# Break the Chain: Large Language Models Can be Shortcut Reasoners

Mengru Ding ∗
Gaode Map, Alibaba Group
<EMAIL_ADDRESS>
&Hanmeng Liu ∗
Zhejiang University
<EMAIL_ADDRESS>
&Zhizhang Fu
Gaode Map, Alibaba Group
<EMAIL_ADDRESS>
Jian Song  Wenbo Xie
Gaode Map, Alibaba Group
james.songjian<EMAIL_ADDRESS>
&Yue Zhang †
Westlake University
<EMAIL_ADDRESS>
###### Abstract

Recent advancements in Chain-of-Thought (CoT) reasoning utilize complex
modules but are hampered by high token consumption, limited applicability, and
challenges in reproducibility. This paper conducts a critical evaluation of
CoT prompting, extending beyond arithmetic to include complex logical and
commonsense reasoning tasks, areas where standard CoT methods fall short. We
propose the integration of human-like heuristics and shortcuts into language
models (LMs) through "break the chain" strategies. These strategies disrupt
traditional CoT processes using controlled variables to assess their efficacy.
Additionally, we develop innovative zero-shot prompting strategies that
encourage the use of shortcuts, enabling LMs to quickly exploit reasoning
clues and bypass detailed procedural steps. Our comprehensive experiments
across various LMs, both commercial and open-source, reveal that LMs maintain
effective performance with "break the chain" strategies. We also introduce
ShortcutQA, a dataset specifically designed to evaluate reasoning through
shortcuts, compiled from competitive tests optimized for heuristic reasoning
tasks such as forward/backward reasoning and simplification. Our analysis
confirms that ShortcutQA not only poses a robust challenge to LMs but also
serves as an essential benchmark for enhancing reasoning efficiency in AI.

Break the Chain: Large Language Models Can be Shortcut Reasoners

Mengru Ding ∗ Gaode Map, Alibaba Group<EMAIL_ADDRESS>Hanmeng
Liu ∗ Zhejiang University<EMAIL_ADDRESS>Zhizhang Fu Gaode Map, Alibaba
Group<EMAIL_ADDRESS>

Jian Song and Wenbo Xie Gaode Map, Alibaba Group james.songjian,
<EMAIL_ADDRESS>Yue Zhang † Westlake University
<EMAIL_ADDRESS>

**footnotetext: These authors contributed equally to this work††footnotetext:
Corresponding author

##  1 Introduction

Figure 1: ChatGPT responses to Chain-of-Thought and "Break the Chain". Our
"Break the Chain" method significantly simplifies the reasoning process.

In the evolving landscape of artificial intelligence, the ability to reason
and solve complex problems symbolizes a cornerstone of intelligence. Language
Models (LMs), particularly those based on transformer Vaswani et al. (2017)
architectures, have revolutionized our approach to natural language processing
(NLP), significantly enhancing capabilities in comprehending and generating
text that bears a striking resemblance to human communication.

Among recent advancements, Chain-of-Thought (CoT) prompting has emerged as a
pivotal technique for utilizing Large Language Models (LLMs) to address
complex reasoning tasks. By methodically eliciting step-by-step reasoning, CoT
prompting has significantly enhanced the problem-solving capabilities of LLMs
across a variety of learning scenarios, including few-shot Wei et al. (2022)
and zero-shot contexts Kojima et al. (2022a). Figure 1 illustrates a zero-shot
example in which the ChatGPT model methodically resolves a mathematical
question. This strategy is further augmented by approaches such as self-
consistency Wang et al. (2022b, 2023c), interactive reasoning Yao et al.
(2022a); Shinn et al. (2024), reflective thinking Ling et al. (2024); Li et
al. (2023), task decomposition Khot et al. (2022); Press et al. (2022), and
strategic planning Wang et al. (2023b); Hu et al. (2023).

Despite its benefits, CoT is also critiqued for its substantial token usage,
as it explores numerous reasoning pathways before arriving at a conclusive
answer. This characteristic is particularly prominent in variants such as
Tree-of-Thought (ToT) Yao et al. (2023), which scrutinize every possible
reasoning chain. Traditionally, CoT has been predominantly applied to
mathematical reasoning, with scant application to commonsense, or complex
logical reasoning tasks. This limited focus may hinder a comprehensive
understanding of CoT’s potential to emulate intricate human-like reasoning
processes. Additionally, instruction fine-tuned (IFT) Ouyang et al. (2022)
large language models like ChatGPT, which are usually capable of reaching the
answers methodically, further question the necessity for explicit CoT
prompting Chen et al. (2023).

Human reasoning uses heuristics to find local rational maximum Karlan (2021);
Neth and Gigerenzer (2015); Lancia et al. (2023), which often relies on
cognitive shortcuts Fernbach and Rehder (2013); Ferrario (2004), a
characteristic that can be mirrored and exploited in LMs. Traditionally, LLMs’
shortcut learning has been viewed as the acquisition of spurious correlations
within datasets Du et al. (2023); Jiang and Bansal (2019); Branco et al.
(2021). However, this perspective fails to capture the nuanced heuristic
reasoning processes inherent in human cognition, both in everyday scenarios
and professional contexts such as clinical decision-making. We argue that
shortcut reasoning, by drastically reducing reasoning steps and computational
demands, offers a valuable means of enhancing LLM efficiency. As depicted in
Figure 1, when prompted with shortcut reasoning, the ChatGPT model swiftly
arrives at answers with minimal token consumption. The ability of LLMs to
employ shortcut reasoning not only mirrors human cognitive strategies but also
has the potential to streamline problem-solving processes, thereby reshaping
computational efficiency and model performance.

The primary goal of our study is to critically evaluate and challenge the
established Chain-of-Thought (CoT) prompting framework used in Large Language
Models (LLMs). Our approach is three-pronged: First, we explore the
effectiveness, limitations, and mechanisms of CoT by comparing it with
different prompts derived from the "break the chain" strategy in both few-shot
and zero-shot scenarios. Second, the study pioneers the use of shortcut
reasoning prompts that encourage LLMs to utilize heuristic shortcuts — akin to
intuitive leaps in human reasoning — to efficiently solve problems. This
method aims to minimize computational demands and token consumption while
maintaining or potentially enhancing performance accuracy. To support this
investigation, we introduce ShortcutQA, a novel dataset meticulously curated
to specifically assess the ability of LLMs to employ heuristic shortcuts. We
conducted experiments on both OpenAI models and open-source models of various
sizes, including Mixtral-8x7B-Instruction, Llama-3-70B-Instruction,
Qwen1.5-72B-Chat, Qwen1.5-14B-Chat, Qwen1.5-1.8B-Chat, to ascertain the
generalizability of our experimental conclusions across different model
configurations.

Our few-shot experiments reveal that Large Language Models (LLMs) are not
adversely affected by disrupted Chain-of-Thought (CoT) demonstrations, casting
doubts on the effectiveness of few-shot CoT methods. To our knowledge, this is
the first series of experiments designed to "break the chain" of in-context
examples. Furthermore, in zero-shot scenarios, models prompted with shortcut
reasoning display robust performance, often surpassing that of traditional CoT
methods. Our evaluations span both OpenAI models and open-source models,
showing consistent results across platforms.

Furthermore, our comparative analysis elucidates distinct performance trends
across various model sizes: smaller models typically experience more
substantial enhancements with Chain-of-Thought (CoT) prompts compared to their
larger counterparts. Notably, as model size increases, the efficacy of "break
the chain" strategies becomes more pronounced, highlighting its effectiveness
in mitigating the impact of disrupted CoT demonstrations.

Most notably, we observe that shortcut reasoning significantly reduces token
consumption, providing a vital advantage in computational efficiency. Under
stringent token constraints, shortcut reasoning strategies not only conserve
resources but also consistently outperform traditional CoT methods. These
benefits are observed across various datasets, underscoring the robustness and
scalability of shortcut reasoning as a superior approach in enhancing LLM
performance.

##  2 Related Work

###  2.1 CoT Prompting in Large Language Models

The evolution of Chain-of-Thought (CoT) prompting, particularly through few-
shot Wei et al. (2022) and zero-shot Kojima et al. (2022a) methodologies, has
markedly advanced Large Language Models’ (LLMs) ability to address complex
reasoning challenges. This field has witnessed the introduction of
sophisticated data structures, such as Tree-of-Thought Yao et al. (2023),
Graph-of-Thought Besta et al. (2024), and Program-of-Thought Chen et al.
(2022), enriching LLMs’ capacity for introspection and nuanced evaluation of
their reasoning paths.

Beyond conventional prompting strategies, the ReAct model Yao et al. (2022b)
integrates reasoning with actionable tasks like data retrieval, whereas the
Selection-Inference framework Creswell et al. (2023) combines context creation
with logical chaining. While pioneering, these approaches rely on the models’
inherent abilities and do not embed explicit logical rules within the
reasoning process.

The adoption of external tools in prompting paradigms, especially for tasks
that demand supplementary knowledge, has also shown considerable progress.
Analogous to the role calculators play in mathematical reasoning, introducing
predefined functions for enforcing inference rules marks a significant step
forward in leveraging external computational aids to bolster reasoning
capabilities.

Moreover, breaking down complex reasoning tasks into more manageable
subproblems or engaging multiple models for collaborative problem-solving has
introduced novel methodologies in LLM prompting. Strategies such as Cumulative
Reasoning Zhang et al. (2023a) focus on an iterative, step-wise approach,
while ScratchPad Nye et al. (2021) emphasizes the articulation of intermediate
steps in multi-step reasoning. Meta-prompting Suzgun and Kalai (2024)
envisions a cooperative framework where LLMs act as orchestrators, decomposing
tasks, delegating them to specialized models, and synthesizing the outcomes,
thereby fostering a holistic approach to problem-solving.

In the specific arena of instruct-tuning LLMs with tailored datasets for
advanced reasoning, initiatives like LogiCoT Liu et al. (2023), which fine-
tunes an LLaMA-7b model with data on logical chaining, demonstrate
considerable improvements in logical reasoning tasks. Similarly, LogicLLM Jiao
et al. (2023) explores a self-supervised learning strategy for logical
reasoning enhancements, and Symbol-LLM Xu et al. (2023) incorporates symbolic
data in a two-stage tuning process to equip a LLaMA-2-chat model with symbolic
reasoning skills. These efforts highlight the potential of fine-tuning with
specialized datasets to significantly enhance the reasoning capabilities of
LLMs, illustrating the dynamic and evolving landscape of CoT prompting in AI
research.

###  2.2 Questioning CoT

Despite the demonstrated effectiveness of Chain-of-Thought (CoT) in enhancing
model performance on complex tasks, the underlying mechanisms by which Large
Language Models (LLMs) generate CoT responses are not fully understood.
Research efforts are increasingly focused on demystifying CoT prompting,
providing empirical insights and developing theoretical frameworks to
comprehend this advanced reasoning capability. However, numerous studies have
highlighted the brittleness of CoT reasoning in various aspects.

Turpin et al. (2023) investigate the faithfulness of CoT reasoning, revealing
systematic misrepresentations in the true rationale behind a model’s
predictions. Lanham et al. (2023) extend this inquiry by introducing errors or
paraphrases within the CoT process to test whether the articulated reasoning
truly reflects the model’s underlying logic, finding that larger models tend
to produce more unfaithful responses. This issue of faithfulness is critical
as it challenges the reliability of CoT explanations. The effectiveness of CoT
is also impacted by the selection and arrangement of demonstrations. Wang et
al. (2023a) find that the accuracy of reasoning chains is less critical than
the relevance of the question and the correctness of the reasoning sequence,
emphasizing the importance of contextual alignment. In contrast, Wang et al.
(2022a) show that CoT can operate even with invalid demonstrations, suggesting
some resilience in the reasoning process. Our research contributes to this
discourse by disturbing the order of the reasoning chain to examine its impact
on CoT consistency.

Dataset | Question Type | # of instances | Avg. # words | Source
---|---|---|---|---
| Analytical shortcuts | 156 | 55.88 | Analytical reasoning tests
ShortcutQA | Logical shortcuts | 108 | 21.76 | Verbal reasoning tests
| Mathematical shortcuts | 185 | 67.19 | Gaokao examinations

Table 1: Dataset statistics of ShortcutQA.

Jin et al. (2024) demonstrate that artificially lengthening the reasoning
steps in prompts — simply by instructing models to "think more steps" — can
enhance LLMs’ performance across various datasets without introducing new
content. This finding suggests that the perceived depth of reasoning may
artificially inflate effectiveness. Conversely, we explore minimalist
prompting strategies where LLMs are instructed to streamline their reasoning
processes.

The sensitivity of LLMs to the ordering of premises is scrutinized by Chen et
al. (2024), who note optimal performance when the order of premises supports
the necessary context in intermediate reasoning steps. This sensitivity is
paradoxical in deductive reasoning contexts where the order of premises should
not logically influence the validity of conclusions. Similarly, Pfau et al.
(2024) Indicates that LLMs solve more problems with meaningless filler tokens
in place of a chain of thought than without meaningless tokens. This finding
suggests that CoT’s effectiveness may sometimes rely solely on the increase in
computational effort, rather than on the literal intermediate reasoning steps.
Our "break the chain" methods experiment with new models and datasets and aim
to illuminate this issue further.

Implicit CoT Deng et al. (2023, 2024) has been introduced to internalize
explicit step-by-step reasoning. Similar to our work, implicit CoT questions
the necessity of step-by-step reasoning. However, we diverge from prior
studies that employed fine-tuning to reduce the need for reasoning steps.

Finally, Chen et al. (2023) question the applicability of CoT in instruction
fine-tuned (IFT) models like ChatGPT, which show inconsistent performance
across various reasoning tasks. Surprisingly, while CoT prompts enhance some
reasoning tasks, they fail in others like arithmetic reasoning, where ChatGPT
can independently generate CoT sequences without specific prompts. This
phenomenon inspires us to abstract a hypothesis that more powerful models
increasingly exhibit a reduced dependency on CoT. Our subsequent experiments
conducted within the Qwen1.5 series of various sizes strive to support this
viewpoint.

##  3 ShortcutQA

The ShortcutQA dataset is designed to evaluate Language Models’ (LMs) ability
to employ heuristic shortcuts in reasoning, addressing a gap in existing
resources that primarily focus on sequential reasoning approaches. Comprising
449 diverse reasoning problems, ShortcutQA spans logical puzzles to real-world
problem-solving scenarios. Each problem is presented with a shortcut-based
solution alongside a detailed step-by-step solution, categorized into three
reasoning types.

Data Collection and Annotation

Data for ShortcutQA were sourced from various online forums and educational
websites, with necessary permissions secured. Annotation was conducted by two
independent domain experts, adhering to strict guidelines for identifying and
categorizing heuristic shortcuts employed in the solutions. A third expert
resolved any discrepancies, ensuring high annotation quality and consistency.

Dataset Categorization

ShortcutQA introduces problems categorized into three distinct types, each
testing different aspects of heuristic reasoning:

  * •

Analytical Shortcuts: Tasks necessitate analyzing situations beyond mere
comprehension, assessing models’ capabilities in efficiently synthesizing and
utilizing key information, and strategic decision-making under time
constraints.

  * •

Logical Shortcuts: Encompassing forms of reasoning such as analogical,
abductive, and forward/backward reasoning, these tasks focus on applying these
logical theories to derive conclusions from provided statements.

  * •

Mathematical Shortcuts: Features problems solvable through approximation
techniques, substitution, simplification, and special-case reasoning,
bypassing traditional sequential thought processes.

Data Statistics are shown in Table 1. We release the data at
https://anonymous.com.

##  4 Method

###  4.1 Break the Chain

To examine the resilience and limitations of Large Language Models (LLMs) in
employing Chain-of-Thought (CoT) reasoning, our research outlines a novel
experimental framework aimed at "breaking the chain" of thought. This approach
seeks to elucidate the conditions under which CoT reasoning may falter,
thereby offering insights into the underlying mechanisms of LLMs’ reasoning
capabilities. Our methodology juxtaposes zero-shot and few-shot scenarios to
delineate the impact of CoT disruption across different prompting contexts.

Few-Shot In the few-shot scenario, our strategy involves perturbing the
sequence of sentences within the in-context examples provided to the LLM. This
disturbance is designed to misalign the logical progression typically
demonstrated in CoT reasoning, thereby testing the model’s ability to maintain
coherent and accurate reasoning despite the disordered presentation of steps.
This manipulation will help ascertain the significance of stepwise logical
progression in the model’s reasoning efficacy and its ability to reorient
itself to reach correct conclusions.

Zero-Shot We initiate probing experiments to assess the efficacy of zero-shot
CoT prompts, aiming to discern whether CoT prompting is essential or merely a
byproduct of longer model responses. Employing controlled experiments, we
craft prompts that obviate the need for reasoning chains, instructing models
to provide either more verbose or minimalist responses. Detailed descriptions
of these prompts are provided in Appendix A. Furthermore, we employ
meticulously designed prompts to stimulate shortcut reasoning, outlined
comprehensively in Appendix A. By directing LLMs to circumvent intermediate
reasoning steps typically associated with CoT, we aim to evaluate the
resilience of their inferential processes and their reliance on detailed
reasoning pathways.

ShortcutQA Probing Parallel to our few-shot and zero-shot experiments, we
introduce the ShortcutQA dataset into our methodology. ShortcutQA is carefully
curated to focus on questions that require shortcut reasoning — a form of
intuitive problem-solving that deviates from traditional step-by-step logical
deduction. The inclusion of ShortcutQA is intended to test the hypothesis that
LLMs can effectively employ heuristic shortcuts, akin to human cognitive
shortcuts, to efficiently resolve complex problems.

###  4.2 Experimental Setup

We evaluate Large Language Models (LLMs) across a variety of commercial and
open-source platforms under both few-shot and zero-shot conditions. Our
methodology includes a diverse array of complex problem-solving tasks
encompassing arithmetic reasoning, commonsense deduction, and logical
reasoning. This design rigorously tests the LLMs’ ability to generalize across
different difficulty levels and domains.

Task | Dataset | Size | Avg #words
---|---|---|---
Arithmetic | SingleEq | 508 | 27.4
AddSub | 395 | 31.5
MultiArith | 600 | 31.8
GSM8K | 1319 | 46.9
AQUA-RAT | 254 | 51.9
SVAMP | 1000 | 31.8
Commonsense | CommonsenseQA | 1221 | 27.8
StrategyQA | 2290 | 9.6
Logic | Date Understanding | 369 | 35.0
Coin Flip | 500 | 37.0
LogiQA | 651 | 146.2
ReClor | 500 | 153.0

Table 2: Statistics of Evaluation benchmarks.

As depicted in Figure 4 in Appendix B, the experimental pipeline begins by
inputting a question and a prompt into an LLM, which then generates a reasoned
response and answer. This output is concatenated with the original question
and prompt, followed by an answer extraction prompt to extract the final
answer.

Task | Dataset | Few-shot | Zero-shot
---|---|---|---
Base | Break the Chain | Base | No Steps | More Tokens
Arithmetic | SingleEq | 92.72 | 92.32 | 86.61 | 90.35 | 88.39
AddSub | 84.05 | 85.32 | 83.80 | 89.62 | 86.58
MultiArith | 99.00 | 98.33 | 83.33 | 91.17 | 93.50
GSM8K | 74.60 | 74.22 | 32.68 | 37.53 | 38.89
AQUA-RAT | 53.15 | 55.51 | 35.43 | 36.61 | 38.97
SVAMP | 76.80 | 79.70 | 71.70 | 81.70 | 76.70
Commonsense | CommonsenseQA | 74.94 | 75.18 | 70.52 | 75.92 | 74.28
StrategyQA | 69.13 | 68.60 | 64.37 | 59.91 | 63.23
Logic | Date Understanding | 81.03 | 82.11 | 64.37 | 64.50 | 63.23
LogiQA | 35.94 | 33.95 | 40.09 | 41.17 | 40.09
ReClor | 51.40 | 50.80 | 52.40 | 51.20 | 54.20

Table 3: ChatGPT performance comparison across tasks. All results are in %,
the best ones are in bold.

#### Benchmarks

For arithmetic reasoning, we assess the models using six datasets: SingleEq
Koncel-Kedziorski et al. (2015), AddSub Hosseini et al. (2014) , MultiArith
Roy and Roth (2015) , GSM8K Cobbe et al. (2021), AQUA-RAT Ling et al. (2017),
and SVAMP Patel et al. (2021). The first three originate from the well-
established Math World Problem Repository Koncel-Kedziorski et al. (2016),
with the remaining datasets presenting more recent and complex challenges.
SingleEq and AddSub feature relatively straightforward problems that can be
solved without multi-step reasoning, whereas MultiArith, AQUA-RAT, GSM8K, and
SVAMP require more intricate, sequential problem-solving.

For commonsense reasoning, we utilize the CommonsenseQA Talmor et al. (2019)
and StrategyQA Geva et al. (2021) datasets. CommonsenseQA tests reasoning
based on general world knowledge Talmor et al. (2019) , while StrategyQA
demands inference of unstated, multi-step reasoning processes Geva et al.
(2021).

For logical reasoning tasks, we select two scenarios from the BIG-bench
Srivastava et al. (2022): Date Understanding and Coin Flip Wei et al. (2022).
Date Understanding challenges models to infer dates from given contexts, and
Coin Flip evaluates the ability to determine the outcome of a series of coin
flips. Additionally, we incorporate LogiQA Liu et al. (2020) and ReClor Yu et
al. (2020), which are reading comprehension tests that require logical
deduction.

#### Language Models

We test both OpenAI commercial models and huggingface open-source models. For
OpenAI models, we choose the ChatGPT (gpt-3.5-turbo-0613) model, an IFT GPT-3
model. For community models, we use Llama-3-70B-Instruct,
Mixtral-8x7B-Instruct, Qwen1.5-72B-Chat, Qwen1.5-14B-Chat, Qwen1.5-1.8B-Chat.

#### Baselines

We run zero-shot CoT Kojima et al. (2022b) and few-shot CoT Wei et al. (2022)
on the datasets to establish our baselines. In the few-shot CoT setup, we
follow Wei et al. (2022) to provide each test with context examples; for the
zero-shot baseline, each question is suffixed with “The answer is ”, following
prior work Kojima et al. (2022b); Zhang et al. (2023b).

##  5 Results

Model | Task | Base | Quick Conclude | Shortcut Reasoning | Effective Shortcut | Innovative Shortcut
---|---|---|---|---|---|---
ChatGPT | Arithmetic | 65.59 | 77.23 | 80.11 | 80.58 | 71.34
Commensense | 67.45 | 73.18 | 73.65 | 72.36 | 67.52
Logical | 51.97 | 53.32 | 57.57 | 56.77 | 56.91
Llama-70B | Arithmetic | 72.47 | 62.29 | 81.59 | 63.37 | 50.96
Commensense | 67.57 | 73.00 | 60.58 | 67.29 | 67.27
Logical | 71.41 | 66.26 | 68.60 | 67.18 | 63.95
Mixtral-8x7B | Arithmetic | 70.80 | 73.22 | 71.70 | 68.77 | 56.63
Commensense | 65.03 | 69.37 | 69.23 | 69.50 | 60.81
Logical | 69.08 | 69.61 | 69.84 | 68.48 | 60.46
Qwen1.5-72B | Arithmetic | 65.28 | 76.00 | 75.51 | 74.52 | 70.83
Commensense | 79.11 | 79.85 | 79.38 | 80.36 | 79.78
Logical | 60.17 | 63.42 | 63.58 | 63.62 | 61.79
Qwen1.5-14B | Arithmetic | 63.30 | 71.97 | 71.57 | 69.94 | 66.85
Commensense | 74.43 | 75.65 | 75.14 | 75.25 | 74.20
Logical | 53.47 | 55.76 | 54.91 | 55.50 | 54.38
Qwen1.5-1.8B | Arithmetic | 39.40 | 39.99 | 37.12 | 31.97 | 28.81
Commonsense | 57.61 | 55.07 | 57.19 | 55.95 | 55.20
Logical | 33.00 | 30.72 | 31.00 | 32.37 | 32.05

Table 4: Experiment results concerning different tasks. Detailed results are
in Appendix C. All results are in %, the best ones are in bold.

#### Few-Shot

Table 3 on the left side shows comparative performance between traditional
few-shot CoT and our "breaking the chain" approach across datasets in
commonsense, arithmetic, and logical reasoning tasks. Notably, in arithmetic
reasoning, performance on the MultiArith dataset decreases slightly from
99.00% to 98.33% with "breaking the chain", while in GSM8K, the decrease is
marginal, from 74.60% to 74.22%. In commonsense reasoning, "breaking the
chain" slightly outperforms the traditional approach on CommonsenseQA (75.18%
vs. 74.94%), but underperforms on StrategyQA, dropping from 69.13% to 68.60%.
LogiQA in logical reasoning shows a more notable performance drop from 35.94%
to 33.95%. These results suggest that while "breaking the chain" generally
performs comparably to the few-shot CoT baseline, it does not significantly
impact the model’s overall performance.

#### Zero-Shot

The right side of Table 3 presents results from our zero-shot probing
experiment, comparing the zero-shot CoT baseline with our "break the chain"
prompts across 11 datasets within three key tasks: arithmetic reasoning,
commonsense reasoning, and logical reasoning. Notably, even when we ablate
step-by-step reasoning, ChatGPT maintains competitive performance across
various tasks. Moreover, prompting with only "More Tokens" leads to the best
performance on several other datasets.

Results for the "Shortcut Reasoning" prompts are detailed in Table 4, where
this approach shows substantial improvements: a 22% increase in arithmetic
tasks, a 9% boost in commonsense tasks, and an 11% enhancement in logical
reasoning tasks. Performance is consistent on the Mixtral and Qwen platforms,
though it varies with the Llama models, underlining the effectiveness of our
approach.

In addition, experiments with Qwen models of varying sizes, both under CoT and
"break the chain" conditions, are documented. Figure 5 in Appendix C
illustrates that smaller models exhibit a more pronounced reliance on CoT,
especially as the model size decreases, narrowing performance gaps from a 16%
deficit in 72B models to parity in 1.8B models for arithmetic tasks. For logic
and commonsense tasks, smaller models transition from underperformance to
outperforming larger counterparts, suggesting less capable models benefit more
from CoT’s structured approach.

These findings question the prevailing assumption that CoT invariably enhances
LLM performance. Our results indicate that specific prompts, even without
detailed reasoning, can yield comparable or superior outcomes. However, the
effectiveness of "break the chain" prompts varies, pointing to a nuanced
interplay between prompt nature and LLM performance that merits further
investigation.

We observe that CoT is particularly adept at tackling questions decomposable
into sub-issues that are solvable in brief sentences. Challenges arise when
generated responses become excessively lengthy, leading to potential task
misalignment and illogical outputs, or when they exceed the maximum length
constraints set in the code, inhibiting the completion of reasoning sequences.

Dataset | Question Type | Base | Quick Conclude | Shortcut Reasoning | Effective Shortcut | Innovative Shortcut
---|---|---|---|---|---|---
ShortcutQA | Analytical Reasoning | 26.79 | 33.93 | 21.43 | 19.64 | 30.36
Verbal Reasoning | 22.73 | 25.00 | 22.73 | 23.86 | 21.59
Mathematical Reasoning | 25.00 | 30.95 | 29.76 | 26.19 | 32.14

Table 5: Performance comparison across tasks within ShortcutQA.

#### ShortcutQA

Table 5 presents a comparative analysis of performance across various task
types within the ShortcutQA dataset. Compared to benchmarks utilized elsewhere
in this study, ShortcutQA poses a greater challenge, making it an ideal
testing ground for advancing model capabilities.

In mathematical reasoning tasks, all "break the chain" prompts outperform the
established baselines. The "Innovative Shortcut" prompt is particularly
effective, achieving a significant relative improvement of 28.56% over the
baseline. "Quick Conclude" also shows substantial gains, with a relative
increase of 23.8% compared to the baseline.

For analytical and verbal reasoning tasks, "Quick Conclude" registers the
highest improvements, with increases of 26.65% and 9.99%, respectively, over
the baseline. "Innovative Shortcut" also posts notable gains in analytical
tasks, while "Effective Shortcut" sees considerable enhancements in verbal
tasks.

Overall, "Innovative Shortcut" and "Quick Conclude" are standout performers on
the ShortcutQA dataset, underscoring the potency of our "break the chain"
strategy. This dataset not only challenges current LLMs but also sets a
benchmark for future enhancements, providing a robust platform for testing and
refining next-generation models.

##  6 Discussion

###  6.1 Reasoning with Token Limits

We investigated the impact of token limits on model performance by
experimenting with different constraints (128, 256 tokens) during the response
generation phase. Figure 2 illustrates how varying token limits affect
outcomes on the mathematical reasoning task within ShortcutQA using different
prompts. We observed that as the token limit increases, so does performance
across all prompts, indicating that constraints on output length significantly
influence the inference process and thus the results. Notably, even at the
minimum limit of 128 tokens, all prompts exceed the baseline performance,
suggesting that our "break the chain" approach is not only efficient but also
effective in conserving computational resources while maintaining or improving
task performance.

Figure 2: Performance comparison of different token limits on the mathematical
reasoning task from ShortcutQA.

###  6.2 Theoretical Analysis

We have developed a qualitative model to formalize the performance dynamics of
Chain-of-Thought (CoT) reasoning and to elucidate the effectiveness of the
"Break the Chain" approach.

In our framework, each CoT step is divided into two phases: analysis and
reasoning. The accuracy of the analysis at step $t$ is denoted as $P(a_{t})$,
and the subsequent reasoning based on this analysis is denoted as $P(r_{t})$.
Therefore, the total accuracy of a CoT sequence depends on the combined
accuracy of these phases across all steps, mathematically expressed as:

$P(\text{CorrectReasoning}_{\text{CoT}})=\prod_{t=1}^{T}P(a_{t})P(r_{t}),$ (1)

where $T$ is the total number of steps in the CoT reasoning chain. To evaluate
the efficacy of different prompting strategies, we define
$P(\text{CorrectReasoning}_{p})$ as the probability of achieving correct
reasoning for a given prompt $p$. A prompt is considered more effective than
the traditional CoT approach if $P(\text{CorrectReasoning}_{p})$ surpasses
$P(\text{CorrectReasoning}_{\text{CoT}})$.

In cases where no explicit analysis or reasoning phase is involved, and both
are integrated by LLMs in each step, Equation 1 simplifies to:

$P(\text{CorrectReasoning}_{\text{CoT}})=\prod_{t=1}^{T}P(i_{t}),$ (2)

where $i_{t}$ signifies the probability of obtaining the correct result in a
single, consolidated inference step.

Figure 3: Relationship between CoT Chain Length and Accuracy.

Our experimental results corroborate the theoretical predictions, as
illustrated in Figure 3. We observe that CoT accuracy generally declines as
chain length increases. Notably, in scenarios like Coin Flip where $P(i_{t})$
approaches 1, accuracy remains stable regardless of chain length. Conversely,
in tasks like SVamp where $P(i_{t})$ is lower, a decrease in accuracy is noted
as the chain lengthens. When comparing "Quick Conclude" on SVamp against
baseline accuracies, the relative CoT accuracy diminishes with increasing
chain length, aligning precisely with our model. Detailed methodologies for
these experiments are available in Appendix D.

##  7 Conclusion

This study critically evaluates Chain-of-Thought (CoT) reasoning in language
models, highlighting limitations such as high token consumption and limited
applicability. Our "break the chain" strategies integrate human-like
heuristics and shortcuts, enhancing efficiency without compromising
performance across various models. The introduction of the ShortcutQA dataset
further advances AI reasoning evaluation by focusing on heuristic tasks,
providing a robust benchmark that challenges traditional methods. Our findings
suggest that adopting more intuitive, efficient reasoning approaches could
significantly improve the problem-solving capabilities of AI systems in real-
world applications.

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##  Appendix A Zero-shot prompts for "break the chain"

The abbreviations of probing prompts and shortcut prompts are shown in the
table 6.

Prompt Type | Abbreviations | Full Prompts
---|---|---
Probing Prompts | Skip Steps | Let’s skip as much as possible.
No Steps | Let’s don’t think step by step.
More Token | Let’s think as much as possible.
Shortcut Prompts | Quick Conclude | Let’s quickly conclude the answer without showing step-by-step reasoning.
Shortcut Reasoning | Let’s quickly conclude the answer with shortcut reasoning.
Effective Shortcut | Rapidly evaluate and use the most effective reasoning shortcut to answer the question.
Innovative Shortcut | Think outside the box and quickly identify an innovative shortcut to solve this problem.

Table 6: The relationship between a prompt abbreviation and its full prompt.

##  Appendix B Pipeline Details

Figure fig:method shows the pipeline of experiments.

Figure 4: Our evaluation pipeline.

##  Appendix C Experiment Results

Table 7 is the original experiment results of diverse model structures.

Figure 5 shows that as model size decreases, CoT’s relative performance
advantage over other prompts increases across all tasks.

Figure 5: The Impact of Model Size on CoT’s Relative Outperformance over Other
Prompts across Datasets

##  Appendix D Detailed Methods

In this section, we introduce our detailed methods for our experiments. For
our experiment in discussion, we generally used GPT4 to evaluate the logs, and
caculate the accuracy of different lengths. First, We used GPT4 to check the
logs of CoT to calculate the length of chain in each question on SVamp and
Coin Flip. Second, we calculated the accuracy at different length of chain.
Third, to exclude the disturbance of various difficulty distributions within
each group, we calculated the accuracy with promptQC in each group of data as
baseline without CoT on SVamp.

model | Task | Dataset | Base | Quick Conclude | Shortcut Reasoning | Effective Shortcut | Innovative Shortcut
---|---|---|---|---|---|---|---
ChatGPT | Arithmetic | SingleEq | 86.61 | 91.14 | 91.73 | 92.32 | 77.36
AddSub | 83.80 | 90.89 | 86.33 | 89.62 | 73.67
AQUA-RAT | 35.43 | 51.97 | 52.76 | 53.94 | 52.36
MultiArith | 83.33 | 91.00 | 94.67 | 94.83 | 89.83
GSM8K | 32.68 | 56.86 | 71.57 | 67.25 | 58.00
SVAMP | 71.70 | 81.50 | 83.60 | 85.50 | 76.80
Commensense | CommonsenseQA | 70.52 | 77.89 | 78.95 | 76.82 | 72.89
| StrategyQA | 64.37 | 68.47 | 68.34 | 67.90 | 62.14
| Logic | LogiQA | 40.09 | 41.32 | 43.32 | 42.70 | 41.01
| ReClor | 52.40 | 52.80 | 51.60 | 52.00 | 51.40
|  | Date Understanding | 63.41 | 65.85 | 77.78 | 75.61 | 78.32
Llama-70B | Arithmetic | SingleEq | 67.91 | 55.91 | 81.10 | 49.80 | 40.16
AddSub | 69.87 | 40.51 | 80.25 | 53.92 | 32.66
AQUA-RAT | 61.02 | 52.36 | 62.20 | 57.87 | 50.79
MultiArith | 79.83 | 71.00 | 93.33 | 73.67 | 60.33
GSM8K | 80.97 | 81.05 | 85.14 | 78.17 | 69.52
SVAMP | 75.20 | 72.90 | 87.50 | 66.80 | 52.30
Commensense | CommonsenseQA | 79.03 | 81.49 | 77.31 | 69.21 | 74.37
| StrategyQA | 56.11 | 64.50 | 43.84 | 65.37 | 60.17
| Logic | LogiQA | 57.60 | 57.30 | 57.45 | 58.83 | 56.07
| ReClor | 71.80 | 69.40 | 68.40 | 69.80 | 70.20
|  | Date Understanding | 84.82 | 72.09 | 79.95 | 72.90 | 65.58
Mixtral-8x7B | Arithmetic | SingleEq | 87.40 | 88.58 | 87.01 | 83.46 | 70.87
AddSub | 85.82 | 86.84 | 84.81 | 83.54 | 72.15
AQUA-RAT | 37.40 | 41.34 | 42.13 | 39.76 | 32.68
MultiArith | 87.50 | 87.33 | 85.83 | 78.17 | 62.17
GSM8K | 48.90 | 55.80 | 54.81 | 49.20 | 39.12
SVAMP | 77.80 | 79.40 | 75.60 | 78.50 | 62.80
Commensense | CommonsenseQA | 71.63 | 72.40 | 72.73 | 71.17 | 65.85
| StrategyQA | 58.43 | 66.33 | 65.72 | 67.82 | 55.76
| Logic | LogiQA | 42.70 | 45.01 | 38.56 | 40.86 | 45.78
| ReClor | 47.40 | 50.60 | 51.80 | 48.60 | 47.60
|  | Date Understanding | 66.40 | 67.48 | 63.96 | 68.02 | 59.08
Qwen1.5-72B | Arithmetic | SingleEq | 80.71 | 87.80 | 88.78 | 88.58 | 86.61
AddSub | 84.56 | 84.81 | 86.33 | 88.61 | 86.84
AQUA-RAT | 37.80 | 47.24 | 48.43 | 46.46 | 35.43
MultiArith | 81.33 | 96.00 | 95.33 | 96.00 | 93.67
GSM8K | 28.96 | 54.06 | 48.98 | 45.26 | 42.30
SVAMP | 78.30 | 86.10 | 85.20 | 82.20 | 80.10
Commensense | CommonsenseQA | 81.98 | 83.54 | 81.98 | 83.37 | 83.7
| StrategyQA | 76.24 | 76.16 | 76.77 | 77.34 | 75.85
| Logic | LogiQA | 46.54 | 50.08 | 51.15 | 50.54 | 47.00
| ReClor | 61.60 | 66.20 | 64.00 | 65.80 | 64.40
|  | Date Understanding | 72.36 | 73.98 | 75.61 | 74.53 | 73.98

Table 7: Comparison of Various Open-Source Large Models’ Performance with
Different Prompts Across Multiple Datasets.


# Modulational instability in large-amplitude linear laser wakefields

A. von Boetticher  Clarendon Laboratory, University of Oxford, Parks Rd,
Oxford OX1 3PU, United Kingdom  Rudolf Peierls Centre for Theoretical Physics,
Parks Road, Oxford OX1 3PU, United Kingdom  R. Walczak  Clarendon Laboratory,
University of Oxford, Parks Rd, Oxford OX1 3PU, United Kingdom  Somerville
College, Woodstock Road, Oxford OX2 6HD, UK.  S. M. Hooker  Clarendon
Laboratory, University of Oxford, Parks Rd, Oxford OX1 3PU, United Kingdom
<EMAIL_ADDRESS>
###### Abstract

We investigate the growth of ion density perturbations in large-amplitude
linear laser wakefields via two-dimensional particle-in-cell simulations.
Growth rates and wave numbers are found to be consistent with a longitudinal
strong-field modulational instability (SFMI). We examine the transverse
dependence of the instability for a Gaussian wakefield envelope and show that
growth rates and wavenumbers can be maximised off-axis. On-axis growth rates
are found to decrease with increasing ion mass or electron temperature. These
results are in close agreement with the dispersion relation of a Langmuir wave
with energy density that is large compared to the plasma thermal energy
density. The implications for wakefield accelerators, in particular multi-
pulse schemes, are discussed.

This article was published in Physical Review E 107, L023201 on 22 February
2023. DOI: 10.1103/PhysRevE.107.L02320 ©2023 American Physical Society.

††preprint: APS/123-QED

Modulational instabilities are a ubiquitous phenomenon, occurring, for
instance, in surface water waves (Benjamin and Feir, 1967), in optical fibers
(Tai  _et al._ , 1986) and fusion plasmas (cf. Diamond  _et al._ , 2005). They
also arise in plasma-based accelerators (Moulin  _et al._ , 1994), which are
capable of generating high-energy particle beams comparable to those of
conventional large-scale accelerators over substantially shorter accelerating
distances. By using a laser pulse to excite a trailing Langmuir wave in a
plasma that acts as an accelerating structure, the laser-wakefield accelerator
(LWFA, Tajima and Dawson, 1979) has demonstrated GeV-scale electron
acceleration in accelerator stages only a few centimetres long (Leemans  _et
al._ , 2006). While such results demonstrate the potential of the technology,
laser wakefield accelerators so far rely on large laser systems to generate
the required short ($\ll$100\text{\,}\mathrm{f}\mathrm{s}$$), high-intensity
($\sim 10^{18}$\text{\,}\mathrm{W}\mathrm{c}\mathrm{m}^{-2}$$) driving pulses.
These laser systems have a low wall-plug efficiency ($<0.1\%$) and low pulse
repetition rates ($\sim$1\text{\,}\mathrm{H}\mathrm{z}$$). In contrast, many
potential applications of LWFAs — such as driving compact sources of
incoherent X-rays for imaging (Ben-Ismaïl  _et al._ , 2011; Kneip  _et al._ ,
2011; Hussein  _et al._ , 2019), free-electron lasers (Wang  _et al._ , 2021),
and, in the longer term, novel particle colliders — require efficient laser
drivers capable of operating at kHz or higher repetition rates. A possible
solution is the multi-pulse laser wakefield accelerator scheme (MP-LWFA, cf.
Umstadter  _et al._ , 1994; Dalla and Lontano, 1994; Umstadter  _et al._ ,
1995; Hooker  _et al._ , 2014; Cowley  _et al._ , 2017), in which the driving
laser energy is delivered over many plasma periods by a train of lower energy
laser pulses, spaced by the plasma period. Recent work has identified a route
to achieving GeV-scale, kHz repetition rate MP-LWFAs driven by existing laser
technology (Jakobsson  _et al._ , 2021).

In laser wakefield accelerators, the dynamics of the plasma ions are not
important when the Langmuir wave is excited by a single, high-intensity pulse,
owing to the small electron-to-ion mass ratio. Ion motion has been shown to be
significant for schemes that use long pulses such as beat-wave accelerators
(Mora  _et al._ , 1988) or in beam driven devices (cf. Rosenzweig  _et al._ ,
2005; Vieira  _et al._ , 2012; Benedetti  _et al._ , 2017). Ion motion can
similarly become important in MP-LWFAs if the plasma wave is excited by a
sufficiently long train of individual laser pulses (Hooker  _et al._ , 2014).

The interaction of Langmuir waves with the ion species of a plasma was first
investigated in the context of weak plasma turbulence, leading to the
discovery of the modulational instability that acts as a dissipation mechanism
for wave energy (Vedenov and Rudakov, 1964). If the energy of the Langmuir
wave is small relative to the plasma thermal energy,
$W:=\epsilon_{0}E^{2}/(2n_{e}k_{\mathrm{B}}T_{e})\ll 1$, where $E$ denotes the
Langmuir wave electric field strength, the modulational instability is
described by the Zakharov equations (Zakharov, 1972). These model the non-
linear interaction of the fast electrostatic Langmuir oscillation with slow
ion motion: small perturbations of the ion density modulate the Langmuir wave
envelope; the resulting ponderomotive force associated with the non-uniform
Langmuir envelope can in turn reinforce the ion perturbations. The ensuing
instability leads to the breakup of the Langmuir wave envelope and eventual
Landau damping of short-wavelength modes. In wakefield accelerators, typically
$W\gg 1$, precluding the applicability of the Zakharov model. Investigations
of this strong-field regime using small mass-ratio approximations by Silin
(1965, 1967) and Sanmartin (1970) showed that the modulational instability
also arises for large-amplitude Langmuir waves. Indeed the development of the
modulational instability in plasma wakefields has been inferred in beat-wave
and self-modulated laser wakefield experiments (Moulin  _et al._ , 1994; Blanc
_et al._ , 1996; Ting  _et al._ , 1996; Marques  _et al._ , 1996; Dyson  _et
al._ , 1996; Kotaki  _et al._ , 2002), but few investigations exist that
combine numerical simulation with theory and, thus far, these have been
restricted to one spatial dimension (Mora  _et al._ , 1988; Héron and Adam,
1993).

Studies of the modulational instability in plasma are therefore of both
fundamental interest and important in determining the performance, and
potential operating regimes, of laser- and beam-drive plasma accelerators. In
this Letter we use two-dimensional particle-in-cell (PIC) simulations to
investigate the modulational instability in large amplitude, linear laser
wakefields. The dominant wave number of the observed perturbations of the ion
density, the rate of growth of this mode, and the transverse variations of
these quantities, are all found to be in good quantitative agreement with a
one-dimensional, analytic theory of the modulational stability in the strong-
field regime developed by Silin (Silin, 1967) and Sanmartin (Sanmartin, 1970),
when the transverse variation of the wakefield amplitude is accounted for. The
dependence on ion mass and electron temperature of the growth rate of the ion
density perturbations observed in the PIC simulations are also found to be in
good agreement with the analytic theory, providing further evidence that the
perturbations arise from the modulational instability. The simulations
performed in this work are, to the best of our knowledge, the first multi-
dimensional PIC simulations of the modulational instability in a laser
wakefield and the first to test the Silin-Sanmartin theory in the high-
amplitude regime.

PIC simulations of laser wakefields were performed with the relativistic PIC
code smilei (Simulation of Matter Irradiated by Light at Extreme Intensities,
Derouillat  _et al._ , 2018). The simulations used ${N}_{\mathrm{PPC}}=64$
particles per cell, a longitudinal grid spacing, $\Delta x$, equal to the
Debye length of the plasma, $\lambda_{D}$, and transverse grid spacing $\Delta
y=5\lambda_{D}$. The simulations were performed in a two-dimensional geometry
containing an electron-proton plasma slab with longitudinal and transverse
extent of 280 $\mu$m and 160 $\mu$m, respectively, at a uniform plasma
density, $n_{e}=9.7\times 10^{17}$ cm-3. The electron temperature was
initialized near the ionisation energy of hydrogen, $T_{e}=15$
eV$k_{\mathrm{B}}^{-1}$; the initial ion temperature was set to $T_{i}=0.075$
eV$k_{\mathrm{B}}^{-1}$, thus $T_{i}\ll T_{e}$ as is typical of a plasma
ionised by a short laser pulse. A high-intensity laser pulse with Gaussian
temporal and transverse spatial envelope was propagated through the plasma,
generating a periodic oscillation of the electron density at the plasma
frequency, $\omega_{pe}=5.59\times 10^{13}\ \mathrm{rad\ s}^{-1}$. The full
width at half-maximum (FWHM) pulse duration and width were set to
$\tau_{\mathrm{fwhm}}=45$ fs and $w_{\mathrm{fwhm}}=45$ $\mu$m, respectively,
with a central laser wavelength of $\lambda_{l}=802.5$ nm and peak intensity
$I_{\mathrm{peak}}=5.3\times 10^{17}$ Wcm-2, corresponding to a normalised
vector potential, $a_{0}=0.5$. These parameters were selected to obtain a
wakefield with a density amplitude variation $\delta n_{e}/n_{e}$ on the order
of $10\%$, corresponding to a regime with large wakefield amplitude but no
strong nonlinear relativistic effects. The simulation properties and laser and
plasma parameters are summarised in Table 1 in the Supplemental Material (Sup,
) to this publication, which includes references (Lifschitz  _et al._ , 2009;
Thornhill and ter Haar, 1978; Brent, 1973; Virtanen  _et al._ , 2020).

Figure 1: (a-b) Longitudinal electric field strength $E_{x}(x,y)$ as a
function of longitudinal ($x$) and transverse ($y$) position, (a) 0.32 ps and
(b) 2.56 ps after wake excitation, showing the development of high-frequency
modulations of the field envelope, magnified in the inset. (c-d) Corresponding
ion density $n_{i}(x,y)$, showing the development of high-frequency
modulations. The red line indicates the axis of initial laser propagation.

The wakefield excited by the laser pulse attains a maximum longitudinal
electric field strength $E_{x}=8.5$ GVm-1 on axis, wavelength
$\lambda_{\mathrm{wake}}=\lambda_{pe}\approx 2\pi c/\omega_{pe}=33.7$ $\mu$m
and oscillates at the plasma frequency, since
$\omega_{k,pe}^{2}=\omega_{pe}^{2}+3k^{2}v_{th,e}^{2}\approx\omega_{pe}^{2}$
for wavelengths $\lambda=2\pi/k$ that are long compared to the Debye length,
$\lambda_{D}=(\epsilon_{0}k_{\mathrm{B}}T_{e}/n_{e}e^{2})^{1/2}=29.1$ nm. Here
$v_{th,e}=(k_{B}T_{e}/m_{e})^{1/2}$ denotes the electron thermal speed. The
longitudinal electric field of the wakefield is shown in Fig. 1(a), at
$t=0.32$ ps after the wakefield excitation. The on-axis longitudinal electric
field strength is associated with a sinusoidal plasma density oscillation
observed in the simulations of amplitude $\delta n_{e}/n_{e}=0.09$. The
transverse profile of the longitudinal electric field envelope is consistent
with the laser pulse profile, with a transverse width $w_{\mathrm{fwhm}}=45$
$\mu$m.

Within two picoseconds of the wakefield excitation, modulations of the
longitudinal electric field and the ion density are observed to develop, with
a characteristic length of $1.4$ $\mu$m on-axis ($y=0$) that is much smaller
than the wakefield wavelength, $\lambda_{pe}=33.7$ $\mu$m. The modulations of
the field and ion density are visible in Fig. 1(b) and 1(d), respectively, at
$t=2.56$ ps after wakefield excitation, and are shown magnified in the insets.
The timescale for the growth of the modulations is thus on the order of the
ion plasma period, $2\pi\omega_{pi}^{-1}=4.8$ ps. The wave number spectrum of
the ion density is shown in Fig. 2 as a function of transverse position,
indicating that modulations of the ion density grow fastest away from the
axis, and that the wavelength of these modulations decreases with distance
from the axis $y=0$. The on-axis spectrum exhibits a peak at a wave number
$k\lambda_{D}\approx 0.13$; a weaker, higher-order mode is seen to develop at
$k\lambda_{D}\approx 0.26$. The modulations of the ion density are
phenomenologically consistent with a modulational instability (Vedenov and
Rudakov, 1964; Zakharov, 1972).

For the plasma and wakefield parameters encountered here, the ratio of the
Langmuir energy to the plasma thermal energy density is $W\approx 150\gg 1$,
precluding the applicability of the Zakharov equations to describe the
modulational instability. The theory of the modulational instability in the
strong-field regime, $W\gg 1$, was developed by Silin (1965, 1967) and
Sanmartin (1970), and is reviewed briefly here and in the Supplemental
Material to this paper (Sup, ). The dispersion relation for a high-frequency
electrostatic oscillation in a Maxwellian plasma obtained by Sanmartin (1970)
simplifies considerably if the frequency of the equilibrium oscillation is
large compared to the ion plasma frequency, $\omega_{0}\gg\omega_{pi}$, and
wavelengths of all modes are long compared to the electron Debye length,
$k\lambda_{D}=kv_{th,e}/\omega_{pe}\ll 1$, as is the case for the wakefield
oscillation and ion density modulations observed in the PIC simulations. Since
the wakefield is in resonance with the plasma frequency, the resonance
parameter
$\tilde{\lambda}:=(\omega_{0}^{2}-\omega_{k,pe}^{2})/\omega_{pe}^{2}\ll 1$ is
small. With $\mu:=m_{e}/m_{i}\ll 1$, and assuming a purely growing mode with
growth-rate $\gamma\in\mathbb{R}$, the dispersion relation is given by
(Sanmartin, 1970)

$\displaystyle 1+\chi_{i}$
$\displaystyle\bigg{[}1-2\frac{J_{0}(a)J_{1}(a)}{a}+k^{2}\lambda_{D}^{2}$
$\displaystyle-2J_{1}(a)^{2}\frac{\tilde{\lambda}-\frac{1}{4}\tilde{\nu}^{2}+2\mu\left[1-J_{0}(a)^{2}\right]a^{-2}}{\tilde{\lambda}^{2}+(2\tilde{\gamma}+\tilde{\nu})^{2}}\bigg{]}=0,$
(1)

where $\gamma$ and $\tilde{\gamma}=\gamma/\omega_{pe}$ denote the growth rate
and normalised growth rate, respectively;
$\tilde{\nu}=\nu/\omega_{pe}\approx(\pi/2)^{1/2}(k\lambda_{D})^{-3}\exp(-1/[2k^{2}\lambda_{D}^{2}])$
is the normalised Landau damping rate, and $J_{0}$ and $J_{1}$ denote Bessel
functions of the first kind. The parameter $a:=ekE_{0}/m_{e}\omega_{0}^{2}$
measures the ratio of the electron excursion distance in a wakefield cycle to
the mode wavelength $2\pi k^{-1}$. The ion susceptibility,
$\chi_{i}(\gamma,k)$, can be related to the plasma dispersion function, $Z$,
(cf. Fried and Conte, 1961), via

$\displaystyle\chi_{i}(\gamma,k)$
$\displaystyle=\frac{\omega_{pi}^{2}}{k^{2}}\int_{C_{L}}\frac{k(dF_{i0}/dv)}{i\gamma-
kv}\ dv$ (2)
$\displaystyle=\frac{\omega_{pi}^{2}}{k^{2}v_{th,i}^{2}}\left[1+\frac{\zeta_{i}}{\sqrt{2}}Z(\zeta_{i}/\sqrt{2})\right]$
(3)

where $C_{L}$ denotes the usual Landau contour,
$F_{0i}=1/(\sqrt{2\pi}v_{th,i})e^{-v^{2}/2v_{th,i}^{2}}$,
$\zeta_{i}:=i\gamma/kv_{th,i}$, and the plasma dispersion function is given by

$\displaystyle
Z(\zeta_{i}):=\frac{1}{\sqrt{\pi}}\int\frac{e^{-u^{2}}}{u-\zeta_{i}}\ du.$ (4)

For calculation it is useful to relate $Z$ to the error function via
$Z_{i}(\zeta_{i})=i\sqrt{\pi}\exp(-\zeta_{i}^{2})\left[1+\mathrm{erf}(i\zeta_{i})\right]$
(Fried and Conte, 1961) so that, with $z:=-i\zeta_{i}=\gamma/kv_{th,i}$, the
ion susceptibility can be expressed as

$\displaystyle\chi_{i}(z,k)=\frac{\omega_{pi}^{2}}{k^{2}v_{th,i}^{2}}\bigg{[}1$
$\displaystyle-z\sqrt{\frac{\pi}{2}}\exp\left(\frac{z^{2}}{2}\right)$
$\displaystyle\times\left(1-\mathrm{erf}\left[z/\sqrt{2}\right]\right)\bigg{]}.$
(5)

This expression can be evaluated using standard implementations of the error
function.

Eq. 1 can be simplified for sufficiently fast growth rates. The assumption
$k\lambda_{D}\ll 1$ together with $T_{i}/T_{e}\ll 1$ implies that
$kv_{th,i}=k\lambda_{D}\sqrt{T_{i}/T_{e}}\omega_{pi}\ll\omega_{pi}$. For the
growth time scales observed in the PIC simulations,
$\gamma^{-1}\sim\omega_{pi}^{-1}$, it follows that, indeed, $z\gg 1$. The ion
susceptibility can then be approximated using the asymptotic expansion of the
plasma dispersion function $Z(\zeta)=-\zeta^{-1}-\zeta^{-3}/2+O(\zeta^{-5})$
(Fried and Conte, 1961). Using this in Eq. 3, one obtains
$\chi_{i}\approx\omega_{pi}^{2}/\gamma^{2}$. The dispersion relation, Eq. 1,
then simplifies to the quartic (Sanmartin, 1970)

$\displaystyle\left[\tilde{\lambda}^{2}+(2\tilde{\gamma}+\tilde{\nu})^{2}\right]\left[\tilde{\gamma}^{2}+\mu\left(1-\frac{2}{a}J_{0}(a)J_{1}(a)\right)\right]$
$\displaystyle-2\mu
J_{1}^{2}(a)\left[\tilde{\lambda}-\frac{1}{4}\tilde{\nu}^{2}+2\mu\left[1-J_{0}(a)^{2}\right]a^{-2}\right]=0.$
(6)

Figure 2: Longitudinal wave number spectrum $\log_{10}[\mathrm{FFT}_{x}(\delta
n_{i}(x,y)/n_{i0})]$ of the perturbed ion density, as a function of transverse
position $y$, at (a) $t=0.64$ ps after wake excitation, (b) $t=1.60$ ps, (c)
$t=2.56$ ps and (d) $t=3.52$ ps. The instability initially grows fastest off-
axis (b). The predicted transverse wave number dependence of the fastest
growing mode is calculated by solving the exact Eq. (1) (EX, black, solid
line) and the approximate Eq. (6) valid for a large driving field amplitude
(LA, blue, dashed).

We numerically solved Eq. (6) and examined the root with $\max\\{\gamma>0\\}$,
corresponding to the fastest growing instability. The dependence of the
maximum growth rate and the associated dominant wave number on the wakefield
electric field strength is shown in Fig. 3(a) and 3(c), respectively. The
growth rate of the modulational instability initially increases with the field
strength, is maximised for $E_{0}\approx 5.5$ GVm-1 and decreases for stronger
fields. This non-monotonic behaviour of the growth-rate is associated with the
electron displacement in the wakefield oscillations. When the electron
excursion distance significantly exceeds the mode wavelength, corresponding to
$a\gg 1$, the coupling between the wakefield and the ion perturbations becomes
weak as the electrons experience a spatially averaged pondermotive force (Mora
_et al._ , 1988); indeed for sufficiently strong fields and large electron
displacements the modulational instability is predicted to be suppressed
entirely (Sanmartin, 1970). By assuming a transverse Gaussian profile of the
wakefield electric field envelope, $E_{x}(y)=E_{0}\exp(-y^{2}/2\sigma^{2})$,
with the parameters $E_{0}$ and $\sigma$ determined from a fit of this
expression to the transverse variation of the initial wake amplitude obtained
from the PIC simulations, the expected transverse scaling of the dominant wave
number can be calculated. To do so, the dispersion relation, Eq. (6), was
solved on a transverse grid, for the wave number, $k(y)$, that maximises the
growth rate. The resulting transverse growth rate and wave number dependence
are shown in Fig. 3(b) and 3(d), respectively. For transverse displacements
$|y|\lesssim 30$ $\mu$m, the predicted wave number dependence is in good
agreement with the transverse variation of the wave number of ion density
modulations observed in the PIC simulations, shown in Fig. 2. For transverse
displacements $|y|\gtrsim 30$ $\mu$m, Eq. (6) predicts a decrease in the wave
number with greater transverse displacement; however at these displacements
the observed growth of the instability is very slow and is suppressed entirely
for $|y|\gtrsim 40$ $\mu$m. Fig. 2(b) shows that modulations of the ion
density first develop at a transverse displacement
$y_{\max\\{\gamma\\}}\approx\pm 20$ $\mu$m, indicating that the growth rate is
maximised off-axis, where the electric field strength of the wakefield is
lower than on axis. This is consistent with the simple model of the transverse
variation of the growth rate which is shown in Fig. 3(b). Fig. 3(b) indicates
that for the wakefield amplitude and transverse profile considered here the
maximum growth rate is indeed expected at $y\approx 20$ $\mu$m.

Figure 3: Growth rates and wave numbers calculated with the exact (EX) Eq. (1)
and the large-amplitude approximation (LA) Eq. (6). (a) Variation of the
maximum growth rate with the wakefield electric field strength, (b) transverse
dependence of the growth rate, for a transverse Gaussian wakefield amplitude
profile, with $E_{x}(y=0)=8.5$ GVm-1 and $w_{\mathrm{fwhm}}=45$ $\mu$m, (c)
variation of the dominant wave number with electric field strength, and (d)
transverse dependence of the wave number for a transverse Gaussian wakefield
profile.

To provide further evidence that the modulational instability is responsible
for the behaviour observed in Fig. 2, PIC simulations were performed for
different ion masses and electron temperatures, with other simulation
parameters held fixed. Growth rates of the ion density perturbations were
determined from the PIC simulations by measuring the amplitude of the fastest
growing mode over time (Sup, ). Figure 4(a) compares the mass dependence of
the growth rates determined this way with that calculated from Eq. (6). The
two calculations are seen to be in excellent agreement. Simulations at
elevated electron temperatures of $T_{e}=100$ eV$k_{\mathrm{B}}^{-1}$ and
$T_{e}=1$ keV$k_{\mathrm{B}}^{-1}$ were performed and growth rates of the ion
density perturbations on axis ($y=0$) were compared to the theoretical
predictions, shown in Fig. 4(b). The growth rates follow a similar trend with
$T_{e}$, and agree within 30% for the temperature range investigated, although
the rates calculated by the PIC simulations are consistently lower than those
deduced from Eq. (6). We hypothesise that dimensionality effects may be
responsible for the observed differences in measured and predicted growth
rates as a function of the electron temperature. The better spatial resolution
of the PIC simulations, relative to the Debye length, obtained at higher
temperatures may also play a role. The analytic theory predicts a weak
resonance in the growth rate near $T_{e}\approx 40$ eV$k_{\mathrm{B}}^{-1}$,
but there is insufficient data to state whether or not this resonance is also
predicted by the PIC simulations. The good quantitative agreement seen in Fig.
4 with the analytic theory provides further confirmation that the ion density
perturbations observed in the PIC simulations arise from the modulational
instability.

Figure 4: (a) Dependence of the modulational instability growth rate on the
ion mass, for $T_{e}=15$ eV$k_{\mathrm{B}}^{-1}$, obtained from Eq. (1) (black
line) and PIC simulations (red squares, for
$m_{i}=\\{0.25,0.5,1.0,2.0,4.0\\}m_{p}$). (b) Dependence of the growth rate on
the electron temperature (black line) and measured growth rates from
simulations (red squares), for $m_{i}=m_{p}$.

The PIC simulations presented here show the development of the strong-field
modulational instability in high-amplitude linear laser wakefields. The growth
rates, wave numbers, and the transverse profiles thereof are found to be in
good agreement with those obtained from the one-dimensional dispersion
relation of Sanmartin (1970) if the transverse dependence of the wakefield
amplitude is accounted for. We note that Vieira  _et al._ (2012) have
investigated the role of ion dynamics in self-modulated proton beam driven
plasma wakefield accelerators. Those accelerators operate in the narrow beam
limit (Vieira  _et al._ , 2012), where the transverse extent of the proton
driver, $\sigma_{r}$, is much smaller than the wavelength of the electron
plasma oscillation. In that regime, ion motion is dominated by a strong
transverse component, since the ponderomotive force of the wakefield envelope
scales with the gradient of the wakefield intensity,
$F_{p,\perp}\sim\nabla_{\perp}E^{2}\sim E^{2}/\sigma_{r}$. In contrast, the
wakefields investigated here have a transverse scale length set by the laser
beam width of $w_{\mathrm{fwhm}}=45$ $\mu$m (FWHM) that is somewhat greater
than the electron plasma wavelength $\lambda_{pe}=33.7$ $\mu$m, such that
longitudinal and transverse effects are comparable. Indeed, the small-scale
transverse variations of the modulations visible in Fig. 1 show that
transverse effects are non-negligible.

We now consider the implications of these results for multi-pulse laser
wakefield accelerators. Since the modulational instability develops on a
timescale comparable to the ion plasma period, it will become relevant for
pulse trains with $N_{\mathrm{pulse}}\gtrsim(m_{i}/m_{e})^{1/2}/Z$, where $Z$
denotes the ion charge number. Indeed, in earlier work we showed that for long
pulse trains with $N_{\mathrm{pulse}}\gg(m_{i}/m_{e})^{1/2}/Z$, the amplitude
of the wakefield did not grow monotonically with pulse number (Hooker  _et
al._ , 2014), indicating a loss of coherence of the electron oscillations.
This earlier finding is explained by the work reported here. The non-monotonic
dependence of the SFMI growth rate on the electric field strength observed
transversely in Fig. 3(b) suggests that, in principle, the intensities of
pulses in the driving train could be tuned to minimise the time for which the
wakefield amplitude is such that the SFMI growth-rate is large. Alternatively,
using a species with a high $m_{i}/Z$ ratio, or heating the electronic
component of the plasma could reduce the onset of the modulational
instability, and allow longer pulse trains to be employed.

Although the effects of the modulational instability could be mitigated by
these methods, they are unlikely to be necessary for MPLWFAs. For example,
Jakobsson  _et al._ (2021) recently introduced a novel method for driving GeV-
scale MPLWFAs with a single picosecond-duration pulse by spectrally-modulating
the pulse in the wakefield driven by a low-energy seed pulse, and compressing
it to generate a train of short pulses. In that work the generated pulse
trains have $N_{\mathrm{pulse}}\sim 10<(m_{i}/m_{e})^{1/2}$. Separate PIC
simulations were performed to confirm that a wakefield excited by such short
pulse trains is indeed unaffected by ion dynamics during the excitation.

This work was supported by the UK Engineering and Physical Sciences Research
Council (EPSRC) [grant nos EP/N509711/1, EP/V006797/1], the UK Science and
Technologies Facilities Council [grant nos. ST/N5044233/1, ST/P002048/1,
ST/V001655/1], the European Union’s Horizon 2020 research and innovation
programme [grant no. 653782], the plasma HEC Consortium [grant no.
EP/R029149/1] and the ARCHER UK national supercomputing service. This material
is based upon work supported by the Air Force Office of Scientific Research
under award number FA9550-18-1-7005.

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