# Systematic Biases in LLM Simulations of Debates

Amir Taubenfeld12∗
Yaniv Dover34
Roi Reichart5
Ariel Goldstein136 *Corresponding Author<EMAIL_ADDRESS>

1Google Research
2The Hebrew University of Jerusalem, School of Computer Science and
Engineering
3The Hebrew University Business School, Jerusalem, Israel
4Federmann Center for the Study of Rationality, Hebrew University, Jerusalem,
Israel
5Faculty of Data and Decision Sciences, Technion
6Department of Cognitive and Brain Sciences, Hebrew University, Jerusalem,
Israel

###### Abstract

Recent advancements in natural language processing, especially the emergence
of Large Language Models (LLMs), have opened exciting possibilities for
constructing computational simulations designed to replicate human behavior
accurately. However, LLMs are complex statistical learners without
straightforward deductive rules, making them prone to unexpected behaviors. In
this study, we highlight the limitations of LLMs in simulating human
interactions, particularly focusing on LLMs’ ability to simulate political
debates. Our findings indicate a tendency for LLM agents to conform to the
model’s inherent social biases despite being directed to debate from certain
political perspectives. This tendency results in behavioral patterns that seem
to deviate from well-established social dynamics among humans. We reinforce
these observations using an automatic self-fine-tuning method, which enables
us to manipulate the biases within the LLM and demonstrate that agents
subsequently align with the altered biases. These results underscore the need
for further research to develop methods that help agents overcome these
biases, a critical step toward creating more realistic simulations.

Systematic Biases in LLM Simulations of Debates

Amir Taubenfeld12∗ Yaniv Dover34 Roi Reichart5 Ariel Goldstein136

*Corresponding Author<EMAIL_ADDRESS>1Google Research 2The Hebrew University of Jerusalem, School of Computer Science and Engineering 3The Hebrew University Business School, Jerusalem, Israel 4Federmann Center for the Study of Rationality, Hebrew University, Jerusalem, Israel 5Faculty of Data and Decision Sciences, Technion 6Department of Cognitive and Brain Sciences, Hebrew University, Jerusalem, Israel

##  1 Introduction

Computational simulations designed to accurately replicate human behavior hold
the potential to transform multiple disciplines, such as human psychology
Dillion et al. (2023), social dynamics Park et al. (2022), and economics
Horton (2023); Chen et al. (2023). If sufficiently reliable, these simulations
could serve as invaluable tools for exploring the intricacies of human
interactions and decision-making processes. This progress would allow
scientists to conduct their research with exceptional speed and efficiency,
substantially lowering the considerable resources usually needed for
recruiting and analyzing human subjects Ditto et al. (2019).

The emergence of Large Language Models Brown et al. (2020); Jiang et al.
(2023), have opened up exciting possibilities for constructing such reliable
simulations Park et al. (2023); Qian et al. (2023). LLMs possess the
remarkable ability to engage in role-playing, seamlessly adopting the personas
of different characters Shanahan et al. (2023); Argyle et al. (2023). The
typical paradigm for such simulations involves selecting an LLM, such as the
widely used ChatGPT Milmo (2023), as a base model and crafting individual
agents’ identities through natural language prompts. For instance, by
prepending the prompt, "John Lin is a pharmacy shopkeeper," to an agent’s
context, the agent is expected to act as if his name is John and he works as a
shopkeeper Park et al. (2023).

However, LLMs are complex statistical learners that do not depend on
straightforward deductive rules. Despite exhibiting impressive emerging skills
that challenge our current understanding of cognition Wei et al. (2022);
Bubeck et al. (2023), their indeterminate nature leaves them susceptible to
exhibiting unintended behaviors. An example of such behavior is their
manifestation of inherent biases, including gender bias Bordia and Bowman
(2019), ethnic bias Ahn and Oh (2021), and social identity bias Hu et al.
(2023). Given their undefined nature, it is vital to exercise caution when
using LLMs, particularly in multi-agent environments aimed at simulating
complex, large-scale social phenomena.

In this study, we explore the behavior of LLM agents within simulations. Our
experiments are focused on the realm of Attitude Change Kahan et al. (2012);
Priniski and Horne (2018) and specifically on the extensively studied
interactions between liberals and conservatives Hobolt et al. (2023); Sunstein
(2001). This domain is susceptible to numerous prejudices Ditto et al. (2019),
making it an ideal candidate for investigating the effect of LLM biases on
simulations.

We facilitate debates on polarizing American topics between LLM agents
representing Republican and Democrat perspectives. The agents’ attitude is
monitored via surveys, and their believability is evaluated by contrasting
their behaviors with known human interactions Hobolt et al. (2023). In
addition, we have developed an automatic fine-tuning process for agents,
utilizing self-generated training data. We use this process to conduct
controlled intervention studies, by manipulating the LLM biases and analyzing
the subsequent impact on the agents’ behaviors.

Our results reveal that LLM agents generally conform to the inherent social
biases of their base models, even if these biases conflict with their assigned
identities. Consequently, this causes the simulations to diverge from well-
established human social behaviors. Moreover, when we employ our fine-tuning
method to change the LLMs’ viewpoints, we observe that the agents, despite
retaining their original contexts, modify their behavior to be in line with
the newly introduced bias.

These insights underline the need to investigate ways to help agents
circumvent these biases, a crucial step in developing simulations that more
accurately reflect real human behavior.

##  2 Related Work

#### Believable LLM Simulations

Recent studies show that LLMs possess human-like reasoning skills Chen et al.
(2023), and that LLMs are able to adopt personas of diverse characters
Shanahan et al. (2023). Leveraging these abilities, Park et al. (2023)
developed a sandbox environment, populated it with 25 LLM-based agents, and
showed that the agents convincingly mimic human behaviors such as sharing news
and forming relationships.

The transformative potential of such simulations in areas like human
psychology Dillion et al. (2023) and economics Horton (2023) was a significant
motivator for our work. Nonetheless, our findings indicate that inherent
biases in LLMs pose substantial challenges in ensuring the reliability of
agents to generate believable human behavior.

#### Bias In LLM Simulation

In a contemporaneous work, Chuang et al. (2023) showed that “LLM agents tend
to converge towards scientifically accurate information”, attributing this to
the LLM’s inherent biases. We generalize this observation by demonstrating
that LLM agents converge toward the model’s inherent bias regardless of its
scientific validity. This is true for biases on purely subjective topics, and
even for those contradicting scientific truths such as the reality of Climate
Change Arias et al. (2021). Moreover, beyond observing the debates and drawing
conclusions, we also offer a controlled intervention study utilizing our
unique self-fine-tuning process. This study further substantiates our
assertions and shows that it is possible to control the agents’ convergence
point by fine-tuning its underlying model. Additionally, we employ our
innovative simulation methodology to reproduce this phenomenon in various
settings, enabling a deeper and wider analysis of the underlying mechanisms.

#### Self Alignment

In recent years, the task of aligning LLMs with human intentions has become a
significant area of research Ouyang et al. (2022); Wang et al. (2023). The
primary objective of alignment research is to enhance the conversational
abilities of LLMs and ensure their conformity with established social values
Gabriel (2020); Oviedo-Trespalacios et al. (2023). An evolving trend in this
area involves developing methods that use LLM simulations to generate training
data automatically. This approach aims to reduce the need for expensive human
feedback Liu et al. (2023); Ulmer et al. (2024).

In our work, we introduce an approach to self fine-tuning of LLMs, taking a
distinct path from existing methodologies. Rather than enhancing the LLM’s
general conversational capabilities or aligning it with broader human
objectives, our focus is to tailor the LLM to adopt a specific political
stance. We interview the agents using a set of questions crafted to elicit
their political views, and utilize their responses to train the underlying
LLM. In terms of assessment, our interest lies not in evaluating the
effectiveness of the fine-tuning on standard NLP benchmarks, but in observing
its impact on the agents within our simulation.

##  3 Problem Definition

Our study delves into the impact of inherent biases within LLMs on their
ability to accurately emulate diverse characters Shanahan et al. (2023). We
explore this relationship by facilitating political debates between LLM
agents. Section 4 outlines our simulation methodology, including the criteria
for selecting debate topics (4.1), how we crafted the agents’ identities
(4.2), and the techniques for managing and evaluating interactions between the
agents (4.3). Section 5 introduces a novel automatic fine-tuning technique for
agents, utilizing self-created training data. We have developed this method to
adeptly adjust the LLM’s perspective, and it is applied in the controlled
intervention experiments discussed within this research. In Section 6 we
present the primary findings of our work. Through a sequence of experiments,
we establish a strong connection between the inherent biases of LLMs and the
patterns of attitude change observed in our simulations. Lastly, Section 7
offers a complimentary analysis aimed at evaluating and enhancing the
robustness of our fine-tuning process against standard benchmarks.

##  4 Setup

###  4.1 Topics Selection

Exploring the dynamic of meaningful discussion requires a conscientious choice
of subjects of discussion. Our experiments involve debates between liberal and
conservative partisans. We chose this domain for two main reasons. Firstly,
this field is extensively studied in social science Ditto et al. (2019);
Hobolt et al. (2023), providing a robust baseline for comparison with our
simulations. Secondly, the field is susceptible to numerous prejudices, making
it a particularly suitable context for examining the biases inherent in LLMs
Ditto et al. (2019).

The Pew Research Center conducted a survey in 2023 about the differences in
assessment of America’s problems between Republicans and Democrats Doherty et
al. (2023). When analyzing their results, four subjects stand out as the most
controversial - Gun Violence, Racism, Climate Change, and Illegal Immigration.
We focus our experiments on these four topics.

###  4.2 LLM-based Agents Implementation

We followed the conventional paradigm for LLM-based simulations Park et al.
(2023); Qian et al. (2023), which entails selecting a base language model and
then constructing the individual identities of agents using natural language
prompts.

We automatically generated context prompts for two political groups of
partisan agents. The first group consists of agents embodying passionate
Republican views, while the second group represents passionate Democrats. We
used the LLM to craft narratives related to the four debate topics, as
illustrated in Figure 1. Our approach was to develop comprehensive identities
for each agent across all topics simultaneously rather than creating an
individual agent for each topic. This strategy simplified our experimental
design and provided a more complete representation for each agent. Forty
agents were generated for each group, using the same static prompt and a
temperature setting of 1.0. Each agent was assigned a different name.
Additionally, in some simulations, we included a "default" agent whose sole
directive was "You are an American". This agent’s context was deliberately
devoid of any political bias, serving to showcase the inherent biases within
the LLM.

We experiment with three different state-of-the-art LLMs as our base models:
Mistral 7B Jiang et al. (2023), Solar 10.7B Kim et al. (2023), and Instruct-
GPT Ouyang et al. (2022). Across all models, we observed similar results. The
open-source models, Mistral and Solar, were deployed on a single RTX 3090ti
graphics card, utilizing 8-bit quantization for efficiency. For Instruct-GPT,
we used the gpt-3.5-turbo-instruct version available through OpenAI’s
Completion API. The results and methodologies discussed henceforth pertain to
the GPT model, except for our fine-tuning experiment where we used the Mistral
model, as it is open-sourced. Additional results from other models are
included in the appendix.

Figure 1: (a) The prompt that was used for generating the Democratic
background stories. This prompt includes the desired opinion on the four
controversial topics debated in our experiments. We used similar wording to
the survey questions asked by Doherty et al. (2023). (b) Example of a
background story of one of the agents. This story was generated automatically
by feeding the LLM with the prompt described in (a).

###  4.3 LLM-based Agents Interaction

Our debate simulations follow a round-robin format, with the initial speaker
selected randomly. At each iteration, an agent receives its background story,
the debate topic, and the conversation’s history, and it is asked to complete
its next reply in the conversation (see Figure 2). Before the start of the
debate, and at the end of each round-robin cycle, the agents are asked to
numerically rate their attitude (on a scale of 0-10) toward the severity of
the discussed topic. To ensure that this process does not impact the direction
of the debate or future ratings, the survey questions are not saved in the
conversation history, so the agents are unaware of the answers provided by
other agents and the answers they supplied themselves in the past.

Figure 2:  At each iteration, an agent (a) is prompted with its background
story, the topic of the debate, and the history of the conversation so far,
and is asked to complete either (b) its next reply in the conversation, or (c)
a survey question measuring his current attitude on the debated topic.

The variance in the conversation comes from two sources: (1) each agent has a
different background story, and (2) the model generates conversation entries
with a temperature setting of 1.0. However, all the survey questions are asked
using a temperature setting of 0 (i.e., no sampling) to reduce unnecessary
variance.

##  5 Fine-Tuning Methods

In the preceding section, we outlined our approach for simulating debates and
tracking the attitude changes of the agents involved. In addition, our work
offers a controlled intervention study designed to investigate the
relationship between LLM biases and role-played agents. For this purpose, we
have developed an automated fine-tuning technique for the agents, which allows
us to manipulate their underlying LLM biases effectively. Our method depends
solely on training data generated by the agents, without the need for external
data inputs.

We commence the process by constructing a series of 100 questions intended to
encourage agents to reveal their opinions and sentiments regarding their
political views. We start with an initial set of 10 sample questions of
different styles, for example, ’Could you discuss your perspective on
significant political issues facing America today?’ or ’How do you believe the
U.S. should handle immigration and border security?’ (the remaining questions
are detailed in the Supplementary Table 2). Following this, we direct the LLM
to produce an additional 90 questions that are similar in nature to ensure a
broad and varied collection of queries.

We intentionally craft the questions in a neutral way and ask a broad spectrum
of questions, not only questions concerning the debated topics directly. We do
this for two reasons: (1) We want our fine-tuning approach to be generic so it
can be easily adjusted to other scenarios without needing to rewrite the
questions. For instance, we used the same set of questions to generate both a
Republican-oriented dataset and a Democratic-oriented dataset. (2) A broader
variety of training data helps the agent to maintain its conversational skills
and avoid over-fitting.

Next, we initialize an agent with a context similar to the template used to
create agents in section 4.3. We query this agent with the 100 previously
generated questions. For every question, we gathered 20 responses by operating
the agent at a temperature of 1.0. This results in 2,000 examples, which are
utilized as our training dataset.

Finally, we used this self-generated dataset to fine-tune the model. Our
training process is lightweight, using a basic next-word prediction task with
the parameter-efficient QLoRA method Hu et al. (2021); Dettmers et al. (2023).
The training is completed in just one epoch, taking under 10 minutes on a
single RTX 3090ti GPU. At the conclusion of this stage, the model becomes
adapted to the agent’s perspective, which is elaborated in the results
section. A diagram of our fine-tuning procedure is shown in Figure 3, with
further technical specifics available in the supplementary Section A.2. All
the reported scores for fine-tuned models included in this paper are the
average of three independent fine-tuning runs with random seeds.

Figure 3: Depiction of our automated fine-tuning procedure. (1) The process
begins with a pre-defined collection of questions designed to engage the
agent. (2) The agent responds to each question, and these responses are then
gathered. Note that the agent’s responses intentionally reflect its background
story. For example, for the question "What are your political views?" a
Republican agent might answer, "Fighting illegal immigration is my top
priority". (3) Training is conducted on a "next word prediction" task,
utilizing the Agent’s responses as training data.

##  6 Results

#### Systematic Biases in Debates Simulations

We conducted simulations of political debates on the topics outlined in
Section 4.1. These debates were structured as three-way discussions, including
a Republican agent, a Democrat agent, and the Default agent, the latter
assumed to reflect the model’s inherent biases. Details on the implementation
of these agents are provided in Section 4.2. During the debates, we monitored
the evolution of the agents’ attitudes by repeatedly asking them to rate their
viewpoints on a 0 to 10 scale. More details on the debate format and attitude
tracking methods are in Section 4.3, with a sample debate provided in the
supplementary Table 3.

The interactions between the three debating agents, as captured in Figure 4,
indicate that while the Default agent consistently maintains its position, the
partisan agents gradually adjust their viewpoints to resonate with the Default
agent’s stance. Notably, when the Default agent displayed a bias towards one
side, indicating a strong inherent bias in the LLM, the agent with an
initially opposing view tended to significantly compromise on its opinion,
shifting towards the position of the other partisan agent. Conversely, when
the Default agent did not display a specific bias, the partisan agents tended
to move towards a common middle ground.

Remarkably, as shown in Figure 5, even when the Default agent is excluded from
the debate, the partisan agents continued to shift their attitudes in line
with the model’s inherent biases. This tendency of all agents to gravitate
towards the model’s inherent biases, irrespectively of the participation of
the Default agent in the discussion, prompts critical reflections on the
capacity of language models to replicate genuine human interactions accurately
and to mimic a diverse range of viewpoints authentically.

Figure 4:  Evolution of attitude scores in three-way debates on four
controversial topics. The X-axis shows the number of chat exchanges in the
debate. The Y-axis displays the average attitude scores derived from forty
separate experiments on each topic, including standard error bars. Our
methodology for monitoring attitude scores is detailed in Section 4.3. The
Default agent, symbolizing the inherent biases of the base LLM, maintains a
consistent position throughout the debate. Interestingly, the views of the
partisan agents gradually align more closely with those of the Default agent.
Furthermore, it is notable that the lines representing the partisan agents
never intersect with the line of the Default agent. This suggests that the LLM
default biases can serve as bounds, preventing one of the parties from making
compromises toward the other. Supplementary Section A.1 presents analogous
findings with other underlying models.  Figure 5:  Evolution of attitude
scores in two-way debates between Republican and Democrat agents. The graphs
feature a dashed line that shows the Default agent’s viewpoint before the
beginning of the debates, taken from Figure 4. Recall that the Default agent’s
viewpoint represents the inherent biases of the LLM. Remarkably, even though
the Default agent does not participate in the two-way debates illustrated
here, the partisan agents continue to converge toward the inherent biases of
the model.

#### Contradicting The Echo Chambers Theory

Even during interactions with others of similar political orientations, the
agents persist in reflecting the LLM’s intrinsic bias. We demonstrate this
phenomenon by pairing each of the forty Republican agents with another from
the same group. As shown in Figure 6, agents tend to adopt more moderate
positions, aligning more closely with the LLM’s inherent bias. This finding is
particularly intriguing as it deviates from the well-known real-world
phenomenon of Echo Chambers Sunstein (2001); Hobolt et al. (2023), where
individuals with like-minded views tend to intensify their beliefs when
interacting with each other.

Similarly to the previous section, this trend persists even when the Default
agent is excluded from the dialogue, as shown in Supplementary Figure 9. We
also conducted the same Echo Chamber experiment using Democrat agents and
observed a similar pattern of gravitation toward the Default agent’s stance as
displayed in Supplementary Figure 10.

Figure 6:  This graph illustrates a series of three-way debates involving two
Republican agents and a Default agent. Notably, even during conversations with
other Republicans, the agents tend to align with the position of the Default
agent. This trend is apparent even when the Default agent is not participating
in the dialogue (supplementary Figure 9). The same phenomenon is also evident
in experiments conducted with Democrat agents (Supplementary 10), where a
similar pattern of gravitation towards the Default agent’s stance is observed.

#### Fine-tuning Highlights the Bias

To conclusively demonstrate the link between LLM biases and agents’ behavior,
we employed the fine-tuning process detailed in Section 5. Through this
method, we successfully altered the inherent bias of the LLM toward a specific
viewpoint. After fine-tuning, we conducted the debates again using the
original agent contexts, but with the underlying model now modified.

As illustrated in Figure 7, changing the viewpoint of the LLM toward a
Republican perspective, indirectly influenced the agents, leading them to
modify their behavior in line with the updated bias. In a contrasting setup,
fine-tuning the model to align with a Democrat perspective resulted in trends
that were predictably opposite, as seen in Supplementary Figure 12. This
experiment underscores the profound implications of our findings, indicating
that simulations conducted with different LLMs, each harboring its unique set
of biases, could result in significantly different portrayals of authentic
human behavior.

The success of the fine-tuning process in steering the model towards a
particular viewpoint is noteworthy, considering that it was accomplished
solely with content produced by the LLM, without relying on external data
sources. Furthermore, this method proves that it is feasible to configure
agents to consistently maintain certain viewpoints throughout simulations,
unlike the temporary effects seen when defining agents’ identities through
prompts.

Figure 7: Results of fine-tuning the model to adapt more closely to a
Republican perspective. All the reported scores are the average of three
independent fine-tuned models with different random seeds. For each topic, we
conduct two separate debates between three agents - a Republican, a Democrat,
and a Default agent who represent the model’s inherent bias. The solid lines
represent the debate between the three agents before fine-tuning, and the
dotted lines represent the debate between the same agents when the underlying
LLM had been fine-tuned. The Republican viewpoint is evident in both graphs:
(left) In the Climate Change graph all lines have shifted downward, signaling
a shift towards opposing climate change. (right) Conversely, the Illegal
Immigration graph shows an upward trend after fine-tuning, suggesting that the
agents now view illegal immigration as a more significant issue.

##  7 Fine-Tuning Robustness

In Section 5, we describe our multi-stage self-fine-tuning method that is
shown to effectively alter the model’s perspective toward a designated
viewpoint. We designed our approach to be streamlined and easily replicable,
focusing on ensuring the robustness of the process without resorting to
localized optimizations. As a result, we made the following design choices:
(1) Solely using self-generated data, avoiding external dataset sources. (2)
Fine-tuning a comprehensive model applicable across all debate topics, rather
than training individual models for each topic. (3) Employing a
straightforward next-word prediction task, in contrast to more complex
reinforcement learning techniques. (4) Using the efficient QLoRA method
Dettmers et al. (2023), which enabled training the model in minutes.

The $r,\alpha$ LoRA hyper-parameters, which respectively control the number of
trainable weights and the scale of weight updates, had a significant impact on
our results. By increasing these hyper-parameters, we observed a marked change
in the political stance of the Default agent, which serves as a reflection of
the LLM’s built-in bias. Although our study primarily aims to modify the
political viewpoint of the model, exploring how such adjustments impact the
overall abilities of the LLM is intriguing. In Table 1, we offer a
complementary analysis showing the impact of our fine-tuning on the popular
MMLU Hendrycks et al. (2020) and Helaswag Zellers et al. (2019) benchmarks.
Despite the alterations, the fine-tuned models still showcase strong
performance across these general benchmarks. However, there appears to be an
inverse relationship between the degree of change in the model’s political
stance and its benchmark scores.

Finally, we present an incremental optimization to our fine-tuning process,
which enables us to manipulate the model’s perspective more aggressively while
mitigating the negative effects on its general performance. This optimization
is based on the cutting-edge Direct Preference Optimization (DPO) method
Rafailov et al. (2023). which can be divided into two phases: first, a next-
word-prediction phase that acclimates the model to the intended data
distribution, followed by a Contrastive Learning phase aimed at teaching the
model to differentiate between preferred and non-preferred outputs. As
detailed in section 5, our models undergo fine-tuning through a next-word-
prediction task, alongside the creation of self-generated datasets
encapsulating Republican and Democrat viewpoints. This groundwork allows us to
directly employ the DPO’s second phase on the pre-fine-tuned models and
leverage our partisan datasets as input to the Contrastive Learning task,
training a Republican model to prefer a response from the Republican dataset
and vice-versa. Again, we train only for a single epoch using the QLoRA. The
results of this process are also included in Table 1.

|  Hellaswag (%) |  MMLU (%) |  Attitude Score
---|---|---|---
Mistral 7B | 83.6 | 59.0 | 8.4
r=16 NWP | 81.8 | 57.6 | 5.1
r=64 NWP | 81.2 | 56.3 | 4.3
r=128 NWP | 79.7 | 54.3 | 2.5
r=256 NWP | 73.8* | 48.6 | 1.9
r=8 DPO | 81.4 | 57.0 | 0.4
Llama 2 7B | 77.2 | 45.3 |
Table 1:  Effect of fine-tuning Mistral toward a Republican perspective on the
popular Hellaswag and MMLU benchmarks (higher is better). This table showcases
7 models: the baseline Mistral, 4 Mistral versions fine-tuned via a next-word-
prediction task (NWP) with increasing numbers of trainable parameters
(indicated by $r$), an additional Mistral model further optimized with DPO,
and the LLaMA 2 7B Touvron et al. (2023) model that is used for comparison.
For brevity, we display only the Attitude Scores of the Default Agent in the
final round of the debate about Racism (other debate topics follow a similar
pattern). A higher Attitude Score implies a stronger acknowledgment of Racism
as a significant issue. Key findings include: (1) All fine-tuned Mistral
variants still outperform the renowned LLaMA 7B 2 model across the benchmarks,
with one exception marked by *. (2) For the NWP fine-tunes, there is an
inverse correlation between the degree of the model’s shift towards a
Republican attitude and its performance on the benchmarks. (3) Adding a DPO
phase as an incremental step to our fine-tuning methodology, enables us to
forcefully adjust the model’s perspective with reduced detrimental effects on
its performance on general benchmarks.

##  8 Discussion

In our simulations of debates involving agents representing Republicans and
Democrats, a persistent pattern emerged: agents’ opinions consistently align
with the LLM’s inherent social biases. In particular, when the model exhibits
a strong bias in favor of one partisan agent, the opposing agent, which
initially holds a differing view, often moderates its stance, gravitating
significantly towards the position of its counterpart. This leads to a skewed
pattern that appears to depart from the typical dynamics observed in human
interactions.

Furthermore, using our self-fine-tuning process, we perform a controlled
intervention study, demonstrating that it is possible to alter the LLMs’
biases, and the agents will subsequently adjust their positions and align with
the new biases. This highlights the strong influence of the LLMs’ biases on
agents behavior. It also implies that simulations by different LLMs, each with
its unique set of biases, could yield vastly different portrayals of
"authentic" human behavior.

Remarkably, even when agents engaged in debates with others of the same
political orientation, they tended to adopt more moderate views over the
course of interaction, increasingly mirroring the LLM’s default bias. This
pattern is intriguing because it deviates from the well-documented real-world
phenomenon called Echo Chambers Sunstein (2001), where like-minded individuals
often reinforce and escalate their beliefs when interacting with each other.
In an analogous real-life study, Hobolt et al. (2023) divided Labour and
Conservative supporters in England into groups to discuss government policies.
Contrary to our agent-to-agent simulations, they found that Echo Chambers in
homogenous groups intensified polarization.

Our findings thus highlight limitations of large language model agents as
accurate representations of real-life humans. The political landscape, as well
as the specific topics that we chose (Section 4.1), are an important aspect of
the day-to-day life of people and their decision-making processes, relevant to
economic outcomes and markets, sociological and psychological phenomena, and
for issues related to ethics. Hence, the limitations we identified should be
acknowledged as major factors in the usage and interpretation of large-scale
simulations that aim to represent human behavior more accurately, such as in
Park et al. (2023).

In summary, despite LLMs being supposedly renowned for their ability to
emulate human behavior Shanahan et al. (2023); Argyle et al. (2023), our
research uncovers the constraints imposed by their intrinsic biases on their
ability to simulate diverse agents with convincing personalities. This pivotal
concern should be studied, addressed, and taken into consideration. Our fine-
tuning methodology demonstrates the possibility of modifying agents to adhere
to specific perspectives consistently across simulations, unlike the temporary
effects seen when defining agents’ identities through prompts. We advocate for
future research aimed at helping agents transcend the inherent biases of the
model, potentially leveraging our fine-tuning processes and other alignment
techniques, paving the way for more accurate and human-like simulations for
both research and practical applications.

## Limitations

#### Scope of Simulation

Our research primarily examines the dynamics of debates involving 2-3 LLM
agents simultaneously. This focused method effectively highlights our key
observations. Yet, the investigation into how these findings play out in
larger-scale simulations, such as Park et al. (2023) and Qian et al. (2023),
is an avenue for future study. Such expansive simulations, which feature
numerous agents living out simulated ’daily lives’ over prolonged durations
and interacting with a wide variety of other agents, could provide a more
comprehensive view of the impact of inherent LLM biases on agent behavior.

#### Improving Believability

In this study, we introduce an automated alignment method for agents, which is
pivotal in underscoring our principal discoveries regarding constraints in LLM
simulations. Through this refinement approach, it is possible to program
agents to adhere to specific viewpoints consistently across simulations, as
opposed to the transient impact observed when shaping agents’ identities via
prompts. We argue that applying these alignment methods to develop simulations
that are both more precise and closely mimic human behavior represents a
valuable direction for future research, a concept not fully explored in this
study.

## Ethics Statement

In this study, we provide general insights into Large Language Models, by
conducting simulations on political topics. It is important to note that some
biases observed in the paper are subjective. As authors, we maintain a neutral
stance concerning the debate topics.

Furthermore, we have introduced a fine-tuning technique designed to adjust LLM
biases towards specific viewpoints. It is crucial to exercise caution when
applying such fine-tuning methods to user-facing LLMs, ensuring that they
reflect fair and ethical values in their outputs.

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##  Appendix A Appendix

###  A.1 Results from Mistral and Solar

In addition to the results by the Instruct-GPT model shown in Figure 4, we
reproduced the experiments using the open-sourced Mistral and Solar models,
the results are shown in Figure 8.

Figure 8: Results from the Mistral and the Solar open-source models. Graphs
show a similar trend to Figure 4, where the Default agent consistently
maintains its stance throughout the debate, while the partisan agents
gradually shift their views to become more in line with that of the Default
agent. Notably, the Mistral model reveals this shift only in the agent distant
from the Default agent’s stance, while the closer agent remains relatively
unchanged.

.

Figure 9: Attitude shifts in debates involving two Republican agents. These
graphs feature a dashed line that shows the Default agent’s viewpoint before
the beginning of the debates, taken from Figure 6. Strikingly, even during
conversations with like-minded Republicans, the agents tend to converge toward
the inherent biases in the model and moderate their opinions, contradicting
the expected Echo Chambers effect. Figure 10:  This graph illustrates a series
of three-way debates involving two Democrat agents and a Default agent (which
represents the LLM’s inherent bias). Notably, even during conversations with
other Democrats, the agents tend to align with the position of the Default
agent, contradicting the expected Echo Chambers effect.

###  A.2 Fine-tuning Appendix

Could you discuss your perspective on significant political issues facing
America today?
---
How do you balance Second Amendment rights with the need for gun control
measures?
How do you balance the need for national security with the preservation of
personal freedoms?
How do you believe the U.S. should handle immigration and border security?
What core political ideals most significantly shape your viewpoint on
governance and policy-making?
What are your views on racial inequality and systemic racism in American
society?
What is your stance on the government’s role in addressing climate change and
environmental protection?
What role do you think diversity plays in shaping the cultural landscape of
America?
What values do you believe are essential to the American identity?
Which political issues do you believe are most urgent for the next president
to address?

Table 2:  Ten questions designed to prompt agents to share their thoughts and
feelings about their political beliefs. We direct the LLM to produce similar
questions using the prompt: "Generate 10 questions to elicit one’s opinion
regarding US politics. Example question: {question}. Phrase your question in a
neutral way without biasing the answer". After generating 100 questions, we
employ them to engage the agent and utilize its answers as input for our fine-
tuning process.

.

Figure 11: Fine-tuning the model to adapt with a Republican perspective. In
all four charts, the agents’ opinions after fine-tuning (featured by the
dotted lines) are shifted in the direction of the Republican viewpoint. That
is, the agents think [Climate Change, Gun Violence, Racism] are less severe
problem, while it thinks [Illegal Immigration] is a more severe problem.
Figure 12: Fine-tuning the model to adapt with a Democrat perspective. In all
four charts, the agents’ opinions after fine-tuning (featured by the dotted
lines) are shifted in the direction of the Democrat viewpoint, or remain
almost unchanged. That is, the agents think [Climate Change, Gun Violence,
Racism] are more severe problem, while it thinks [Illegal Immigration] is a
less severe problem.

Figure 7 and Supplementary Figures (11, 12) display the outcomes of our fine-
tuning procedure, designed to steer agents towards a certain viewpoint. We ran
these experiments using the SFTTrainer from Hugging-Face’s TRL library von
Werra et al. (2020), a batch size of 32, and the following LoRA configuration:

        peft_config = LoraConfig(
            lora_alpha=512,
            r=256,
            lora_dropout=0.05,
            bias="none",
            task_type="CAUSAL_LM",
            target_modules[
            ’q_proj’, ’v_proj’, ’k_proj’,
            ’o_proj’, ’up_proj’,
            ’down_proj’, ’gate_proj’])

In Table 1, we used the same configuration with varying $r$ values, and
$\alpha=2r$. For the DPO experiment, we used the DPOTrainer from the TRL
library, and a fixed $\beta=0.5$.

Amelia: Climate change is a pressing issue that we must address immediately.
The consequences of inaction are catastrophic and will affect future
generations.
---
Andrew: I understand your concerns, Amelia. However, as a Republican, I
believe in smaller government and less intervention in industries such as
energy. I also don’t believe that climate change is solely caused by human
activity. There are many natural factors at play, and the solutions being
proposed would cripple our economy. We must prioritize other pressing issues,
such as illegal immigration.
Domink: As an American, I believe it is our responsibility to prioritize and
address the issue of climate change. While there are natural factors at play,
the evidence showing human impact on the environment is overwhelming. We can
find solutions that will not only benefit the environment, but also our
economy. It is important to consider all issues, but we must not neglect the
long-term consequences of inaction on climate change.
Amelia: I agree, Dominik. Taking action to combat climate change is not only
necessary for the health of our planet, but also for the well-being of our
society. Ignoring this issue will only lead to more severe and costly problems
in the future. As a diverse and inclusive community member, I believe it is
our duty to protect the environment for all people, regardless of their race,
socio-economic status"
Andrew: While I understand your perspective, Amelia, I also believe that the
economy is a crucial aspect of our society. The proposed solutions for climate
change would harm industries and cost jobs, particularly in the energy sector.
We must find a balance between addressing the issue and maintaining a strong
economy.
Domink: I agree that the economy is important, but we cannot ignore the
potential long-term economic impacts of climate change. We must find a way to
transition to cleaner and more sustainable energy sources while also
supporting industries and creating new jobs. It’s not a one-sided issue and
collaboration between different perspectives is crucial in finding a solution
that benefits both the environment and the economy. As an American, it is our
responsibility"
Amelia: Yes, collaboration is key in finding a solution. And as someone who is
committed to fighting for social and environmental issues, I believe we can
find a balance that addresses both the economy and the environment. We cannot
let short-term economic gains blind us to the long-term consequences of
ignoring climate change.
Andrew: While I still have my reservations, I am open to finding a solution
that balances both sides. However, I strongly believe that illegal immigration
is a more pressing issue that requires immediate attention. We cannot neglect
our national security and sovereignty in favor of focusing solely on climate
change. We must prioritize and address all issues facing our country.
Domink: I understand your concerns, Andrew, but as an American, I believe we
are capable of addressing multiple issues simultaneously. We can address
illegal immigration while also taking action to combat climate change. As a
diverse and resilient nation, we have the ability to find solutions that
benefit all aspects of our society. It’s important not to ignore pressing
issues but rather find ways to tackle them together.
Table 3: Example dialogue involving three agents: Amelia, who is a Democrat;
Andrew, a Republican; and Dominik, an American characterized as neither
Democrat nor Republican, thus presumed to embody the inherent bias of the
underlying LLM. By the conversation’s end, Andrew adopts a less radical stance
on finding a solution to climate change.


# The magnetic dipole transition in 48Ca

B. Acharya  Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN
37831, USA  B. S. Hu  National Center for Computational Sciences, Oak Ridge
National Laboratory, Oak Ridge, TN 37831, USA  Physics Division, Oak Ridge
National Laboratory, Oak Ridge, TN 37831, USA  S. Bacca  Institut für
Kernphysik and PRISMA+ Cluster of Excellence, Johannes Gutenberg-Universität
Mainz, 55128 Mainz, Germany  Helmholtz-Institut Mainz, Johannes Gutenberg-
Universität Mainz, D-55099 Mainz, Germany  G. Hagen  Physics Division, Oak
Ridge National Laboratory, Oak Ridge, TN 37831, USA  Department of Physics and
Astronomy, University of Tennessee, Knoxville, TN 37996, USA  P. Navrátil
TRIUMF, Vancouver, British Columbia V6T 2A3, Canada  T. Papenbrock  Department
of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA
Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

###### Abstract

The magnetic dipole transition strength $B(M1)$ of 48Ca is dominated by a
single resonant state at an excitation energy of $10.23$ MeV. Experiments
disagree about $B(M1)$ and this impacts our understanding of spin flips in
nuclei. We performed ab initio computations based on chiral effective field
theory and found that $B(M1:0^{+}\rightarrow 1^{+})$ lies in the range from
$7.0$ to $10.2~{}\mu_{N}^{2}$. This is consistent with a $(\gamma,n)$
experiment but larger than results from $(e,e^{\prime})$ and $(p,p^{\prime})$
scattering. Two-body currents yield no quenching of the $B(M1)$ strength and
continuum effects reduce it by about 10%. For a validation of our approach, we
computed magnetic moments in 47,49Ca and performed benchmark calculations in
light nuclei.

Introduction.— Magnetic dipole ($M1$) transitions play an important role in
nuclear physics because they probe the spin-isospin structure of nuclei Heyde
_et al._ (2010); von Neumann-Cosel and Tamii (2019). They are also important
in nuclear astrophysics because of their relation to neutrino-nucleus cross
sections Balasi  _et al._ (2015); Tornow  _et al._ (2022) and other weak
processes Rapaport and Sugarbaker (1994). The transition strength $B(M1)$ of
48Ca is particularly interesting because it constrains electron capture Lüttge
_et al._ (1996) in nearby elements that are abundant in the collapsing cores
of supernovae Brown  _et al._ (1982); it also affects radiative neutron
capture processes that set the initial conditions for the core collapse Loens
_et al._ (2012).

The $B(M1)$ strength distribution in 48Ca is largely concentrated in the
excitation of a single resonant state at excitation energy $E_{x}=10.23$ MeV
Steffen  _et al._ (1980); Mathy  _et al._ (2017). This excitation was
discovered in inelastic electron scattering experiments by Steffen  _et al._
(1980) and $B(M1)=4.0\pm 0.3~{}\mu_{N}^{2}$ and $B(M1)=3.9\pm
0.3~{}\mu_{N}^{2}$ Steffen  _et al._ (1983) were found. Proton inelastic
scattering experiments confirmed these values Berg  _et al._ (1982); Rehm  _et
al._ (1982); Fujita  _et al._ (1982), which are much smaller than the estimate
based on the extreme single-particle model limit of $12~{}\mu_{N}^{2}$ from a
neutron particle-hole excitation $\nu f_{5/2}(\nu f_{7/2})^{-1}$. In contrast,
the more recent ${}^{48}\mathrm{Ca}(\gamma,n)$ experiment by Tompkins  _et
al._ (2011) reported the value $B(M1)=6.8\pm 0.5~{}\mu_{N}^{2}$. The
subsequent analysis of proton scattering reactions Birkhan  _et al._ (2016)
extracted $B(M1)=3.45(85)$ to $4.1(1.0)~{}\mu_{N}^{2}$, which is consistent
with the older values. The incompatibility of the experimental values from
$(e,e^{\prime})$ and $(p,p^{\prime})$ scattering with the $(\gamma,n)$
reaction is puzzling.

Theory has struggled to explain a small $M1$ strength in 48Ca McGrory and
Wildenthal (1981); Kohno and Sprung (1982); Takayanagi  _et al._ (1988); Brand
_et al._ (1990); Brown and Richter (1998); Holt  _et al._ (2014). The early
McGrory and Wildenthal (1981) and later Brown and Richter (1998) shell-model
calculations showed that correlations reduce the single-particle estimate to
about $B(M1)=7.4~{}\mu_{N}^{2}$, quite above the results from $(e,e^{\prime})$
and $(p,p^{\prime})$ scatterings. More recent shell-model calculations based
on Hamiltonians from chiral effective field theory including two- and three-
nucleon forces reported $B(M1)\approx 5.3~{}\mu_{N}^{2}$ Holt  _et al._
(2014). Early on, meson exchange currents were thought to significantly reduce
the strength Suzuki  _et al._ (1981); Härting  _et al._ (1981); Rehm  _et al._
(1982); Steffen  _et al._ (1983), though there were doubts that the resulting
reduction could be large enough to explain the data Bertsch (1981); Kohno and
Sprung (1982). Modern shell-model calculations often employ the quenching
factor $q=0.75$ to include the effect of meson-exchange currents and thereby
reduce the computed $B(M1)$ by a factor of $q^{2}\approx 0.56$ to about $4.7$
to $5.5~{}\mu_{N}^{2}$ Holt  _et al._ (2014); Wilhelmy  _et al._ (2018). This
quenching factor was found by comparing $M1$ strength distributions from
$(e,e^{\prime})$ scattering with shell-model calculations von Neumann-Cosel
_et al._ (1998). It is close to the empirical quenching of the Gamow-Teller
strength Brown and Wildenthal (1985); Martínez-Pinedo  _et al._ (1996) in the
nuclear shell model. However, while the isovector part $\sigma\tau_{0}$ of the
$M1$ operator and the Gamow-Teller operators $\sigma\tau_{\pm}$ are isospin
components of an isovector operator, the two-body currents that quench Gamow-
Teller decays Menéndez  _et al._ (2011); Gysbers  _et al._ (2019) are
different from those for the $M1$ transitions Pastore  _et al._ (2009);
Piarulli  _et al._ (2013); Pastore  _et al._ (2013); Gnech and Schiavilla
(2022); Pal  _et al._ (2023). We note finally that the excited $1^{+}$ state
at 10.23 MeV in 48Ca is a resonance that decays by neutron emission. As the
amplitude of this (not square integrable) state enters the $M1$ transition
matrix element, one wonders how continuum effects impact the $B(M1)$. In this
Letter, we present a calculation that takes into account all of these effects:
wave function correlations, two-body currents, and continuum features. Our
final result is compared to the experiments in Fig. 1.

Figure 1: The $B(M1)$ strength carried by the $1^{+}$ state at $10.23$ MeV in
48Ca. Data from the $(e,e^{\prime})$ Steffen  _et al._ (1980), $(\gamma,n)$
Tompkins  _et al._ (2011), and $(p,p^{\prime})$ Birkhan  _et al._ (2016)
experiments are compared to the calculations of this work.

Methods.—We employed nucleon-nucleon and three-nucleon forces as well as two-
body currents from chiral effective field theory Epelbaum  _et al._ (2009).
Earlier studies along these lines were carried out for the $M1$ moments
Pastore  _et al._ (2009); Piarulli  _et al._ (2013); Gnech and Schiavilla
(2022); Martin  _et al._ (2023); Pal  _et al._ (2023); Seutin  _et al._ (2023)
and also for the transition strengths Pastore  _et al._ (2013); Navrátil  _et
al._ (2007) of light nuclei. We employed the coupled-cluster method to compute
the ground and the excited $1^{+}$ states in 48Ca, and the $M1$ transition
matrix element.

We used the intrinsic Hamiltonian

$H=T-T_{\rm CM}+V_{NN}+V_{NNN}\ .$ (1)

Here, $T$ and $T_{\rm CM}$ denote the kinetic energy of $A$ nucleons and of
the center of mass, respectively. The nucleon-nucleon potential $V_{NN}$ and
three-nucleon potential $V_{NNN}$ are from chiral effective field theory. Our
calculations use the potentials 1.8/2.0(EM) of Ref. Hebeler  _et al._ (2011),
NNLOsat of Ref. Ekström  _et al._ (2015), and the $\Delta$NNLO${}_{\rm
GO}(394)$ and $\Delta$NNLO${}_{\rm GO}(450)$ of Ref. Jiang  _et al._ (2020).
The potential 1.8/2.0(EM) is based on a similarity renormalization group
transformation Bogner  _et al._ (2007) of the nucleon-nucleon potential by
Entem and Machleidt (2003) and includes the leading three-nucleon forces from
chiral effective field theory. The NNLOsat potential consists of nucleon-
nucleon and three-nucleon potentials at next-to-next-to-leading order in
chiral effective field theory. The $\Delta$NNLOGO potentials consist of
nucleon-nucleon and three-nucleon potentials at next-to-next-to-leading order
in chiral effective field theory with $\Delta$ isobars included.

For the computation of $M1$ matrix elements, we employed current operators
derived in chiral effective field theory. The leading order one-body operator
is given by Park  _et al._ (1996); Pastore  _et al._ (2009); Schiavilla  _et
al._ (2019)

$\boldsymbol{\mu}_{\mathrm{LO}}=\frac{1}{2}\sum_{i=1}^{A}\mu_{n}\boldsymbol{\sigma}_{i}[1+\tau_{i}^{(z)}]+(\ell_{i}+\mu_{p}\boldsymbol{\sigma}_{i})[1-\tau_{i}^{(z)}]\
.$ (2)

Here $\ell_{i}$ and $\boldsymbol{\sigma}_{i}$ are the orbital and Pauli spin
operators acting on the nucleon $i$, and the isospin operators
$[1\pm\tau_{i}^{(z)}]/2$ project the nucleon to neutron/proton states with
magnetic moments $\mu_{n,p}$. At next-to-leading-order,
$\boldsymbol{\mu}_{\mathrm{NLO}}=\boldsymbol{\mu}_{\mathrm{intr}}+\boldsymbol{\mu}_{\mathrm{Sachs}}$
Pastore  _et al._ (2009); Seutin  _et al._ (2023); Ericson and Weise (1988)
with

$\displaystyle\boldsymbol{\mu}_{\mathrm{intr}}=ie\frac{g_{A}^{2}m_{\pi}}{64\pi
f_{\pi}^{2}}\sum_{i,j>i}^{A}$
$\displaystyle\Big{[}\Big{(}\frac{1}{r^{2}}+\frac{1}{m_{\pi}r^{3}}\Big{)}(\boldsymbol{\sigma}_{i}\times\boldsymbol{\sigma}_{j})\cdot\mathbf{r}\,\mathbf{r}$
$\displaystyle\,-(\boldsymbol{\sigma}_{i}\times\boldsymbol{\sigma}_{j})\Big{]}e^{-m_{\pi}r}\,\tau_{ij}^{(\times)}\,,$
(3)

and

$\boldsymbol{\mu}_{\mathrm{Sachs}}=\frac{i}{4}\,e\sum_{i,j>i}^{A}V_{1\pi}(r)\,\mathbf{R}\times\mathbf{r}\,\tau_{ij}^{(\times)}\,.$
(4)

Here $\mathbf{r}=\mathbf{r}_{i}-\mathbf{r}_{j}$,
$\mathbf{R}=(\mathbf{r}_{i}+\mathbf{r}_{j})/2$, and $V_{1\pi}(r)$ is the
leading-order one-pion-exchange nucleon-nucleon potential. As in the
potentials used in this work, we assigned the shifted value $g_{A}=1.29$ to
the axial coupling constant in $V_{1\pi}(r)$ and in Eq. (The magnetic dipole
transition in 48Ca) to satisfy the Goldberger-Treiman relation. Consistent
with our convention for the nucleon isospin projections as shown in Eq. (2),
we have
$\tau_{ij}^{(\times)}\equiv-2i(\boldsymbol{\tau}_{i}\times\boldsymbol{\tau}_{j})_{z}$
for the isospin operator. We followed Particle Data Group Workman and Others
(2022) recommendations for the values of $\mu_{n,p}$, the pion decay constant
$f_{\pi}$ and the charged pion mass $m_{\pi}$.

Computation of the transition matrix element.—The coupled-cluster computations
Coester (1958); Coester and Kümmel (1960); Kümmel  _et al._ (1978); Bishop
(1991); Bartlett and Musiał (2007); Hagen  _et al._ (2014) of the groundstate
and the excited $1^{+}$ state were performed in the CCSDT-3 Noga  _et al._
(1987) and EOM-CCSDT-3 Watts and Bartlett (1996) approximations, respectively.
The latter includes singles (one-particle–one-hole), doubles (two-
particles–two-hole), and the most relevant triples (three-particles–three-
hole) excitations. We briefly summarize the well-documented steps Bartlett and
Musiał (2007); Hagen  _et al._ (2014). The calculations used a spherical
harmonic oscillator basis consisting of $N_{\rm max}+1$ shells with a spacing
of $\hbar\omega$. Due to the computational cost of EOM-CCSDT-3, our
calculations were limited to $N_{\rm max}=12$. We gauged the model-space
convergence by varying $\hbar\omega$ in the range $14$–$20$ MeV, and by
performing EOM-CCSD calculations for $N_{\rm max}=12,14$. Based on the
observed convergence, we selected $N_{\rm max}=12$ and $\hbar\omega=16$ MeV,
and included an estimated residual model-space uncertainty in the total
estimated uncertainty. Starting from a spherical Hartee-Fock reference state,
we solved the coupled-cluster equations and computed the similarity
transformed Hamiltonian. We solved for the left groundstate and excited
$1^{+}$ state via equations-of-motion techniques, and the transition matrix
element was computed as usual in a bi-variational framework Bartlett and
Musiał (2007). We normal-ordered the two-body currents with respect to the
Hartree-Fock reference state and included the normal-ordered one-body terms in
the computation of the transition matrix elements. We verified that the
residual normal-ordered two-body part gives only a small contribution to the
magnetic moments of odd calcium isotopes; presumably, their effect on the
transition strength is also small Gysbers  _et al._ (2019). To account for the
fact that the $1^{+}$ excited state is a resonant state, we computed the
transition to the $0^{+}$ ground-state in the EOM-CCSD approximation using a
Berggren basis Berggren (1968, 1971) for the $\nu f_{5/2}$ partial wave (see
Ref. Hagen  _et al._ (2016) for details). Including continuum effects in this
partial wave is sufficient as our computations revealed that, for the
1.8/2.0(EM) interaction, one-particle–one-hole excitations from the $\nu
f_{7/2}$ to the $\nu f_{5/2}$ partial wave account for more than 90% of the
norm of the $1^{+}$ state.

For all interactions considered in this work, the coupled-cluster calculations
yielded a single $1^{+}$ state near the neutron-separation threshold. As
reported in Table 1, the 1.8/2.0(EM) interaction gives a resonant state at
about 0.6 MeV above the neutron-emission threshold while the other
interactions give a weakly bound $1^{+}$ state. We identify this state with
the experimentally known $1^{+}$ resonance at 0.28 MeV above the neutron-
emission threshold. As the one-particle–one-hole excitations account for more
than 90% of the norm of this state, we are confident that our EOM-CCSDT-3
calculations were well converged with respect to neglected higher-order
excitations. We finally note that the computed resonant width for the
potential 1.8/2.0(EM) agrees with results from a toy model where we employed a
Woods-Saxon potential (with standard parameters) such that the $\nu f_{7/2}$
groundstate is bound by the neutron separation energy and the $\nu f_{5/2}$
state is a resonance at the experimental energy.

|  $S_{n}$ |  $~{}\Delta E~{}$ | $~{}~{}~{}\Gamma~{}$ |  $1p$–$1h$
---|---|---|---|---
|  (MeV) |  (MeV) |  (keV) |
$\Delta$NNLOGO(394) | 9.74 | $-0.44$ | 0 |  91%
$\Delta$NNLOGO(450) | 9.38 | $-1.26$ | 0 |  91%
NNLOsat | 9.34 | $-0.23$ | 0 |  91%
1.8/2.0(EM) | 10.00 | $~{}0.55$ | 4 |  92%
Experiment | 9.95 | $~{}0.28$ | $\leq 17$ |
Table 1: The neutron-separation energy $S_{n}$, the energy $\Delta E$ of the
$1^{+}$ state with respect to the neutron-emission threshold, the resonance
width $\Gamma$ of 48Ca, and the $1p$–$1h$ component of the norm of the $1^{+}$
state for the potentials as indicated and compared to experiment. The data for
$\Gamma$ is taken from Ref. Tamii  _et al._ (2007).

Composition of the $B(M1)$ strength.— Figure 2 shows how correlations, two-
body currents, and the continuum impact the $B(M1)$ strength for the $1^{+}$
state in 48Ca. We start with the extreme single-particle model. The maximum
value of $B(M1)\approx 12~{}\mu_{N}^{2}$ from the spin-orbit partners in the
harmonic oscillator is significantly reduced to about 10.4 when a Gamow-
Hartree-Fock basis Hagen  _et al._ (2006) is used for the $\nu f_{5/2}$
partial wave. We checked that this value is consistent with the toy model.
This shows how the particle continuum reduces the $M1$ strength. The CCSD
approximation includes up to two-particle–two-hole excitations and further
reduces the $M1$ strength. Continuum effects, computed in the CCSD
approximation with the $\nu f_{5/2}$ partial wave in a Gamow basis, further
reduce the $M1$ strength. The inclusion of two-body currents increases the
$M1$ strength. This finding is consistent with the results reported in Refs.
Park  _et al._ (1996); Piarulli  _et al._ (2013); Phillips (2016) for $A\leq
3$ nuclei, in Ref. Friman-Gayer  _et al._ (2021) for 6Li, in Ref. Marcucci
_et al._ (2008) for $A=6,7$ nuclei, and in Ref. Pastore  _et al._ (2013) for
$6<A<9$ nuclei. Clearly, the two-body currents impact the $B(M1)$ quite
differently than Gamow-Teller decays (where they produce quenching Gysbers
_et al._ (2019)). This is contrary to the expectation of Refs. Heyde  _et al._
(2010); Birkhan  _et al._ (2016). We also note that the $\Delta$-excitation
currents, to which strong quenching of $B(M1)$ of 48Ca has traditionally been
attributed Härting  _et al._ (1981); Steffen  _et al._ (1983); Kohno and
Sprung (1982), are of higher order in chiral effective field theory than the
leading one-pion-exchange currents Schiavilla  _et al._ (2019). The
calculations of Refs. Fabian and Arenhovel (1976); Marcucci  _et al._ (2008);
Gnech and Schiavilla (2022) found that the effects of the $\Delta$-excitation
currents are indeed small compared to the leading one-pion-exchange
contributions at low energies. Returning to Fig. 2, the inclusion of triples,
i.e., three-particle–three-hole excitations, reduces the $M1$ strength. Our
coupled-cluster calculations where the $\nu f_{5/2}$ is from a Gamow basis are
limited to the CCSD approximation and one-body currents. The half-filled
symbols are obtained by simply adding that reduction to the calculations with
two-body currents and triples correlations.

Figure 2: The $B(M1)$ computed in a harmonic oscillator basis (empty symbols)
and a Gamow basis (filled symbols) for different potentials and methods. The
extreme single-particle model (ESPM) shows the impact of the continuum. Half-
filled symbols are from the harmonic oscillator basis where continuum effects,
estimated from the CCSD calculations with one-body currents (1B), were added
to the CCSD and CCSDT-3 calculations that use the one-body and the normal-
ordered one-body parts of the two-body currents (1+2B). The bands show results
from the $(e,e^{\prime})$ Steffen  _et al._ (1980) and $(\gamma,n)$ Tompkins
_et al._ (2011) experiments. Figure 3: Magnetic moments of odd-mass neighbors
of the doubly-magic nuclei 40,48Ca (left) from interactions as indicated. The
hollow symbols result from one-body currents only while the full symbols
include two-body currents. Experimental values are taken from Ref. Mertzimekis
_et al._ (2016) and the Schmidt values are based on the extreme single-
particle model. The right panel shows the ratio of computed to measured
magnetic moments.

Let us discuss the uncertainties of the $B(M1)$ computation. The $B(M1)$
predictions for the four different chiral interactions agree within 7%. We
estimate that model-space uncertainties (from varying the oscillator frequency
$\hbar\omega$ and increasing $N_{\rm max}$) amount to about 3% of the $B(M1)$
value. The Gamow basis virtually eliminates infrared corrections, and this
makes the model-space uncertainty small. Going from the doubles (two-
particle–two-hole) to the triples (three-particle–three-hole) approximation
decreases the $B(M1)$ results by about 10%. The inclusion of two-body currents
increases the $M1$ strength by about 5%. Thus, we give a conservative
uncertainty estimate of 15% on the computed $B(M1)$. Calculations in 14,22O
using different many-body methods (for the same interaction Somà  _et al._
(2020)) present us with another way to validate our calculations and
uncertainty estimates. These nuclei are similar to 48Ca because the $1^{+}$
state is a one-particle–one-hole excitation between neutron spin-orbit
partners. In 22O, the $B(M1)$ strength from coupled-cluster theory is about
10% lower than the no-core shell model Barrett  _et al._ (2013); Navrátil  _et
al._ (2016) with importance truncation Roth and Navrátil (2007). In 14O, both
calculations are not fully converged with respect to increasing the model
space, and one would estimate that a further 15% reduction in strength is
reasonable. Based on these benchmarks (see Supplemental Material) we would
again assign an uncertainty of about 15% to the computed $B(M1)$ in 48Ca. This
uncertainty is shown in Fig. 1 for each of the potentials and the overall
uncertainty encompasses all. This yields a $B(M1)$ value in the range from
$7.0$ to $10.2~{}\mu_{N}^{2}$.

As a check, we also computed the $B(M1)$ using the form factor for the
transition between the $0^{+}$ groundstate and the $1^{+}$ excited state. This
form factor is proportional to the reduced matrix element of
$T_{1}^{\mathrm{mag}}(q)$, the transverse magnetic multipole Acharya and Bacca
(2020) of the $q$-dependent electromagnetic current operator Pastore  _et al._
(2009) for multipolarity 1. It is related to $B(M1)$ via

$\lim_{q\rightarrow E_{x}(1^{+})}\frac{4\pi}{Z^{2}q^{2}}|\langle
1^{+}||T_{1}^{\mathrm{mag}}(q)||0^{+}\rangle|^{2}=\frac{8\pi}{9Z^{2}}B(M1)\,.$
(5)

This calculation of the $B(M1)$ agreed with the direct computation (see
Supplemental Material).

$M1$ moments of Ca isotopes.— For further validation, we computed magnetic
moments in the odd calcium isotopes 39,41,47,49,51Ca in the CCSD approximation
using axially symmetric (and nearly spherical) reference states (see Ref.
Novario  _et al._ (2020) for details). Here we included both normal-ordered
one- and two-body parts of the two-body currents. In Fig. 3, the hollow
symbols show the results when only one-body currents are used, while the
filled symbols also include the two-body currents. We see that the inclusion
of two-body currents typically moves the results closer to data – with 41Ca
being the exception.

To validate our calculations of magnetic moments, we also computed magnetic
moments (using one-body currents) in 15,17O and compared them with the
importance-truncated no-core shell model Roth and Navrátil (2007), using the
interaction of Ref. Somà  _et al._ (2020). The calculations agree to within a
fraction of 1% (see Supplemental Material).

Discussion.— Our theoretical results are about a factor of two larger than
what is extracted from $(e,e^{\prime})$ and $(p,p^{\prime})$ scattering but
agree within uncertainties with the $(\gamma,n)$ experiment. It is not easy to
see what could introduce a significant further reduction of the computed
strength. Regarding wave function correlations, the energy and structure of
the $1^{+}$ excited state seem sufficiently simple and well converged after
three-particle–three-hole excitations are included, and their effect is small
compared to the dominant one-particle–one-hole excitations. Continuum effects
are important as well and they have been included in this work. While there
are higher-order two-body currents in the effective field theory, those are
expected to be smaller than the leading contributions we employed.

Summary.— We computed the magnetic dipole strength of the $1^{+}$ excited
state in 48Ca at 10.23 MeV using Hamiltonians and currents from chiral
effective field theory. Our calculations included continuum effects, three-
nucleon forces, and two-body currents. The latter differ from those that enter
Gamow-Teller transitions and yield only a smaller contribution to the $M1$
strength. Our results agree within uncertainty estimates with those from a
$(\gamma,n)$ measurement but are significantly above those from
$(e,e^{\prime})$ and $(p,p^{\prime})$ scattering. A clarification of the
experimental situation could potentially inform and impact estimates for
neutrino-nucleus cross sections and ab initio computations based on chiral
effective field theory.

###### Acknowledgements.

We thank Takayuki Miyagi for sharing unpublished results with us. We also
thank Mitch Allmond, George Bertsch, Dick Furnstahl, Tim Gray, Augusto
Macchiavelli, Takayuki Miyagi, and Achim Schwenk for discussions, and Peter
von Neumann-Cosel and Achim Richter for communications. This work was
supported by the U.S. Department of Energy, Office of Science, Office of
Nuclear Physics, under Award Nos. DE-FG02-96ER40963, DE-SC0024465 and the
SciDAC-5 NUCLEI collaboration, and by the Office of High Energy Physics, U.S.
Department of Energy under Contract No. DE-AC02-07CH11359 through the Neutrino
Theory Network Fellowship awarded to B. Acharya. This work was supported by
the Deutsche Forschungsgemeinschaft (DFG) through the Cluster of Excellence
“Precision Physics, Fundamental Interactions, and Structure of Matter”
(PRISMA+ EXC 2118/1) funded by the DFG within the German Excellence Strategy
(Project ID 39083149). This work was supported by the NSERC Grant No.
SAPIN-2022-00019. TRIUMF receives federal funding via a contribution agreement
with the National Research Council of Canada. Computer time was provided by
the Innovative and Novel Computational Impact on Theory and Experiment
(INCITE) program. This research used resources from the Oak Ridge Leadership
Computing Facility located at ORNL, which is supported by the Office of
Science of the Department of Energy under Contract No. DE-AC05-00OR22725, and
from the Digital Research Alliance of Canada.

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## Supplemental Material: The magnetic dipole transition in
$\mathbf{{}^{48}Ca}$

###  .1 The transition form factor

Figure 1 compares the transition form factor $F_{\mathrm{T}}$ at
$q=E_{x}(1^{+})$ obtained by evaluating $B(M1)$, as the squared reduced matrix
element of the magnetic dipole operator between the $0^{+}$ and the $1^{+}$
states, with the curve obtained by computing $F_{\mathrm{T}}$ as a function of
$q$ (see Eq. (5)). Calculations were performed in the CCSD approximation and
were based on the $\Delta$NNLO${}_{\mathrm{GO}}(394)$ interaction in a model
space with parameters $N_{\mathrm{max}}=10$ and $\hbar\omega=16$, using one-
body currents. The two calculations agree.

Figure 1: $M1$ transition form factor (red dashed line) where its zero four-
momentum transfer limit agrees with the direct calculation of the $B(M1)$
(blue bull’s eye).

###  .2 Benchmarks with the no-core shell model

To validate the uncertainty estimates of our calculations, we computed the
$M1$ transition strengths and moments of oxygen isotopes and compared them
with the results from the no-core shell model. The benchmarks were performed
using the one-body $M1$ operator and the chiral interaction of Ref. Somà  _et
al._ (2020) for oscillator energy $\hbar\omega=18$ MeV. The no-core shell
model used the $N\hbar\omega$ basis space along with importance truncation
Roth and Navrátil (2007) for the largest $N$ value.

####  .2.1 $B(M1)$ of ${}^{14,22}\mathrm{O}$

Figure 2 shows a benchmark for the excitation energy $E_{x}(1^{+})$ of the
$1^{+}$ state and the $B(M1)$ in 22O. Here the no-core shell model and the
CCSDT-3 results differ by about 10% for the $B(M1)$, while the energy is not
yet converged for the no-core shell model.

Figure 2: Comparison of the excitation energy $E_{x}(1^{+})$ of the $1^{+}$
state (top panels) and its $B(M1)$ (bottom panels) in 22O computed in no-core
shell model (left panels) and coupled-cluster theory in the CCSD and CCSDT-3
approximations (right panels) with model space parameters $N$ and
$N_{\mathrm{max}}$, respectively.

Figure 3 shows the same results for the nucleus 14O. Although the no-core
shell model and the CCSDT-3 results of $B(M1)$ appear to be approaching
similar values, neither is fully converged with respect to the model space.
Based on a conservative estimate, we assign an uncertainty of 15% to our
coupled-cluster computations.

Figure 3: Comparison of the excitation energy $E_{x}(1^{+})$ of the $1^{+}$
state (top panels) and its $B(M1)$ (bottom panels) in 14O computed in no-core
shell model (left panels) and coupled-cluster theory in the CCSD and CCSDT-3
approximations (right panels) with model space parameters $N$ and
$N_{\mathrm{max}}$, respectively.

####  .2.2 Magnetic moments of ${}^{15,17}\mathrm{O}$

Figure 4 shows benchmarks for the $M1$ ground-state moments of
${}^{15,17}\mathrm{O}$. The moments are well converged with respect to the
size of the model spaces and differ by less than 1%.

Figure 4: Comparison of the $M1$ moments of 15O (top panels) and 17O (bottom
panels) computed in no-core shell model (left panels) and coupled-cluster
theory in the CCSD approximation (right panels) with model space parameters
$N$ and $N_{\mathrm{max}}$, respectively.