# Ethical Reasoning and Moral Value Alignment of LLMs Depend on the Language
we Prompt them in

###### Abstract

Ethical reasoning is a crucial skill for Large Language Models (LLMs).
However, moral values are not universal, but rather influenced by language and
culture. This paper explores how three prominent LLMs – GPT-4, ChatGPT, and
Llama2-70B-Chat – perform ethical reasoning in different languages and if
their moral judgement depend on the language in which they are prompted. We
extend the study of ethical reasoning of LLMs by Rao et al. (2023) to a
multilingual setup following their framework of probing LLMs with ethical
dilemmas and policies from three branches of normative ethics: deontology,
virtue, and consequentialism. We experiment with six languages: English,
Spanish, Russian, Chinese, Hindi, and Swahili. We find that GPT-4 is the most
consistent and unbiased ethical reasoner across languages, while ChatGPT and
Llama2-70B-Chat show significant moral value bias when we move to languages
other than English. Interestingly, the nature of this bias significantly vary
across languages for all LLMs, including GPT-4.

Ethical Reasoning and Moral Value Alignment of LLMs Depend on the Language we
Prompt them in

Utkarsh Agarwal∗1 Kumar Tanmay∗1 Aditi Khandelwal∗1 Monojit Choudhury†2
---
∗Microsoft Corporation
†MBZUAI
{t-utagarwal, t-ktanmay<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract content

11footnotetext: Equal contribution.22footnotetext: Work done while at
Microsoft.

## 1\. Introduction

Large Language Models (LLMs) like ChatGPT have gained popularity all over the
world for their ability to generate fluent and engaging natural language texts
Schulman et al. (2022); OpenAI (2023). However, the widespread and rapid use
of LLMs has brought about ethical concerns and potential problems, especially
when we consider using them in different languages Blodgett et al. (2021);
Choudhury and Deshpande (2021); Wang et al. (2023); Ahuja et al. (2023). As
LLMs become more prevalent and find applications in everyday life, they must
confront complex moral dilemmas rooted in the existence of multiple
conflicting values, commonly dubbed as the problem of value pluralism James
(1891); Dai and Dimond (1998); Ramesh et al. (2023). Several researchers (see
for instance Rao et al. (2023) and Zhou et al. (2023)) have argue that instead
of being firmly aligned to a specific set of values, LLMs should be trained to
function as generic ethical reasoners, adaptable to different contexts and
languages. The final moral judgments, of course, should be made by the
stakeholders at different stages of the application life-cycle. LLMs should be
able to reason ethically in a generic way, given a situation and a moral
stance, and should soundly resolve the dilemma when possible, or else ask for
more clarity on the stance.

In a recent study, Rao et al. (2023) has demonstrated that LLMs, especially
GPT-4, are capable of carrying out sound ethical reasoning. They showed that
when the LLMs are presented with a moral dilemma and a moral stance presented
at different levels of abstraction and following different formalisms of
normative ethics in the prompt, they are often capable of resolving the
dilemma in a way that is consistent with the moral stance. They further argue
that this is a promising direction towards solving the issues of value
pluralism at a global scale, because the different stake-holders in the
development and use of an AI system can specify their moral stance which can
be meaningfully consumed in the prompt by the LLM to arrive at a sound moral
judgment. The alternative approach of aligning LLMs to moral values is bound
to fail due to the absence of a universal value hierarchy. However, this study
was conducted only for English.

It is a well established fact that the abilities of the LLMs in languages
beyond English are often poor and unpredictable (see for example Ahuja et al.
(2023), Zhao et al. (2023) and Wang et al. (2023)). Moreover, an intriguing
human phenomenon known as the Foreign Language effect Costa et al. (2014a)
comes into play when people face moral dilemmas presented in a foreign
language (L2). People often make different moral judgments in L2 when compared
to when they encounter the same dilemmas in their native language (L1). This
suggests that language can significantly shape our emotional and cognitive
responses to moral situations, influencing our choices and beliefs.

Given this complex interplay of culture, language and values in making moral
judgments, one might ask: Do LLMs also exhibit a Foreign Language effect,
changing their behavior when confronted with moral dilemmas in different
languages? In this work, we extend the study by Rao et al. (2023) to five
languages other than English, namely Spanish, Russian, Chinese, Hindi, Arabic,
and Swahili. We probe three popular LLMs: GPT-4 OpenAI (2023), ChatGPT
(September 2023) Schulman et al. (2022) and Llama2-70B-Chat Touvron et al.
(2023) systematically to assess their ethical reasoning abilities in different
languages by prompting them to resolve an ethical dilemma reflecting conflicts
between interpersonal, professional, social and cultural values, and a set of
ethical policies (i.e., moral stances) that can help arrive at a clear
resolution of the dilemma. These policies are drawn from three branches of
normative ethics: deontology Alexander and Moore (2021), virtue Hursthouse and
Pettigrove (2022) and consequentialism Sinnott-Armstrong (2022) and have three
different levels of abstractions (similar to Rao et al. (2023)).

Our study clearly demonstrates that indeed the LLMs exhibit different biases
while resolving the moral dilemmas in different languages. This bias is
minimal for English, in the sense that the resolution of the dilemma depends
on the ethical policy rather than the model’s own judgment, and it is maximum
for low-resource languages such as Hindi and Swahili. This observation can
also be interpreted as a reduced ethical reasoning capability of the LLMs in
languages beyond English. However, as we shall see in this paper, the ethical
reasoning ability (or conversely the bias) is dilemma-specific, which makes us
conclude that the LLMs have strong value alignment biases in languages beyond
English. Other salient findings are: (1) Across all languages, GPT-4 has the
highest ethical reasoning ability, while Llama2-70B-Chat has the poorest; (2)
across all models, the reasoning is poorest for Hindi and Swahili, while best
for unsurprisingly, English, and also Russian; and (3) across all models,
English and Spanish, and Hindi and Chinese have similar bias patterns.

## 2\. Background

The topic of right and wrong has been a subject of ongoing discussion among
philosophers, psychologists, and other social scientists. Each field brings
its unique perspectives and worries into this debate. In this section, we take
these concerns as a guide to offer an overview of this ongoing discussion and
how it connects with the field of machine ethics. Our main focus is on how
these conversations affect the Natural Language Processing (NLP) community,
and we also explore how Large Language Models (LLMs) can advance the frontiers
of machine ethics, particularly in multilingual contexts.

### 2.1. Ethics and Moral Philosophy

Ethics is the branch of philosophy concerned with determining what is morally
good or bad and what is considered right or wrong. It encompasses systems or
theories of moral values and principles Kant (1977, 1996). Within the realm of
ethics, normative ethics plays a central role by seeking to establish
standards of conduct for human actions, institutions, and ways of life.
Normative ethics branches into deontology, which evaluates the inherent
rightness or wrongness of actions based on moral rules or duties Alexander and
Moore (2021); virtue ethics, which focuses on an individual’s character and
virtues rather than specific rules or consequences Hursthouse and Pettigrove
(2022); and consequentialism, which emphasizes the goodness or value of the
outcomes or goals of actions Sinnott-Armstrong (2022).

Ethical dilemmas are situations characterized by conflicts between two or more
moral values or principles, posing challenges for moral judgment and decision-
making Slote (1985). The question of whether moral dilemmas can coexist with a
consistent system of moral values is a subject of debate. Philosopher Williams
argues that ethical consistency doesn’t eliminate the possibility of moral
dilemmas, as some actions that ought to be done may be incompatible Williams
(1988). Resolving these dilemmas often requires making new value judgments
within the existing ethical framework.

Value pluralism is a key component of ethical dilemmas, suggesting that there
are multiple values that can be equally correct and yet in conflict with each
other James (1891). Different individuals or cultures may prioritize these
values differently, resulting in varying resolutions of ethical dilemmas, each
of which is ethically sound and consistent. Within the realm of pluralism,
there exist various sub-schools of thought, including Rossian Pluralism Ross
and Stratton-Lake (2002) and Particularism Hare (1965), each offering distinct
viewpoints. Rossian pluralists advocate for the evaluation of moral principles
based on their merits and demerits. Conversely, particularists contend that
the assessment of moral pros and cons should be context-dependent.
Nevertheless, both schools of thought share a fundamental conviction that
there is no one-size-fits-all principle capable of resolving all moral
conflicts. They also reject the notion of a rigid hierarchy of moral
principles that could facilitate such resolutions. This perspective implies
that there is no universally applicable set of moral values or principles that
can address all situations and apply uniformly to all individuals.

Inglehart and Welzel (2010) introduced a framework for mapping global cultures
which employs a two-dimensional axis system, where the x-axis represents a
spectrum that stretches from survival ethics on the left to self-expression on
the right and y-axis covers a range from tradition-based or ethnocentric moral
views at the bottom to democratic and rational principles at the top. This
visual representation illustrates the tendency of societies to move diagonally
from the lower-left corner to the upper-right corner as they progress through
industrialization and development.

### 2.2. Foreign Language Effect on Morality

Recent studies such as Costa et al. (2014b); Hayakawa et al. (2017); Corey et
al. (2017) have found a fascinating link between moral judgment and what’s
known as the "Foreign-Language Effect." This effect shows that people tend to
make more practical choices when they face moral dilemmas in a foreign
language (L2) compared to their native language (L1). This shift seems to be
because using a foreign language makes people less emotionally connected to
the situation, which, in turn, reduces the influence of emotions on their
moral decisions. Čavar and Tytus (2018) also highlights that being better at
and more comfortable with the foreign language (L2) can decrease this tendency
to make practical decisions. This means that the language you use can
significantly affect how you make moral choices, impacting many people.
Furthermore, bilingual individuals’ moral decision-making process is quite
complex, involving factors like the type of dilemma, emotional excitement, and
the language they’re using Chan et al. (2016).

### 2.3. Ethics in LLMs

In the field of Ethics in NLP, most approaches assume a deontological
perspective, with developers setting moral rules, but these may not readily
apply to various contexts or Large Language Models (LLMs) Talat et al. (2022).
Awad et al. (2022) introduce the Computational Reflective Equilibrium (CRE)
framework for AI-based ethics, emphasizing moral intuitions and principles.
Sambasivan et al. (2021), Bhatt et al. (2022) and Ramesh et al. (2023) have
raised questions of value-pluralism in AI and the need for recontextualizing
fairness and AI ethics, particularly in global contexts. Diddee et al. (2022)
explore ethical concerns in Language Technologies for social good, emphasizing
stakeholder interactions and strategies. Choudhury and Deshpande (2021)
advocates the Rawlsian principle over utilitarianism in multilingual LLMs for
linguistic fairness.

AI alignment seeks to ensure that AI systems conform to human goals and
ethical standards, as highlighted by Piper (Oct 15, 2020). Various initiatives
have put forth ethical frameworks, guidelines, and datasets to train and
evaluate Language Models (LLMs) in terms of ethical considerations and
societal norms Hendrycks et al. (2020); Zhou et al. (2023); Jiang et al.
(2021); Rao et al. (2023); Tanmay et al. (2023). Moreover, Tanmay et al.
(2023) introduces an ethical framework utilizing the Defining Issues Test to
assess the ethical reasoning abilities of LLMs. However, it’s worth noting
that these efforts may be susceptible to biases based on the backgrounds of
those providing annotations, as pointed out by Olteanu et al. (2019). Recent
research has placed a growing emphasis on in-context learning and supervised
tuning to align LLMs with ethical principles, as demonstrated by the studies
conducted by Hendrycks et al. (2020); Zhou et al. (2023); Jiang et al. (2021);
Rao et al. (2023); Sorensen et al. (2023). These methodologies aim to
accommodate a range of ethical perspectives, recognizing the multifaceted
nature of ethics. Rao et al. (2023) posits that the generic ethical reasoning
abilities can be infused into the LLMs so that they can handle value pluralism
at a large scale. However, the authors have considered the ethical policies
only in English.

In this study, we will be extending the work of Rao et al. (2023) with
different languages to explore how LLMs behave when the multilingual ethical
policies are infused into these LLMs . As far as we know, this study is the
first of it’s kind dealing with the ethics of LLMs in multilingual settings.

### 2.4. Multilingual Performance of LLMs

Language Models (LLMs) exhibit remarkable multilingual capabilities in natural
language processing tasks, although their proficiency varies among languages
Zhao et al. (2023). Their primary training data is in English, but they also
incorporate data from various other languages, contributing to their
generalisability Brown et al. (2020); Chowdhery et al. (2022); Zhang et al.
(2022); Zeng et al. (2022). Nevertheless, significant challenges arise when
LLMs interact with non-English languages, especially in low-resource contexts
Bang et al. (2023); Jiao et al. (2023); Hendy et al. (2023); Zhu et al.
(2023). Several studies suggest that enhancing their multilingual performance
is possible through in-context learning and the strategic design of prompts
Huang et al. (2023); Nguyen et al. (2023). Experiments conducted by Ahuja et
al. (2023) and Wang et al. (2023) have brought to light an interesting aspect.
They benchmarked LLMs across a range of Natural Language Processing (NLP)
tasks, including Machine Translation, Natural Language Inference, Sentiment
Analysis, Text Summarization, Named Entity Recognition, and Natural Language
Generation. The results indicate that, while LLMs excel for a few well-
resourced languages, they generally under-perform for most languages.
Furthermore, Kovač et al. (2023) have shown that LLMs exhibit context-
dependent values and personality traits that can vary across different
perspectives. This is in contrast to humans, who typically maintain more
consistent values and traits across various contexts.

Importantly, the current body of research has primarily concentrated on the
technical capabilities of multilingual LLMs. There has been a relative lack of
exploration into their moral reasoning within diverse linguistic and cultural
contexts. Recognizing the significant impact of LLMs on real-world
applications and domains, there is a growing need to delve into the ethical
dimensions surrounding these multilingual language models.

## 3\. Methodology

### 3.1. Language-based Ethical Framework

We extend the existing framework for defining ethical policies in the LLM
prompt, as initially presented by Rao et al. (2023) and reformulate it to be
used in multilingual settings. Let us consider an LLM, denoted as
$\mathcal{L}$, which accepts a prompt $p$ in language $\lambda$ and produces a
textual output  through the function $\mathcal{L}(p)$. We establish the
definition of $p$ as an arbitrary composition (for example, concatenation), an
ethical policy $\pi$, and user input $x$, in language $\lambda$. In original
formulation, the task definition is defined as $\tau$. Since we are defining
this specifically for ethically reasoning task, variable $p$ can be expressed
as a function of the variables $\pi$, $x$, and $\lambda$, denoted as

> $p=P(\pi,x,\lambda)$.

Hence we extend the definition of ethical consistency to following:

Definition Ethical Consistency. The output produced by the model $\mathcal{L}$
is considered ethically consistent with the policy $\pi$ if and only if is a
valid response or resolution to the input $x$ in the language $\lambda$
according to the policy $\pi$. We represent this as:

> $x\wedge\pi~{}\vdash_{e}\ y$

where, similar to logical entailment, $\vdash_{e}$ represents ethical
entailment.

If a policy is ambiguous for the resolution of $x$, it can lead to
discrepancies in the system Williams (1988). In such instances, LLMS should
refrain from definitively resolving the dilemma in either direction. In this
framework, it is anticipated that the LLM will assert that a definitive
resolution, but when it is unattainable, the response should be $\phi$ for
such responses. Therefore, in the case when $\pi$ is not fully specified, the
likelihood function:

> $\mathcal{L}(P(\pi,x,\lambda))\rightarrow\phi$.

### 3.2. Ethical Policies

As explained in Rao et al. (2023), ethical policies are distinctly
characterized as expressions of preference pertaining to either moral values
or ethical principles. The absence of a universally agreed-upon set of ethical
principles necessitates the flexibility to define policies based on various
ethical formalisms or their combinations. In the context of a specific ethical
formalism, denoted as $F$, there exists a set of fundamental moral principles,
represented as

> $R^{F}={r^{F}_{1},r^{F}_{2},\dots r^{F}_{n_{F}}}$.

In our work, we extend these principles to the unique challenges posed by LLMs
in multilingual settings.

It is important to underscore the definition of an ’Ethical Policy’ as
described in this preceding work. An ethical policy, denoted as $\pi$, is
essentially a partial order applied to a subset of elements within $R^{F}$.
Specifically, $\pi$ is represented as $(R^{F}_{s},\leq^{F}_{s})$, where
$R^{F}_{s}$ denotes a subset of $R^{F}$, and $\leq^{F}_{s}$ signifies a non-
strict partial order relation governing the importance or priority of these
ethical principles. This level of policy abstraction is herein referred to as
a ’Level 2 policy,’ and it serves as an illustrative example of how virtue
ethics may manifest, for instance, as ’prioritizing loyalty over objective
impartiality.’

Furthermore, the refinement of policies is also elaborated upon in this
preceding work, wherein they are classified into ’Level 1’ and ’Level 0
policies.’ Level 1 policies, such as ’favoring loyalty towards a friend over
professional impartiality,’ provide specificity by designating the variables
to which ethical virtues apply. Meanwhile, Level 0 policies, like
’prioritizing loyalty towards her friend Aisha over objectivity towards
scientific norms of publishing,’ delve into even finer details by specifying
the values to which these virtues are to be applied.

The systematic approach to ethical policies described here underscores their
pragmatic applicability, depending on the level of abstraction and specificity
desired. It is important to note that these policies, as described in the
previous work, are predominantly conveyed in natural language. Nevertheless,
it is conceivable that future developments may explore alternative means of
policy representation, including symbolic, neural, or hybrid models, drawing
from various ethical formalisms. This nuanced understanding of ethical
policies in different languages, as derived from the prior research, forms the
basis for the discussion in this present paper .

## 4\. Evaluating Ethical Reasoning Across Languages

Here, we describe the design of our experiment to study the change in the
ethical reasoning abilities of three popular Large Language Models with well
known multilingual capabilities with prompts in different languages. In the
experiment, the models were prompted with moral dilemmas (x’s) in language $L$
that needed to be resolved for a predetermined ethical policy ($\pi$).

We evaluate two OpenAI’s models, ChatGPT (GPT-3.5-turbo) Schulman et al.
(2022) and GPT-4 OpenAI (2023). ChatGPT is a finetuned version of the GPT-3.5
model, optimized for dialog using RLHF. We use the September 2023 preview of
ChatGPT for our experiments. GPT-4 is a larger and more recent model by
OpenAI. We also evaluate Meta’s publicly available Llama2-70B-Chat-hf model
Touvron et al. (2023). It is a 70B parameter model optimized for dialog use
cases.

We set temperature equal to 0 for all the experiments. Others parameters are
set as follows: top probability is 0.95 and the presence penalty is 1.

### 4.1. Dataset

We examine the moral dilemmas and value statements proposed in Rao et al.
(2023), each of the four dilemmas contain nine pairs of contrasting policies
with at three different levels of abstraction. These policies are related to
three branches of normative ethics: Virtue, Deontology, and Consequentialism.
The dilemmas highlights the clash between different ethical values. The three
dilemmas created by Rao et al. (2023) emphasize conflicts of interpersonal
versus professional and community versus personal values. Each of these
dilemmas consists of nine policies, denoted as $\pi=(r^{F}_{i}\geq
r^{F}_{j})$, and their corresponding complementary forms,
$\bar{\pi}=(r^{F}_{j}\geq r^{F}_{i})$. This gives us a total of eighteen
distinct policies for each dilemma.

To evaluate the ethical consistency of the Language Models’ outputs, we use
the ideal resolutions for each dilemma under each policy, as annotated by Rao
et al. (2023). None of these ideal resolutions equate to $\phi$.

Since all the proposed dilemmas are in english, we translate these dilemmas
and the corresponding policies to six different languages (Spanish, Chinese,
Russian, Hindi, Arabic and Swahili) using Google Translation API
333https://translate.google.com/. We back-translated them into English to
check the consistency of the meanings of the values by manual inspection. The
prompt instruction is also translated similarly.

### 4.2. Experiments

First, we conduct a baseline experiment in which the models are prompted to
respond to a moral dilemma without any given policy, and they provide their
moral judgment or resolution from the three options: $y=$ "he/she should",
$\neg y=$ "he/she shouldn’t" and $\phi=$ "can’t decide". This process is
repeated for all languages. In order to eliminate the effect of potential
positional biases, we utilize six different permutations of these three
options and each permutation is run 5 times. This experiment is designed to
reveal the models’ inherent biases and moral stances, and to study how they
vary across different languages. For each language, a total of 120 (4 $\times$
6 $\times$ 5) experiments are conducted, including four dilemmas and six
permutations of options. We note the baseline resolution of the model for each
of the dilemmas per language in Table 1.

In the next part, there is a policy statement given along with the dilemma
instructing the model to resolve the dilemma strictly based on the policy.
This results in each model being probed a total of 432 times for each language
(18 $\times$ 4 $\times$ 6). The prompt structure used here is the same as the
one proposed with the dilemmas.

### 4.3. Metrics

The following metrics were used to study the models’ behaviors across the
dilemmas in different languages.

Accuracy here is defined as the percentage of number of times the model
correctly resolves the dilemma given the policy as per the proposed
resolution.

Bias and Confusion are two key metrics calculated to assess model behavior.
Bias is defined as the fraction of times the model sticks to its baseline
stance, even when the provided policy dictates otherwise. Confusion is the
fraction of times the model deviates from it’s baseline stance when the policy
prompts it to stick with the same stance. We calculate both of these as
follows:

$bias\ =\ \frac{\sum\nolimits_{i}(1\ |\ x_{i}\neq A,\ y_{i}=A)}{\sum_{i}(1\ |\
x_{i}\neq A)}$ $confusion=\frac{\sum\nolimits_{i}(1\ |\ \ x_{i}=A,\ y_{i}\neq
A)}{\sum_{i}(1\ |\ x_{i}=A)}$

Here, $x_{i}$ represents the ground truth, $y_{i}$ represents the model
prediction, and $A$ represents the model’s baseline stance.

A higher bias value illustrates a strong alignment of the model to it’s
preferred resolution which it still tries to reason for despite an opposing
policy. A high confusion score illustrates the possibility that the model is
perhaps not able to understand the dilemma and it’s associated values very
well and thus deviates from expected resolution for no clear reason.

## 5\. Results and Observations

Table 1 shows the baseline performance of all three models across various
languages. GPT-4 resolves dilemmmas with a remarkably high agreement across
all cases, with one notable exception observed in the Heinz dilemma when
probed in Hindi, resulting in a 50% agreement rate. An intriguing observation
emerges when analyzing the GPT-4 results for the Rajesh dilemma; a distinct
pattern of opposite resolutions (highlighted in red) is observed for most
languages when compared to the English context (highlighted in green).
Furthermore, ChatGPT exhibits conflicting behaviors between English and Hindi
for all dilemmas in its resolution process. Llama2-70B-Chat exhibits the least
consistent behavior among the three models, primarily due to the lower degree
of agreement among all possible permutations of choices. Notably,
Llama2-70B-Chat demonstrates a remarkable departure from the behavior of
ChatGPT and GPT-4 in most dilemmas and languages. Llama2-70B-Chat tends to opt
for affirmative resolutions, such as "should share" and "should steal the
drug," more frequently, showcasing a distinct and contrasting behavioral
pattern when compared to its counterparts. These findings hint at the
potential presence of bias within the models towards specific dilemmas (and
consequently, while resolving certain kinds of value conflicts) in specific
languages, warranting further investigation.

Figure 2 provides a comprehensive overview of the results obtained from
policy-based resolution by the models across various languages, comparing them
to the ground-truth resolutions. GPT-4 consistently demonstrates superior
ethical reasoning abilities across most languages, with the notable exception
of Hindi. In stark contrast, Llama2-70B-Chat exhibits the least ethical
reasoning capability across the board.

Table 2 lists the accuracy of each model across the different levels of
abstraction of the policies. We can see the trend that models tend to perform
slightly better on average on lower abstraction levels.

When considering the different policies, it becomes evident from Figure 2 that
Level 2 policies, aligned with the consequentialist framework, are where the
models predominantly excel, except for ChatGPT for Russian. For Level 1
deontological policies, Llama2-70B-Chat and ChatGPT perform well. These models
perform well for Level 1 policies in virtue ethics except for Russian and
Spanish. This highlights the nuanced interaction between policy levels,
ethical frameworks, and model performance.

GPT-4 exhibits improved reasoning ability when Level 2 policies are applied in
Arabic, Russian, and Spanish, as compared to English within the
consequentialist framework. Deontological policies in Level 2 work better for
Russian and Spanish than for English. However, for virtue ethics-based
policies, GPT-4 shows a distinct advantage for English. Conversely,
Llama2-70B-Chat demonstrates notably superior performance with all ethical
policies when expressed in English, as compared to all other languages.
Overall, a general trend emerges where all models tend to perform less
effectively in Hindi and Swahili, while achieving their best results in
English and Russian. These observations provide valuable insights into the
models’ ethical reasoning abilities in various linguistic and ethical
contexts.

Figure 1 provides a visual representation of our comparative analysis,
illustrating the bias and confusion scores of each model across dilemmas and
languages. From the figure, it becomes evident that GPT-4 exhibits the lowest
levels of bias among the models, while ChatGPT demonstrates the highest bias
levels. Interestingly, all models exhibit similar bias scores in both the
English and Spanish, as well as in the case of Hindi and Chinese.

On comparing the confusion scores, Llama2-70B-Chat consistently shows the
highest scores among all models, with GPT-4 displaying the lowest confusion
scores. Notably, GPT-4’s behavior varies across languages in the Rajesh
dilemma, highest confusion score being observed for Swahili. In the context of
Hindi, all models exhibit significantly divergent behavior when compared to
their performance in English.

| English | Arabic | Chinese | Hindi | Russian | Spanish | Swahili
---|---|---|---|---|---|---|---
ChatGPT
Heinz | 100% | 100% | 76.6% | 100% | 83.3% | 66.6% | 96.6%
Monica | 100% | 66.6% | 100% | 100% | 100% | 100% | 100%
Rajesh | 100% | 66.6% | 100% | 100% | 100% | 66.6% | 66.6%
Timmy | 100% | 83.3% | 100% | 50% | 96.6% | 83.3% | 83.3%
GPT-4
Heinz | 100% | 100% | 100% | 50% | 100% | 100% | 100%
Monica | 100% | 100% | 100% | 100% | 100% | 100% | 100%
Rajesh | 100% | 100% | 100% | 66.6% | 63.3% | 100% | 56.6%
Timmy | 66.7% | 100% | 86.6% | 100% | 100% | 100% | 100%
Llama2-70B-Chat
Heinz | 100% | 66.7% | 83.3% | 66.6% | 66.6% | 100% | 50%
Monica | 100% | 66.7% | 50% | 83.3% | 66.7% | 100% | 50%
Rajesh | 83.3% | 66.7% | 66.7% | 66.7% | 100% | 66.7% | 100%
Timmy | 66.7% | 66.7% | 66.7% | 66.7% | 83.3% | 83.3% | 57.1%
Table 1: Baseline resolutions percentage of the times the majority resolution
was chosen, Green - $y$ majority, red - $\neg y$ majority and yellow - equal

(a) ChatGPT bias

(b) ChatGPT confusion

(c) GPT-4 bias

(d) GPT-4 confusion

(e) Llama2-70B-Chat bias

(f) Llama2-70B-Chat confusion

Figure 1: Bias and Confusion scores for the three models for each language-
dilemma pair

## 6\. Discussion and Conclusion

In this paper, we presented a study on the multilingual ethical reasoning
capability of three popular LLMs, in the spirit of “ethical policy in prompt"
over value alignment as originally suggested by Rao et al. (2023). Our study
shows that while for some languages, notably English and Russian, LLMs,
especially GPT-4, has superior ethical reasoning abilities, for low-resource
languages - Hindi and Swahili, all models fail to perform well. Thus, along
the lines of many other studies on multilingual evaluation, our work provides
further evidence in support of the performance gap across languages for LLMs,
and brings out yet another dimension – that of ethical reasoning – where the
gap is prominently evident. Why this gap exists, and how it can be bridged are
two important problems we would like to consider for future studies.

Languages and values are strongly intertwined, as is language and culture. The
World Value Survey Inglehart and Welzel (2010) shows how the values vary by
countries, and therefore, in the languages spoken there. While our study
brings out distinct biases of the LLMs across languages, it is not clear
whether these biases are a reflection of cultural differences across the
languages, or simply an artifact of poor performance. Similar bias patterns
between English and Spanish, and Chinese and Hindi across models provide a
hint that there might be more to this than just performance disparity. This is
an interesting question that calls for further investigation.

Recent studies in neuroscience and psychology has shown that humans, most of
the time, arrive at a moral judgment akin to aesthetic judgments shaped by
their past experiences and cultural biases, rather than by reasoning Haidt
(2001). This

Model | Level | Arabic | Chinese | English | Hindi | Russian | Spanish | Swahili
---|---|---|---|---|---|---|---|---
ChatGPT | Level 0 | 66.0 | 54.2 | 60.4 | 55.9 | 68.1 | 50.0 | 50.0
Level 1 | 58.3 | 52.1 | 59.0 | 49.3 | 55.6 | 50.7 | 50.7
Level 2 | 54.2 | 53.5 | 56.9 | 50.2 | 56.3 | 48.6 | 48.6
Average | 59.5 | 53.3 | 58.8 | 51.8 | 60.0 | 49.8 | 49.8
GPT-4 | Level 0 | 81.3 | 61.8 | 95.8 | 61.1 | 85.6 | 90.9 | 66.7
Level 1 | 84.0 | 79.9 | 95.8 | 68.8 | 95.5 | 91.7 | 75.0
Level 2 | 72.9 | 68.1 | 88.2 | 58.3 | 80.6 | 82.6 | 72.9
Average | 79.4 | 69.9 | 93.3 | 62.7 | 87.2 | 88.4 | 71.5
Llama2 | Level 0 | 47.2 | 61.8 | 81.9 | 51.4 | 73.6 | 63.9 | 40.5
Level 1 | 48.9 | 60.4 | 79.9 | 50.0 | 73.6 | 68.8 | 42.5
Level 2 | 45.8 | 59.7 | 72.2 | 50.7 | 63.9 | 54.9 | 40.6
Average | 47.3 | 60.6 | 78.0 | 50.7 | 70.4 | 62.5 | 41.2
Table 2: Accuracy (%) (wrt ground truth) of resolution for policies averaged
over types of ethics and abstraction levels.

(a) ChatGPT

(b) GPT-4

(c) Llama2-70B-Chat

Figure 2: Accuracy(%) (wrt ground truth) of resolution for policies of
different types and levels of abstraction across different languages

is also known to be the reason behind implementation of unfair policies by
governments and organizations, even if the people involved in making these
decisions had the right intentions. In this light, use of LLMs for ethical
reasoning support across cultures, values and languages can be a very
promising use-case with significant large scale positive impact.

## Broader Impact Statement

Our framework is subject to some key limitations. Firstly, it relies on the
latest models, such as ChatGPT, GPT-4, and Llama2-70B-Chat, for ethical
reasoning and the results cannot be generalized to all current models and
primarily supports an ’in context’ ethical policy approach. However, we
anticipate that forthcoming language models will enhance this capability.
Another limitation pertains to the construction of dilemmas, moral policies,
and ideal resolutions which is designed by Rao et al. (2023) who mention that
these may include some bias due to their ethnically homogenous background,
potentially limiting the diversity of representation. Our study focuses on a
limited set of languages, primarily emphasizing linguistic diversity, which
may restrict the generalizability of our findings to languages not included.
Additionally, using Google Translator for multilingual dilemma translation
introduces the potential for translation errors. Despite these constraints,
our research provides valuable insights into the cross-cultural ethical
decision-making of Large Language Models (LLMs) across diverse languages,
underscoring the importance of addressing these limitations in future
investigations to enhance the strength and robustness of our findings. An
ethical concern stemming from our research is the potential misinterpretation
that GPT-4’s superior ethical reasoning capabilities could imply its readiness
for real-life ethical decision-making. This assumption can be perilous, as the
model’s testing is confined to just seven languages, and caution should be
exercised against generalizing its performance to untested languages. It’s
important to emphasize that our current study doesn’t offer a robust
foundation for employing LLMs in moral judgment processes, and further
research and considerations are warranted.

## Acknowledgements

We would like to thank the following people and their corresponding native
languages for their help in validation of the translations: Abdelrahman Atef
Mohamed Ali Sadallah (Arabic, MBZUAI), Hongyi Zhang (Chinese, Microsoft
Corporation), Roman Kazakov (Russian, MBZUAI), Emilio Cueva (Spanish, MBZUAI),
Jesus Ortiz Barajas (Spanish MBZUAI), Millicent Ochieng (Swahili, Microsoft
Corporation)

##

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# From the chiral model of TBG to the Bistritzer–MacDonald model

Simon Becker<EMAIL_ADDRESS>ETH Zurich, Institute for Mathematical
Research, Rämistrasse 101, 8092 Zurich, Switzerland  and  Maciej Zworski
<EMAIL_ADDRESS>Department of Mathematics, University of California,
Berkeley, CA 94720, USA

###### Abstract.

We analyse the splitting of exact flat bands in the chiral model of the
twisted bilayer graphene (TBG) when the $AA^{\prime}/BB^{\prime}$ coupling of
the full Bistritzer–MacDonald model is taken into account. The first-order
perturbation caused by the $AA^{\prime}/BB^{\prime}$ potential the same for
both bands and satisfies interesting symmetries (see (1.3)), in particular it
vanishes on the line defined by the $K$ points. The splitting of the flat
bands is governed by the quadratic term which vanishes at the $K$ points.

##  1\. Introduction and statement of the main result

The Bistritzer–MacDonald Hamiltonian (BMH) [BiMa11] is considered a good model
for the study of twisted bilayer graphene (TBG) and is famous for an accurate
prediction of the twisting angle at which superconductivity occurs [Ca*18]
(see [CGG22, Wa*22] for its rigorous derivation).

Figure 1.  Plots of $k\mapsto E_{\pm 1}(\alpha,\lambda,k)$ for $\alpha$ the
first real magic element of $\mathcal{A}$ and
$\lambda=10^{-3},10^{-2},10^{-1}$. We see that for very small coupling the
flat bands “move together” and split only when the coupling gets larger; the
quadratic term controls the splitting of the bands, see Figure 2. For animated
versions see https://math.berkeley.edu/~zworski/Chiral2BM.mp4 and
https://math.berkeley.edu/~zworski/1band_1D.mp4.

In our notation it takes the form

$H(\alpha,\lambda):=\begin{pmatrix}\lambda C&D(\alpha)^{*}\\\
D(\alpha)&\lambda C\end{pmatrix}:H^{1}(\mathbb{C};\mathbb{C}^{4})\to
L^{2}(\mathbb{C};\mathbb{C}^{4}),\ \ \alpha\in\mathbb{C},\ \
\lambda\in\mathbb{R},$ (1.1)

where $D(\alpha)$ and $C$ are defined in (1.4). The bands for (1.1) are given
as the eigenvalues of $e^{-i\langle k,z\rangle}H(\alpha,\lambda)e^{i\langle
k,z\rangle}$ acting on $L^{2}_{0}$ (a subspace of
$L^{2}_{\rm{loc}}(\mathbb{C};\mathbb{C}^{4})$ satisfying a periodicity
condition – see (2.3)). The chiral limit of BMH corresponds to $\lambda=0$ and
its elegant mathematical properties have been exploited first by
Tarnopolsky–Kruchkov–Vishwanath [TKV19], then mathematically by Watson–Luskin
[WaLu21], and the authors and their collaborators [Be*21, Be*22, BHZ22a,
BHZ22b, BHZ23, Ya23, GZ23, Be*23, HZ23].

The bands in the chiral model, that is the eigenvalues of $e^{-i\langle
k,z\rangle}H(\alpha,0)e^{i\langle k,z\rangle}$ on $L^{2}_{0}$, are symmetric
with respect to $0$ and we can label them as $E_{\pm\ell}(\alpha,0,k)$,
$E_{-\ell}=-E_{\ell}$, $E_{1}\geq 0$. $\ell\in\mathbb{Z}\setminus\\{0\\}$.
Following [Be*22, BHZ22b], there exists a discrete set of magic $\alpha$’s,
$\mathcal{A}\subset\mathbb{C}$, such that $E_{\pm 1}(\alpha,0,k)\equiv 0$ for
all $k\in{\mathbb{C}}$, if and only if $\alpha\in\mathcal{A}$ (the angle of
twisting is proportional to $1/\alpha$). A magic $\alpha$ is simple if
$E_{\ell}(\alpha,0,k)>0$ for $\ell>1$ – see [BHZ22b] and, for the existence
and properties of degenerate magic $\alpha$’s, [BHZ23].

Since flat bands in the chiral limit are uniformly gapped from the rest of the
spectrum [BHZ22b, BHZ23], we can still consider, for small $\lambda$, the
bands $E_{\pm\ell}(\alpha,\lambda,k)$, where for $\alpha\in\mathcal{A}$,
$E_{\pm\ell}(\alpha,0,k)\equiv 0$. The purpose of this note is to use
symmetries of BHM (1.1), some properties of theta functions, and standard
degenerate perturbation theory to show

Theorem. Suppose that $\alpha\in\mathcal{A}\cap\mathbb{R}$ is simple and that
$k\mapsto E_{\pm 1}(\alpha,\lambda,k)$ are the two lowest bands (in absolute
value) of BMH in (1.1). Then there exist
$e(\alpha,\bullet),f(\alpha,\bullet)\in C^{\infty}(\mathbb{C}/\Lambda^{*})$
such that

$E_{\pm
1}(\alpha,\lambda,k)=e(\alpha,k)\lambda\pm|f(\alpha,k)|\lambda^{2}+\mathcal{O}(\lambda^{3}),\
\ \lambda\to 0,$ (1.2)

$f(\pm K)=0$, ($\omega K\equiv K\\!\\!\\!\mod\\!\Lambda^{*}$, $K\neq 0$), and

$e(\alpha,k)=-e(\alpha,-k)=-e(\alpha,\bar{k})=e(\alpha,\omega k),\ \
\omega=e^{{2\pi i}/3}.$ (1.3)

Remarks. 1\. A surprising feature, which does not seem to have been observed
before, is that the linear term does not depend on the band: the two bands
move together for small coupling constants – see Figure 1.

2\. We observe numerically that the coefficient of the linear term is much
smaller than the coefficient of the quadratic term and that the approximation
(1.2) is reasonable for physically relevant coupling $\lambda\simeq
0.7\alpha$, for first real magic $\alpha$ – see Figure 2.

3\. The situation is different for complex $\alpha$’s showing that our
assumption $\alpha\in\mathcal{A}\cap\mathbb{R}$ is necessary to have purely
quadratic behaviour in band splitting – see Figure 3.

4\. Similar analysis can be performed for double magic $\alpha$ but to keep
things simple we restrict ourselves to numerical illustrations – see §5.

We now review the definitions needed for the notation of (1.1) and the
theorem.

We recall that general version of $D(\alpha)$ (and the corresponding self
adjoint Hamiltonian) and of $C$:

$D(\alpha)=\begin{pmatrix}2D_{\bar{z}}&\alpha U(z)\\\ \alpha
U(-z)&2D_{\bar{z}}\end{pmatrix},\ \ \ \ C:=\begin{pmatrix}0&V(z)\\\
V(-z)&0\end{pmatrix},$ (1.4)

where the parameter $\alpha$ is proportional to the inverse relative twisting
angle. With $\omega=e^{2\pi i/3}$ and $K:=\frac{4}{3}\pi$, we assume that

$U(z+\gamma)=e^{i\langle\gamma,K\rangle}U(z),\ \gamma\in\Lambda,\quad U(\omega
z)=\omega U(z),\ \ \overline{U(\bar{z})}=-U(-z),$ (1.5)

$\Lambda:=\mathbb{Z}\oplus\omega\mathbb{Z}$, and

$V(z)=V(\bar{z})=\overline{V(-z)},\ \ V(\omega z)=V(z),\ \
V(z+\gamma)=e^{i\langle\gamma,K\rangle}V(z).$ (1.6)

The specific potentials (see [Be*21] and references given there) and the ones
we use in numerical experiments are

$U(z)=-\tfrac{4}{3}\pi i\sum_{\ell=0}^{2}\omega^{\ell}e^{i\langle
z,\omega^{\ell}K\rangle},\ \ V(z)=\sum_{\ell=0}^{2}e^{i\langle
z,\omega^{\ell}K\rangle},\ \ K=\tfrac{4}{3}\pi.$ (1.7)

Figure 2.  On the left a log-log plot of
$\lambda\mapsto\max_{k}|E_{1}(\alpha,\lambda,k)|$ which shows the transition
from the linear to quadratic dominant behaviour for the first two magic
$\alpha\simeq\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0.586},\,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2.221}\\};$
on the right a log-log plot of $\lambda\mapsto|E_{1}(\alpha,\lambda,0)|$ for
two degenerate magic angles
$\alpha\simeq\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0.8538},\,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2.701}\\}$.
The choice $\lambda\approx 0.7\alpha$ is indicated by vertical dashed lines.
The potentials used on the left are given in (1.7) and correspond to the
standard BMH. On the right we use $U_{1}(z):=(U(z)-U(2z))/\sqrt{2}$ and
$V(z)$. That potential exhibits double magic $\alpha$’s on the real axis – see
[BHZ23].

Figure 3.  Left: log-log plot of
$\lambda\mapsto\max_{k}|E_{1}(\alpha,\lambda,k)|$ which shows the simple
complex magic angles
$\alpha\simeq\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1.121+1.57i},\,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1.312+2.862i}\\}$;
Right: log-log plot of $\lambda\mapsto\max_{k}|E_{1}(\alpha,\lambda,k)|$ for
two two-fold degenerate complex magic
$\alpha\simeq\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0.963+0.987i},\,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1.226+2.268i}\\}$
both with $U_{1}$ and $V$ as in Figure 2. Vertical lines at
$|\lambda/\alpha|=0.7$

We conclude this introduction by describing the relation between
$H(\lambda,\alpha)$ given in (1.1) and the representation of that Hamiltonian
in the physics literature [BiMa11], [TKV19]. The Bistritzer-MacDonald
Hamiltonian for twisted bilayer graphene reads

$\mathscr{H}(\beta,\zeta)=\begin{pmatrix}-i\boldsymbol{\sigma}\cdot\nabla&T(\beta,\zeta)\\\
T(\beta,\zeta)^{\dagger}&-i\boldsymbol{\sigma}\cdot\nabla\end{pmatrix},$ (1.8)

where we defined $\boldsymbol{\sigma}=(\sigma_{x},\sigma_{y})$,
$\mathbf{r}=(x,y)$,
$\boldsymbol{\sigma}\cdot\nabla=\sigma_{x}\partial_{x}+\sigma_{y}\partial_{y}$
($\sigma_{\bullet}$ are the Pauli matrices) and interlayer tunnelling matrix

$T(\beta,\zeta\mathbf{r})=\begin{pmatrix}\tilde{\beta}V(\zeta\mathbf{r})&\beta\overline{U(-\zeta\mathbf{r})}\\\
\beta U(\zeta\mathbf{r})&\tilde{\beta}V(\zeta\mathbf{r})\end{pmatrix}.$

The honeycomb graphene lattice has a non-equivalent pair of atoms in its
fundamental domain labelled by $A,B$ respectively with $A^{\prime},B^{\prime}$
used for the second layer in case of TBG. In the matrix $T$, the potentials
$U$ and $V$ facilitating $AB^{\prime}$ tunnelling and
$AA^{\prime}/BB^{\prime}$ respectively, see Figure 4, are given for
$q_{i}=R^{i}(0,-1)$ with $2\pi/3$ rotation matrix
$R=\frac{1}{2}\begin{pmatrix}-1&-\sqrt{3}\\\ \sqrt{3}&-1\end{pmatrix}$ by

$\begin{gathered}U(\mathbf{r})=\sum_{i=0}^{2}\omega^{i}e^{-iq_{i}\cdot\mathbf{r}},\
\ V(\mathbf{r})=\sum_{i=0}^{2}e^{-iq_{i}\cdot\mathbf{r}},\ \
q_{i}:=R^{i}\begin{pmatrix}\ \ 0\\\ -1\end{pmatrix},\ \
R:=\tfrac{1}{2}\begin{pmatrix}-1&-\sqrt{3}\\\
\sqrt{3}&-1\end{pmatrix}.\end{gathered}$

($R$ is the $2\pi/3$ rotation matrix.)

Let $K=\operatorname{diag}(1,\sigma_{x},1)$ then upon making the change of
variables $\zeta\mathbf{r}\mapsto\mathbf{r}$, $\lambda=\beta/\zeta$, and
$\alpha=\beta/\zeta$, we find

$K\mathscr{H}(\beta,\zeta)K=\zeta\begin{pmatrix}\lambda
C&D(\alpha)^{\dagger}\\\ D(\alpha)&\lambda C\end{pmatrix}$

where

$\begin{split}D(\alpha)&=\begin{pmatrix}D_{x}+iD_{y}&\alpha U(\mathbf{r})\\\
\alpha U(-\mathbf{r})&D_{x}+iD_{y}\end{pmatrix}\text{ and
}C=\begin{pmatrix}0&V(\mathbf{r})\\\
V(-\mathbf{r})&0\end{pmatrix}.\end{split}$

The physics coordinates $\mathbf{r}=(x,y)$ are related to our Hamiltonian
(1.1) by the change of coordinates

$x+iy=\tfrac{4}{3}\pi iz,\quad\tfrac{1}{3}\Gamma=\tfrac{4}{3}\pi
i\Lambda,\quad 3\Gamma^{*}=\frac{3}{4\pi i}\Lambda^{*}$

where $\tfrac{1}{3}\Gamma$ is the moiré lattice and $\Gamma$ is the lattice of
periodicity of the potentials $U(\mathbf{r})$ and $V(\mathbf{r})$. (In the
figures we use coordinates based on $(x,y)$ and the corresponding $k$
coordinates.)

Notation. In this paper we use the physics notation: for an operator $A$ on
$L^{2}(M,dm)$, $\langle u|A|v\rangle:=\int_{M}Av\,\bar{u}\,dm$. Also,
$|u\rangle$ denotes the operator $\mathbb{C}\ni\mu\to\mu u\in L^{2}$ and
$\langle u|$, its adjoint $L^{2}\ni v\to\langle u|v\rangle\in\mathbb{C}$. For
$z,w\in\mathbb{C}\simeq\mathbb{R}^{2}$, we use the real inner product,
$\langle z,w\rangle:=\operatorname{Re}z\bar{w}$.

Acknowledgements. MZ gratefully acknowledges partial support by the NSF grant
DMS-1901462 and by the Simons Foundation under a “Moiré Materials Magic”
grant.

Figure 4. $|V(\mathbf{r})|$ (left) peaked at $AA^{\prime}/BB^{\prime}$
regions, $|U(\mathbf{r})|$ (center) concentrated at $AB^{\prime}$ regions, and
figure of moiré fundamental cell with regions of particle-type overlap
(right).

##  2\. Symmetries of BMH

In this section we review the symmetries of BMH given in (1.1) and the spaces
used in the definition of the bands $k\mapsto E_{\pm\ell}(\alpha,\lambda,k)$,
$\ell\in\mathbb{Z}\setminus\\{0\\}$. We follow the presentation of [Be*21,
Supplementary Materials], with modifications due to the variables used here,
see [BHZ22b, Appendix A].

###  2.1. Translation and rotation symmetries

The translation symmetry for $u\in
L^{2}_{\rm{loc}}(\mathbb{C};\mathbb{C}^{2})$ is given by

$L_{\gamma}u:=\begin{pmatrix}e^{i\langle\gamma,K\rangle}&0\\\
0&e^{-i\langle\gamma,K\rangle}\end{pmatrix}u(z+\gamma),\ \ \
\gamma\in\Lambda,\ \ K=\tfrac{4}{3}\pi.$ (2.1)

We extend this action diagonally for $w\in
L^{2}_{\rm{loc}}(\mathbb{C};\mathbb{C}^{4})$:

$\mathscr{L}_{\gamma}w=\begin{pmatrix}L_{\gamma}w_{1}\\\
L_{\gamma}w_{2}\end{pmatrix},\ \ \ w=\begin{pmatrix}w_{1}\\\
w_{2}\end{pmatrix},\ \ w_{j}\in L^{2}_{\rm{loc}}(\mathbb{C};\mathbb{C}^{2}).$

We then have have, in the notation of (1.1), (1.4) with $U$, $V$ satisfying
(1.5) and (1.6), $L_{\gamma}D(\alpha)=D(\alpha)L_{\gamma}$, and
$\mathscr{L}_{\gamma}H(\alpha,\lambda)=H(\alpha)\mathscr{L}_{\gamma}$. We also
define

$\begin{gathered}\Omega:L^{2}_{\rm{loc}}(\mathbb{C};\mathbb{C}^{2})\to
L^{2}_{\rm{loc}}(\mathbb{C};\mathbb{C}^{2}),\ \ \
\mathscr{C}:L^{2}_{\rm{loc}}(\mathbb{C};\mathbb{C}^{4})\to
L^{2}_{\rm{loc}}(\mathbb{C};\mathbb{C}^{4}),\\\ \Omega u(z):=u(\omega z),\ \
\mathscr{C}\begin{pmatrix}w_{1}\\\ w_{2}\end{pmatrix}:=\begin{pmatrix}\Omega
w_{1}\\\ \bar{\omega}\Omega w_{2}\end{pmatrix},\end{gathered}$ (2.2)

so that $\Omega D(\alpha)=\omega D(\alpha)$ and
$\mathscr{C}H(\alpha,\lambda)=H(\alpha,\lambda)\mathscr{C}$.

The natural subspaces of $L^{2}_{\rm{loc}}(\mathbb{C};\mathbb{C}^{p})$,
$p=2,4$, are given by

$\begin{gathered}L^{2}_{k}(\mathbb{C};\mathbb{C}^{4}):=\\{u\in
L^{2}_{\rm{loc}}(\mathbb{C},\mathbb{C}^{4}):\mathscr{L}_{\gamma}u=e^{i\langle
k,\gamma\rangle}u\\},\end{gathered}$ (2.3)

with $L^{2}_{k}(\mathbb{C};\mathbb{C}^{2})$ defined by replacing
$\mathscr{L}_{\gamma}$ by $L_{\gamma}$. When the context is clear we simply
write $L^{2}_{k}$ for either space. We note that these spaces depend only on
the congruence class of $k$ in $\mathbb{C}/\Lambda^{*}$,

$\Lambda^{*}:=\frac{4\pi i}{\sqrt{3}}\Lambda,\ \ \ k\mapsto
z(k):=\frac{\sqrt{3}k}{4\pi i},\ \ \ \Lambda^{*}\to\Lambda,\ \ \langle
p,\gamma\rangle\in 2\pi\mathbb{Z},\ \ p\in\Lambda^{*},\ \gamma\in\Lambda.$

The points of high symmetry, $\mathcal{K}$, are defined by demanding that
$\omega p\equiv p\\!\mod\\!\Lambda^{*}$. They are given by
$\mathcal{K}=\\{K,-K,0\\}+\Lambda^{*}$, $K=\frac{4}{3}\pi$ (see (1.7), (2.1)).
For $k\in\mathcal{K}/\Lambda^{*}$ and $p\in\mathbb{Z}_{3}$ we also define

$L_{k,p}^{2}(\mathbb{C};\mathbb{C}^{4}):=\\{u\in
L^{2}_{k}(\mathbb{C};\mathbb{C}^{4}):\mathscr{C}^{p}u=\bar{\omega}^{p}u\\},$
(2.4)

with the definition of $L_{k,p}^{2}(\mathbb{C};\mathbb{C}^{2})$ obtained by
replacing $\mathscr{C}$ by $\Omega$. We have orthogonal decompositions
$L^{2}_{k}=\bigoplus_{p\in\mathbb{Z}_{3}}L_{k,p}$,
$k\in\mathcal{K}/\Lambda^{*}$. Also, the actions of $\mathscr{L}_{\gamma}$ and
$\mathscr{C}$ on $L^{2}_{p,k}$ commute. (In general,
$\mathscr{L}_{\gamma}\mathscr{C}=\mathscr{C}\mathscr{L}_{\omega\gamma}$.)

As recalled in §1, the Floquet spectrum is obtained by taking the spectrum of

$H_{k}(\alpha,\lambda):=e^{-i\langle z,k\rangle}H(\alpha,\lambda)e^{i\langle
z,k\rangle}=\begin{pmatrix}\lambda C&D(\alpha)^{*}+\bar{k}\\\
D(\alpha)+k&\lambda C\end{pmatrix},$ (2.5)

on $L^{2}_{0}$ with the domain given by $H^{1}_{\rm{loc}}\cap L^{2}_{0}$.

The spaces $L^{2}_{k,p}$ will be crucial in showing that $f(\alpha,\pm K)=0$
in (1.2). That, and the proof of (1.3), requires additional symmetries, well
known in the physics literature.

###  2.2. Additional symmetries

We start with the symmetries of $D(\alpha)$, a non-self-adjoint building block
of BMH (1.4) for $U$ satisfying the conditions in (1.5). We start with

$\mathscr{E}D(\alpha)\mathscr{E}^{-1}=-D(\alpha),\ \
\mathscr{E}v(z):=Jv(-z),\quad J:=\begin{pmatrix}\ \ 0&1\\\ -1&0\end{pmatrix},\
\ \mathscr{E}:L^{2}_{0}\to L^{2}_{0}.$ (2.6)

If $\alpha\in\mathcal{A}$ is simple then the kernel, $\mathbb{C}u_{0}$, of
$D(\alpha)$ on $L^{2}_{0}$ is one dimensional (see [BHZ22b, Theorem 2]) and,
as the spectrum of $\mathscr{E}$ given by $\\{\pm i\\}$
($\mathscr{E}^{2}=-I$), $\mathscr{E}u_{0}=\pm iu_{0}$ (for one choice of
sign). Hence,

$u_{0}(z)=\begin{pmatrix}\psi(z)\\\ \pm i\psi(-z)\end{pmatrix},\ \ \
\psi(z+\gamma)=e^{-i\langle\gamma,K\rangle}\psi(z),\ \ \gamma\in\Lambda.$
(2.7)

(This observation about the Bloch wave function associated to the $\Gamma$
point was made in – see the discussion before [TKV19, (26)].) Interestingly,
numerical evidence suggests that the signs alternate, with $-$ at the first
real magic angle, then $+$, and so on. For future reference we also recall
from [BZ23, Proposition 3.6] that $u_{0}\in L^{2}_{0,2}$, that is

$\psi(\omega z)=\omega\psi(z),\ \ \psi(z)=z\psi_{1}(z),\ \ \psi_{1}(0)\neq
0\text{ with }\psi_{1}\in C^{\omega}({\mathbb{C}};{\mathbb{C}}).$ (2.8)

We next recall an anti-linear symmetry,

${\mathscr{Q}}v(z):=\overline{v(-z)},\ \ \ \
{\mathscr{Q}}(D(\alpha)+k){\mathscr{Q}}=(D(\alpha)+k)^{*},$ (2.9)

and two linear symmetries,

$\mathscr{H}\begin{pmatrix}u_{1}(z)\\\
u_{2}(z)\end{pmatrix}:=\begin{pmatrix}-iu_{2}(-z)\\\ iu_{1}(-z)\end{pmatrix},\
\ \ \ \mathscr{H}(D(\alpha)+k)\mathscr{H}=-(D(\alpha)-k),$ (2.10)

(We have, in the notation of (2.6), $i\mathscr{H}=\mathscr{E}$; the factor $i$
is there for $\mathscr{H}^{2}=I$.)

$\mathscr{N}u(z):=\begin{pmatrix}u_{2}(-\bar{z})\\\
u_{1}(-\bar{z})\end{pmatrix},\ \ \ \
\mathscr{N}(D(\alpha)+k)\mathscr{N}=-(D(\bar{\alpha})-\bar{k})^{*}.$ (2.11)

All of these operators satisfy, $A^{2}=I$ and $A:L^{2}_{0}\to L^{2}_{0}$.
Remarkably, for potentials in (1.4) satisfying (1.5) and (1.6) these
symmetries give symmetries of BMH (acting on
$L^{2}_{\rm{loc}}(\mathbb{C};\mathbb{C}^{4})$). They have the following quaint
physical names:

$\mathcal{P}\mathcal{T}$/$C_{2z}T$ symmetry:
$\displaystyle\mathcal{P}\mathcal{T}:=\begin{pmatrix}0&\mathscr{Q}\\\
\mathscr{Q}&0\end{pmatrix},$ (2.12) particle-hole symmetry:
$\displaystyle\mathscr{S}:=\begin{pmatrix}\mathscr{H}&0\\\
0&\mathscr{H}\end{pmatrix},$ mirror symmetry:
$\displaystyle\mathscr{M}:=\begin{pmatrix}\ 0&i\mathscr{N}\\\
-i\mathscr{N}&0\end{pmatrix}.$

These operators all satisfy $A^{2}=I$ but the mapping properties are more
complicated, see (2.14) below. The commuting components of the
$\mathcal{P}\mathcal{T}$ symmetry are given by the parity-inversion and time-
reversal operators:

$\mathcal{P}u:=\begin{pmatrix}0&I_{{\mathbb{C}}^{2}}\\\ \
I_{{\mathbb{C}}^{2}}&0\end{pmatrix}u(-z),\ \ \ \
\mathcal{T}u(z):=\overline{u(z)}.$

We also remark that the antilinear symmetry $\mathscr{A}$ which holds for more
general $D(\alpha)$ is given by $\mathscr{A}=-i\mathscr{H}\mathscr{Q}$, see
[BHZ23, §6.1] and [BZ23, §3.5].

We summarize the basic properties of these symmetries in the following

######  Proposition 1.

With the definitions given in (1.1),(2.12) and for potentials satisfying
(1.5), (1.6), we have, for $\alpha\in\mathbb{C}$, $\lambda\in\mathbb{R}$,

$\begin{gathered}\mathcal{P}\mathcal{T}H(\alpha,\lambda)=H(\alpha,\lambda)\mathcal{P}\mathcal{T},\
\ \ \ \mathscr{M}H(\alpha,\lambda)=H(\bar{\alpha},\lambda)\mathscr{M},\\\
\mathscr{S}H(\alpha,\lambda)=-H(\alpha,\lambda)\mathscr{S}.\end{gathered}$
(2.13)

In addition, in the notation of (2.4), and for $k=\\{0,\pm K\\}+\Lambda^{*}$,
$p\in\mathbb{Z}_{3}$, $K=\frac{4}{3}\pi$,

$\mathcal{P}\mathcal{T}:L^{2}_{k,p}\to L^{2}_{k,-p+1},\ \
\mathscr{S}:L^{2}_{k,p}\to L^{2}_{k+K,p},\ \ \mathscr{M}:L^{2}_{k,p}\to
L^{2}_{-k,1-p},$ (2.14)

where
$L^{2}_{\bullet,\bullet}=L^{2}_{\bullet,\bullet}(\mathbb{C};\mathbb{C}^{4})$.

###### Proof.

To establish (2.13) we start with the $\mathcal{P}\mathcal{T}$ symmetry where
in view of (2.9) we have

$\begin{pmatrix}0&\mathscr{Q}\\\
\mathscr{Q}&0\end{pmatrix}\begin{pmatrix}\lambda C&D(\alpha)^{*}\\\
D(\alpha)&\lambda
C\end{pmatrix}=\begin{pmatrix}\lambda\mathscr{Q}C\mathscr{Q}&D(\alpha)^{*}\\\
D(\alpha)&\lambda\mathscr{Q}C\mathscr{Q}\end{pmatrix}\begin{pmatrix}0&\mathscr{Q}\\\
\mathscr{Q}&0\end{pmatrix}.$

Since $\overline{V(-z)}=V(z)$ (see (1.6)) we have $\mathscr{Q}C\mathscr{Q}=C$.

For the mirror symmetry in (2.12) we use (2.11) to see that

$\begin{pmatrix}\ 0&i\mathscr{N}\\\
-i\mathscr{N}&0\end{pmatrix}\begin{pmatrix}\lambda C&D(\alpha)^{*}\\\
D(\alpha)&\lambda
C\end{pmatrix}=\begin{pmatrix}-\mathscr{\lambda}\mathscr{N}C\mathscr{N}&\ \
D(\bar{\alpha})^{*}\\\
D(\bar{\alpha})&-\lambda\mathscr{N}C\mathscr{N}\end{pmatrix}\begin{pmatrix}\
0&i\mathscr{N}\\\ -i\mathscr{N}&0\end{pmatrix}.$

From $V(\bar{z})=V(z)$ (see (1.6), we conclude $\mathscr{N}C\mathscr{N}=C$,
proving $\mathscr{M}H(\alpha,\lambda)=H(\bar{\alpha},\lambda)\mathscr{M}$.
Finally, $\mathscr{S}H(\alpha,\lambda)=-H(\bar{\alpha},\lambda)\mathscr{S}$
follows from (2.10) and $\mathscr{H}C\mathscr{H}=-C$.

To obtain (2.14), we first note that
$L_{\gamma}\mathscr{Q}=\mathscr{Q}L_{-\gamma}$,
$L_{\gamma}\mathscr{H}=e^{i\langle\gamma,K\rangle}\mathscr{H}L_{-\gamma}$ and
$L_{\gamma}\mathscr{N}=L_{-\bar{\gamma}}\mathscr{N}$. Hence,

$\mathscr{L}_{\gamma}\mathcal{P}\mathcal{T}=\mathcal{P}\mathcal{T}\mathscr{L}_{-\gamma},\
\ \ \ \mathscr{L}_{\gamma}\mathscr{S}=e^{i\langle
K,\gamma\rangle}\mathscr{S}\mathscr{L}_{\gamma},\ \ \ \
\mathscr{L}_{\gamma}\mathscr{M}=\mathscr{M}\mathscr{L}_{-\bar{\gamma}}.$
(2.15)

In the notation of (2.2), $\Omega\mathscr{Q}=\mathscr{Q}\Omega$,
$\Omega\mathscr{H}=\Omega\mathscr{H}$ and
$\Omega\mathscr{N}=\mathscr{N}\Omega^{*}$ and thus

$\mathscr{C}\mathcal{P}\mathcal{T}=\bar{\omega}\mathcal{P}T\mathscr{C},\ \ \ \
\mathscr{C}\mathscr{S}=\mathscr{S}\mathscr{C},\ \ \ \
\mathscr{C}\mathscr{M}=\bar{\omega}\mathscr{M}\mathscr{C}^{*}.$ (2.16)

We can now check the mapping properties (2.14): for $u\in L^{2}_{p,k}$,

$\mathscr{L}_{\gamma}\mathscr{C}\mathcal{P}\mathcal{T}u=\bar{\omega}\mathcal{P}\mathcal{T}\mathscr{C}\mathscr{L}_{-\gamma}u=\bar{\omega}\mathcal{P}\mathcal{T}(\bar{\omega}^{p}e^{-i\langle\gamma,k\rangle}u)=e^{i\langle\gamma,k\rangle}\bar{\omega}^{1-p}\mathcal{P}Tu,$

that is, $\mathcal{P}\mathcal{T}u\in L^{2}_{k,1-p}$. Then

$\mathscr{L}_{\gamma}\mathscr{C}\mathscr{S}u=e^{i\langle
K,\gamma\rangle}\mathscr{S}\mathscr{C}\mathscr{L}_{\gamma}u=e^{i\langle
K+k,\gamma\rangle}\bar{\omega}^{p}\mathscr{S}u,$

that is $\mathscr{S}u\in L^{2}_{k+K,p}$. Finally,

$\mathscr{L}_{\gamma}\mathscr{C}\mathscr{M}u=\bar{\omega}\mathscr{M}\mathscr{C}^{*}\mathscr{L}_{-\bar{\gamma}}u=\bar{\omega}^{1-p}e^{-i\langle\gamma,k\rangle}\mathscr{M}u,$

(since $k=\pm K,0$, $\langle\gamma,k\rangle=\langle\bar{\gamma},k\rangle$)
which means that $\mathscr{M}u\in L^{2}_{-k,1-p}$. This completes the proof of
(2.14). ∎

###  2.3. Protected states of BMH

We now use a modification of the argument from [Be*21] to obtain symmetry
protected states for $H(\alpha,\lambda)$. In the case of the chiral limit
$H(\alpha,0)$, the same conclusion is easier and is based on the more
immediate symmetry of eigenvalues of $H(\alpha,0)$ about $0$ [TKV19, Be*22].

######  Proposition 2.

In the notation of (2.5) and for all $\alpha,\lambda\in\mathbb{R}$,

$\dim\ker_{L^{2}_{0}}H_{\pm K}(\alpha,\lambda)\geq 2,\ \ \
K:=\tfrac{4}{3}\pi,$ (2.17)

$\omega K\\!\equiv K\\!\\!\mod\\!\Lambda^{*}$.

###### Proof.

Since $H_{\pm K}(\alpha,\lambda)=e^{\mp i\langle
z,K\rangle}H(\alpha,\lambda)e^{\pm i\langle z,k\rangle}$, and multiplication
by $e^{\pm i\langle z,K\rangle}$ takes $L^{2}_{0}$ to $L^{2}_{\pm K}$, (2.17)
is equivalent to $\dim\ker_{L^{2}_{\pm K}}H(\alpha,\lambda)\geq 2$, for all
real $\alpha$ and $\lambda$. We now observe that Proposition 2 gives

$\begin{split}&U_{+}H(\alpha,\lambda)U_{+}^{-1}=-H(\alpha,\lambda),\ \ \
U_{+}:=\mathscr{S}\mathscr{M}\mathcal{P}\mathcal{T},\\\
&U_{-}H(\alpha,\lambda)U_{-}^{-1}=-H(\alpha,\lambda),\ \ \
U_{-}:=\mathscr{M}\mathscr{S}\mathcal{P}\mathcal{T}.\end{split}$ (2.18)

Also

$U_{\pm}:L^{2}_{\pm K,p}\to L^{2}_{\pm K,p}.$ (2.19)

In fact, from (2.14) we see that

$\begin{split}&L^{2}_{K,p}\xrightarrow{\mathscr{S}}L^{2}_{2K,p}\xrightarrow{\mathscr{M}}L^{2}_{-2K,1-p}\xrightarrow{\mathcal{P}\mathcal{T}}L^{2}_{-2K,p}=L^{2}_{K,p}\\\
&L^{2}_{-K,p}\xrightarrow{\mathscr{M}}L^{2}_{K,1-p}\xrightarrow{\mathscr{S}}L^{2}_{2K,1-p}\xrightarrow{\mathcal{P}\mathcal{T}}L^{2}_{2K,p}=L^{2}_{-K,p}.\end{split}$

The mapping property (2.19) and (2.18) show that

$\operatorname{Spec}_{L^{2}_{\pm
K,p}}H(\alpha,\lambda)=-\operatorname{Spec}_{L^{2}_{\pm
K,p}}H(\alpha,\lambda).$ (2.20)

If $\\{\mathbf{e}_{j}\\}_{j=1}^{4}$ is the standard basis of $\mathbb{C}^{4}$
then we easily check that

$\displaystyle\ker_{L^{2}_{K,0}}H(0,0)=\mathbb{C}\mathbf{e}_{1},\ \ \ \
\ker_{L^{2}_{-K,0}}H(0,0)=\mathbb{C}\mathbf{e}_{2},$
$\displaystyle\ker_{L^{2}_{K,1}}H(0,0)=\mathbb{C}\mathbf{e}_{3},\ \ \ \
\ker_{L^{2}_{-K,1}}H(0,0)=\mathbb{C}\mathbf{e}_{4},$

and in particular all these spaces are one dimensional. In view of (2.20),
these dimensions have to remain odd for all $\alpha,\lambda\in\mathbb{R}$ and
that shows that $\dim\ker_{L^{2}_{\pm K}}H(\alpha,\lambda)\geq 2$, proving
(2.17). ∎

##  3\. Perturbation theory

Suppose that $\alpha\in\mathcal{A}\cap\mathbb{R}$ is simple (see §1). Then by
[BHZ22b, Theorem 2], $\ker(D(\alpha)+k)=\mathbb{C}u(k)$, $u(k)\neq 0$ for all
$k\in\mathbb{C}$. We also have
$\ker(D^{*}(\alpha)+\bar{k})=\mathbb{C}u^{*}(k)$ and the fact that the kernels
are nonempty for all $k$ implies that $\langle u(k)|u^{*}(k)\rangle=0$.
(Otherwise perturbation theory, as in the Grushin problem below, would give us
invertibility of $D(\alpha)+k$.) We normalize $u(k),u^{*}(k)$ to have norm
one. To simplify notation we temporarily drop $\alpha$ in $D(\alpha)$ and
consider $H_{k}(\alpha,0)$, a self-adjoint Hamiltonian corresponding to the
chiral limit:

$H(k):=\begin{pmatrix}0&D^{*}+\bar{k}\\\ D+k&0\end{pmatrix}.$

We then define

$\begin{gathered}\mathcal{H}(k):=\begin{pmatrix}H(k)&R_{-}(k)\\\
R_{+}(k)&0\end{pmatrix},\ \ \ R_{-}(k):\mathbb{C}^{2}\to L^{2}_{0},\ \ \
R_{+}(k)=R_{-}(k)^{*}:L^{2}_{0}\to\mathbb{C}^{2},\\\
R_{-}(k):=\begin{pmatrix}0&|u(k)\rangle\\\ |u^{*}(k)\rangle&0\end{pmatrix}\ \
\ R_{+}(k)=\begin{pmatrix}0&\langle u^{*}(k)|\\\ \langle
u(k)|&0\end{pmatrix},\end{gathered}$

so that

$\begin{gathered}\mathcal{H}(k)^{-1}=\begin{pmatrix}E(k)&E_{+}(k)\\\
E_{-}(k)&E_{-+}(k)\end{pmatrix},\ \ E_{+}(k)=R_{-}(k),\ \ E_{-}(k)=R_{+}(k)\\\
E(k):=\begin{pmatrix}0&E_{0}(k)^{*}\\\ E_{0}(k)&0\end{pmatrix},\ \ \
E_{-+}(k)\equiv 0,\\\
E_{0}(k):=\left((D(\alpha)+k)|_{|u(k)\rangle^{\perp}\to|u^{*}(k)\rangle^{\perp}}\right)^{-1}(I-|u^{*}(k)\rangle\langle
u^{*}(k)|).\end{gathered}$ (3.1)

We label the eigenvalues of $H(k)$ as $E_{\ell}(k)$,
$\ell\in\mathbb{Z}\setminus\\{0\\}$, $E_{\ell}(k)=-E_{-\ell}(k)$. Under our
assumptions we have $E_{\pm 1}(k)=0$, $E_{\ell}(k)\neq 0$ $|\ell|>1$, see
[BHZ22b, Theorem 2]. We now consider the BMH (1.1) as a perturbation of
$H(k)$:

$H(k,\lambda):=\begin{pmatrix}\lambda C&D^{*}+\bar{k}\\\ D+k&\lambda
C\end{pmatrix},\ \ C=C^{*}.$

For $|\lambda|\ll 1$, the two smallest (in modulus) eigenvalues of
$H(k,\lambda)$, $E_{\pm 1}(k,\lambda)$, are given by the eigenvalues of

$E_{-+}(k,\lambda)=-\sum_{\ell=0}^{\infty}(-1)^{\ell}\lambda^{\ell+1}E_{-}(k)C(E(k)C)^{\ell}E_{+}(k):\mathbb{C}^{2}\to\mathbb{C}^{2},$

see [TaZw23, Proposition 2.12] for this standard fact. Using (3.1) we then
obtain

$\begin{gathered}E_{-+}(k,\lambda)=\begin{pmatrix}\lambda
e_{+}(k)&\lambda^{2}f(k)\\\
\lambda^{2}\overline{f(k)}&e_{-}(k)\end{pmatrix}+\mathcal{O}(\lambda^{3})_{\mathbb{C}^{2}\to\mathbb{C}^{2}},\\\
e_{+}(k)=-\langle u(k)|C|u(k)\rangle,\ \ \ e_{-}(k)=-\langle
u^{*}(k)|C|u^{*}(k)\rangle,\\\ f(k)=\langle
u(k)|\bar{V}E_{0}(k)^{*}\bar{V}|u(k)\rangle.\end{gathered}$ (3.2)

We should stress (see [BHZ22b, §5.2]) that $u(k)$ cannot be made continuous in
$k$ and $\Lambda^{*}$-periodic (even modulo a $k$-independent unitary
transformation, see (3.4) below). However $E_{-+}(k,\lambda)$ is periodic. We
present a quick argument relevant to (1.2).

######  Lemma 3.

In the notation of (3.2), $e_{\pm},f\in C^{\infty}(\mathbb{C})$ and

$e_{\pm}(k+p)=e_{\pm}(k),\ \ f(k+p)=f(k),\ \ \ k\in\mathbb{C},\ \
p\in\Lambda^{*}.$ (3.3)

###### Proof.

We recall that

$\begin{gathered}\tau(p)^{*}(D(\alpha)+k)\tau(p)=D(\alpha)+k+p,\\\
\tau(p)v(z):=e^{i\langle z,p\rangle}v(z),\ \ p\in\Lambda^{*},\ \
\tau(p):H^{s}(\mathbb{C}/\Lambda)\to H^{s}(\mathbb{C}/\Lambda),\ \
s\in\mathbb{R}.\end{gathered}$ (3.4)

We know (see (3.8) for an explicit formula) that $k\mapsto u(k)$ is a smooth
function of $k\in\mathbb{C}$. Simplicity of $\alpha$ then shows that

$|u(k+p)\rangle=\rho_{p}(k)\tau(p)^{*}u(k)\rangle,\ \ \ |\rho_{p}(k)|=1,\ \ \
k\in\mathbb{C},\ \ p\in\Lambda^{*}.$

(We have an explicit formula for $\rho_{p}(k)$ coming from (3.8), see [BHZ22b,
§5], but that is of no importance here.) This and (3.4) also shows that
$\tau(p)^{*}:|u(k)\rangle^{\perp}\to|u(k+p)\rangle^{\perp}$,
$\tau(p):|u^{*}(k)\rangle^{\perp}\to|u^{*}(k+p)\rangle^{\perp}$ and hence, in
the notation of (3.1),

$\begin{gathered}\langle u(k+p)|=\overline{\rho_{p}(k)}\langle u(k)|\tau(p)|,\
\ \ |u(k+p)\rangle=\rho_{p}(k)|\tau(p)^{*}|u(k)\rangle,\\\
E_{0}(k+p)=\tau(p)E_{0}(k)\tau(p)^{*},\ \ \ k\in\mathbb{C},\
p\in\Lambda^{*}.\end{gathered}$

Since $C$ and $\bar{V}$ are multiplication operators which commute with
$\tau(p)$ and $\tau(p)^{*}$, (3.3) follows. ∎

We also have

######  Lemma 4.

In the notation of (3.2) we have

$e_{+}(k)=e_{-}(k)=:e(k).$

###### Proof.

We calculate

$e_{+}(k)=-\langle u(k)|C|u(k)\rangle=-\langle
u_{1}(k)|V|u_{2}(k)\rangle-\langle u_{2}(k)|\bar{V}|u_{1}(k)\rangle.$ (3.5)

In view of (2.9) we can take

$u^{*}(k)={\mathscr{Q}}u(k)=\begin{pmatrix}\overline{u_{1}(k,-z)}\\\
\overline{u_{2}(k,-z)}\end{pmatrix},$

so that the definition of $e_{-}(k)$ gives, (recall from (1.6) that
$V(-z)=\overline{V(z)}$)

$\begin{split}e_{-}(k)&=-\langle u^{*}(k)|C|u^{*}(k)\rangle=-\langle
u_{1}^{*}(k)|V|u_{2}^{*}(k)\rangle-\langle
u_{2}^{*}(k)|\bar{V}|u_{1}^{*}(k)\rangle\\\
&=-\langle\overline{u_{1}(k,-\bullet)}|V|\overline{u_{2}(k,-\bullet)}-\langle\overline{u_{2}(k,-\bullet)}|\bar{V}|\overline{u_{1}(k,-\bullet)}\rangle\\\
&=-\langle u_{2}(k)|V(-\bullet)|u_{1}(k)\rangle-\langle
u_{1}(k)|\overline{V(-\bullet)}|u_{2}(k)\rangle\\\ &=-\langle
u_{2}(k)|\bar{V}|u_{1}(k)\rangle-\langle
u_{1}(k)|V|u_{2}(k)\rangle=e_{+}(k),\end{split}$

which is the desired conclusion. ∎

Hence the eigenvalues of (3.2) are given by (1.2). From Proposition 2 we see
that $E_{\pm}(\alpha,\lambda,\pm K)=0$ and hence $e(\pm K)=f(\pm K)=0$. It
remains to establish (1.3). This will be done in §4.

We conclude this section with an expression for $e(k)$ in terms of $\psi$ in
(2.7). To that we recall the notation of [BZ23]:

$F_{k}(z)=e^{\frac{i}{2}(z-\bar{z})k}\frac{\theta(z-z(k))}{\theta(z)},\ \ \
z(k):=\frac{\sqrt{3}k}{4\pi i},\ \ z:\Lambda^{*}\to\Lambda,$ (3.6)

where $\theta(\zeta)=\theta_{1}(\zeta|\omega)$ is a Jacobi theta function –
see [KhZa15, §2.2]. We recall that

$\begin{gathered}F_{k}(z+m+n\omega)=F_{k}(z),\\\
(2D_{\bar{z}}+k)F_{k}(z)=a(k)\delta_{0}(z),\ \ a(k):=2\pi
i{\theta(z(k))}/{\theta^{\prime}(0)}.\end{gathered}$ (3.7)

We can then use (2.7) and take

$u(k)=c(k)F_{k}(z)u_{0}(z)=c(k)F_{k}(z)\begin{pmatrix}\ \ \psi(z)\\\ \pm
i\psi(-z)\end{pmatrix},$ (3.8)

where $c(k)>0$ is the normalizing constant. This gives

$\begin{split}e_{+}(k)&=-\langle u(k)|C|u(k)\rangle=-\langle
u_{1}(k)|V|u_{2}(k)\rangle-\langle u_{2}(k)|\bar{V}|u_{1}(k)\rangle\\\
&=-c(k)^{2}\int_{\mathbb{C}/\Lambda}(V(z)u_{2}(k,z)\overline{u_{1}(k,z)}+\overline{V(z)}u_{1}(k,z)\overline{u_{2}(k,z)}dm(z)\\\
&=\mp
ic(k)^{2}\int_{\mathbb{C}/\Lambda}|F_{k}(z)|^{2}(V(z)\psi(-z)\overline{\psi(z)}-\overline{V(z)}\overline{\psi(-z)}\psi(z))dm(z)\\\
&=\pm
2c(k)^{2}\int_{\mathbb{C}/\Lambda}|F_{k}(z)|^{2}\operatorname{Im}(V(z)\psi(-z)\overline{\psi(z)})dm(z).\end{split}$
(3.9)

##  4\. Symmetries of $e(k)$

We will now use different symmetries of $u_{0}$ and $F_{k}(z)$ to obtain
symmetries of $e(k)$ listed in (1.3).

###  4.1. The reflection $k\mapsto-k$ symmetry

To see that $e$ is an odd function of $k$ we use (2.10): from the simplicity
assumption $u(-k)=\rho(k)\mathscr{H}u(k)$, $|\rho(k)|=1$, and, as
$V(-\bullet)=\bar{V}$,

$\begin{split}e(-k)&=-\langle
u_{1}(-k)|V|u_{2}(-k)\rangle{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}-}\langle
u_{2}(-k)|\bar{V}|u_{1}(-k)\rangle\\\ &=\langle
u_{2}(k,-\bullet)|V|u_{1}(k,-\bullet)\rangle+\langle
u_{1}(k,-\bullet)|\bar{V}|u_{2}(k,-\bullet)\rangle\\\ &=\langle
u_{2}(k,)|V(-\bullet)|u_{1}(k)\rangle+\langle
u_{1}(k)|\bar{V}(-\bullet)|u_{2}(k)\rangle\\\ &=\langle
u_{2}(k,)|\bar{V}|u_{1}(k)\rangle+\langle u_{1}(k)|V|u_{2}(k)\rangle\\\
&=-e(k).\end{split}$ (4.1)

###  4.2. The rotation $k\mapsto\omega k$ symmetry

Using the definition (3.6), we consider $F_{k}(\omega z)$:

$a(k)\delta_{0}(z)=[(2D_{\bar{z}}+k)F_{k}](\omega
z)=\omega((2D_{\bar{z}}+\bar{\omega}k)[F_{k}(\omega z)].$

The uniqueness of the fundamental solution of $2D_{\bar{z}}+k$ on the torus
shows that

$\omega a(k)^{-1}F_{k}(\omega z)=a(\bar{\omega}k)^{-1}F_{\bar{\omega}k}(z),$

that is

$F_{k}(\omega
z)=\frac{\bar{\omega}a(k)}{a(\bar{\omega}k)}F_{\bar{\omega}k}(z)=\frac{\bar{\omega}\theta(z(k))}{\theta(\bar{\omega}z(k))}F_{\bar{\omega}k}(z).$

Hence,

$\begin{split}c(\bar{\omega}k)^{-2}&=\int_{\mathbb{C}/\Lambda}|F_{\bar{\omega}k}(z)|^{2}(|\psi(z)|^{2}+|\psi(-z)|^{2})dm(z)\\\
&=\left|\frac{\theta(\bar{\omega}z(k))}{\theta(z(k))}\right|^{2}\int_{\mathbb{C}/\Lambda}|F_{k}(\omega
z)|^{2}(|\psi(z)|^{2}+|\psi(-z)|^{2})dm(z)2=\left|\frac{\theta(\bar{\omega}z(k))}{\theta(z(k))}\right|^{2}c(k)^{-2},\end{split}$

which implies $c(\omega k)/{c(k)}=|{\theta(z(k))}/{\theta(\omega z(k))}|$.
Using (3.9) we obtain

$\begin{split}\pm e_{\pm}(k)&=2c(k)^{2}\int_{\mathbb{C}/\Lambda}|F_{k}(\omega
z)|^{2}\operatorname{Im}(V(\omega z)\overline{\psi(\omega z)}\psi(-\omega
z))dm(z)\\\ &=2c(k)^{2}\int_{\mathbb{C}/\Lambda}|F_{k}(\omega
z)|^{2}\operatorname{Im}(V(z)\overline{\psi(z)}\psi(-z))dm(z)\\\
&=2c(k)^{2}\left|\frac{\theta(z(k)}{\theta(\bar{\omega}z(k))}\right|^{2}\int_{\mathbb{C}/\Lambda}|F_{\bar{\omega}k}(z)|^{2}\operatorname{Im}(V(z)\overline{\psi(z)}\psi(-z))dm(z)\\\
&=\frac{c(k)^{2}|\theta(z(k))|^{2}}{c(\bar{\omega}k)^{2}|\theta(\bar{\omega}z(k))|^{2}}e(\bar{\omega}k)=\pm
e(\bar{\omega}k),\end{split}$

or

$e(\omega k)=e(k).$ (4.2)

###  4.3. The reflection $k\mapsto\bar{k}$ symmetry

We first claim that

$F_{-\bar{k}}(z)=-\frac{\theta(z(\bar{k}))}{\overline{\theta(z(k))}}\overline{F_{k}(\bar{z})}\
\ \text{ and }\ c(k)=c(\bar{k}).$ (4.3)

To see the first part, we note that

$(2D_{\bar{z}}-\bar{k})[\overline{F_{k}(\bar{z})}]=\overline{(-2D_{z}-k)[F_{k}(\bar{z})]}=-\overline{[(2D_{\bar{z}}+k)F_{k}](\bar{z})}=-\overline{a(k)}\delta_{0}(z),$

so that (3.7) and the uniqueness of the Green function give the first part of
(4.3). Hence,

$\begin{split}c(\bar{k})^{-2}&=\left|\frac{\theta(z(\bar{k}))}{\theta(z(k))}\right|^{2}\int_{\mathbb{C}/\Lambda}|F_{k}(\bar{z})|^{2}(|\psi(\bar{z})|^{2}+|\psi(-\bar{z})|^{2})dm(z)\\\
&=\left|\frac{\theta(z(\bar{k}))}{\theta(z(k))}\right|^{2}c(k)^{-2}.\end{split}$

The second part of (4.3) then follows from the following fact:

$\theta(z)=e^{-i\pi/4}\,\overline{\theta(\bar{z})}.$ (4.4)

In fact, we can use the product representation of $\theta$ (see [Mu83, §I.14]
or [KhZa15, (2.10a)]):

$\theta(z)=2q^{\frac{1}{4}}\sin\pi
z\prod_{n=1}^{\infty}(1-q^{2n})(1-q^{2n}e^{2\pi iz})(1-q^{2n}e^{-2\pi iz}),\ \
\ q=e^{\pi i\omega},\ \ \bar{q}=-q.$

Since

$q^{\frac{1}{4}}=e^{\frac{1}{4}\pi i\omega},\ \ \
\overline{q^{\frac{1}{4}}}=e^{-\frac{1}{4}i\bar{\omega}}=e^{-\frac{1}{4}\pi
i}e^{\frac{1}{4}i\omega}=e^{-\frac{1}{4}\pi i}q,$

we have

$\overline{\theta(\bar{z})}=2(-q)^{\frac{1}{4}}\sin\pi
z\prod_{n=1}^{\infty}(1-q^{2n})(1-q^{2n}e^{-2\pi iz})(1-q^{2n}e^{2\pi
iz})=e^{-\frac{1}{4}\pi i}\theta(z),$

as claimed in (4.4).

Assuming that $\alpha\in\mathcal{A}\cap\mathbb{R}$ is simple we can take
$u^{*}(k)=\mathscr{N}u(-\bar{k})$ in the definition of $e(k)=e_{-}(k)$ in
(3.2): This means that

$\begin{split}u_{1}^{*}(k,z)&=u_{2}(-\bar{k},-\bar{z})=\pm
iF_{-\bar{k}}(-\bar{z})\psi(\bar{z}),\\\
u_{2}^{*}(k,z)&=u_{1}(-\bar{k},-\bar{z})=F_{-\bar{k}}(-\bar{z})\psi(-\bar{z}).\end{split}$

Using (1.6) again we have

$\begin{split}e(k)&=c(k)^{2}\int_{\mathbb{C}/\Lambda}V(z)u_{2}^{*}(k,z)\overline{u_{1}^{*}(k,z)}+V(-z)u_{1}^{*}(k,z)\overline{u_{2}^{*}(k,z)}dm(z)\\\
&=\mp
ic(k)^{2}\int_{\mathbb{C}/\Lambda}|F_{-\bar{k}}(-\bar{z})|^{2}\left(V(z)\psi(-\bar{z})\overline{\psi(\bar{z})}-V(-z)\psi(\bar{z})\overline{\psi(-\bar{z})})\right)dm(z)\\\
&=\mp
ic(k)^{2}\int_{\mathbb{C}/\Lambda}|F_{-\bar{k}}(z)|^{2}\left(\overline{V(z)}\psi(z)\overline{\psi(-z)}-V(z)\psi(-z)\overline{\psi(z)})\right)dm(z)\\\
&=c(k)^{2}c(\bar{k})^{-2}e(-\bar{k})=c(k)^{2}c(\bar{k})^{-2}e(-\bar{k}).\end{split}$

We now recall (4.1) and (4.3) to see that $e(k)=-e(\bar{k})$.

Remark. We could avoid using (4.3) by considering (see (2.9) and (2.11))

$\mathscr{R}:={\mathscr{Q}}\mathscr{N},\ \ \
\mathscr{R}(D(\alpha)+k)\mathscr{R}=-(D(\bar{\alpha})-\bar{k}).$

This means that for a simple $\alpha\in\mathbb{R}\cap\mathcal{A}$,

$\mathscr{R}u(k)=\rho(k)u(-\bar{k}),\ \ \ |\rho(k)|=1.$

or that

$\overline{u_{2}(k,\bar{z})}=\rho(k)u_{1}(-\bar{k},z),\ \
\overline{u_{1}(k,\bar{z})}=\rho(k)u_{2}(-\bar{k},z).$

Using that in the definition of $e(k)$ we obtain $e(k)=e(-\bar{k})$.

##  5\. Comments on degenerate magic angles

We now consider degenerate magic angles of multiplicity two – see [BHZ23,
(1.7) and Theorem 1]. The analogue of Figure 1 is shown in Figure 5. For small
$|\lambda|\leq 10^{-3}$, we observe that two out of the four flat bands
undergo an upward shift in energy, while the other two undergo an equal and
opposite downward shift in energy. This is different compared with simple
magic angles where all flat bands move together for small $\lambda.$

Another difference is that the bands are only anchored at
$K,K^{\prime},\Gamma$ at zero energy, rather than on the line through these
points, as was the case for simple magic angles. For larger perturbations
$|\lambda|\geq 10^{-2}$, the two upper and two lower bands undergo a splitting
into separate bands, respectively. In addition, the two Dirac cones at
$\Gamma$ start to overlap (center figure in Fig. 5). In the right Figure 5, we
see that the eigenvalue perturbation for $|\lambda|\leq 10^{-2}$ small is
linear and for $|\lambda|\geq 10^{-1}$ quadratic terms are dominant.

Figure 5.  Plots of $k\mapsto E_{\pm\ell}(\alpha,\lambda,k)$ with
$\ell\in\\{1,2\\}$ for $\alpha$ the first real magic element with potential
$AB^{\prime}$ tunnelling potential $U_{1}(z):=(U(z)-U(2z))/\sqrt{2}$ and
$AA^{\prime}/BB^{\prime}$ potential $V(z)$ as in (1.7). In the chiral limit,
the potential $U_{1}$ exhibits double magic $\alpha$’s on the real axis – see
[BHZ23]. We then study $\lambda=10^{-3},10^{-2},10^{-1}$ (from left to right).
We see that for very small coupling two of the four bands ”move together” and
split only when the coupling gets larger. For an animated version see
https://math.berkeley.edu/~zworski/2bands_1D.mp4

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