# Learning-at-Criticality in Large Language Models for Quantum Field Theory and Beyond

arXiv: https://arxiv.org/abs/2506.03703 · HTML: https://mlsanity.com/paper/2506.03703/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 31 references that are in the map.
- First posted on arXiv 4 Jun 2025.

## Builds on (most cited first, 15 of 15)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Beyond the 80/20 Rule: High-Entropy Minority Tokens Drive Effective Reinforcement Learning for LLM Reasoning](https://mlsanity.com/paper/2506.01939/) (2025)
- [Reinforcement Learning for Reasoning in Large Language Models with One Training Example](https://mlsanity.com/paper/2504.20571/) (2025)
- [Benchmarking Reasoning Robustness in Large Language Models](https://mlsanity.com/paper/2503.04550/) (2025)
- [Enhancing Mathematical Reasoning in Large Language Models with Self-Consistency-Based Hallucination Detection](https://mlsanity.com/paper/2504.09440/) (2025)
- [Self-Consistency Training for Density-Functional-Theory Hamiltonian Prediction](https://mlsanity.com/paper/2403.09560/) (2024)
- [Overcoming Data Scarcity in Generative Language Modelling for Low-Resource Languages: A Systematic Review](https://mlsanity.com/paper/2505.04531/) (2025)
- [Refining Integration-by-Parts Reduction of Feynman Integrals with Machine Learning](https://mlsanity.com/paper/2502.05121/) (2025)

## Built on by (most cited first, 5 of 5)

- [Emergent Slow Thinking in LLMs as Inverse Tree Freezing](https://mlsanity.com/paper/2509.23629/) (2025)
- [Inverse Knowledge Search over Verifiable Reasoning: Synthesizing a Scientific Encyclopedia from a Long Chains-of-Thought Knowledge Base](https://mlsanity.com/paper/2510.26854/) (2025)
- [Grading the Unspoken: Evaluating Tacit Reasoning in Quantum Field Theory and String Theory with LLMs](https://mlsanity.com/paper/2604.14188/) (2026)
- [Kohn-Sham Hamiltonian from Effective Field Theory: Quasiparticle Band Narrowing from Frozen Core Dynamics](https://mlsanity.com/paper/2604.25199/) (2026)
- [Do Quantum AIs Dream in Paths? Path-Integral Slow Thinking through Grover Interference](https://mlsanity.com/paper/2609.05842/) (2026)

## Abstract (arXiv)

Fundamental physics often confronts complex symbolic problems with few guiding exemplars or established principles. While artificial intelligence (AI) offers promise, its typical need for vast datasets to learn from hinders its use in these information-scarce frontiers. We introduce learning at criticality (LaC), a reinforcement learning (RL) scheme that tunes Large Language Models (LLMs) to a sharp learning transition, addressing this information scarcity. At this transition, LLMs achieve peak generalization from minimal data, exemplified by 7-digit base-7 addition -- a test of nontrivial arithmetic reasoning. To elucidate this peak, we analyze a minimal concept-network model (CoNet) designed to capture the essence of how LLMs might link tokens. Trained on a single exemplar, this model also undergoes a sharp learning transition. This transition exhibits hallmarks of a second-order phase transition, notably power-law distributed solution path lengths. At this critical point, the system maximizes a ``critical thinking pattern" crucial for generalization, enabled by the underlying scale-free exploration. This suggests LLMs reach peak performance by operating at criticality, where such explorative dynamics enable the extraction of underlying operational rules. We demonstrate LaC in quantum field theory: an 8B-parameter LLM, tuned to its critical point by LaC using a few exemplars of symbolic Matsubara sums, solves unseen, higher-order problems, significantly outperforming far larger models. LaC thus leverages critical phenomena, a physical principle, to empower AI for complex, data-sparse challenges in fundamental physics.