# Mixture of Public and Private Distributions in Imperfect Information Games.

Jérôme Arjonilla  Paris Dauphine University - PSL
Paris, France
<EMAIL_ADDRESS>Abdallah Saffidine  University of New South Wales
Sydney, Australia
<EMAIL_ADDRESS>Tristan Cazenave  Paris Dauphine University -
PSL
Paris, France
<EMAIL_ADDRESS>
###### Abstract

In imperfect information games (e.g. Bridge, Skat, Poker), one of the
fundamental considerations is to infer the missing information while at the
same time avoiding the disclosure of private information. Disregarding the
issue of protecting private information can lead to a highly exploitable
performance. Yet, excessive attention to it leads to hesitations that are no
longer consistent with our private information. In our work, we show that to
improve performance, one must choose whether to use a player’s private
information. We extend our work by proposing a new belief distribution
depending on the amount of private and public information desired. We
empirically demonstrate an increase in performance and, with the aim of
further improving performance, the new distribution should be used according
to the position in the game. Our experiments have been done on multiple
benchmarks and in multiple determinization-based algorithms (PIMC and IS-
MCTS).

###### Index Terms:

Imperfect Information Games, Search Algorithms, Belief Distributions

##  I Introduction

Search in artificial intelligence has been constantly evolving over the last
few decades, and game-oriented research has always been a cornerstone of this
success. Chess, Go [1, 2, 3], Poker [4], Skat, Contract Bridge or Dota [5] are
among the most famous ones.

Perfect information games (Chess, Go) — where all information is available for
each player — have been the most studied, and many algorithms have been able
to achieve a level far beyond the level of a human professional player. On the
other hand, Imperfect Information Games (IIGs) (Poker, Skat, Bridge) — where
some information is hidden — have been less studied, and only a few algorithms
are capable of beating professional human player [6, 4].

In IIG, the complexity is heightened by the missing information, as one must
try to infer the missing information of the opponents and, at the same time,
be wary to not reveal private hidden information to opponents. Among the
methods used in IIG, determinization-based algorithms — where the hidden
information is fixed according to a belief distribution — such as Perfect
Information Monte Carlo (PIMC) [7], Recursive PIMC [8], Information Set MCTS
[9] or AlphaMu [10] achieve state-of-the-art performance in many trick-taking
card games (Contract-Bridge, Skat).

In the work cited above, the determinization operates by sampling the hidden
information according to the private information of a given player, i.e. what
has happened since the beginning, from the point of view of a given agent.
However, by doing so, one can indirectly reveal private information to
opponents, which can lead to a highly exploitable performance.

Recently, the concept of public knowledge [11] — where a distinction is made
between observations accessible to everyone and those accessible individually
— has emerged in IIGs. This concept has resulted in many breakthroughs thanks
to the decomposition, which made the calculations feasible [12, 13]. Despite
this large benefit, there are limitations to its use, especially in the
context of belief distribution. By doing so, we completely remove the
knowledge observed by the acting player, and one might wonder whether not
using the private information was useful.

In this work, we analyze the impact of using one method rather than another
and present a new belief distribution, which is a mixture of both public and
private belief distribution. We extend the study by analyzing different
mixtures, depending on the position within the game. Our experiments are
carried out on determinization-based algorithms, which use the belief
distribution to fix the incertitude.

The paper is organized as follows: Section II presents notation and current
determinization-based algorithms; Section III explains the different belief
distributions used with their advantages and drawbacks, and presents our new
belief distribution; Section IV empirically shows that using the new belief
distribution allows us to improve past performance; and the last section
summarizes our work and future work.

##  II Notation and Background

###  II-A Notation

We use the notation based on factored-observation stochastic games (FOSGs
[11]). This formalism distinguishes between private and public observations.

A Game G is composed of the following elements. The set of agents
$\mathcal{N}=\\{1,2,\dots,N\\}$ agents, the set of world state possible
$\mathcal{W}$. In each world state $w\in\mathcal{W}$, the acting player $i$
chooses an action $a\in\mathcal{A}(w)$, where $\mathcal{A}(w)$ denotes the
legal actions at $w$. After an action $a$, we reach the next world state
$w^{\prime}$ from the probability distribution of playing $a$ in $w$.

During the transition from $w$ to $w^{\prime}$ by playing $a$, two
observations are received: a public observation and a private observation.
Public observation is the observation visible by every player noted
$o_{pub}\in\mathcal{O}_{pub}(w,a,w^{\prime})$ where
$\mathcal{O}_{pub}(w,a,w^{\prime})$ refers to all the possible public
observations. Private observation is the observation visible by a precise
player $i$, noted $o_{priv(i)}\in\mathcal{O}_{priv(i)}(w,a,w^{\prime})$ where
$\mathcal{O}_{priv(i)}(w,a,w^{\prime})$ refers to all the possible private
observations.

A _history_ is a finite sequence of legal actions and world states, denoted
$h^{t}=(w^{0},a^{0},w^{1},a^{1},...,w^{t})$. For describing the point of view
of an agent $i$ of a history $h$, we introduce an infostate $s_{i}(h)$. An
infostate for agent $i$ is a sequence of an agent’s observations and actions
$s^{t}_{i}$ = ($o_{i}^{0}$, $a^{0}_{i}$, $o_{i}^{1}$, $a^{1}_{i}$, …,
$o_{i}^{t}$) where $o_{i}^{k}$ = ($o_{pub}^{k}$, $o^{k}_{priv(i)})$. A public
infostate is a sequence of public observations
$s^{t}_{pub}=(o_{pub}^{0},o_{pub}^{1},...,o_{pub}^{t})$.

Determinization refers to the fact that we sample a world state according to a
belief distribution of the world states possible. Determinizing the belief
distribution is not new and a similar concept exists in other formalisms such
as belief state in Partially Observable Markov Decision Process (POMDP)
problems [14] or occupancy-state in Decentralised-POMDPs problems [15].

###  II-B Determinization-based algorithms

Each determinization-based algorithm has its own characteristics.
Nevertheless, they share some common features such as (i) sampling a world
state according to a belief distribution over the possible world states; and
(ii) using a perfect information algorithm for estimating the value of the
sampled world state.

The algorithms are simple and, in practice, they achieve great results, mainly
due to the use of perfect information algorithms (AlphaBeta [16], MCTS [17] or
Value Network) that are fast and efficient.

In the following, we present two determinization-based algorithms that are
baseline and will, at a later stage, be used in our experiments.

####  II-B1 PIMC

Perfect Information Monte Carlo (PIMC) is the state of the art of many IIG
problems such as Contract-Bridge, Skat, and many others.

The algorithm is defined in Algorithm 1 and works as follows: (i) samples a
world state by using the player’s private information; (ii) plays each action
of the sampled world state; (iii) estimates the reward of the new world state
by using an algorithm available in perfect information setting; (iv) repeats
until the budget is over; and, (v) selects the action that produces the best
result in average. In practice, PIMC often uses AlphaBeta as the perfect
information evaluator.

Function  _PIMC(_$\mathrm{s}$_)_:

for  _$\mathrm{m}$ $\in$ $\mathrm{Moves}$ ($\mathrm{s})$_ do

$\mathrm{score}$[$\mathrm{m}$] $\leftarrow$ $0$;

end for

while  _$\mathrm{budget}$_ do

$\mathrm{w}$ $\leftarrow$ $\mathrm{InfoSampling}$($\mathrm{s}$);

for  _$\mathrm{m}$ $\in$ $\mathrm{Moves}$ ($\mathrm{w}$)_ do

$\mathrm{score}$ [$\mathrm{m}$] $\leftarrow$ $\mathrm{score}$[$\mathrm{m}$] +
$\mathrm{PerfectAlgo}$ ($\mathrm{w}$, $\mathrm{m}$);

end for

end while

return _Best action on average_

Algorithm 1 PIMC

####  II-B2 IS-MCTS

Information Set Monte Carlo Tree Search [9] uses Monte Carlo Tree Search
(MCTS) [17] according to a sampled world state.

MCTS is a state-of-the-art tree search algorithm in perfect information games.
It works as follows (i) selection — selects a path of nodes based on an
exploitation policy; (ii) expansion — expands the tree by adding a new child
node; (iii) playout — estimates the child node by using an exploration policy;
and, (iv) backpropagation — backpropagates the result obtained from the
playout through the nodes chosen during the selection phase. In practice, MCTS
often uses random playout as the perfect information evaluator, and UCB1 in
the selection phase.

IS-MCTS works by using MCTS according to a sampled world state, i.e. the
selection and playout are done on the sampled world state.

Function  _IS-MCTS(_$\mathrm{s}$_)_:

while  _$\mathrm{budget}$_ do

$\mathrm{w}$ $\leftarrow$ $\mathrm{InfoSampling}$($\mathrm{s}$);

$\mathrm{MCTS}$ conditioned on $\mathrm{w}$.;

end while

return _Normalise visit count for each action_

Function  _MCTS(_$\mathrm{w}$_)_:

$\mathrm{u}$ $\leftarrow$ $\mathrm{Selection}$($\mathrm{w}$);

$\mathrm{u}$ $\leftarrow$ $\mathrm{Expansion}$ ($\mathrm{u}$,$\mathrm{w}$);

$\mathrm{u}$ $\leftarrow$ $\mathrm{Simulation}$ ($\mathrm{u}$,$\mathrm{w}$);

$\mathrm{Backpropagation}$($\mathrm{u})$;

Algorithm 2 IS-MCTS

##  III Belief Distributions

To present the different belief distributions, with their advantages and
drawbacks, we use the following example throughout the section to facilitate
understanding.

The example is based on the famous game ‘Liar’s Dice’ (an explanation of the
game is given in Subsection IV-A2). In our case, two players, each with $1$
die of $2$ sides. We denote $\\{P_{1}:X;P_{2}:Y\\}$ for player $1$ has $X$ and
player $2$ has $Y$. There are four world states possible
($w_{1}=\\{P_{1}:1;P_{2}:1\\}$, $w_{2}=\\{P_{1}:1;P_{2}:2\\}$;
$w_{3}=\\{P_{1}:2;P_{2}:2\\}$, $w_{4}=\\{P_{1}:2;P_{2}:1\\}$).

For each player, there are two infostates possible and one public infostate
$s_{pub}=\\{o^{1}_{pub}=\emptyset,o^{2}_{pub}=\emptyset\\}$ (no observation).
For the player $1$ we have
$s_{1}=\\{o^{1}_{priv(1)}=1,o^{2}_{priv(1)}=\emptyset\\}$ or
$s^{\prime}_{1}=\\{o^{1}_{priv(1)}=2,o^{2}_{priv(1)}=\emptyset\\}$ (i.e.
Player $1$ observes the die rolled but not the die rolled by the other
player), and for the player $2$, we have
$s_{2}=\\{o^{1}_{priv(2)}=\emptyset,o^{2}_{priv(2)}=1\\}$ or
$s^{\prime}_{2}=\\{o^{1}_{priv(2)}=\emptyset,o^{2}_{priv(2)}=2\\}$ (i.e.
Player $2$ observes the die rolled but not the die rolled by the other
player).

In the following, we suppose that the world state of this example is $w_{2}$.
Therefore, for the player $1$, the infostate is $s_{1}$ with two world states
possible ($\\{w_{1},w_{2}\\}$) and for the player $2$, the infostate is
$s^{\prime}_{2}$ with two world states possible ($\\{w_{2};w_{3}\\}$). Fig. 1
represents the different belief distributions presented throughout the
section.

Figure 1: Multiple belief distributions for the game Liar’s Dice with $1$ dice
of $2$ sides each. Four world states possible $w_{1}$, $w_{2}$, $w_{3}$ and
$w_{4}$. The Public-Private belief uses the mixture distribution with
$\lambda=0.5$.

###  III-A Private Distribution

As previously introduced, current determinization-based algorithms work by
sampling world states according to the player’s private information
distribution, i.e. knowing a player’s private and public observation, we
sample a world state.

Let $S_{j}(s_{i})$ be the set of possible infostates for player $j$
conditioned on the infostate $s_{i}$ of the player $i$. In our example, the
infostate possible for the player $2$ when the player $1$ has $s_{1}$ is
$S_{2}(s_{1})=\\{s_{2};s^{\prime}_{2}\\}$, i.e. having the die $1$ for the
player $1$ does not exclude the player $2$ to have a 1 or a 2. Depending on
the game $S_{j}(s_{i})$ can be restrictive, e.g. in trick-taking card games if
the player $i$ has the card ‘Queen of Hearts’, no opponent can have it.

######  Definition (Private Belief Distribution).

Let $S_{j}(s_{i})$ be the set of possible infostates for player $j$
conditioned on the infostate $s_{i}$. Let $\Delta S_{j}(s_{i})$ denotes the
probability distribution over the elements of $S_{j}(s_{i})$. We define the
private belief distribution as $\Delta_{i}(s_{i})=(\Delta
S_{1}(s_{i}),\dots,\Delta S_{i}(s_{i}),\dots,\Delta S_{N}(s_{i}))=(\Delta
S_{1}(s_{i}),\dots,s_{i},\dots,\Delta S_{N}(s_{i}))$ .

In Fig. 1, using Player 1’s private belief state provides the following belief
distribution
$\Delta_{1}(s_{1})=(\\{s_{1}:100\%\\},\\{s_{2}:50\%;s^{\prime}_{2}:50\%\\})$,
which results in two equiprobable world states ($w_{1}$, $w_{2}$).

When using the private distribution for determinization, the algorithm samples
a world state ($w_{1}$ or $w_{2}$) consistent with the current player’s
information ($s_{1}$) and, as the state-of-the-art in trick-taking game shows,
great performance is obtained. Yet, by doing so, 3 problems arise.

(i) It is not consistent with the other player’s belief, e.g. if we use it
with the first player, the algorithm samples $w_{1}$ or $w_{2}$ but never
$w_{3}$, which is nevertheless, a world state possible from the point of view
of the player $2$.

(ii) It is not able to mislead others. In our example, two actions are
possible for the first player, ‘I have a one’ and ‘I have a two’. The action
‘I have a two’ is a lie, however, one may want to play this action with the
aim of deceiving the opponent. However, in our case only $w_{1}$ or $w_{2}$
can be sampled and, in each world, the action ‘I have a two’ results in a
defeat because the second player will say ‘This is a lie’. Therefore, lying is
never an option, as it never succeeds.

(iii) It, indirectly, allows the opponents to infer our private information,
e.g. after playing multiple matches, the second player understands that, if
the first player plays ‘I have a two’, it is because he really has a two as it
can not lie, and therefore, play to counter it.

Trying to infer missing information is one of the key components of IIG, and
using the private belief distribution could result in a highly exploitable
performance. To remove this problem, one can use public belief distribution,
as presented in the next section.

###  III-B Public Distribution

Recently in IIG, many algorithms [12, 13] have been using the concept of
public observation. This concept has resulted in many breakthroughs thanks to
decomposition, which made the calculations feasible. One application of public
observation is the creation of a public belief distribution over the world
states possible according to the public observations observed so far.

######  Definition (Public Belief Distribution [13]).

Let $S_{j}(s_{pub})$ be the set of possible infostates for player $j$
conditioned on the public infostate $s_{pub}$. Let $\Delta S_{j}(s_{pub})$
denote the probability distribution over the elements of $S_{j}(s_{pub})$. We
define the public belief distribution as $\Delta_{pub}(s_{pub})=(\Delta
S_{1}(s_{pub}),...,\Delta S_{N}(s_{pub}))$.

In our example, using the public belief distribution from the point of view of
the player $1$ would result in the following belief distribution
$\Delta_{pub}=(\\{s_{1}:50\%;s^{\prime}_{1}:50\%\\},\\{s_{2}:50\%;s^{\prime}_{2}:50\%\\}$.
In other world, every world state are equiprobable, this is due to the public
infostate that does not contain any information.

Using a public belief distribution instead of a private belief distribution
removes the problem defined in Section III-A.

(i) It is consistent with the other player’s doubts, e.g. it samples the world
$w_{3}$ which is a world state possible of the second player.

(ii) It is capable of misleading others, e.g. when sampling $w_{3}$ or $w_{4}$
the action ‘I have a two’ does not result in a defeat for the first player,
therefore, allows the first player to play the action ‘I have a two’.

(iii) It no longer reveals private information, i.e. as the reasoning is no
longer biased toward the private information, it can not be used against it.

Nevertheless, using public distribution has a significant drawback as it does
not consider a player’s private information, and one might wonder whether it
is useful to not use private information. It is straightforward to consider
that the extent to which private information should be kept hidden depends on
the game being played and, in certain games, it is not necessary to keep the
information concealed.

In addition, by using public distribution, one must be aware as there are more
world states possible (e.g. by using private distribution, we have two world
states possible and by using public distribution, we have four world states
possible), which can be intractable in large games.

###  III-C Mixture between public and private distribution

To solve both of the problems defined in Section III-A and in Section III-B,
we propose to use a mixture of private and public distribution.

######  Definition (Mixture Belief Distribution).

Let $s_{pub}$ be the public infostate associated with the infostate $s_{i}$.
We define the mixture belief distribution as
$\Delta_{\lambda}(s_{i})=(1-\lambda)\Delta_{i}(s_{i})+\lambda\Delta_{pub}(s_{pub})$

The mixture belief distribution allows us to be consistent with the problem
encountered. When care must be taken not to reveal information, one can
increase $\lambda$. In contrast, when it is not appropriate to withhold
information, one can decrease $\lambda$. The private belief distribution is
obtained when $\lambda=0$ and the public belief distribution is obtained when
$\lambda=1$.

In our example, when using the mixture with $\lambda=0.5$ for the player $1$,
we obtain the following belief distribution
$\Delta_{0.5}(s_{1})=(\\{s_{1}:75\%;s^{\prime}_{1}:25\%\\},\\{s_{2}:50\%;s^{\prime}_{2}:50\%\\}$.
$w_{1}$ and $w_{2}$ are more probable ($37.5\%$ each) than $w_{3}$ and $w_{4}$
($12.5\%$ each). Nevertheless, their probabilities are not zero, which makes
it consistent with the other player’s belief.

It is possible to expand this concept by considering that $\lambda$ depends on
the progress of the game. As an example, in trick-taking card games, it may be
important to keep the private information hidden at the beginning of the game
(so as not to reveal information) but, as the game progresses, the focus
shifts to accumulating points before the end, where the importance of
concealing this information may decrease.

###  III-D Adaptation of algorithms

PIMC and IS-MCTS have been created with private belief distribution in mind.
Therefore, it is necessary to modify the algorithms to use the public or a
mixture belief distribution. Instead of starting at an infostate $s_{i}$, the
algorithms must be adapted to start at $s_{pub}$, where $s_{pub}$ is the
public infostate associated with $s_{i}$.

####  III-D1 PIMC

In the case of PIMC, one must use a distinct PIMC for each infostate possible
($S_{i}(s_{pub})$), and combine the final result by aggregating the scores
using the distribution of possible infostates ($\Delta S_{i}(s_{pub})$).

In our example, when using the mixture belief distribution, two infostates are
possible for the first player ($s_{1}$ and $s^{\prime}_{1}$). If $w_{2}$ or
$w_{1}$ are sampled, the algorithm used is the one defined for $s_{1}$, on the
other hand, if $w_{3}$ or $w_{4}$ are sampled, the algorithm used is the one
defined for $s^{\prime}_{1}$. In the end, if $s_{1}$ has been visited $75\%$
(corresponding to the mixture belief distribution with $\lambda=0.5$), the
action chosen in $s_{1}$ will have more impact than the action chosen in
$s^{\prime}_{1}$.

####  III-D2 IS-MCTS

$s_{1}$$s^{\prime}_{1}$$s_{2}$$s^{\prime}_{2}$$s^{\prime\prime}_{2}$$s^{\prime\prime\prime}_{2}$
(a) Constructed with the mixture belief distribution.

$s_{1}$$s_{2}$$s^{\prime}_{2}$$s^{\prime\prime}_{2}$$s^{\prime\prime\prime}_{2}$
(b) Constructed with private belief distribution.

Figure 2: Example of the tree constructed by IS-MCTS. The first player is
acting in the red square, the second player is acting in the green diamond and
the blue circle refers to the chance node.

With IS-MCTS, a singular algorithm is feasible as IS-MCTS creates a tree where
the nodes represent infostates, and an infostate for player $j$ may come from
several infostates of player $i$.

An example is provided in Fig. 2. For the first player, two infostates are
possible ($s_{1}$ and $s^{\prime}_{1}$) and four infostates are possible for
the second player after the first player’s action
($s_{2}=\\{o^{1}_{priv(2)}=\emptyset,o^{2}_{priv(2)}=1,o^{3}_{priv(2)}=a_{1}\\}$,
$s^{\prime}_{2}=\\{o^{1}_{priv(2)}=\emptyset,o^{2}_{priv(2)}=1,o^{3}_{priv(2)}=a_{2}\\}$,
$s^{\prime\prime}_{2}=\\{o^{1}_{priv(2)}=\emptyset,o^{2}_{priv(2)}=2,o^{3}_{priv(2)}=a_{1}\\}$
or
$s^{\prime\prime\prime}_{2}=\\{o^{1}_{priv(2)}=\emptyset,o^{2}_{priv(2)}=2,o^{3}_{priv(2)}=a_{2}\\}$).

For the second player, all infostates are achievable through any infostate of
the first player. For example, $s_{2}$ is achievable when sampling $w_{1}$
(from $s_{1}$) or when sampling $w_{2}$ (from $s^{\prime}_{1})$ and playing
the action $a_{1}$.

##  IV Experimentation

###  IV-A Benchmarks

For our experiments, the following benchmarks are tested ‘Liar’s Dice’ (LD),
‘Card Games (CG)’, and ‘Leduc Poker’ (LP). Each of them is described below.

####  IV-A1 Card game

For the purpose of the experimentation, we use a smaller version of classic
trick-taking card games. The game is played with two players, $10/20$ cards
known by all, $2/6$ are hidden and the rest is distributed to each player.

The playing phase is decomposed into tricks, the player starting the trick is
the one who won the previous trick. The starting player of a trick can play
any card in his hand, but the other players must follow the suit of the first
player. If they can not, they can play any card they want but, without the
possibility of winning the trick. The winner of the trick is the one with the
highest-ranking card. At the end of the game, the points of each player are
counted (plain version of trick-taking card game). The count is defined by the
number of tricks won. A player wins if it has at least half of the points.

####  IV-A2 Liar’s Dice

Liar’s dice is a dice game played with two or more players, where each player
possesses $N$ dice of $K$ sides and in which a player must deceive and be able
to detect an opponent’s deception.

In the beginning, each player rolls his dice and observes the values. After
that, players take turns guessing the number of dice of a particular type held
by everyone. The game continues until a player accuses another of lying. If
the player who made the assumption is right, he wins the game, on the
opposite, if the challenged player did not lie, the challenged player wins.

During the game, a player can not bid less than previously, i.e. he must at
least bid more dice than the previous player’s bid, or the same number of dice
but with a higher value. Lastly, the highest face is a wild card, i.e. the
value can be used to count for any other face.

####  IV-A3 Leduc Poker

Leduc Poker, as described in [18], is a variation of poker that uses a deck
with only two suits, each containing three cards.

The game consists of two rounds. In the first round, each player is dealt a
single private card. In the second round, a single board card is revealed. The
maximum number of bets allowed is two, with the first round allowing raises of
$2$ and the second round allowing raises of $4$. Both players begin the first
round with $1$ already in the pot.

###  IV-B Experimentation

In our experiments, our objective is (i) to observe the extent to which an
algorithm $X$ reveals information according to mixture belief distribution;
(ii) to analyze how the mixture belief distribution impacts the performance
against an opponent that uses the revealed information; and (iii) to analyze
how the mixture belief distribution impacts the performance against an
opponent that does not use the revealed information.

Our code is based on OpenSpiel [19]. This is a collection of environments and
algorithms for research in general reinforcement learning and search/planning
in games.

PIMC and IS-MCTS are used with their basic version, i.e. PIMC uses AlphaBeta
and IS-MCTS uses random rollouts as the perfect information evaluator and an
exploration constant of $0.7$. For both, $1000$ world states are sampled.

To achieve a stable policy (as PIMC and IS-MCTS are online algorithms), we run
the algorithm multiple times for every infostate until the policy obtained has
less than $1\%$ of variation.

The experiments were conducted according to the player’s playing position
(each position reveals more or less information). In the following part, the
experiments are carried out for the first player and in the appendix for the
second player.

####  IV-B1 How much information is revealed

$0$$0.2$$0.4$$0.6$$0.8$$1$$2$$4$$6$$8$$10$Lambda valueTSSR valuePIMCIS-MCTS

(a) Liar’s dice with 2 dice

$0$$0.2$$0.4$$0.6$$0.8$$1$$2$$4$$6$$8$Lambda valueTSSR valuePIMCIS-MCTS

(b) Liar’s dice with 3 dice

$0$$0.2$$0.4$$0.6$$0.8$$1$$1$$1.2$$1.4$$1.6$$1.8$$2$$2.2$Lambda valueTSSR
valuePIMCIS-MCTS

(c) Leduc poker

$0$$0.2$$0.4$$0.6$$0.8$$1$$1$$1.2$$1.4$$1.6$$1.8$Lambda valueTSSR valuePIMCIS-
MCTS

(d) Card Game with 10 cards

Figure 3: Average TSSR for IS-MCTS and PIMC on multiple benchmarks according
to $\lambda$ of the mixture distribution.

For analyzing the impact of the revealed information according to the
distribution used, we use the formula called True State Sampling Ratio (TSSR)
[20]. TSSR measures how much more likely it is for the opponent to guess the
current world state when using an algorithm $X$ rather than using a uniform
function.

The formula is $TSSR(w)=\eta(w\mid s_{i})\cdot|S_{i}(s_{i})|$ where $s_{i}$ is
the infostate corresponding to $w$, $\eta(w\mid s_{i})$ is the probability
that the true state is guessed given the information set $s_{i}$. The closer
the result is to $1$, the less likely it is to know the real world state. Fig.
3 presents the TSSR value obtained according to $\lambda$ of the mixture
distribution.

As expected, playing closer to the public belief distribution greatly reduces
the probability of knowing the real-world state. In ‘Liar’s Dice’ with $2$
dice with PIMC, it is up to $10$ fold more likely to guess the real world
state when using the private instead of the public belief distribution.

In terms of information revealed, we observe that PIMC reveals more
information than IS-MCTS in every benchmark. In ‘Leduc Poker’, it’s up to
$2.2$ times more likely to deduce the true state with PIMC at $\lambda=0.0$
whereas, with IS-MCTS, it is ‘only’ $1.3$ times more likely to deduce the true
state.

In addition, ‘Liar’s Dice’ is the game that reveals the most information with
the algorithm revealing up to $10$ times more likely than random, whereas in
‘Leduc Poker’ or ‘Card Game’, it is only up to $2$ times more likely than
random.

For the following experiments, it is expected to observe $\lambda$ closer to 1
for PIMC in ‘Liar’s Dice’, as it reveals more information, and therefore,
could be exploited by the opponent.

####  IV-B2 How does the mixture impact the performance

tableExpected utility against best responder when playing at the first player
position.  Algo Game $\lambda$ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PIMC
LD 2D 0.300 0.298 0.297 0.292 0.294 0.288 0.281 0.290 0.336 0.382 0.382 LD 3D
0.313 0.276 0.265 0.269 0.235 0.283 0.324 0.356 0.359 0.393 0.458 LP 0.622
0.616 0.660 0.767 0.797 1.481 1.626 1.480 1.532 1.599 1.611 IS-MCTS LD 2D
0.513 0.512 0.517 0.528 0.539 0.547 0.552 0.554 0.555 0.562 0.562 LP 0.797
0.890 0.966 0.959 1.158 1.226 1.402 1.673 1.786 2.083 2.326

To measure how the mixture impacts the performance, we compute the expected
utility against the best responder. The best responder is the worst possible
enemy of all algorithms, i.e. it knows exactly the policy our algorithm will
execute, and therefore, can infer the true infostate and plays the best action
against it.

The results are available in Table IV-B2 where the values represent the
expected utility of the best responder and must be minimized. The results
obtained are exact utility (without variation), as the best responder computes
the best strategy knowing all the distributions in every infostate of the
game.

We observe that the private belief distribution performs better than the
public belief distribution, i.e. for all benchmarks and algorithms (better
results are obtained when $\lambda=0.0$ than when $\lambda=1.0$).

In ‘Liar’s Dice’ with PIMC, the best performances are obtained when $\lambda$
is close to $0.5$ (with $2$ dice, we obtain the best value when
$\lambda=0.6$). These results were expected, as PIMC reveals a lot of
information with Liar’s Dice which is then exploited by the best responder.

On the other hand, when the algorithm reveals less information (as observed in
‘Leduc Poker’ or IS-MCTS), it is preferable to use the private belief
distribution or very close, as it is not sufficient for the best responder to
exploit the revealed information.

####  IV-B3 Can the use of multiple mixture belief distributions throughout
the game improve performance

In this experiment, we analyze the use of multiple mixtures throughout the
game to improve performance. For this purpose, we compute multiple mixture
distributions against the best responder.

(a) Leduc Poker with PIMC

(b) Liar’s Dice 2 dice with PIMC

(c) Leduc Poker with IS MCTS

(d) Liar’s Dice 2 dice with IS MCTS

Figure 4: Heatmap of the expected utility against the best response when
playing at the first position.

Fig. 4 represents heatmaps for ‘Leduc Poker’ and ‘Liar’s Dice’ according to
the position throughout the game when using PIMC (resp. IS-MCTS). For both
games, we have a mixture distribution for the first action and another for the
second action.

In all experiments, we observe that using multiple mixtures throughout the
game has an impact on the performance. In ‘Leduc Poker’ for both algorithms,
not using our private belief distribution is more punished in the second round
than in the first round (e.g. $\\{0.0,1.0\\}$ has a value of $1.17$ whereas
$\\{1.0,0.0\\}$ has a value of $1.88$ for IS-MCTS). On the other hand, for
‘Liar’s Dice’, we observe that the first round is the most important one.

In addition, we observe that playing multiple mixtures improve performance. In
‘Liar’s Dice’, the best value for IS MCTS is obtained when we have
$\\{0.0,0.6\\}$ and for PIMC when we have $\\{0.6,0.2\\}$.

####  IV-B4 How does the mixture impact the winning rate

tableWinning rate when the opponent uses ‘PIMC’ when playing at the first
player position.  Our Game $\lambda$ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
PIMC LD 3D 48.6 50.4 47.9 47.4 44.6 42.6 39.9 37.5 36.1 28.9 27.8 LD 5D 43.1
43.4 42.2 43.4 42.5 41.4 39.8 36.3 37.1 29.8 23.6 CG 10C 48.2 47.9 47.7 47.7
47.6 47.4 46.7 46. 45.5 39.8 31.6 CG 20C 53.7 53.8 54.2 54.5 53.9 53.2 52.8
52. 47.4 36.3 23.5 IS MCTS LD 3D 23.7 23.7 24.7 27. 23.1 23.1 21.7 20. 19.3
15.4 16.4 LD 5D 22. 20.9 21.9 22.2 21.9 20.8 21.6 21. 16.9 15.5 13.4 CG 10C
45.3 46.3 45.4 45.1 43.8 45.1 45. 43.1 42.7 37.6 30. CG 20C 36.5 38.5 38.2
36.2 36.4 36.6 35.5 34.9 33.3 33.1 20.8

As observed in the previous experiments, when using a $\lambda$ closer to the
public belief distribution, we obtain a distribution of action less relevant
but with the advantage of disclosing less information. Therefore, when faced
with an opponent who does not infer on our private information, it is expected
to lose the benefit of using a $\lambda$ closer to the public belief
distribution. Nevertheless, using a $\lambda$ closer to the public belief
distribution not only reveals less information but allows it to be more
consistent with the other player’s doubts.

To measure the impact of being more consistent with the other player’s doubts,
we evaluate the performance against an algorithm that does not try to infer
our private information. To do this, we compute the winning rate against
‘PIMC’ over $1000$ games which results in $3.1\%$ variation ($95\%$ of
confidence interval). The scores are available in Table IV-B4.

As before, we observe that it is preferable to use private belief distribution
instead of public belief distribution. In ‘Liar’s Dice’ with 3 dice with PIMC,
we observe a drop of $20.8$ in the winning rate between the private and public
belief distribution. In addition, we observe that in every benchmark tested
and for both algorithms, using a $\lambda$ between $0.0$ to $0.5$ does not
produce a drop in performance, but provides equivalent results.

These results are surprising, as we could have expected a drop in performance
as the actions are less relevant to the current infostate (as we have sampled
less often the true infostate). This implies that being more consistent with
the doubts of the other players compensates for the loss of the player’s
private information.

##  V Conclusion

In this paper, we study the strengths and weaknesses of probability
distributions (private and public) in which particular attention has been paid
to the revealed information and the impact of this revealed information on
performance. Our study has been carried out on determinization-based
algorithms and on multiple imperfect information games.

We complete the study by proposing a new probability distribution, a mixture
of the two previous ones, which solves problems encountered by other
distributions. We show that using the mixture is beneficial to reduce the
revealed information and improve performance. We also show that using multiple
mixtures throughout the game improves performance. In addition, we observed
that using the mixture against an opponent that does not use our private
information revealed results in a good performance as we are being more
consistent with the other player’s doubt.

An avenue for improvement would be to extend the utilization of using multiple
mixtures throughout the game. For example, by using the mixture at each public
infostate instead of a fixed time step or using a different lambda for the
opponent player. Another area for improvement would be to extend the study of
algorithms that do not use determinization or even, without probability
distributions but bearing in mind that one should not always use one’s private
information at the risk of revealing information and, on the contrary, that
one should not always use one’s public information in order to be more
consistent to one’s private knowledge. Lastly, it would be interesting to
extend the results at a larger scale, either by using more games or by using
larger games.

###  -A Complementary experiments

The following experiments are identical to those in the primary paper, with
the exception that they are conducted for the second player position.

Similar results are observed, i.e. PIMC reveals more information than IS-MCTS,
the private belief distribution obtains better performance than the public
belief distribution against the best responder, using multiple mixtures is
useful to improve the performance and it is all as well to play the mixture as
the private against an opponent that does not try to infer.

Yet, we also observe some differences, especially that less information is
revealed when playing in the second position, which results in $\lambda$
closer to the private belief distribution against the best responder.

tableExpected utility for best responder against our algorithm being the
second player.  Algo Game $\lambda$ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
PIMC LD 2D 0.678 0.695 0.703 0.707 0.716 0.718 0.711 0.741 0.779 0.836 0.836
LP 0.398 0.400 0.459 0.612 0.796 1.461 1.450 1.509 1.593 1.615 1.632 IS-MCTS
LD 2D 0.697 0.687 0.697 0.716 0.727 0.732 0.740 0.751 0.759 0.768 0.787 LP
0.784 0.784 0.898 0.800 1.017 1.078 1.186 1.324 1.561 1.728 2.002

$0$$0.2$$0.4$$0.6$$0.8$$1$$1$$1.5$$2$Lambda valueTSSR valuePIMCIS-MCTS

(a) Liar’s dice with 2 dice

$0$$0.2$$0.4$$0.6$$0.8$$1$$1$$1.1$$1.2$$1.3$$1.4$$1.5$Lambda valueTSSR
valuePIMCIS-MCTS

(b) Leduc poker

Figure 5: Average TSSR according to $\lambda$ value of the mixture
distribution.

(a) Leduc poker with PIMC

(b) Liar’s dice 2 dice with PIMC

(c) Leduc poker with IS-MCTS

(d) Liar’s dice 2 dice with IS-MCTS

Figure 6: Heatmap of expected utility against the best response when playing
at the second position.

tableWinning rate when the opponent uses ‘PIMC’ when playing at the second
player position.  Our Game $\lambda$ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
PIMC LD 3D 51.9 53. 51.3 49.8 49.7 51.3 48.7 48.6 46.9 46.1 42.7 LD 5D 56.7
55.5 56. 56.2 54.8 56.1 55.3 53. 51.9 44.7 42.3 IS MCTS LD 3D 48.4 51.3 49.9
49. 50.1 51. 47.4 44. 39.7 36.9 33.3 LD 5D 48.4 47.1 48. 46.7 47.8 45. 46.5
40.7 34.4 23.2 14.7

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# Multifold topological semimetals

Iñigo Robredo  Max Planck Institute for Chemical Physics of Solids, 01187,
Dresden, Germany  Donostia International Physics Center, 20018, Donostia - San
Sebastian, Spain  Niels Schröter  Max Planck Institut für Mikrostrukturphysik,
06120, Halle, Germany  Claudia Felser  Max Planck Institute for Chemical
Physics of Solids, 01187, Dresden, Germany  Jennifer Cano  Department of
Physics and Astronomy, Stony Brook University, 11794, Stony Brook, USA  Center
for Computational Quantum Physics, Flatiron Institute, 10010, New York, USA
Barry Bradlyn  Department of Physics, University of Illinois, 61820, Urbana-
Champaign, USA  Maia G. Vergniory  Max Planck Institute for Chemical Physics
of Solids, 01187, Dresden, Germany  Donostia International Physics Center,
20018, Donostia - San Sebastian, Spain

###### Abstract

The discovery of topological semimetals with multifold band crossings has
opened up a new and exciting frontier in the field of topological physics.
These materials exhibit large Chern numbers, leading to long double Fermi arcs
on their surfaces, which are protected by either crystal symmetries or
topological order. The impact of these multifold crossings extends beyond
surface science, as they are not constrained by the Poincaré classification of
quasiparticles and only need to respect the crystal symmetry of one of the
1651 magnetic space groups. Consequently, we observe the emergence of free
fermionic excitations in solid-state systems that have no high-energy
counterparts, protected by non-symmorphic symmetries. In this work, we review
the recent theoretical and experimental progress made in the field of
multifold topological semimetals. We begin with the theoretical prediction of
the so-called multifold fermions and discuss the subsequent discoveries of
chiral and magnetic topological semimetals. Several experiments that have
realized chiral semimetals in spectroscopic measurements are described, and we
discuss the future prospects of this field. These exciting developments have
the potential to deepen our understanding of the fundamental properties of
quantum matter and inspire new technological applications in the future.

##  I Introduction

The prediction of Dirac and Weyl fermions dates back to 1928 Dirac  and 1929
Weyl  respectively, when Dirac proposed a four-component wave function to
describe massive spin-1/2 fermions and Weyl simplified it to a two-component
spinor to describe massless particles. Despite decades of searching by
particle physicists, Weyl fermions remained elusive for almost a century.
However, in recent years, the search has shifted to condensed matter systems,
where Weyl and Dirac fermions are observed as protected crossings in
electronic Bloch bands near the Fermi level, in two (Dirac) and three
dimensions (Dirac and Weyl). This shift in research has opened up new avenues
for investigating the properties and applications of these exotic particles.

The first attempt to find a Dirac fermion in a solid state system was made in
1987, when Boyanovsky et al. Boyanovsky_1987  proposed PbTe as a platform to
realize the parity anomaly of 2 + 1 dimensional QED, after the prediction of a
Dirac crossing in its band structure. Unfortunately, the model was incomplete
and the theoretical prediction proved incorrect PbTe . A successful
identification of a solid state Dirac fermion was delayed until many years
later in graphene RevModPhys2009 , a two-dimensional (2D) material consisting
of a single layer of carbon atoms arranged in a hexagonal lattice. The two
low-energy electronic bands in graphene are doubly degenerate and meet at two
linearly dispersing crossing points, known as the Dirac cones or Dirac points
because near these points electrons behave as massless Dirac fermions. Later,
the 3D generalizations of graphene, known as Dirac semimetals, were predicted
Wang2013 ; Wang2012  and observed YulinNatMat2014 ; YulinScience2014  in
Cd3As2 and Na3Bi.

It took longer to discover Weyl fermions in solid state systems. The first
prediction of the ubiquity of perturbatively stable, ungappable 2-fold
crossings in noncentrosymmetric or non-TR-invariant crystals (Weyl nodes) was
made by Herring in 1937 Herring . The first proposal on a realistic material,
though, came much later. They were predicted in pyrochlore iridates Wan_2011 ;
yang2011quantum , which was followed by a roadmap to engineer Weyl fermions in
topological insulator heterostructures BurkovWeyl2011 . Unlike Dirac fermions,
Weyl fermions are unique to 3D systems, and they exhibit twofold band
crossings near the Fermi level with linear dispersion in all three dimensions.
They are protected by a topological invariant known as the Chern number, which
allows them to appear at any point within the Brillouin zone (BZ). Soon after,
the first manifestation of a Weyl node in a crystal was pointed out
theoretically in HgCr2Se4 by Wan et alXu2011ChernSA  and a Weyl node in
Hg1-x-yCdxMny by Bulmash et alBulmash . Unfortunately, none of these systems
came to fruition experimentally. Weyl semimetals became finally a reality with
the experimental discovery DingPRX2015 ; SYXuScience2015  confirming the
theoretically predicted XDaiPRX2014 ; SMHuangNatComm2015  Weyl nodes in the
TaAs family of compounds. However, finding new materials exhibiting Weyl
fermions remained a challenge.

A big step forward was taken when it was realized that the co-representations
(coreps) of the little group BC-book  at the high-symmetry points in the BZ
could correspond to nodal fermions manes2012 ; Young2012 . This discovery set
the stage for the machinery to systematically identify new materials using
first-principles calculations combined with crystal symmetry and group theory
Steinberg2014 ; Wieder2016 . In 2016 Bradlyn et al NewFermions  and Wieder et
al Wieder8fold  recognized that non-symmorphic crystalline symmetries could
stabilize previously undiscovered crossings with no analogs in high-energy
physics. In these systems, the low-energy $\boldsymbol{k}\cdot\boldsymbol{p}$
Hamiltonians do not resemble the dispersion relations of any relativistic
particle. Yet, they exhibit an isotropic limit resembling a Weyl fermion,
$H=\boldsymbol{k}\cdot\boldsymbol{S}$, with $S_{i}$ a set of $n$-dimensional
spin matrices. Initially dubbed the multifold fermions, these excitations
introduced the field of multifold topological semimetals. The newly introduced
band crossings can be 3-, 4-, 6- and 8-fold degenerate at the crossing point,
which may or may not exhibit a topological charge depending on their crystal
symmetry. In current parlance, these $n$-dimensional fermions are referred to
as _$n$ -folds_, with _n_ denoting the dimension of the band crossing, which
is the notation we will use throughout this work. This work reviews the
classification of these n-fold fermions, along with recent theoretical and
experimental advances.

The manuscript is organized in the following way: in Section II we will give a
full classification of high-order crossings in the 1651 Shubnikov groups and
provide a general procedure to identify these nodes in materials. Section III
will review the recent progress in materials realization with particular focus
in chiral crystals. In Section IV we provide the computational details of the
band structure calculations in Section III. We conclude the manuscript with a
discussion of future perspectives.

##  II Classification of all Shubnikov groups and general procedure

In this section, we classify the symmetry-protected nodal points that can
exist at high symmetry points within magnetic space groups by first reviewing
the Shubnikov groups that describe crystalline materials and then discussing
the dimensionality of their coreps.

###  II.1 Magnetic Space Group Types

Multifold fermions, unlike Weyl fermions, need symmetry protection to be
stabilised. In a particular symmetry group, the symmetry operations and their
commutation relations dictate the multifold fermions that the group can
protect. We thus start by reviewing the classification of Shubnikov groups,
which will serve as a basis for the classification of multifold fermions in
crystalline materials.

There are 1651 Shubnikov groups, also named magnetic space groups (MSG), which
can be classified into four types. First, the type I groups are those that
contain only unitary crystalline symmetries, i.e., no time reversal symmetry
(TRS) or combinations of crystalline symmetries and TRS. There are 230 such
groups. They can describe some materials with magnetic order, but only a
reduced subset of all commensurate magnetic orderings. Nonetheless, the type I
groups provide the underlying structure from which the remaining MSGs can be
constructed. Starting from a type I MSG, $H$, adding TRS as a group element
creates a type II MSG (non-magnetic, often referred to as ‘gray’ groups),
$G=\\{E,\mathcal{T}\\}\times H=H\,\cup\mathcal{T}H$, with $\mathcal{T}$ the
TRS operator. These groups describe non-magnetic crystals, such as the ones
analysed in the topological quantum chemistry database TQC_databse ; Alltopo ;
Catalogue_Zhang ; Tang2019 . The remaining two types (type III and IV), the so
called black-white groups, deal with the combination of TRS and unitary
symmetries and can describe the remaining commensurate magnetic orderings.
These MSGs can be written as $G=H\cup\mathcal{T}g_{0}H$, with $H$ a type I MSG
and $g_{0}\notin H$ a unitary symmetry. In this context, type IV groups are
those that contain fractional translations in $g_{0}H$, while type III MSGs do
not.

This exhausts the classification of magnetic symmetry groups, which can be
used to describe all commensurate magnetic orderings. A further sub-
classification of MSGs is obtained by distinguishing between the presence or
absence of improper symmetries, that is, symmetries with determinant $-1$. If
a MSG does not contain elements such as rotoinversions or mirrors that reverse
spatial orientation it is called a Sohnke MSG and it gives rise to a
_structurally chiral_ crystal, while if it contains inversion symmetry,
rotoinversions or mirrors it is non-chiral. This distinction is relevant
because _only Sohnke MSGs can host topologically charged multifold fermions_.
In non-Sohnke MSGs, symmetry prevents a multifold fermion from carrying a
topological charge.

We now turn to identifying the MSGs that can host $n$-fold crossings in the
presence of spin-orbit coupling (SOC). The first step is to find the
$n$-dimensional ($n=3,4,6,8$) irreducible coreps of each of the 1651 MSGs: an
irreducible corep corresponds to an $n$-dimensional symmetry enforced
degeneracy. This search can now be easily carried out by using the Bilbao
Crystallographic Server (BCS) BCS1 ; BCS2 ; BCS3 . Preliminary work on this
classification was carried out by Bradlyn et al BradlynEA17 , Chang et al
chang2018topological  and Wieder et al Wieder8fold  for type II MSGs and
generalized to all MSGs by Cano et al cano2019multifold .

###  II.2 Multifold Fermions

Table 1: List of MSGs containing 3-fold, 6-fold and 8-fold fermions in systems with significant SOC. Each cell lists the MSG and the $\mathbf{k}$-point at which the multifold fermion resides. | Type I | Type II | Type III | Type IV
---|---|---|---|---
3-fold | 199.12, 214.67, 220.89 | P | 199.13, 214.68, 220.90 | P | 214.69, 220.91, 230.148 | P |  |
206.37, 230.145 |  |  |  |  |  |
198.9, 205.33, 212.59 | R |  |  | 212.61, 213.65 | R | 198.11, 212.62, 213.66 | R
213.63 |  |  |  |  |  |
6-fold |  |  | 206.37, 230.145 | P | 206.39, 230.147, 230.149 | P |  |
|  | 198.10, 205.34, 212.60, 213.64 | R | 205.35 | R | 205.36 | R
8-fold |  |  | 130.424, 135.484 | A |  |  | 125.374, 126.385, 129.420 | A
|  |  |  |  |  | 131.445, 132.458, 136.504
|  | 218.82, 222.99, 223.105 | R | 222.102, 223.108 | R | 215.73, 221.97, 224.115 | R
|  | 220.90, 230.146 | H | 230.149 | H |  |

We start by analysing the 3-fold fermion, which was first predicted in the
type I MSG I$2_{1}31^{\prime}$ (199.13) BradlynEA17  as the
$\boldsymbol{k}\cdot\boldsymbol{p}$ Hamiltonian in the vicinity of the
$P=(1/4,1/4,1/4)$ point. This turns out to be the most general 3-fold fermion,
to which all 3-fold fermions in all MSGs are equivalent. The continuum
Hamiltonian reads:

$H_{\text{3-fold}}(\phi,\boldsymbol{k})=\begin{pmatrix}0&e^{i\phi}k_{x}&e^{-i\phi}k_{y}\\\
e^{-i\phi}k_{x}&0&e^{i\phi}k_{z}\\\
e^{i\phi}k_{y}&e^{-i\phi}k_{z}&0\end{pmatrix},$ (1)

where $\phi$ is a free parameter in the type II MSGs but can be symmetry-
restricted in other MSGs. In the particular case of $\phi=\pi/2$, the
Hamiltonian in Eq. 1 takes the familiar form
$H_{\text{3-fold}}(\pi/2,\boldsymbol{k})=\boldsymbol{k}\cdot\boldsymbol{S}$,
with $\boldsymbol{S}_{x,y,z}$ the generators of the spin $s=1$ representation
of $SU(2)$. Notice that it resembles the Weyl Hamiltonian,
$H_{\text{Weyl}}=\boldsymbol{k}\cdot\boldsymbol{\sigma}$ (with
$\boldsymbol{\sigma}$ the vector of Pauli matrices), which is the special case
of $s=1/2$. Hence the 3-fold fermions are referred to as spin-1 Weyl fermions.
The $H_{\text{3-fold}}$ Hamiltonian is gapped away from the degeneracy point,
so the Chern number of each band can be computed by computing the Chern number
of a sphere enclosing the degeneracy point. This calculation reveals that the
Chern number is $-2$/$+2$ for the lower/upper bands and 0 for the middle band
(see Fig. 1a). Noticeably, the 3-folds in type IV MSGs have an additional
symmetry generator
$g=\\{E|\frac{1}{2}\frac{1}{2}\frac{1}{2}\\}\circ\mathcal{T}$, which forces
$\phi=\pi/2$. This Hamiltonian, which can only be achieved by fine-tuning in
type I/II/III MSGs, provides an ideal situation where the energy gap away from
the degeneracy point is maximized, thus allowing maximum range for Fermi arcs
cano2019multifold . In Table 1 we list all MSGs which can host 3-fold fermions
in systems with significant SOC and at which high symmetry $k$-point.

The 6-fold fermions can be constructed by using the 3-fold fermion as a
building block. There are two classes of 6-fold fermions depending on whether
the MSG is chiral: double spin-1 fermions and spin-1 Diracs, respectively. In
chiral type II MSGs, TRS pins two symmetry-related copies of the 3-fold
fermion in the same $k$-point. The bands are gapped in generic directions away
from the degeneracy point and the topological charge distribution can be seen
in Fig. 1b. The two lower bands have topological charges of -2 each, while the
middle ones have 0 charge, which amounts to a total of -4 at half-filling, the
maximum topological charge of multifold fermions in the MSGs. In non-chiral
type II MSGs, inversion (or rotoinversion) symmetry pins two 3-folds with
opposite topological charge at the same $k$-point. In this case, the bands are
doubly degenerate in generic directions away from the degeneracy point and the
topological charge is 0 for all bands. These 6-folds are usually called spin-1
Dirac fermions, since the original Dirac fermion can be decomposed into two
spin-1/2 Weyl fermions of opposite charge. In type III MSGs, all 6-folds are
of spin-1 Dirac type, due to the presence of $\mathcal{I}\circ\mathcal{T}$
symmetry. In type IV MSG P${}_{I}a\bar{3}$ (205.36), the 6-fold is also spin-1
Dirac, but the extra symmetries in the little co-group force the parameters to
be pinned at an exactly solvable point in parameter space, similarly to what
happens to the 3-folds in type IV MSGs.

Figure 1: Topologically charged multifold fermion dispersions. a) 3-fold
spin-1, b) 6-fold double spin-1 and c) 4-fold RS fermion. The color code shows
the topological charge (Chern number) of each band.

It is natural to ask whether there are higher-dimensional effective
Hamiltonians of the form $H=\boldsymbol{k}\cdot\boldsymbol{S}$. An exhaustive
search of all MSGs revealed a 4-dimensional version of this Hamiltonian,
usually referred to as a Rarita-Schwinger (RS) or spin-3/2 fermion, due to its
analogy to high-energy physics111Technically, the RS fermion Hamiltonian is
not analytic in $k$ (Eq. (7) in Ref. cano2019multifold ) because the second
term contains $k^{2}$ in the denominator. The authors decided to rename only
the analytic part as RS∗ fermion, which is the Hamiltonian of the form
$H_{3/2}=\boldsymbol{k}\cdot\boldsymbol{S^{3/2}}$ with $S^{3/2}_{x,y,z}$ the
generators of the spin $s=3/2$ representation. cano2019multifold ;
liang2016semimetal ; tang2017multiple . In this case, the $S_{x,y,z}$ matrices
are the generators of the spin $s=3/2$ representation (also denoted as
$S^{3/2}_{x,y,z}$) and the resulting Chern numbers of the bands are $-3$,
$-1$, $1$ and $3$, respectively (see Fig. 1c). At half filling (two occupied
bands) these quasiparticles have maximum topological charge, $1+3=4$.
Noticeably, they can only be hosted in Sohnke MSGs, as rotoinversions pin two
copies with opposite charge at the same point. In Ref. BradlynEA17  the
authors list the type II MSGs where these multifolds can occur. However, there
has been no extensive search for spin-3/2 multifold fermions in type III or
type IV MSGs.

Analogous to the 3-fold, for certain material parameters the 4-fold
Hamiltonian is adiabatically connected to the
$H_{3/2}=\boldsymbol{k}\cdot\boldsymbol{S^{3/2}}$ Hamiltonian described
before, with Chern numbers -3,-1,1,3. This phase is realized in most known
materials with fourfold fermions (see Sec. III. However, the 4-fold
$\boldsymbol{k}\cdot\boldsymbol{p}$ Hamiltonian also admits phases not
adiabatically connected to $H_{3/2}$ BradlynEA17 . It was recently shown
Schnyder_other_multifold  that for certain parameter values the fourfold can
exhibit Chern numbers $-3,+5,-5,+3$ in the four bands. In the same work, they
propose BaAsPt as a material displaying this phase of the 4-fold fermion.
Finding experimental prove of a material that realizes this interesting phase
is an active area of study.

Similar to how (spin-1/2 or spin-1) Dirac fermions can be decomposed into two
(spin-1/2 or spin-1) Weyl fermions with opposite chirality, which are pinned
to the same energy and momentum by symmetry, certain 8-fold fermions can be
decomposed into two copies of 4-folds. The 8-fold degeneracy is the highest
possible in crystalline solids TQC . The 8-fold fermions hosted in cubic MSGs
are formed by two spin-3/2 4-fold fermions with opposite charge that are
pinned together by symmetry. The result is doubly degenerate bands in generic
directions and no net topological charge.

The 8-fold fermions in the tetragonal MSGs, though, are fundamentally
different. In that case, if inversion symmetry is removed as a group
generator, a 4-dimensional corep yields a fermion isomorphic to a spin-1/2
Dirac. Inversion symmetry doubles this corep to create the 8-fold fermion.
Thus, the tetragonal 8-fold can be thought of as a double spin-1/2 Dirac.

These two types of 8-fold fermions – the doubled spin-3/2 fermion and doubled
spin-1/2 Dirac – are the only types of 8-fold fermions protected by
crystalline symmetry. The two of them have in common that all bands are doubly
degenerate in arbitrary directions from the node and have 0 topological
charge.

We now turn to the type III and type IV MSGs. In type III MSGs the same
results apply, since a type III MSG with an 8-fold degeneracy is a subgroup of
a type II MSG with that same 8-fold degeneracy. In type IV MSGs, all 8-folds
are of the double spin-1/2 Dirac form. However, similar to the 3-fold and
6-fold fermions in type IV MSGs, the little cogroup symmetries pin the
Hamiltonian to a particularly simple form. In this case, the result is 8-fold
fermions that are perfectly isotropic double spin-1/2 Dirac fermions. In Table
1 we show all MSGs that can host 8-fold fermions.

##  III Material realizations

Figure 2: Crystal structures and electronic bands of non-magnetic and magnetic
materials displaying multifold fermions. a) Pd3Bi2S2, a chiral material
displaying a 3-fold fermion at the P point Alltopo ; TQC_databse . b)
Elemental Cs, in non-Sohnke MSG P$4_{2}$/mbc1’, displaying an 8-fold fermion
close to the Fermi level at A. c) Mn3IrSi in MSG P213 with chiral magnetic
ordering displays 3-fold spin-1 fermions at R. d) PtGa in Sohnke MSG P2131’,
displaying the maximum charge RS 4-fold at $\Gamma$ and double spin-1 6-fold
at R. e) ARPES measurements reported in Ref. PtGa_exp  showing the bulk band
dispersion of both multifold fermions. f) Surface ARPES measurements showing
Fermi arcs and their spin polarization. g) PdSb2 in non-Sohnke MSG Pa-31’
presenting a spin-1 Dirac 6-fold fermion at R. h) ARPES measurements reported
in Ref. 6fold_Dirac_exp_PdSb2  showing the bulk dispersion of the spin-1 Dirac
6-fold (extracted from Ref. 6fold_Dirac_exp_PdSb2 ). Notice that ARPES
measurements for PdSb2 were performed at the VUV range 6fold_Dirac_exp_PdSb2 ,
while they were performed at the SX range for PtGa PtGa_exp .

In this section we will describe the material realisations of the multifolds
described in previous sections.

###  III.1 Non-magnetic materials

The theoretical prediction of multifold fermions led to a huge advance in
material realisations, particularly in nonmagnetic systems. 3-fold fermions
have been reported in chiral Pd3Bi2S2 in SG I$2_{1}3$1’ (199.13) Alltopo ;
TQC_databse  (see Fig. 2a). 4-fold and 6-fold double-spin-1 fermions can be
found in the B20 material family in the presence of SOC. Materials such as the
previously mentioned (Rh,Co)(Si,Ge) TMSi_multis , AlPt AlPt , PdGa PdGa  and
PtGa PtGa  (see Fig. 2d) are excellent examples of materials hosting these
fermions, with extensive theoretical predictions and experimental confirmation
of topological properties such as very long Fermi arcs connecting the surface
projections of the multifold fermions. 6-fold Dirac fermions have been
reported in non-chiral PdSb2 6fold_Dirac_exp_PdSb2  (see Fig. 2g and Fig. 2h),
PtBi2 6fold_Dirac_exp_PtBi2 , Rb4O6 6fold_Dirac_theo_Rb4O6 , Li${}_{1}2$Mg3Si4
6fold_Dirac_theo_Li12Mg3Si4  and SrGePt, which also displays a 4-fold fermion
multiSrGePt .

Finally, 8-fold fermions have also been predicted and found in non-magnetic
materials. The 8-fold double-Dirac fermions can exist in seven type II SGs and
were predicted in Bi2AuO5 in MSG P4/ncc1’ (130.424) 8fold_theo_Bi2AuO5 .
Elemental Cs crystals in MSG P$4_{2}$/mbc1’ (135.484) also host an 8-fold
crossing close to the Fermi level TQC_databse ; Alltopo  (see Fig. 2b).

###  III.2 Magnetic materials

Multifolds in magnetic materials are inherently harder to find, due to the
relative scarcity of high quality magnetic materials’ crystal and magnetic
structures reported in the literature.

The most notable example is that of the Mn3IrSi family (Mn3IrGe, Mn3
Ir1-yCoySi, and Mn3CoSi1-x Gex) with type I MSG P$2_{1}3$ (198.9), which
displays a rare example of chiral magnetism. It was predicted to host 3-fold
fermions at the R point below the Néel temperature cano2019multifold  (see
Fig. 2c). Above the critical temperature, the magnetic order is destroyed and
we recover the full gray group P$2_{1}3$1’ (198.10), in which the 3-folds are
forced to occur in pairs due to TRS.

Following recent advances in the development of high-quality databases for
magnetic materials (for instance, MAGNDATA MAGNDATA ), we are now able to
predict several materials in type III MSGs that can host 3-fold fermions, such
as BaCuTe2O6 in MSG P$4_{1}^{\prime}32^{\prime}$ (213.65) at the $R$ point
3fold_TypeIII  and the family of La3X5O${}_{1}2$ (where La is a lanthanide and
X=Ga,Al) in MSG I$a\bar{3}d^{\prime}$ (230.148) at the $P$ point
3fold_TypeIII_2 ; 3fold_TypeIII_3 ; 3fold_TypeIII_4 ; 3fold_TypeIII_5 . 6-fold
fermions can be found in the TRh (T=Dy, Ho, Er) family of compounds in MSG
$P_{C}2_{1}/m$ (11.57)6fold_TypeIV , as well as in Eu3PbO with MSG
$P_{I}a\bar{3}$ (205.36)6fold_TypeIV_2 , both type IV MSGs. Finally, we
predict 8-fold fermions in BaNd2ZnS5, with MSG $P_{C}4/nnc$ (126.385)
8fold_TypeIV .

###  III.3 Experimental evidence

There are two fundamental experimental signatures of multifold fermions: their
bulk band dispersion and Fermi arc surface states. Both can be measured by
means of angle-resolved photoemission spectroscopy (ARPES). Due to the short
inelastic mean free path photoelectrons in solids of $\lambda\approx
0.5\text{nm}$ when excited with extreme ultraviolet light (photon energies of
$\approx$ 10-150 eV), this technique has been very successful in imaging
topological surface states ARPES1 ; ARPES2 . In particular, it has been used
to visualize the predicted long Fermi arcs connecting the projection of the
4-fold and 6-fold fermions in the B20 material family PtGa ; PtGa_exp ;
RhSi_exp ; RhSi_multi ; CoSi_exp  (see Fig. 2f). These multifold fermions were
predicted to carry the maximum topological charge of four, requiring four
Fermi arcs to emerge from the surface projection of the multifolds. The four
Fermi arcs can be observed in materials where the SOC is large enough to lead
to a sizable lifting of the spin-degeneracy, for example, close to the
$\Gamma$ point in PtGa PtGa_exp  or in PdGa PdGa_exp . When SOC is small, it
is difficult to resolve the spin-splitting, as in CoSi CoSi_exp .

To study the bulk electronic structure of solids with ARPES, higher photon
energies in the soft X-ray (SX) regime (approx. 150-1500 eV) need to be
employed. Here, the inelastic mean free path can typically extend to multiple
unit cells, which reduces the effect of momentum broadening in the out of
plane direction strocov2003intrinsic . Soft X-ray ARPES is therefore more
suitable to extract information about the three-dimensional bulk band
structure. An example of this can be seen in Fig. 2e, which shows the bulk
dispersion of 4-fold and 6-fold multifold fermions in PtGa PtGa_exp  in ARPES
and as predicted by DFT (Fig. 2d), as well as in Fig. 2h, which shows the bulk
bands of the 6-fold Dirac in PdSb2. Notice that for the visualization of the
sixfold in PdSb2 in Fig.42, lower photon energies in the Vacuum-ultraviolet
(VUV) range were employed 6fold_Dirac_exp_PdSb2 .

Even though most of the experimental research has been focused on materials in
the B20 family RhSi_exp ; CoSi_exp ; PdGa_exp ; RhSi_exp_2 ; CoSi_exp_2 ;
CoSi_exp_3 ; CoSi_exp_4 ; CoSi_exp_5 ; CoSi_exp_6 ; FeSi_exp ; RhSi_exp_3 ;
CoSiRhSi_exp , PdBiSe, also in MSG P$2_{1}3$1’ (198.10), exhibits 4-fold and
6-fold fermions which have been experimentally observed PdBiSe_exp . In non-
chiral crystals, 6-fold Dirac fermions have been recently found in cubic SG
Pa$\bar{3}$1’ (205.34) in PdSb2 6fold_Dirac_exp_PdSb2 ;
6fold_Dirac_exp_PdSb2_2 ; 6fold_Dirac_exp_PdSb2_3  and in PtBi2
6fold_Dirac_exp_PtBi2 . Finally, ‘practical’ 8-fold fermions have been
reported in TaCo2Te28fold_exp_TaCo2Te2  in MSG Pnma1’ (62.442) without SOC.

##  IV Computational methods

The calculations presented in Figure 2 have been computed for the purpose of
this review based on the structural parameters reported in their respective
publications Alltopo ; AlPt ; 6fold_Dirac_exp_PdSb2 ; TQC_databse ;
cano2019multifold . We performed density functional theory (DFT) calculations
as implemented in the Vienna Ab Initio Simulation Package (VASP) VASP1 ; VASP2
; VASP3 ; VASP4 . The interaction between the ion cores and valence electrons
was treated by the projector augmented wave (PAW) method PAW , the generalized
gradient approximation (GGA) was employed for the exchange-correlation
potential with the Perdew–Burke–Ernzerhof for solid parameterization PBE  and
the SOC was considered on the second variation method DFT-SOC . We used a
$\Gamma$-centered Monkhorst-Pack of $9\times 9\times 9$ k-point grid for
reciprocal space integration and we used a 500eV energy cutoff for plane wave
expansion. The electronic ground state was self-consistently converged with an
accuracy of $10^{-5}$ eV/unit cell.

##  V Conclusion

This work has reviewed the recent progress in material realizations of
multifold fermions from a three-pronged perspective, namely, symmetry
arguments, DFT predictions of realistic materials, and experimental
realizations. In addition to the basic bulk and surface properties described
here, multifold fermions have been predicted to display other interesting
physical responses, such as second harmonic generation in B20 RhSi
Second_Harmonic_RhSi , quantized circular photogalvanic effect in multifold
fermions CPGE , non-linear Hall conductivity in CoSi Nonlinear_Hall_CoSi  and
a new hydrodynamic responses in the B20 family 3fold_Hallvisco . There has
also been research on unconventional superconductivity, with an unusual s-wave
triplet pairing proposed in topological semimetals hosting 3-fold fermions
Spin1_superconduct . Finally, the robustness of multifold fermions to disorder
is an important question that deserves future study, as Fermi arcs from 3-fold
fermions retain their sharpness at weak disorder 3fold_disorder .

With the rapid growth of material predictions in recent years, we expect to
see more experimental realizations of these exotic multifold fermions in the
future. In particular magnetic systems have recently been the focus of many
high-throughput searches MagneticHT ; HighThrougput_collinear . These new
materials will prove to be excellent platforms to study multifold fermions and
their phenomenology.

##  VI Acknowledgements

M.G.V. and C.F. thank support from the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) GA3314/1-1 -FOR 5249 (QUAST). M.G.V and I.R. thank
support to the Spanish Ministerio de Ciencia e Innovacion grant
PID2022-142008NB-I00 and the Ministry for Digital Transformation and of Civil
Service of the Spanish Government through the QUANTUM ENIA project call -
Quantum Spain project, and by the European Union through the Recovery,
Transformation and Resilience Plan - NextGenerationEU within the framework of
the Digital Spain 2026 Agenda. This project was partially supported by the
European Research Council (ERC) under the European Union’s Horizon 2020
Research and Innovation Programme (Grant Agreement No. 101020833). B.B.
acknowledges the support of the Alfred P. Sloan foundation, and the National
Science Foundation under grant DMR-1945058. J.C. acknowledges the support of
the Alfred P. Sloan foundation, the National Science Foundation under grant
DMR-1942447, and the Flatiron Institute, a division of the Simons Foundation.
N.B.M.S. was funded by the European Union (ERC Starting Grant ChiralTopMat,
project number 101117424). Views and opinions expressed are however those of
the author(s) only and do not necessarily reflect those of the European Union
or the European Research Council Executive Agency. Neither the European Union
nor the granting authority can be held responsible for them. C.F. was
financially supported by Deutsche Forschungsgemeinschaft (DFG) under SFB1143
(Project No. 247310070) and Würzburg-Dresden Cluster of Excellence on
Complexity and Topology in Quantum Matter—ct.qmat (EXC 2147, project no.
390858490).

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