in Quantum Matter–$ct.qmat$ (EXC 2147, Project No. 390858490), and
SFB 1143, as well as by the HLD at HZDR, member of the European Magnetic Field
Laboratory (EMFL). The work in Augsburg was supported by the DFG via the
Project No. 107745057 (TRR80). M.E.Z. acknowledges financial support from ANR,
France, Grant No. ANR-18-CE05-0023.

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