A research team led by Rintaro Masaoka (Master’s Program, 2nd year) and Associate Professor Haruki Watanabe from the Department of Applied Physics, Graduate School of Engineering, the University of Tokyo, in collaboration with Dr. Tomohiro Soejima from Harvard University, has succeeded in rigorously proving for the first time a universal lower bound z ≥ 2 on the dynamical critical exponent z of the Ising model—one of the most fundamental models in classical statistical mechanics—valid in any spatial dimension. This theoretical breakthrough addresses a century-old unresolved problem in our understanding of dynamical critical phenomena.
The Ising model was originally introduced in 1925 in the doctoral dissertation of Ernst Ising to describe ferromagnetic phase transitions. Despite its simple structure, the model captures complex phenomena such as criticality and phase transitions, and it has found applications across physics, information science, biology, and even the social sciences. In particular, the two-dimensional Ising model, exactly solved by Lars Onsager in 1944, stands as a landmark in the history of statistical mechanics. The year 2025 marks the 100th anniversary of the model’s inception.
The study focused on the dynamical Ising model, which describes the time evolution—or relaxation dynamics—of Ising spins. In this model, spin flips occur probabilistically according to prescribed stochastic rules, forming a Markov process. Near the critical point, the system exhibits critical slowing down, where the relaxation time τ diverges with increasing system size L. The scaling relation τ∼Lz defines the dynamical critical exponent z.
Prior to this work, theoretical approaches had only established a bound z ≥1.75 for the two-dimensional Ising model, whereas numerical studies suggested a value of z≈2.1667(5), leaving a significant gap between theory and simulation.
To close this gap, the researchers employed a novel approach rooted in recent advances in the theory of frustration-free quantum many-body systems. By mapping classical Markov processes to corresponding quantum systems, they rigorously proved that the relaxation time must grow at least as fast as L2/(logL )2, thereby establishing the universal inequality z ≥ 2(see Fig. 1). This proof ingeniously combines the Simon–Lieb inequality, a classical result on Ising correlation functions, with a modern detectability lemma from quantum information theory, recently formulated by Gosset and Huang. The result exemplifies a novel synthesis of quantum and classical methodologies.
The significance of this work extends far beyond the derivation of a single inequality. The dynamical critical exponent is a foundational concept that informs the evaluation of spin relaxation times in materials science, convergence estimates in Monte Carlo methods, and theoretical frameworks for nonequilibrium systems. Establishing a universal bound provides essential guidance for both future theoretical development and experimental design. Furthermore, this result highlights the potential of insights from quantum information theory and many-body quantum physics to contribute in a reductionist manner to classical statistical mechanics, underscoring the interdisciplinary impact of the research.
Figure 1: Simulation of time evolution at the critical temperature in a magnetic system
Results from a Monte Carlo simulation of the two-dimensional dynamical Ising model at the critical temperature T=Tc, performed using the Markov chain Monte Carlo method. Black and white pixels represent spin orientations in the magnetic material; the vertical axis corresponds to a one-dimensional spatial slice, and the horizontal axis denotes time. The dynamical critical exponent z quantifies the scaling relation τ∼Lz, where τ is the relaxation time and L the system size. This study rigorously establishes a universal lower bound z ≥ 2.
Papers
Journal: Journal of Statistical Physics
Title: Rigorous lower bound of the dynamical critical exponent of the Ising model
Authors: Rintaro Masaoka, Tomohiro Soejima, Haruki Watanabe*
DOI：10.1007/s10955-025-03456-3
URL：https://link.springer.com/article/10.1007/s10955-025-03456-3 (Open Access）
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