# Zero, Finite, and Infinite Belief History of Theory of Mind Reasoning in
Large Language Models

Weizhi Tang
University of Edinburgh
<EMAIL_ADDRESS>
Vaishak Belle
University of Edinburgh
<EMAIL_ADDRESS>

###### Abstract

Large Language Models (LLMs) have recently shown a promise and emergence of
Theory of Mind (ToM) ability and even outperform humans in certain ToM tasks.
To evaluate and extend the boundaries of the ToM reasoning ability of LLMs, we
propose a novel concept, taxonomy, and framework, the ToM reasoning with Zero,
Finite, and Infinite Belief History and develop a multi-round text-based game,
called Pick the Right Stuff, as a benchmark. We have evaluated six LLMs with
this game and found their performance on Zero Belief History is consistently
better than on Finite Belief History. In addition, we have found two of the
models with small parameter sizes outperform all the evaluated models with
large parameter sizes. We expect this work to pave the way for future ToM
benchmark development and also for the promotion and development of more
complex AI agents or systems which are required to be equipped with more
complex ToM reasoning ability111Our source code of the study and benchmark is
open source and available at https://anonymous.4open.science/r/Pick-the-Right-
Stuff-B1F0..

Zero, Finite, and Infinite Belief History of Theory of Mind Reasoning in Large
Language Models

Weizhi Tang University of Edinburgh<EMAIL_ADDRESS>Vaishak Belle
University of Edinburgh<EMAIL_ADDRESS>

##  1 Introduction

Theory of Mind (ToM) is the ability to understand, attribute, and infer the
mental states of oneself and others to be able to represent how others
perceive, interpret, reason, and respond to the surrounding world (Baron-Cohen
et al., 1985; Premack and Woodruff, 1978; Quesque and Rossetti, 2020). In
order for individuals to understand and reason about others’ beliefs,
intention, desires, emotions, obligations, and so on, the ability of ToM is
essential (Premack and Woodruff, 1978; Frith and Frith, 2006; Quesque and
Rossetti, 2020). And with the emergence and consistent advancement of Large
Language Models (LLMs), they have demonstrated impressive and near-human-level
capabilities in various tasks (Naveed et al., 2024; Wan et al., 2024; Wang et
al., 2023b; Chang et al., 2023) and some studies have shown a promise and even
an emergence of ToM reasoning ability in LLMs (Strachan et al., 2024;
Kosinski, 2023; Jamali et al., ; Moghaddam and Honey, ). Furthermore, since
the essence of ToM is understanding and representing the mental states of
others, for LLMs within its frequently mentioned and used situation, the
question and answering (QA), correctly understanding the mental states, such
as the intention of the users, is necessary and the key to efficiently
generate appropriate, meaningful, and beneficial responses. Additionally, not
only in the QA situation, LLMs are mentioned and used in various other
scenarios, such as using LLMs as embodied agents (Zhang et al., 2024; Liu et
al., 2023; Gallotta et al., 2024; Wang et al., 2023a), understanding the
beliefs of other agents in environments is essential to act in a beneficial,
safe, and helpful way for both themselves and other agents. Therefore, further
exploration of the ToM ability of LLMs is necessary and meaningful.

In order to understand the ToM ability of LLMs, various benchmarks have been
developed in different ways to handle different evaluation perspectives of the
ToM ability of LLMs in previous work, such as that MindGames leverages dynamic
epistemic logic to generate complex ToM problems including the k-order ToM
reasoning and different problem contexts (Sileo and Lernould, 2023), Hi-ToM
introduces a complex k-order ToM problems based on Sally-Anne-like stories,
which further contain communications between agents (Wu et al., 2023), and
FANToM introduces the ToM benchmark in a manner of interactions of agents and
aligns the ToM problems more with real-world scenarios requiring ToM reasoning
(Kim et al., 2023).

In this study, in order to further understand the ToM ability of LLMs, besides
providing a text-based and multi-round game as a new benchmark to evaluate the
ToM reasoning ability of LLMs in a complex and dynamic way, we have extended
the current boundaries of ToM ability evaluation range by introducing a novel
concept, taxonomy, and framework of ToM evaluation, which is the ToM reasoning
with Zero, Finite, and Infinite Belief History. This concept and taxonomy aim
to provide a clear idea and structure for evaluating a specific aspect of ToM,
the beliefs, in a more advanced and nuanced manner.

We give a brief explanation of the concept here and discuss it in more detail
in Section 3. The Zero Belief History means that the test subject can identify
the latest beliefs of others without needing to reason from the available or
known belief history while the Finite Belief History means that the test
subject should and have to utilize and reason with the known finite belief
history to distinguish and identify the latest beliefs of others. In addition,
since recent ToM benchmarks introduce more complex ToM problems which are even
not comfortable for humans, such as the high k-order ToM problems mentioned by
Sileo and Lernould (2023) and Wu et al. (2023), and because Strachan et al.
(2024) shows LLMs, especially GPT-4 (OpenAI et al., 2024), sometimes can even
outperform humans in ToM problems, we propose to not make the ToM evaluation
of LLMs be constrained in human-level but to extend the boundaries to more
complex ToM problems so that we may have confidence to put LLMs in situations
which need more complex ToM reasoning. Thus, while we consider Zero and Finite
Belief History to be reasonably solvable for humans, in addition to that, we
further introduce the Infinite Belief History in which the test subject should
maintain an infinite belief history, typically in the way of implicit
patterns, and propose to discover and study it in future work.

In this study, we explain Zero, Finite, and Infinite Belief History in detail
with examples. In addition, we have developed a game called Pick the Right
Stuff as both a proof-of-concept benchmark and example for future work and an
out-of-box benchmark to evaluate the ToM ability of LLMs in the case of Zero
Belief History and Finite Belief History. We have also evaluated 6 LLMs, in
which there are 3 models in large parameter sizes and 3 models in small
parameter sizes. Unsurprisingly, all LLMs perform better under the condition
of the Zero Belief History than the Finite Belief History case. In addition,
an unexpected and supervising finding is that two of LLMs with the small
parameter sizes outperform all the three LLMs with large parameter sizes in
both of Zero Belief History and Finite Belief History case, which may further
bring up evidence and an answer to the question, whether increasing the sizes
of the parameter of LLMs is necessary and effective to enhance model
capabilities, at least in the context of ToM. Our overall contributions are
listed as follows:

  1. 1.

We propose a novel concept, taxonomy, and framework for evaluating ToM ability
of LLMs.

  2. 2.

Along with the concept we proposed, we have developed a multi-round text-based
game as a new ToM benchmark to be both a proof-of-concept and example
benchmark for future benchmark development and an out-of-box benchmark to
evaluate the ToM ability of LLMs.

  3. 3.

We have shown that LLMs perform better in the case of Zero Belief History than
in the case of Finite Belief History and also shown that LLMs in small
parameter sizes can even outperform LLMs in large parameter sizes when
tackling ToM problems.

##  2 Related Work

###  2.1 ToM in LLMs

Although there has been debating around whether the ToM ability is really
emerged in LLMs among previous work (Kosinski, 2023; Jamali et al., ; Sap et
al., 2023; Trott et al., 2023; Bubeck et al., 2023), recent work has shown a
hard-to-deny promise of emergence of ToM in LLMs (Strachan et al., 2024;
Kosinski, 2023; Jamali et al., ), especially in the study of Strachan et al.
(2024) which compares GPT-4 (OpenAI et al., 2024) and Llama 2 (Touvron et al.,
2023) with human performance and demonstrates their superior performance
compared to humans in certain tasks such as false beliefs and misdirection
while suggesting that GPT’s poor performance stems from a hyperconservative
approach to committing to conclusions rather than from a genuine failure of
inference. In this work, instead of focusing on discussing and debating the
emergence of ToM in LLMs, we propose a novel concept and taxonomy of an
evaluation direction of the ToM ability of LLMs and provide a clear framework
for evaluating the ToM ability of LLMs in this direction.

###  2.2 ToM Benchmarks

The benchmarks for evaluating ToM ability of LLMs are recently in active
development such as MindGames (Sileo and Lernould, 2023), FANToM (Kim et al.,
2023), Hi-ToM (Wu et al., 2023), ToMChallenges (Ma et al., 2023a), SymmToM
(Sclar et al., ), ToMi (Le et al., 2019), and so on. Each of them is proposed
and developed to enhance the complexity or diversity of the ToM evaluation. In
addition, Ma et al. (2023b) has proposed to taxonomize machine ToM into
individual components such as belief, intentions, desires, emotions, percepts,
and more, and also suggested treating LLMs as agents that are physically
situated in environments and socially engaged in interactions with humans or
intelligent agents. In this paper, we propose to further extend the evaluation
of beliefs in terms of Zero, Finite, and Infinite Belief History and develop a
benchmark as a game in which an LLM is a player to play the multi-round text-
based game for evaluating its ToM reasoning ability.

##  3 ToM Reasoning with Zero, Finite, and Infinite Belief History

###  3.1 Zero Belief History

Figure 1: An example of Zero Belief History in which the blue shadowed block
represents the belief history which is not necessary, the red shadowed block
represents a gap of beliefs between the LLM and Agent A.

The ToM reasoning with Zero Belief History refers to a scenario where LLMs can
directly search and find associated information from the context to perform
ToM reasoning to directly distinguish and identify the beliefs of others
without requiring any additional reasoning with prior knowledge and beliefs
including but not limited to leveraging social rules, culture context,
mathematical reasoning, and so on. For example, consider three agents
discussing a certain topic in a room. If one agent leaves the room, the
remaining two agents understand that the agent who left is unaware of any new
content they will be discussing.

We have also demonstrated an example in Figure 1. The gap depicted in a red
shadow block represents the belief difference between the LLM and Agent A. In
order for the LLM to identify the belief of Agent A, the LLM only needs to
find associated information provided in the context which is the information
of the gap, without needing to use a belief history represented as a blue
shadow block.

###  3.2 Finite Belief History

Figure 2: An example of Finite Belief History in which the blue shadowed block
representing a finite belief history is necessary for the LLM to reason with
to identify the latest belief of the Agent A.

In contrast to the ToM reasoning with the Zero Belief History, under the
condition of Finite Belief History, in addition to searching and finding
available and associated information from the context, due to the information
is not sufficient to enable it to directly identify the latest beliefs of
others, it needs and has to utilize a belief history and reason with it in a
certain way, such as levering mathematical reasoning, to be able to correctly
identify the latest beliefs of others. In addition, a worth noting is that, in
this case, the LLM only needs to reason with a finite belief history, such as
a limited number and complexity of cultural background information, social
rules, or mathematical formulations.

As an example shown in Figure 2, besides the condition of Zero Belief History
demonstrated in Figure 1, the LLM needs to utilize and reason with a finite
belief history represented in the blue shadowed block to identify the correct
latest belief of Agent A.

###  3.3 Infinite Belief History

As a complex variant of the Finite Belief History, under the condition of
Infinite Belief History, the only difference is the LLM needs and has to
reason with an infinite belief history and everything else remains the same as
in the Finite Belief History. An infinite belief history is typically
implicitly held by the LLM in a certain pattern such as a function which
represents the beliefs of the possible responses and actions towards different
situations of environments in which since there could be infinite situations,
responses, and actions, and the generated belief history could not be logged
finitely, this function can be as a pattern to represent an infinite belief
history.

As depicted in Figure 3, the only difference between the Finite Belief History
and Infinite Belief History is that the belief history in the case of the
Infinite Belief History colored in the blue shadowed block is infinite rather
than finite and the LLM needs and have to utilize and reason with it to
identify the latest belief of Agent A.

Figure 3: An example of Infinite Belief History in which the LLM needs to
leverage an infinite belief history represented in the blue shadowed block to
reason with to identify the latest belief of the Agent A.

##  4 Methodology

###  4.1 Pick the Right Stuff

In order to evaluate the ToM reasoning ability of LLMs with each type of
belief history, we develop a multi-round text-based game as a benchmark,
called Pick the Right Stuff, in which an LLM plays as a warehouse manager
required to correctly pick and deliver the right stuff belonging to its
original owner in the condition that the position of the stuff is not the same
as its original position while the users of the warehouse do not know. This
game focuses on testing ToM reasoning in the cases of Zero and Finite Belief
History, while leaving the Infinite Belief History as the future work. We
expect this benchmark not only as an out-of-box benchmark for ToM evaluation
but also as a proof-of-concept example for future benchmark development along
the concept and taxonomy of Zero, Finite, and Infinite Belief History.

####  4.1.1 Game Logic for Zero History Belief

Figure 4: The game logic of Pick the Right Stuff under the condition of Zero
Belief History in which there are two rooms, (1) Room 1 is the Locker Room
containing an opaque locker used for storing the items of the users, and (2)
Room 2 is the Monitoring Room where the LLM is located.

The game, Pick the Right Stuff, for the case of Zero Belief History, operates
under the following logic222An example of the game instance under the
condition of Zero Belief History is shown in Appendix A.1..

There are two rooms in the game, Room 1 and Room 2. In Room 1, there is an
opaque locker where the items are stored. The positions of the items in this
opaque locker are randomly shuffled and reset throughout the game. Room 2
contains a monitor that allows observation of all items in the opaque locker
in Room 1. An LLM is situated in this room, meaning it can observe and know
the positions and owners of items at all times.

At the onset of the game, users place their items in Room 1 and both they and
the LLM are aware of the initial positions of the items. During the game, some
users may randomly enter Room 2 to view the monitor and they may enter the
room and view the monitor more than once times, and the LLM is also able to
observe the users since in the same room. Furthermore, users may enter Room 1
at any time and expect to retrieve their items. The LLM needs to think about
and provide the positions of the items as believed by the users, so that the
system can automatically swap the real items with those at the given
positions. If the swap is correct, the user successfully retrieves their item
and the LLM scores a point. If incorrect, the LLM scores no points and the
system will automatically intervene to ensure the user successfully obtains
their item. After retrieval, the position of the item becomes vacant, and the
retrieving user will leave. The game will continue until all positions are
empty. We have also demonstrated the game logic in Figure 4.

In this game under the condition of Zero Belief History, the users’ beliefs
are updated when they place their items at the onset of the game and when they
enter Room 2 to observe the positions of the items, which means their beliefs
are updated dynamically and may be changed multiple times during the game. The
LLM can always know the correct positions of the items and is required to
identify the false beliefs of the users in order to correctly pick and deliver
the correct items to their owners to score points. Additionally, it is
considered under the condition of Zero Belief History because the LLM does not
need to identify the latest beliefs of the users based on a belief history but
can utilize the available and associated information around the context which
is whether the users enter the Room 2 to view the monitor to determine the
latest beliefs of the users.

####  4.1.2 Game Logic For Finite Belief History

Figure 5: The game logic of Pick the Right Stuff under the condition of Finite
Belief History in which there are three rooms, (1) Room 1 is the Locker Room
containing an opaque locker, (2) Room 2 is the Monitoring Room where the LLM
is located and enables the LLM to know the situations from both of Room 1 and
Room 3, and (3) Room 3 is the Snapshots Room which shows a certain state of
positions of items to users.

Under the condition of Finite History Belief, the game logic is similar to
that of under the condition of Zero History Belief333An example of the game
instance under the condition of Finite Belief History is shown in Appendix
A.2.. The key difference lies in the addition of a third room, Room 3. In this
room, there is a screen that randomly shows a snapshot representing a random
previous state of the monitor located in Room 2 which is the previous state of
the positions of items. Instead of entering Room 2 to observe the current
monitor, some users may randomly enter Room 3 to view the snapshot on the
screen during the game. In addition, the LLM located in Room 2 is only
informed about the specific previous state relative to the current state in an
indirect way such as "nth-to-last" instead of directly being presented.
Therefore, the LLM must utilize and reason with a finite belief history to
identify the latest beliefs of the users. The game logic under the condition
of Finite Belief History is shown in Figure 5.

###  4.2 Experiment Settings

In order to have a comprehensive view of the ToM reasoning ability of LLMs
under Zero and Finite Belief History, we have conducted evaluations with six
Large Language Models, comprising three models with large parameter sizes and
three models with small parameter sizes.

For the models with large parameter sizes, we choose GPT-3.5-Turbo noted as
gpt-3.5-turbo, Llama 3 for which we select its instruct model in 70 billion
parameter size noted as llama3:70b-instruct, and Qwen (Bai et al., 2023) for
which we use its chat model in 72 billion parameter size noted as
qwen:72b-chat.

For the models with small parameter sizes, we select Gemma (Team et al., 2024)
for which we use its instruct model in 7 billion parameter size noted as
gemma:7b-instruct, Mistral (Jiang et al., 2023) for which we choose its
instruct model in 7 billion parameter size noted as mistral:7b-instruct, and
Qwen (Bai et al., 2023) for which we use its chat model in 7 billion parameter
size noted as qwen:7b-chat.

We then fix the temperature to 0 for all the selected LLMs to ensure the
consistency of the responses. And we also fix the random seed of the game to
ensure each model faces the same game states and questions. In addition, we
run each model to play 60 turns of the game in the case of 5 users444Each turn
consists of multiple ToM questions..

##  5 Results and Analyses

The average scores of the gpt-3.5-turbo, llama3:70b-instruct, qwen:72b-chat,
gemma:7b-instruct, mistral:7b-instruct, and qwen:7b-chat playing the game,
Pick the Right Stuff, under the conditions of Zero and Finite Belief History
are depicted in Figure 6.

For all the evaluated models, the average score under the condition of the
Zero Belief History is 32.06, while it is 26.33 under the condition of the
Finite Belief History, indicating that although LLMs show an impressive
promise and performance on ToM reasoning, they still struggle with it.

In addition, from the difference between the two conditions, which is 5.73,
and also as shown in Figure 6, the performance of LLMs under the condition of
Finite Belief History is consistently lower than the performance under the
condition of Zero Belief History, which indicates that the Finite History
Belief case is indeed harder for them than the Zero History Belief.

Figure 6: The average scores of each model for playing the game under the
condition of Zero or Finite Belief History.

Furthermore, among the selected LLMs, the model with small parameter size,
gemma:7b-instruct, emerged as the best-performing, achieving an average score
of 43.00 under the Zero Belief History condition and 34.33 under the Finite
Belief History condition, thereby outperforming all the other LLMs. In
addition, mistral:7b-instruct also shows better performance over all the other
LLMs with large parameter sizes, like gpt-3.5-turbo. Although there may be
differences in the training data and architectures leveraged by these models,
the outstanding performance of the models with smaller parameter sizes
presents evidence and an answer to a thought-provoking question, whether
increasing model parameter size may effectively enhance LLMs capabilities, at
least discussed in the context of ToM reasoning ability.

##  6 Implications

We propose a novel concept, taxonomy, and framework to evaluate the ToM
reasoning ability of LLMs under the conditions of Zero, Finite, and Infinite
Belief History. Although we focus primarily on LLMs as the subjects of our
evaluation, this concept and evaluation approach is equally applicable to
other AI agents and systems.

Additionally, since the essence and necessity of complex ToM reasoning ability
in many applications such as autonomous driving, complex multi-actor story
writing, embodied agents under the multi-agent case, and more, we expect this
work to pave the way for more intelligent AI agents or systems which are
required to equipped with more complex ToM reasoning.

##  7 Conclusion

In this study, we propose a novel concept, taxonomy, and framework to extend
the boundaries of evaluation of the ToM reasoning ability of LLMs, by
introducing Zero, Finite, and Infinite Belief History. In addition, we develop
a multi-round text-based game as both of a proof-of-concept and example
benchmark for future benchmark development along the concept proposed and an
out-of-box benchmark to evaluate the ToM reasoning ability of LLMs under the
conditions of Zero and Finite Belief History. We choose six LLMs of which
three of them are in large parameter sizes and three of them are in small
parameter sizes. We have demonstrated the evaluated LLMs perform better under
the condition of Zero Belief History than under the condition of Finite Belief
History. In addition, we have also found that two of the models with small
parameter sizes outperform all the selected models with large parameter sizes.
We hope this work can pave the way for future benchmark development and also
the development of more complex AI agents and systems which are required to
tackle more complex ToM problems.

##  8 Limitations

Due to this study focusing on proposing a novel evaluation concept of ToM
reasoning rather than presenting an exhaustively comprehensive comparison of
all available LLMs and since the unavoidable heavy computation costs incurred
by GPT-4, we do not run an evaluation of GPT-4 and other LLMs. Nevertheless,
future work can leverage this out-of-box game to evaluate GPT-4 and other
LLMs.

In addition, the game we developed does not cover all possibilities and
variants under the conditions of Zero and Finite Belief History such as
replacing mathematical reasoning in the case of Finite Belief History with
social rules or cultural background. We expect the future work can work along
it to develop various benchmarks to evaluate ToM reasoning ability under the
conditions of Zero, Finite, and Infinite Belief History with different kinds
of belief history to prompt comprehensive evaluation.

Furthermore, since presenting implementation examples of Zero and Finite
Belief History is sufficient to express and explain the concept and due to
that the LLMs still struggle with handling ToM reasoning in the game under the
conditions of the Zero and Finite Belief History, we leave the implementation
of Infinite Belief History for future work.

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##  Appendix A Appendix

###  A.1 Game Instance of Zero Belief History

Listing 1: Example of the Pick the Right Stuff game under the condition of
Zero Belief History.

⬇

Welcome to, Pick the Right Stuff!

In this game, you will play the role of a warehouse manager. The warehouse
contains two rooms. Room 1 is used for storing items, with each item stored in
a certain position inside the opaque locker. You are situated in the Room 2,
which contains a monitor that allows you to see the content of the opaque
locker located in the Room 1 through the camera inside the opaque locker. Due
to malfunctions in the locker system, it randomly resets the positions of the
items in the opaque locker from time to time. To ensure that each user
retrieves their stored item correctly, when a user comes to retrieve an item,
you are required to predict the position of the item the user believes (the
user will always retrieve their item based on the position they last
believed). You only need to tell the system which position inside the locker
the user will go to retrieve their item and then the locker system will
automatically swap the item at that location with the one belonging to the
user. During the game, users may or may not enter the Room 2 to observe the
monitor. By observing the monitor, users will update their beliefs about the
position of their item.

If a user successfully retrieves their item, you score a point and the item is
removed from the locker.

If a user retrieves the wrong item, the item is returned, the user contacts
the system administrator to take the correct item, and you score no points.

Indeed, this is a problematic locker system, but you are hoped to be an
excellent warehouse manager!

============

Game Begins!

There are 3 users. User 0 stores its item at the position 0th of the locker.
User 1 stores its item at the position 1st of the locker. User 2 stores its
item at the position 2nd of the locker.

Now they leave the room.

The locker is malfunctioning and randomly resetting the positions of the items
in the locker...

The locker has returned to normal.

From the monitor, you can see the content of the locker:

The position 0th stores the item of User 1.

The position 1st stores the item of User 2.

The position 2nd stores the item of User 0.

The locker is malfunctioning and randomly resetting the positions of the items
in the locker...

The locker has returned to normal.

From the monitor, you can see the content of the locker:

The position 0th stores the item of User 2.

The position 1st stores the item of User 0.

The position 2nd stores the item of User 1.

User 1 is coming to Room 1 to take his/her item...

You should only answer the position of the item the user will go to retrieve
their item (e.g. 0 for the 0th, 1 for the 1st, 2 for the 2nd...).

For example, if you think the user will go to position 0th to retrieve their
item, you should only answer in single number ’0’.

Please make your prediction:

1

Your prediction is correct! Item in the position 1st is exchanged with the
correct item in the position 2nd. User 1 successfully retrieved the item from
the correct position. You score a point!

Now, from the monitor, you can see the content of the locker:

The position 0th stores the item of User 2.

The position 1st box is empty.

The position 2nd stores the item of User 0.

User 0 walks into the Room 2 and leaves the room after observing the monitor.

The locker is malfunctioning and randomly resetting the positions of the items
in the locker...

The locker has returned to normal.

From the monitor, you can see the content of the locker:

The position 0th stores the item of User 0.

The position 1st stores the item of User 2.

The position 2nd box is empty.

The locker is malfunctioning and randomly resetting the positions of the items
in the locker...

The locker has returned to normal.

From the monitor, you can see the content of the locker:

The position 0th stores the item of User 2.

The position 1st box is empty.

The position 2nd stores the item of User 0.

The locker is malfunctioning and randomly resetting the positions of the items
in the locker...

The locker has returned to normal.

From the monitor, you can see the content of the locker:

The position 0th box is empty.

The position 1st stores the item of User 0.

The position 2nd stores the item of User 2.

User 0 is coming to Room 1 to take his/her item...

You should only answer the position of the item the user will go to retrieve
their item (e.g. 0 for the 0th, 1 for the 1st, 2 for the 2nd...).

For example, if you think the user will go to position 0th to retrieve their
item, you should only answer in single number ’0’.

Please make your prediction:

2

Your prediction is correct! Item in the position 2nd is exchanged with the
correct item in the position 1st. User 0 successfully retrieved the item from
the correct position. You score a point!

Now, from the monitor, you can see the content of the locker:

The position 0th box is empty.

The position 1st stores the item of User 2.

The position 2nd box is empty.

The locker is malfunctioning and randomly resetting the positions of the items
in the locker...

The locker has returned to normal.

From the monitor, you can see the content of the locker:

The position 0th box is empty.

The position 1st box is empty.

The position 2nd stores the item of User 2.

The locker is malfunctioning and randomly resetting the positions of the items
in the locker...

The locker has returned to normal.

From the monitor, you can see the content of the locker:

The position 0th stores the item of User 2.

The position 1st box is empty.

The position 2nd box is empty.

The locker is malfunctioning and randomly resetting the positions of the items
in the locker...

The locker has returned to normal.

From the monitor, you can see the content of the locker:

The position 0th box is empty.

The position 1st box is empty.

The position 2nd stores the item of User 2.

User 2 is coming to Room 1 to take his/her item...

You should only answer the position of the item the user will go to retrieve
their item (e.g. 0 for the 0th, 1 for the 1st, 2 for the 2nd...).

For example, if you think the user will go to position 0th to retrieve their
item, you should only answer in single number ’0’.

Please make your prediction:

2

Your prediction is correct! Item in the position 2nd is exchanged with the
correct item in the position 2nd. User 2 successfully retrieved the item from
the correct position. You score a point!

Now, from the monitor, you can see the content of the locker:

The position 0th box is empty.

The position 1st box is empty.

The position 2nd box is empty.

Correct: 3

Final score: 100

Game Over!

Do you want to play another turn?(Y/n)

###  A.2 Game Instance of Finite Belief History

Listing 2: Example of the Pick the Right Stuff game under the condition of
Finite Belief History.

⬇

Game client is running!

Welcome to, Pick the Right Stuff!

In this game, you will play the role of a warehouse manager. The warehouse
contains three rooms. Room 1 is used for storing items, with each item stored
in a certain position inside the opaque locker. You are situated in the Room
2, which contains a monitor that allows you to see the content of the opaque
locker located in the Room 1 through the camera inside the opaque locker. Due
to malfunctions in the locker system, it randomly resets the positions of the
items in the opaque locker from time to time. To ensure that each user
retrieves their stored item correctly, when a user comes to retrieve an item,
you are required to predict the position of the item the user believes (the
user will always retrieve their item based on the position they last
believed). You only need to tell the system which position inside the locker
the user will go to retrieve their item and then the locker system will
automatically swap the item at that location with the one belonging to the
user. Additionally, Room 3 contains a screen which will randomly show a
certain previous snapshot of the monitor located in Room 2. During the game,
users may or may not enter the Room 3 to observe a certain snapshot of the
monitor. By observing the snapshot, users will update their beliefs about the
position of their item.

If a user successfully retrieves their item, you score a point and the item is
removed from the locker.

If a user retrieves the wrong item, the item is returned, the user contacts
the system administrator to take the correct item, and you score no points.

Indeed, this is a problematic locker system, but you are hoped to be an
excellent warehouse manager!

============

Game Begins!

There are 3 users. User 0 stores its item at the position 0th of the locker.
User 1 stores its item at the position 1st of the locker. User 2 stores its
item at the position 2nd of the locker.

Now they leave the room.

The locker is malfunctioning and randomly resetting the positions of the items
in the locker...

The locker has returned to normal.

From the monitor, you can see the content of the locker:

The position 0th stores the item of User 1.

The position 1st stores the item of User 2.

The position 2nd stores the item of User 0.

The locker is malfunctioning and randomly resetting the positions of the items
in the locker...

The locker has returned to normal.

From the monitor, you can see the content of the locker:

The position 0th stores the item of User 2.

The position 1st stores the item of User 0.

The position 2nd stores the item of User 1.

User 1 is coming to Room 1 to take his/her item...

You should only answer the position of the item the user will go to retrieve
their item (e.g. 0 for the 0th, 1 for the 1st, 2 for the 2nd...).

For example, if you think the user will go to position 0th to retrieve their
item, you should only answer in single number ’0’.

Please make your prediction:

1

Your prediction is correct! Item in the position 1st is exchanged with the
correct item in the position 2nd. User 1 successfully retrieved the item from
the correct position. You score a point!

Now, from the monitor, you can see the content of the locker:

The position 0th stores the item of User 2.

The position 1st box is empty.

The position 2nd stores the item of User 0.

User 0 walks into the Room 3 and is observing the snapshot of the monitor...

User 0 observes the snapshot which depicts the 2nd-to-last state of the
monitor and leaves the room.

User 2 is coming to Room 1 to take his/her item...

You should only answer the position of the item the user will go to retrieve
their item (e.g. 0 for the 0th, 1 for the 1st, 2 for the 2nd...).

For example, if you think the user will go to position 0th to retrieve their
item, you should only answer in single number ’0’.

Please make your prediction:

2

Your prediction is correct! Item in the position 2nd is exchanged with the
correct item in the position 0th. User 2 successfully retrieved the item from
the correct position. You score a point!

Now, from the monitor, you can see the content of the locker:

The position 0th stores the item of User 0.

The position 1st box is empty.

The position 2nd box is empty.

User 0 is coming to Room 1 to take his/her item...

You should only answer the position of the item the user will go to retrieve
their item (e.g. 0 for the 0th, 1 for the 1st, 2 for the 2nd...).

For example, if you think the user will go to position 0th to retrieve their
item, you should only answer in single number ’0’.

Please make your prediction:

1

Your prediction is correct! Item in the position 1st is exchanged with the
correct item in the position 0th. User 0 successfully retrieved the item from
the correct position. You score a point!

Now, from the monitor, you can see the content of the locker:

The position 0th box is empty.

The position 1st box is empty.

The position 2nd box is empty.

Correct: 3

Final score: 100

Game Over!

Do you want to play another turn?(Y/n)


# Valley-polarized quantum anomalous Hall effect in van der Waals
heterostructures based on monolayer jacutingaite family materials

Xudong Zhu§ ICQD, Hefei National Research Center for Physical Sciences at the
Microscale, University of Science and Technology of China, Hefei, Anhui
230026, China  CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics,
and Department of Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China  Yuqian Chen§ ICQD, Hefei National Research Center
for Physical Sciences at the Microscale, University of Science and Technology
of China, Hefei, Anhui 230026, China  CAS Key Laboratory of Strongly-Coupled
Quantum Matter Physics, and Department of Physics, University of Science and
Technology of China, Hefei, Anhui 230026, China  Zheng Liu
<EMAIL_ADDRESS>ICQD, Hefei National Research Center for Physical
Sciences at the Microscale, University of Science and Technology of China,
Hefei, Anhui 230026, China  CAS Key Laboratory of Strongly-Coupled Quantum
Matter Physics, and Department of Physics, University of Science and
Technology of China, Hefei, Anhui 230026, China  Yulei Han<EMAIL_ADDRESS>Department of Physics, Fuzhou University, Fuzhou, Fujian 350108, China
Zhenhua Qiao<EMAIL_ADDRESS>ICQD, Hefei National Research Center for
Physical Sciences at the Microscale, University of Science and Technology of
China, Hefei, Anhui 230026, China  CAS Key Laboratory of Strongly-Coupled
Quantum Matter Physics, and Department of Physics, University of Science and
Technology of China, Hefei, Anhui 230026, China

###### Abstract

We numerically study the general valley polarization and anomalous Hall effect
in van der Waals (vdW) heterostructures based on monolayer jacutingaite family
materials Pt2AX3 ($A$ = Hg, Cd, Zn; $X$ = S, Se, Te). We perform a systematic
study on the atomic, electronic, and topological properties of vdW
heterostructures composed of monolayer Pt2AX3 and two-dimensional
ferromagnetic insulators. We show that four kinds of vdW heterostructures
exhibit valley-polarized quantum anomalous Hall phase, i.e., Pt2HgS3/NiBr2,
Pt2HgSe3/CoBr2, Pt2HgSe3/NiBr2, and Pt2ZnS3/CoBr2, with a maximum valley
splitting of 134.2 meV in Pt2HgSe3/NiBr2 and sizable global band gap of 58.8
meV in Pt2HgS3/NiBr2. Our findings demonstrate an ideal platform to implement
applications on topological valleytronics.

##  I Introduction

The honeycomb lattice materials, such as graphene [1, 2, 3, 4] and transition
metal dichalcogenides (TMDs) [5, 6, 7], host two inequivalent valleys $K$ and
$K^{\prime}$ in momentum space located at the corners of the hexagonal
Brillouin zone. In addition to the spin of electron, valley is another degree
of freedom in two-dimensional honeycomb-like materials. The valley degree of
freedom can be tuned by electric fields [8, 9, 10], external magnetic fields
[11, 12, 13], and circularly polarized light excitation [14, 15, 16]. Similar
to spintronics, valleytronics is mainly focused on manipulating valley degree
of freedom in designing functional nanodevices [17, 18, 19, 20], such as
electron beam splitter, valley filter, and valley valves[21, 22, 23, 24, 25].

Quantum anomalous Hall effect (QAHE) is the quantized version of Hall effect
without applying external magnetic fields [26, 27, 28, 29], and has been
experimentally realized in various material systems [30, 31, 32, 33]. The
valley polarization, e.g., in TMD materials, is often topologically trivial
[34, 35, 36, 37], whereas QAHE is usually a valley degenerated state. To
implement valley-polarized QAHE, the inversion symmetry and time-reversal
symmetry should be simultaneously broken. One rational way to realize valley-
polarized QAHE is by inducing the magnetic proximity effect in the
$\mathbb{Z}_{2}$ topological insulator/ferromagnetic insulator
heterostructures [38, 39, 40, 41, 42, 43]. Although several valley-polarized
QAHE systems have been theoretically proposed [38, 39, 40, 41, 42, 43, 44, 45,
46, 47], more experimentally realizable candidates are still highly desired to
observe the valley-polarized QAHE.

In this article, we propose an ideal platform to realize valley polarization
and valley-polarized QAHE based on monolayer jacutingaite family materials.
Jacutingaite (Pt2HgSe3) is a natural mineral with large intrinsic spin-orbit
coupling (SOC). The atomic structure of jacutingaite can be treated as a
quarter of $X$ atoms ($X=$ S, Se, Te) of the 1$T$-phase TMD Pt$X_{2}$ replaced
by $A$ atoms ($A=$ Cd, Hg, Zn), and the monolayer is crystallized in
$P\overline{3}m1$ space group (No. 164) with a buckled honeycomb lattice. The
dynamically stable monolayer Pt2HgSe3 and its family materials
Pt${}_{2}AX_{3}$ ($A$ = Hg, Cd, Zn; $X$ = S, Se, Te) are $\mathbb{Z}_{2}$
topological insulators [48, 49, 50, 51, 53, 52, 54, 55]. The monolayer
ferromagnetic substrates, i.e. $MY_{2}$ ($M=$ Fe, Co, Ni; $Y=$ Cl, Br, I),
CI3, VI3, and MnBi2Te4, are used to induce magnetic exchange interactions [56,
57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71]. These two-
dimensional magnets possess the ferromagnetic ground states in freestanding
monolayer under the Curie temperatures as summarized in Table S2 [72]. By
performing first-principles calculations, we systematically investigate the
atomic, electronic, and topological properties of over 100 kinds of
Pt${}_{2}AX_{3}$/ferromagnetic substrate van der Waals (vdW) heterostructures.
By examining the lattice mismatch, stacking configuration, and band alignment,
44 kinds of well-matched vdW heterostructures are selected. Further analyses
on band structures and topological properties demonstrate that these systems
host valley polarization and exhibit different Berry curvature distributions
at $K/K^{\prime}$ valleys.

In particular, we observe valley-polarized QAHE in Pt2HgS3/NiBr2,
Pt2HgSe3/CoBr2, Pt2HgSe3/NiBr2, and Pt2ZnS3/CoBr2, with the Chern number of
$\mathcal{C}=\pm 1$. We find a large valley splitting of 134.2 meV in
Pt2HgSe3/NiBr2 heterostructure and a considerable topological band gap of 58.8
meV in Pt2HgS3/NiBr2 heterostructure. Both the monolayer jacutingaite family
materials and the two-dimensional ferromagnetic substrates are energetically
and dynamically stable [53, 56, 57, 58, 59, 60], greatly facilitating the
experimental implementation of valley-polarized QAHE. Moreover, the structural
stabilities of these Pt${}_{2}AX_{3}$-based vdW ferromagnetic heterostructures
are also verified by molecular dynamic simulations [72].

##  II Calculation methods

Our first-principles calculations were performed by using the projected
augmented-wave method [73] as implemented in the Vienna $ab~{}initio$
simulation package (VASP) [74, 75]. The generalized gradient approximation of
the Perdew-Burke-Ernzerhof (PBE) type was used to describe the exchange-
correlation interaction [76]. The vdW interaction was treated by using DFT-D3
functional [77]. All atoms were fully relaxed until the Hellmann-Feynman force
on each atom was less than 0.01 eV/Å. A vacuum buffer layer of 20 Å was used
to avoid unnecessary interaction along $z$ direction between adjacent slabs.
The plane-wave energy cutoff was set to be 520 eV. The
$\mathrm{\Gamma}$-centered Monkhorst-Pack $k$-point grid of $\mathrm{11\times
11\times 1}$ was adopted in all our calculations. To deal with the strong
correlation effect of 3$d$ magnetic elements (i.e., Fe, Co, Ni, Cr, V, and
Mn), the GGA+$U$ method [78] was used with the corresponding on-site repulsion
energy $U$ and exchange interaction $J$ [60, 61, 62, 63, 64] as described in
Table S1 in supplemental materials [72]. Topological properties were
calculated by using maximally-localized Wannier functions as implemented in
Wannier90 and WannierTools software packages [79, 80, 81]. Spin-resolved band
structures are extracted by using vaspkit software [82] and pyprocar code
[83]. Atomic structure and charge density difference are visualized by using
vesta software package [84].

##  III Atomic structures

Monolayer Pt${}_{2}AX_{3}$ ($A$ = Hg, Cd, Zn; $X$ = S, Se, Te) is crystallized
in $P\overline{3}m1$ space group (No. 164) with a buckled honeycomb structure
as shown in Fig. 1(a). The $A$ atoms are located at two different sublattices
$A1$ and $A2$. The jacutingaite family monolayer Pt${}_{2}AX_{3}$ belongs to
Kane-Mele type topological insulator [52, 53], with the maximum band gap of
178 meV for Pt2ZnSe3 [53]. The typical band structures of monolayer
Pt${}_{2}AX_{3}$ are displayed in Figs. 1(b) and 1(c), where the Dirac points
can be observed at $K$ and $K^{\prime}$ valleys in the absence of SOC [see
Fig. 1(b)]. When the SOC is considered, as shown in Fig. 1(c), a topological
nontrivial band gap opens up around the two valleys. The valley degeneracy of
monolayer Pt${}_{2}AX_{3}$ is protected by the time-reversal symmetry and
inversion symmetry, leading to $\Delta_{K}=\Delta_{{K}^{\prime}}$.

Figure 1: (a) Top and side views of Pt${}_{2}AX_{3}$/$MY_{2}$ heterostructure
where $2\times 2$ $MY_{2}$ is adopted to match $1\times 1$ Pt${}_{2}AX_{3}$.
$d$ denotes the optimized vdW gap between the ferromagnetic substrate and
jacutingaite family monolayer. $A$1 and $A$2 represent different sublattices
of $A$ atoms, where $A$1 locates closer to the ferromagnetic substrate. (b),
(c) Band structures for the pristine jacutingaite Pt2HgSe3 monolayer (b)
without and (c) with SOC. $\Delta_{K}$ and $\Delta_{K^{\prime}}$ are local
band gaps at valley $K$ and $K^{\prime}$ valleys, respectively. Figure 2: Band
alignments and lattice mismatch of jacutingaite family monolayer
Pt${}_{2}AX_{3}$/ferromagnetic substrate heterostructures. Gray region stands
for the intrinsic band gap of each jacutingaite family monolayer
Pt${}_{2}AX_{3}$. The blue (white) area represents the presence (absence) of
electronic states. Band edges are relative to the vacuum level by calculating
work functions (dashed red line). Orange star (right axis) denotes the lattice
mismatch of corresponding vdW heterostructure. (a)-(c), (d)-(f), and (g)-(i)
represent the cases with $X$ = S, Se, Te, respectively. (a)(d)(g), (b)(e)(h),
and (c)(f)(i) represent the cases with $A$ = Hg, Cd, Zn, respectively.

In our calculations, we adopt $1\times 1$ Pt${}_{2}AX_{3}$ and different
ferromagnetic substrates with suitable supercell to match the lattice
constant. For example, the vdW heterostructure as shown in Fig. 1(a) consists
of $1\times 1$ Pt${}_{2}AX_{3}$ and $2\times 2$ $MY_{2}$. To find the well-
matched vdW heterostructures, we carry out the lattice mismatch calculations
as plotted in Fig. 2 with orange stars and summarized in Table S3 in
supplemental materials [72]. Considering the experimental feasibility, we
mainly explore the physical properties of 62 kinds of heterostructures with
lattice mismatch less than $5\%$ as highlighted in Table S3 [72]. We then
investigate the most stable stacking orders of these vdW heterostructures. For
Pt${}_{2}AX_{3}$/$MY_{2}$, we consider four stacking orders according to the
atom site directly below A1 atoms: (i) $M$, (ii) $Y$(top), (iii) $Y$(hollow),
and (iv) $MY$(bridge), respectively. Fig. 1(a) displays the type-i stacking
configuration and the other three configurations are displayed in Fig. S3 in
supplemental materials [72]. After geometrical optimization and total energy
computation for different stacking orders, we summarize the most stable
stacking configurations for every well-matched vdW heterostructure in Table S5
in supplemental materials [72].

##  IV Band alignments

To design functional devices based on heterostructures, band alignment is a
crucial strategy that is widely used to analyze band edges of the two
different materials forming a heterostructure [85, 86]. According to the
different band gap positions, band alignment can be specified into three
types: (I) straddling gap, (II) staggered gap, and (III) broken gap. Due to
the type-III band alignment leading to the absence of global band gap in a
heterostructure [87, 88], we apply the band alignment strategy to screen out
the type-I and type-II vdW heterostructures in order to obtain a global band
gap.

Table 1: Transferred charge $\Delta Q$, vdW gap $d$, and magnetic moments of 8 kinds of typical Pt${}_{2}AX_{3}$/$MY_{2}$ vdW heterostructures. $\Delta Q$ is extracted from the Bader charge analysis to evaluate the transferred electrons from the ferromagnetic substrate to Pt${}_{2}AX_{3}$ layer. $d$ represents the interlayer distance after full relaxation. $M_{\text{jac}}$, $M_{\text{tot}}$, $M_{A1}$, and $M_{A2}$ denote the magnetic moment of Pt${}_{2}AX_{3}$ layer, total vdW heterostructure, $A1$ atom, and $A2$ atom, respectively. vdW Heterostructures |  Pt2HgS3/ CoBr2 |  Pt2HgSe3/ CoBr2 |  Pt2HgTe3/ CoBr2 |  Pt2ZnS3/ CoBr2 |  Pt2HgS3/ NiBr2 |  Pt2HgSe3/ NiBr2 |  Pt2CdS3/ NiBr2 |  Pt2ZnSe3/ NiBr2
---|---|---|---|---|---|---|---|---
$\Delta Q$ ($e$) | -0.17 | -0.20 | -0.09 | -0.32 | -0.03 | -0.11 | -0.18 | -0.17
$d$ (Å) | 2.51 | 2.74 | 2.90 | 2.70 | 2.28 | 2.46 | 2.18 | 2.51
$M_{\text{jac}}$ $(\mu_{\text{B}})$ | 0.330 | 0.061 | 0.068 | 0.050 | 0.219 | 0.204 | 0.276 | 0.167
$M_{\text{tot}}$ $(\mu_{\text{B}})$ | 11.525 | 11.530 | 11.490 | 11.503 | 7.453 | 7.446 | 7.392 | 7.420
$M_{A1}$ $(\mu_{\text{B}})$ | 0.005 | -0.009 | -0.005 | -0.014 | -0.016 | -0.021 | -0.020 | 0.009
$M_{A2}$ $(\mu_{\text{B}})$ | 0.056 | 0.016 | 0.007 | 0.024 | 0.067 | 0.045 | 0.104 | 0.027

Figure 2 displays the band alignments of jacutingaite family monolayer
Pt${}_{2}AX_{3}$ with different ferromagnetic substrates, which are obtained
by calculating the work function and band structures of each freestanding
material. We first take Pt2HgS3 as an example to analyze the band alignment.
The first column in Fig. 2(a) represents the band edge of monolayer Pt2HgS3,
and the Fermi level (dashed red line) with respect to the vacuum level (0 eV)
is about $-5$ eV lying inside the intrinsic band gap as displayed in gray
region. The blue (white) area represents the presence (absence) of electronic
states. One can find that the band gap of Pt2HgS3 layer lies in the band gap
of NiBr2, resulting in a type-I heterostructure of Pt2HgS3/NiBr2. Similarly,
the Pt2HgS3/FeBr2 system follows the type-II band alignment with the staggered
gap. However, the Pt2HgS3/NiCl2 system demonstrates the type-III nature
because of the broken gap alignment, which will be excluded in our following
discussions. By combing the band alignment and lattice mismatch analyses, we
can filter out 44 kinds of vdW heterostructures from over 100 combinations as
summarized in Table S5 in supplemental materials [72], in which we hopefully
obtain global band gaps with Chern insulating phase after these
heterostructures are formed.

##  V Interfacial Electronic and Magnetic Properties

When two different materials form heterostructures, the charge transfer
process usually exists and further modifies the electronic and magnetic
properties of the heterostructure [89, 90, 91]. To explore the interfacial
electronic properties of the Pt${}_{2}AX_{3}$/ferromagnetic insulator vdW
heterostructures, we calculate the charge density difference and the planar-
averaged electrostatic potential along $z$ direction. Here, we take eight
typical systems with CoBr2 and NiBr2 substrates to illustrate the charge
transfer process as shown in Fig. 3, Table 1 and Figs. S10-S11 in supplemental
materials [72].

For Pt2HgSe3/NiBr2, as displayed in Fig. 3(a), the planar-averaged
electrostatic potential alone $z$ direction in Pt2HgSe3 layer is no longer
symmetric with respect to the Pt atomic layer, indicating the presence of
charge transfer process at the interface. More quantitatively, the Bader
charge analysis demonstrates the transferred charge $\Delta Q$ from NiBr2
layer to Pt2HgSe3 layer is $-0.11e$. The charge density difference, as
illustrated in Fig. 3(b), also demonstrates that the NiBr2 layer accumulates
electrons whereas the Pt2HgSe3 layer loses electrons after they form a
heterostructure. As a result, a built-in electric field $\Delta V_{z}$ appears
at the interface as displayed in Fig. 3(a), which can further enhance the
Rashba SOC of the Pt2HgSe3/NiBr2 system.

Figure 3:  (a) The planar-averaged electrostatic potential $V_{z}$ and (b)
charge density difference of the Pt2HgSe3/NiBr2 heterostructure. $V_{z}$ in
Pt2HgSe3 layer is asymmetric with respect to the Pt atomic layer, indicating
the charge transfer at the interface. Highlighted regions in (a) correspond to
associated horizontal atomic layers in (b). Cyan and yellow contours represent
charge depletion and accumulation, respectively. Figure 4: (a)-(b) Spin-
resolved band structures of the Pt2HgSe3/NiBr2 heterostructure (a) without and
(b) with SOC, respectively. (d)-(e) Spin-resolved band structures of the
Pt2ZnS3/CoBr2 heterostructure (d) without and (e) with SOC, respectively.
(g)-(h) Spin-resolved band structures of the Pt2HgS3/NiBr2 heterostructure (g)
without and (h) with SOC, respectively. (j)-(k) Spin-resolved band structures
of the Pt2HgSe3/CoBr2 heterostructure (j) without and (k) with SOC,
respectively. The red (blue) color denotes the spin-up (spin-down) state. The
Chern number $\mathcal{C}$ and the global band gap for every system is
demonstrated in (b), (e), (h), and (k), respectively. (c), (f), (i), (l)
Layer-resolved band structures of (c) Pt2HgSe3/NiBr2, (f) Pt2ZnS3/CoBr2, (i)
Pt2HgS3/NiBr2, and (l) Pt2HgSe3/CoBr2. The pink bubble represents the element
projection of jacutingaite family layer whereas the light blue bubble
represents the element projection of ferromagnetic substrate layer.

Table 1 displays the transferred charge $\Delta Q$, vdW gap $d$, and magnetic
moments of the eight typical Pt${}_{2}AX_{3}$/$MY_{2}$ vdW heterostructures.
One can observe that all the interlayer vdW gaps $d$ become smaller than the
initial state after full structural relaxation. Due to the smaller atomic
radius of S atom compared to Se atom, Pt2HgS3 tends to move closer to the
substrate than Pt2HgSe3 when constructing a heterostructure with the same
substrate. Moreover, $d$ in Pt${}_{2}AX_{3}$/NiBr2 is generally smaller than
that in Pt${}_{2}AX_{3}$/CoBr2. The shorter $d$ in Pt${}_{2}AX_{3}$/NiBr2
leads to a stronger magnetic proximity effect and large magnetization
$M_{\text{jac}}$ in jacutingaite layer. The magnetic proximity effect also
induces different magnetic moments $M_{A1}$ and $M_{A2}$ on $A1$ and $A2$
because of the different distances to the substrate. The presence of a the
ferromagnetic substrate breaks both inversion and time-reversal symmetries and
leads to inequivalent exchange fields on $A1$ and $A2$ atoms, which lifts the
spin degeneracy and valley-dependent band gaps. Nonzero Berry curvature and
anomalous Hall responses can also appear [38]. For example, in Pt2HgSe3/NiBr2,
the magnetic moment of Hg2 atom is 0.045 $\mu_{B}$ and is parallel to that of
NiBr2, whereas an antiparallel magnetic moment of -0.021 $\mu_{B}$ is induced
in Hg1 atom. Consequently, the valley degeneracy of Pt${}_{2}AX_{3}$ can be
lifted by the proximity effect induced broken time-reversal symmetry and
Rashba SOC. Hereinbelow, we explore the electronic properties of these
Pt${}_{2}AX_{3}$/ferromagnetic substrate vdW heterostructures.

##  VI Electronic and Topological Properties

After calculating band structures and topological properties of the 44 kinds
of well-matched Pt${}_{2}AX_{3}$/ferromagnetic substrate vdW heterostructures,
we find four systems are topologically nontrivial, whereas the remaining
systems are topologically trivial (see Figs. S12-S22 in supplemental materials
[72] for band structures of all 44 systems). Figure 4 shows the spin-resolved
and layer-resolved band structures of the four topologically nontrivial
systems, i.e., Pt2HgSe3/NiBr2, Pt2ZnS3/CoBr2, Pt2HgS3/NiBr2, and
Pt2HgSe3/CoBr2.

We start from Pt2HgSe3/NiBr2 as an example [see Figs. 4(a)-4(c)]. In the
absence of SOC, as shown in Fig. 4(a), we can observe that the magnetic
proximity effect of ferromagnetic substrates induces sizable Zeeman splitting
but keeps the valley degeneracy. Further calculations on the band structures
with and without magnetization also demonstrate that the band structures of
the Pt2AX3 near the Fermi level are modified by the magnetic exchange
interaction [72]. When SOC is considered, as displayed in Fig. 4(b), a band
inversion appears around $K^{\prime}$ point with a band gap of 6.3 meV induced
by Rashba SOC, whereas the bands close to the Fermi level around $K$ point
move far away from each other in a repulsive manner, forming a sizable local
band gap of $\Delta_{K}$ of 182.8 meV. The difference around $K/K^{\prime}$
valleys results in a large valley splitting of 134.2 meV. We can also observe
from the layer-resolved band structures as shown in Fig. 4(c) that the
electronic states near the Fermi level are mainly dominated by the topological
Pt2HgSe3 layer, indicating the electronic structures of the Pt2HgSe3 monolayer
are modified after coupled to the NiBr2 substrate.

The above band analysis can be applied to the remaining three nontrivial
systems. For Pt2ZnS3/CoBr2 system, as shown in Figs. 4(d)-4(f), the band
structures are similar to that of Pt2HgSe3/NiBr2 with a band inversion and a
global band gap of 7.8 meV around $K^{\prime}$ point induced by Rashba SOC.
For Pt2HgS3/NiBr2 and Pt2HgSe3/CoBr2 systems, as illustrated in Figs.
4(g)-4(l), the bands are different from that of Pt2HgSe3/NiBr2, where the
local gap at $K$ valley is smaller than that at $K^{\prime}$ valley. Table 2
summarizes the extracted valley splitting $\Delta$, global band gaps
$E_{\text{g}}$, local band gaps $\Delta_{K}$/$\Delta_{K^{\prime}}$ at valleys
$K/K^{\prime}$, and Chern numbers $\mathcal{C}$ for above four vdW
heterostructures. One can see that the valley polarization generally exists in
these systems (also see Table S5 in supplemental materials [72] for the
remaining 40 kinds of heterostructures). Among the 44 systems, the
Pt2HgSe3/NiBr2 exhibits the maximum valley splitting of 134.2 meV, which is an
order of magnitude larger than experimental observations in TMD/ferromagnetic
substrate systems [92, 93], providing an ideal platform to realize topological
valleytronics.

Table 2: Valley splitting $\Delta$, global band gap $E_{\text{g}}$, local band gaps $\Delta_{K}$/$\Delta_{K^{\prime}}$ at $K/K^{\prime}$ valleys, and Chern number $\mathcal{C}$ for nontrivial jacutingaite family monolayer/ferromagnetic substrate vdW heterostructures. Valley splitting is evaluated via the difference between valence band maximum at valley $K/K^{\prime}$, i.e. $\Delta=E_{\text{V}}({K^{\prime}})-E_{\text{V}}({K})$.  Nontrivial vdW Heterostructrues |  $\Delta$ (meV) |  $E_{\text{g}}$ (meV) |  $\Delta_{K}$ (meV) |  $\Delta_{K^{\prime}}$ (meV) | $\mathcal{C}$
---|---|---|---|---|---
Pt2HgSe3/NiBr2 | 134.2 | 6.3 | 182.8 | 73.5 | -1
Pt2ZnS3/CoBr2 | 36.3 | 7.8 | 185.9 | 59.9 | -1
Pt2HgS3/NiBr2 | 15.4 | 58.8 | 74.2 | 78.8 | 1
Pt2HgSe3/CoBr2 | 3.5 | 19.3 | 22.8 | 124.0 | 1
Figure 5: Berry curvature distributions in the first Brillouin zone of four
nontrivial vdW heterostructures (a) Pt2HgSe3/NiBr2, (b) Pt2ZnS3/CoBr2, (c)
Pt2HgS3/NiBr2, and (d) Pt2HgSe3/CoBr2, respectively. Berry curvature is
centered around the ${K}/{K}^{\prime}$ valleys and is absent elsewhere but
dissimilarly distributes in the two inequivalent valleys $K/K^{\prime}$. For
clarity, (a) and (b) plot $-\Omega_{z}$ whereas (c) and (d) plot $\Omega_{z}$.
Figure 6: Anomalous Hall conductance $\sigma_{xy}$ and local density of states
(LDOS) of four nontrivial vdW heterostructures (a) Pt2HgSe3/NiBr2, (b)
Pt2ZnS3/CoBr2, (c) Pt2HgS3/NiBr2, and (d) Pt2HgSe3/CoBr2. Insets correspond to
enlarged views of edge states. In all LDOS plots, there are chiral edge states
connecting conduction bands and valence bands.

To evaluate the topology of the above band structures, we compute the Berry
curvature of all occupied bands in each system by using maximally localized
Wannier functions [79, 80]. We carry out orbital projections for every
insulating system to guide the Wannier functions. For example, by analyzing
the orbital projection on each kind of elements Pt2HgSe3/NiBr2, we find that
Pt-$d$, Hg-$s$, Se-$p$, Ni-$d$, and Br-$p$ orbitals are distributed around
Fermi level (see Figs. S1-S2 in supplemental materials [72]). Figure 5
displays the Berry curvature distributions of above four systems in the first
Brillouin zone. One can see that the Berry curvature is centered around the
${K}/{K}^{\prime}$ valleys but is absent elsewhere. The value of the Berry
curvature around $K/K^{\prime}$ valleys depends on the specific material
combinations, i.e., the Berry curvature only appears in $K^{\prime}$ valley
for Pt2HgSe3/NiBr2 as displayed in Fig. 5(a), while it exists in both $K$ and
$K^{\prime}$ valleys with different magnitudes for other three systems as
shown in Figs. 5(b)-5(d). By integrating the Berry curvature all over the
first Brillouin zone, we can obtain the Chern numbers as listed in Table 2.
The nonzero Chern number indicates the nontrivial band topology in the four
systems, i.e., the formation of QAHE. Combining the valley polarizations and
different Berry curvature distributions at the two inequivalent valleys, we
can find that these four systems can implement valley-polarized QAHE. More
strikingly, the global band gap of 58.8 meV in Pt2HgS3/NiBr2 is over three
times larger than that in Pt2HgSe3/CrI3 heterostructure (17.2 meV) as
predicted in our previous report [38], implying the potential for high-
temperature QAHE [94, 95, 96, 97, 98, 99]. Surprisingly, the
magnetocrystalline anisotropy energy (MAE) calculation shows that the MAE of
Pt2ZnS3/CoBr2 heterostructure is 1.795 meV/Co, suggesting the magnetic easy
axis of CoBr2 can be tuned by Pt2ZnS3 to out-of-plane direction [72], which is
beneficial to the experimental realization of valley-polarized QAHE.

The nontrivial topology of the four systems can also be confirmed by the
anomalous Hall conductance $\sigma_{xy}$ and the local density of states
(LDOS) calculations, as displayed in Fig. 6. The anomalous Hall conductance in
Figs. 6(a) and 6(b) demonstrates a quantized Hall plateau of
$\sigma_{xy}=-e^{2}/h$ in Pt2HgSe3/NiBr2 and Pt2ZnS3/CoBr2 systems when the
Fermi energy lies inside the global band gap, which is a strong evidence for
QAHE. Similarly, in Figs. 6(c) and 6(d), we can also observe that the
anomalous Hall conductance is quantized as $\sigma_{xy}=e^{2}/h$ in
Pt2HgS3/NiBr2 and Pt2HgSe3/CoBr2 systems. Such a Chern insulating topological
phase is also confirmed by the LDOS of these nontrivial systems since there
are chiral edge states in the global band gaps connecting conduction bands and
valence bands.

##  VII Summary and conclusion

In this work, we proposed an excellent platform to implement valley
polarization and valley-polarized QAHE based on the vdW heterostructure
constructed by monolayer jacutingaite family materials Pt${}_{2}AX_{3}$ and
two-dimensional ferromagnetic substrates. We systematically investigated
atomic structures, band alignments, interfacial characteristics, electronic
and topological properties of over 100 kinds of Pt${}_{2}AX_{3}$/ferromagnetic
substrate heterostructures. By using the band alignment strategy, we filtered
out 44 kinds of well-matched systems with potential for global band gaps. We
evaluated the interfacial characteristics by analyzing the charge density
difference, Bader charge transferring, planar-averaged electrostatic
potential, and magnetic properties. The interlayer charge transfer leads to a
built-in electric field and the asymmetric atomic structure results in a
broken inversion symmetry. Furthermore, the reducing interlayer vdW gap leads
to a strong magnetic proximity, which induces large magnetization in Kane-Mele
type topological insulator Pt${}_{2}AX_{3}$. Consequently, the valley
degeneracy is lifted by the breaking of time-reversal symmetry and inversion
symmetry. We found the general valley polarization in the 44 kinds of well-
matched systems and observed a sizable valley splitting of 134.2 meV in
Pt2HgSe3/NiBr2 system.

Particularly, we identified four Chern insulators in Pt2HgS3/NiBr2,
Pt2HgSe3/CoBr2, Pt2HgSe3/NiBr2, and Pt2ZnS3/CoBr2 due to their nontrivial band
topology. In addition to the valley polarization, we also observed different
Berry curvature distributions in the two inequivalent valleys $K/K^{\prime}$
in these four systems, indicating the presence of valley-polarized QAHE.
Eventually, we obtained valley-polarized QAHE with total Chern numbers
$\mathcal{C}=\pm 1$ in the vdW heterostructures based on monolayer
jacutingaite family materials. Most surprisingly, we observed a sizable
nontrivial global band gap of 58.8 meV in the Pt2HgS3/NiBr2 system with
potential implementation for the high-temperature QAHE.

In conclusion, valley polarization universally exists in the ferromagnetic vdW
heterostructures based on monolayer jacutingaite family materials and several
valley-polarized QAHE vdW heterostructures can be achieved when the valley
splitting is strong enough to lead to a band inversion. Our work proposes an
excellent platform based on monolayer jacutingaite family materials
Pt${}_{2}AX_{3}$ to implement topological valleytronics and may stimulate more
applications on the two-dimensional valley-polarized topological systems.

###### Acknowledgements.

We are grateful to Prof. Yang Gao for helpful advice and discussions. This
work was financially supported by the National Natural Science Foundation of
China (No. 11974327 and No. 12004369), Fundamental Research Funds for the
Central Universities (WK3510000010, WK2030020032), Anhui Initiative in Quantum
Information Technologies (Grant No.AHY170000). We also thank for the high-
performance supercomputing services provided by AM-HPC and the Supercomputing
Center of University of Science and Technology of China.

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