Universal Symmetry-Breaking Dynamics at Continuous Phase Transitions: Evidence for a New Dynamical Critical Exponent

Abstract: Uncovering and understanding universal dynamics in matter far from equilibrium remains a key challenge. In this work, we identify a so far unrecognized form of universal behavior that emerges after a sudden symmetry-breaking quench at continuous phase transitions. Our key observation is that the order-parameter fluctuations in Ising models exhibit a compelling temporal collapse across a wide range of system sizes and quench strengths, indicative of an emergent single-variable scaling form. This phenomenon can be explained by introducing a so far unknown dynamical critical exponent for the underlying continuous phase transition. We find evidence for a lower critical effective dimension of this universal regime: it is observed in the 2D quantum and 3D and 4D classical Ising models, but not in the 1D quantum or 2D classical cases. Our results suggest that our observed universal far-from-equilibrium scaling may extend beyond the Ising models studied here and could more broadly characterize systems with non-conserved order parameters, opening new avenues for exploring universal dynamics both theoretically and in current experimental platforms.

• The paper provides evidence for a new non-equilibrium dynamical critical exponent in Ising transitions.
• It uses Neural Quantum States & variational principles for simulating dynamics in large-scale quantum and classical Ising models.
• Findings illustrate universal dynamics, expanding critical behavior understanding in non-linear regimes beyond traditional frameworks.

Universal Out-of-Equilibrium Scaling after Symmetry-Breaking Quenches at Continuous Ising Transitions

Introduction and Context

Symmetry-breaking dynamics following quenches at criticality are central to the understanding of universal far-from-equilibrium behavior in many-body systems. While the universality of static and near-equilibrium critical phenomena is well-established—resident within finite sets of universal exponents and scaling forms—such comprehensive classification far from equilibrium remains elusive, especially when considering strongly fluctuating situations beyond linear response. Existing frameworks address slow ramps (Kibble-Zurek mechanism), initial-slip and short-time scaling following quenches from disordered states, and non-thermal fixed points. However, a universal characterization of symmetry-breaking protocol initiated from a state at the continuous phase transition, with a symmetry-breaking field coupling directly to the order parameter, has not been developed outside of the near-equilibrium linear regime.

The work "Universal Symmetry-Breaking Dynamics at Continuous Phase Transitions: Evidence for a New Dynamical Critical Exponent" (2605.07753) rigorously addresses this deficit. The analysis uncovers a previously unrecognized universal dynamical scaling emerging after such critical symmetry-breaking quenches in Ising systems with non-conserved order parameters. Importantly, the observed scaling is governed not only by standard equilibrium exponents but also necessitates the introduction of a new, genuinely non-equilibrium dynamical critical exponent.

Protocol and Simulation Methodology

The paper formalizes the following protocol: The system (quantum or classical) is initialized at the critical point—either as the ground state of a quantum system or as the Gibbs state at the thermal transition. At time $t = 0$, a uniform symmetry-breaking field $h$ is switched on, suddenly quenching the system to an explicitly symmetry-broken Hamiltonian. The post-quench evolution is then governed by the Schrödinger equation or Glauber dynamics, depending on quantum or classical context.

For the quantum 2D transverse-field Ising model, the numerical challenge of accessing real-time dynamics from a critical initial state is surmounted using the Neural Quantum States (NQS) variational ansatz combined with time-dependent variational principle (TDVP) propagation. This approach allows reliable tracking of dynamics for system sizes up to $16\times16$ spins, which is well beyond the scope of exact diagonalization or matrix-product state approaches for 2D critical systems.

In classical 3D and 4D Ising models, Glauber dynamics are simulated from the equilibriated thermal critical state following the same quench protocol. All models have non-conserved order parameters, ensuring nontrivial quench dynamics.

Main Results: Emergent Universal Scaling and Novel Dynamical Exponent

A singular finding of the study is the compelling collapse of the order-parameter fluctuations $\langle M^2 (t) \rangle$ onto a single scaling function when properly rescaled by both system size and the symmetry-breaking field, provided one introduces an additional dynamical exponent $\omega$ that is not present in traditional equilibrium or near-equilibrium analyses.

For the 2D quantum Ising model, the optimal data collapse is realized by plotting the rescaled fluctuation $L^{-\kappa}\langle M^2 (t) \rangle$ (where $\kappa$ is the known equilibrium exponent) versus the universal variable $h t^{\omega}$, with $\omega = 1.86 \pm 0.05$. Analogous scaling forms are found in the 3D and 4D classical Ising models (with $\omega$ values $h$0 and $h$1, respectively), showing that the phenomenon is robust across both quantum and classical universality classes above a certain effective lower critical dimension. Notably, this scaling is absent in the quantum 1D and classical 2D Ising models, supporting the assertion of a lower critical dimension for the emergence of this far-from-equilibrium scaling regime.

The study provides quantitative evidence for a new universal non-equilibrium dynamical exponent. This exponent governs how the interplay of the quench field and system size organizes dynamics deep in the non-linear response regime, beyond what is captured by equilibrium RG analysis or linear-response theory.

Dynamic Finite-Size Scaling Analysis and Non-Equilibrium Organization

The observed single-variable data collapse can be framed within the dynamic finite-size scaling (DFSS) paradigm. At equilibrium, DFSS yields a two-variable form for observable scaling functions, but the evidence presented demonstrates that, for the order-parameter fluctuations in these quench protocols, the form reduces empirically to a one-variable function:

$h$2

where $h$3 is empirically determined and is not reducible to existing RG exponents for dynamics. This single-parameter scaling, which holds robustly across large system sizes and field strengths, points to emergent temporal coherence in post-quench fluctuations, driven by the symmetry-breaking perturbation and the underlying critical correlations of the initial state.

Importantly, the effect is restricted to systems and protocols with non-conserved order parameters. The emergent scaling cannot be subsumed under hydrodynamic or linear response descriptions, nor under initial-slip analyses used for quenches directly from disordered states.

Theoretical and Practical Implications

The work reveals that novel universal organization principles may govern strongly non-equilibrium critical dynamics, expanding beyond the established universality classes encoded in equilibrium RG or the Hohenberg-Halperin scheme. The identification of a new dynamical exponent that organizes post-quench order parameter fluctuations indicates the necessity of developing a more general non-equilibrium RG or kinetic theory capable of predicting such exponents and scaling forms.

From a practical perspective, the quench protocols examined are directly implementable in quantum simulation experiments (e.g., Rydberg arrays, trapped ions), where both the initial preparation at criticality and the real-time monitoring of magnetization fluctuations are accessible. This strongly motivates experimental tests aiming to measure the identified universal collapse and the associated exponent.

On the theoretical side, multiple avenues open: elucidation of the microscopic mechanism giving rise to the exponent $h$4, generalization to other universality classes and order parameter symmetries, and investigation of whether similar single-parameter collapse emerges for observables beyond order-parameter variance.

Conclusions

This research establishes the existence of a universal far-from-equilibrium scaling regime after critical symmetry-breaking quenches, which is characterized by the collapse of order-parameter fluctuations across system sizes and field strengths and is governed by an emergent, previously unknown dynamical critical exponent. The effect displays a dimensional threshold, indicating a lower critical dimension for its appearance, and is robust in both quantum and classical Ising models with non-conserved order parameters. These findings broaden the conceptual framework for universal dynamics in many-body physics and point toward new organizing principles for non-equilibrium criticality potentially applicable across a range of platforms and models. The microscopic origin and field-theoretic description of the new exponent remain open and significant questions for future research.