🔬 condensed matter

Rare Events Govern Defect Formation under Weak Symmetry Breaking

Using large-deviation theory, this paper explains how weak explicit symmetry breaking leads to an exponential suppression of topological defect formation during non-equilibrium phase transitions by showing that defects arise from rare fluctuations, a mechanism validated by simulations of stochastic Ginzburg-Landau models.

Original authors: Jiang Liu, Peng Yang, Matteo Baggioli

Published 2026-06-29

📖 3 min read☕ Coffee break read

CC BY 4.0

Original authors: Jiang Liu, Peng Yang, Matteo Baggioli

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to organize a massive crowd of people into two distinct teams: Team Red and Team Blue. In a perfect, fair world, the rules are completely neutral. As the "game" starts, different groups of people (who can't talk to each other) independently decide which team to join. Some pick Red, some pick Blue. Wherever a Red group meets a Blue group, a "clash" or a "defect" happens at the boundary.

This is the classic story of how topological defects form during a phase transition (like water turning to ice), a concept known as the Kibble-Zurek mechanism. Scientists have long known that if you slow down the game, the number of clashes follows a predictable, mathematical pattern.

The Twist: The Biased Coin
However, in the real world, things are rarely perfectly fair. Imagine someone secretly puts a tiny weight on the coin, making it slightly more likely to land on "Team Red." This is what physicists call weak symmetry breaking.

Previous studies noticed that when this tiny bias exists, the number of clashes (defects) drops much faster than the old math predicted. But nobody knew why this extra drop happened or how to calculate it.

The New Discovery: Rare Fluctuations
This paper, by Liu, Yang, and Baggioli, solves the mystery using a concept called Large-Deviation Theory. Here is their explanation in simple terms:

The Normal Process: Usually, when a group decides on a team, it's a 50/50 coin toss.
The Biased Process: With the tiny weight (the external field), the coin is now 99% likely to land on Red and only 1% on Blue.
The Rare Event: For a "clash" (defect) to happen now, a group of people has to do something incredibly unlikely: they must all accidentally ignore the weight and choose Blue anyway.
- Think of it like a calm lake. Usually, the water is flat. But if a massive, rare wave suddenly crashes into a specific spot, it can push the water in the "wrong" direction.
- The paper argues that defects only form when these rare, massive fluctuations (the giant waves) are strong enough to overcome the tiny bias and force a region into the "disfavored" state.

The Formula
The authors developed a new mathematical formula that acts like a "rare-event calculator." It predicts that the number of defects isn't just a simple power law anymore; it has an exponential suppression.

In plain English: Because it is so incredibly hard for a whole group to accidentally choose the "wrong" team against the bias, the number of clashes drops off like a cliff. The stronger the bias or the quieter the background noise, the fewer the clashes.

How They Proved It
To test this, the researchers ran computer simulations (like a digital video game) of these phase transitions in one and two dimensions.

They watched the "order parameter" (the team choice) evolve over time.
They saw that defects only appeared when the system experienced those rare, strong fluctuations.
When they plugged their data into their new formula, everything matched perfectly. The "rare event" theory explained the data exactly, whereas the old "fair coin" theory failed.

The Bottom Line
This paper tells us that when a system is slightly biased, defect formation stops being a game of chance and becomes a game of rare miracles. Defects only appear when the universe throws a rare, massive fluctuation that overcomes the bias. This gives scientists a precise tool to predict how many defects will form in real-world materials where perfect symmetry is almost never found.

Technical Summary: Rare Events Govern Defect Formation under Weak Symmetry Breaking

Problem Statement
The formation of topological defects during continuous phase transitions driven out of equilibrium is traditionally described by the Kibble–Zurek (KZ) mechanism. This framework predicts that defect density ( $n$ ) follows a universal power-law scaling with the quench time ( $\tau_Q$ ), determined by the system's spatial dimensionality ( $d$ ) and critical exponents ( $\nu, z$ ). However, recent numerical studies have identified systematic deviations from this scaling in systems subject to weak explicit symmetry breaking. In these scenarios, an additional exponential suppression of defect formation is observed. While deviations arising from weakly first-order transitions have been explained via classical nucleation theory, the physical origin of the exponential correction in the context of weak symmetry breaking has remained theoretically elusive.

Methodology
The authors develop a theoretical framework based on large-deviation theory to explain defect formation under weak explicit symmetry breaking. The approach combines the standard KZ picture with stochastic dynamics:

Conceptual Model: The authors posit that while the KZ mechanism establishes independent correlated domains (KZ domains) of size $\hat{\xi}$ , the presence of a weak symmetry-breaking field ( $h$ ) biases the selection of the symmetry-broken state within each domain.
Rare Event Mechanism: Defect formation is reinterpreted not as a random mismatch between domains, but as a rare fluctuation event. For a domain to select the disfavored (metastable) state against the external bias, a coherent thermal fluctuation of sufficient magnitude must persist throughout the freeze-out spacetime volume ( $\Omega_{KZ} \sim \hat{t}\hat{\xi}^d$ ).
Theoretical Derivation: Using the probability distribution of Gaussian thermal noise, the authors derive the probability ( $P_{wrong}$ ) of such a rare fluctuation. This leads to a closed-form expression for the defect density that incorporates an exponential action cost term ( $S_h$ ).
Numerical Verification: The theoretical predictions are tested via direct simulations of stochastic Ginzburg–Landau models in one spatial dimension (real scalar field, $Z_2$ symmetry, domain walls) and two spatial dimensions (complex scalar field, $U(1)$ symmetry, vortices). The simulations utilize a linear quench protocol ( $b(t) = t/\tau_Q$ ) and vary the symmetry-breaking field strength ( $h$ ), noise strength ( $\theta$ ), and damping coefficients ( $\eta$ ).

Key Contributions and Results

Closed-Form Expression: The paper derives a general expression for defect density in arbitrary dimensions under weak symmetry breaking:
$n^{(d)}_{\text{defects}} \sim \left( \frac{\eta}{\tau_Q} \right)^{\frac{d\nu}{1+z\nu}} \exp \left[ -A_d \frac{h^2}{\theta} \left( \frac{\tau_Q}{\eta} \right)^{\frac{\nu(d+z)}{1+z\nu}} \right]$
This formula captures the standard KZ power-law prefactor multiplied by an exponential suppression factor dependent on the field strength, noise, and quench time.
Scaling of the Action Cost: The authors demonstrate that the action cost ( $S_h$ ) governing the exponential suppression scales as $S_h \propto \tau_Q^{(d+2)/4}$ for mean-field critical exponents ( $\nu=1/2, z=2$ ).
Universal Collapse: Numerical results for both 1D and 2D systems confirm the theoretical scaling. When the suppression coefficient $\lambda$ is plotted against the scaling variable $h^2 / (\theta \eta^{(d+2)/4})$ , data from various parameters collapse onto a single master curve, validating the derived scaling form.
Distinction from Nucleation Theory: The authors clarify that this mechanism differs fundamentally from nucleation theory corrections seen in weakly first-order transitions. Here, defect formation is driven by rare fluctuations that coherently bias an entire KZ domain during freeze-out, rather than by the activated crossing of a free-energy barrier via critical droplet nucleation.

Significance
The paper establishes that under weak explicit symmetry breaking, the problem of defect formation transitions from a standard Kibble–Zurek process to a rare-event problem governed by large-deviation theory. This provides a general theoretical framework to describe the exponential suppression of defects observed in numerical studies. The findings are significant for understanding realistic phase transitions where perfect symmetry is rarely realized, offering a precise description of how weak external fields and thermal noise jointly determine defect densities in nonequilibrium systems. The agreement between the analytical predictions and stochastic simulations in both one and two dimensions provides strong evidence for the validity of this rare-fluctuation mechanism.

More like this