Machine learning topological defect formation

TL;DR

This paper explores how machine learning, specifically Recurrent Neural Networks, can predict the final configuration of topological defects during phase transitions by analyzing early-time order parameter evolution, aligning with Kibble-Zurek scaling.

Contribution

It introduces a novel approach using ML to predict defect formation during phase transitions based on early dynamics, extending KZM insights.

Findings

ML predictions follow KZM power law scaling

Recurrent Neural Networks successfully predict defect configurations

Early-time order parameter data suffices for accurate predictions

Abstract

According to the Kibble-Zurek mechanism (KZM), the density of topological defects created during a second-order phase transition is determined by the correlation length at the freeze-out time. This suggests that the final configuration of topological defects in such a transition is largely established during the impulse regime, soon after the critical point is traversed. Motivated by this, we conjecture that machine learning (ML) can predict the final configuration of topological defects based on the time evolution of the order parameter over a short interval in the vicinity of the critical point, well before the order parameter settles into the emerging new minima resulting from spontaneous symmetry breaking. Furthermore, we show that the predictability of ML also follows the power law scaling dictated by KZM. We demonstrate these using a Recurrent Neural Network.

Equations

$$\ddot{\varphi }+\eta \dot{\varphi }-{\partial }_{xx}\varphi +{\partial }_{\varphi }V(\varphi )=\vartheta (x,t)$$

Full text

The Kibble-Zurek mechanism (KZM) combines the scenario of topological defect formation in cosmological phase transitions proposed by Kibble Kibble (1976, 1980) with the theory Zurek (1985, 1993, 1996); Del Campo and Zurek (2014) that relates their density in the broken symmetry phase to the critical slowing down and, hence, to the universality class of the second order phase transition. KZM has found wide-ranging applications across diverse fields, from cosmology Kibble (1976, 1980); Zurek (1985, 1993, 1996); Chuang et al. (1991) and condensed matter Lin et al. (2014); Libál et al. (2020); Sadhasivam et al. (2024); Shu et al. (2025); Schaller et al. (2025); Balducci et al. (2024); Suzuki et al. (2025); Du et al. (2023) to quantum computing Keesling et al. (2019); Zhang et al. (2025); Visuri et al. (2025); Schützhold (2025); Xu et al. (2018).

The key insight [3] that leads to KZM scaling is the realization that, near the critical point of the second order phase transition, critical slowing down will result in an “impulse” time interval $[-\hat{t},+\hat{t}]$ where the order parameter is too sluggish to adjust to the potential that is changing faster than its reaction time. Thus, while outside this interval the order parameter can be in approximate equilibrium, within the interval $[-\hat{t},+\hat{t}]$ its evolution “cannot keep up”. Fluctuations imparted after $\sim-\hat{t}$ seed topological defects that germinate at $\sim+\hat{t}$ in ways that depend on the nature of the system. In KZM, the density of topological defects created during a second-order phase transition is estimated using the freezeout conjecture. In this scenario, the dynamics is divided into three stages: adiabatic, nonadiabatic, and adiabatic again, as the order parameter undergoes spontaneous symmetry breaking. Far away from the critical point, where the equilibrium relaxation time $\tau$ of the system is much shorter than the timescale over which the potential changes, the dynamics is “adiabatic” as the order parameter follows the minimum of the effective potential. By contrast, near the critical point, when spontaneous symmetry breaking is initiated, the dynamics is essentially “frozen” or in an “impulse” regime because of the critical slowing down due to the divergence of the equilibrium relaxation time, which makes it impossible for the order parameter to keep up with the instantaneous equilibrium. The system is then incapable of adjusting to the external changes in the evolving potential, resulting in the seeding of defects. This continues until the dynamics reaches the post-transition boundary between the adiabatic and impulse regimes, where the equilibrium relaxation time $\tau$ is approximately equal to the time elapsed since crossing the critical point, given by $\dot{\epsilon}/\epsilon$, with $\epsilon(t)=t/\tau_{Q}$ representing the dimensionless distance from the critical point and $\tau_{Q}$ denoting the quench timescale. The time $\hat{t}$ that satisfies this relation is referred to as the freeze-out time. The average size of domains between defects can be estimated as the equilibrium correlation length at time $+\hat{t}$, which leads to the well-known KZM scaling $\hat{\xi}\propto\tau_{Q}^{\frac{\nu}{1+\nu z}}$ Zurek (1985, 1993, 1996). The universality class and associated critical exponents then determine the dependence of defect density on the quench rate in a second-order phase transition.

The KZM account of the dynamics of symmetry breaking suggests that the final configuration of topological defects is determined during the impulse regime before the freeze-out time. This in turn motivates the conjecture that machine learning (ML) can predict the final configuration of topological defects from the time evolution of the order parameter over a short interval in the impulse regime before the order parameter settles into the emerging new minima associated with the spontaneous symmetry breaking (Fig. 1). This would enable prediction of the final locations of topological defects well before the order parameter reaches local equilibrium following the phase transition. In this paper, we demonstrate that this is indeed the case by using a Recurrent Neural Network (RNN). Furthermore, we show that the predictability of ML also exhibits universal behavior rooted in the power law scaling of KZM.

RNNs are a type of artificial neural network specifically designed to process sequential or time series data Rumelhart et al. (1986); Williams and Zipser (1989); Hibat-Allah et al. (2020). They incorporate recurrent connections, allowing the output from one time step to be fed back as input at the next. This feedback mechanism enables RNNs to capture temporal dependencies within the data, facilitating the development of ML models capable of predicting future states based on sequential inputs. In our case, the sequential inputs are short time series of the time evolution of the order parameter during a second-order phase transition, and RNN is employed to predict the final configuration of topological defects once the order parameter settles into the emerging minima of the symmetry-breaking potential. Remarkably, we will show that the ML model can accurately predict even the locations of topological defects—i.e., the points where the order parameter changes sign in our one-dimensional Landau-Ginzburg model—using input data consisting of the short-time evolution of the order parameter in the impulse regime.

The training data is prepared as follows. The formation of topological defects (kinks) during a second-order phase transition can be described by the Langevin equation in (1+1) dimensions for a real scalar field $\phi$ representing the order parameter that evolves under a Landau-Ginzburg potential Laguna and Zurek (1997, 1998); Suzuki and Zurek (2024a, b); Dziarmaga et al. (1999):

[TABLE]

where $V(\phi)=(\phi^{4}-2\epsilon\phi^{2})/8$. As dimensionless distance from the critical point $\epsilon(t)=t/\tau_{Q}$ where $\tau_{Q}$ is the quench timescale, changes from a negative to a positive value, spontaneous symmetry breaking occurs. It then leads to the formation of topological defects. $\vartheta$ represents the noise characterized by correlation function, $\langle\vartheta(x,t),\vartheta(x^{\prime},t^{\prime})\rangle=2\eta\theta\delta(x^{\prime}-x)\delta(t^{\prime}-t)$. In this paper, we set the overall damping constant $\eta=1$, and temperature $\theta=10^{-8}$. Note that we can use different parameters, such as a temperature of $\theta=10^{-2}$ that implies more fluctuations, and still obtain results similar to those presented in this paper. In our simulation, the spatial computational domain has size $L=512$, discretized into $1024$ grid points with periodic boundary conditions (i.e., $dx=0.5$).

We numerically computed the time evolution of the order parameter using Eq. (1) in which $\epsilon$ changes from –1.5 to 1, and obtained training data by repeating the process for multiple random noise realizations (see Methods).

In Fig. 2, we show the time evolution of $\langle|\phi(x,t)|\rangle$ where spatial position $x$ is averaged over, and the quench timescale $\tau_{Q}=128$. As suggested by the freezeout paradigm, $\phi$ fluctuates around the old symmetric minimum at $\phi=0$ for a while even after the critical point is crossed at $t=0$ (i.e., the order parameter ceases to follow its equilibrium value since it is unable to adjust to the externally imposed change in the potential due to the divergence of the relaxation time). Then the order parameter catches up with the emerging new minima at $\phi=\pm\sqrt{\epsilon}$ represented by the dashed grey line around the freeze-out time $+\hat{t}$. The red line indicates the time at which $\langle|\frac{d}{dt}\phi(x,t)|\rangle$ reaches its maximum after the phase transition at $t=0$. The freeze-out of the defect configuration occurs around this point Suzuki and Zurek (2024b).

Fig. 3 illustrates examples of the prediction process performed by the RNN for quench timescale $\tau_{Q}=128$. The input data consists of time series $\phi(x,t^{\prime})$, where each series spans from $t^{\prime}=t-5$ to $t+5$ with a time step $\Delta t=1$ (11 sequential data points). An RNN is employed to predict the final configuration of the order parameter $\phi$ at $t=\tau_{Q}$ with $\epsilon=1$. The predicted locations of topological defects (i.e., points where $\phi=0$) are then compared with the true final defect locations. It shows that, even from the short-time evolution of small fluctuations in $\phi$ during the impulse regime—between the critical point at $t=0$ and before the freeze-out time $+\hat{t}$—the model can predict the final locations of topological defects with notable accuracy.

The upper panels in Fig. 4 show the training loss (dashed) and validation loss (solid) plotted as a function of epoch. The loss function is the mean squared error (MSE) between the predicted and true $\phi$ at the time when $t=\tau_{Q}$ with $\epsilon=1$. It can be seen that ML cannot predict the final topological defect configuration from the input data taken before the critical point at $t=0$. However, after the critical point, it begins to effectively predict the final topological defect configuration, even though the inputs are from a time series well before the freeze-out time $+\hat{t}$. The lower panels show the normalized root mean squared error (RMSE) between the predicted and true defect locations, where RMSE is divided by the size of the spatial computational domain $L$. Both of them are evaluated using 100 testing samples after training the ML model for 60 epochs on 3000 samples of the time evolution of the order parameter. The RMSE of the defect locations is computed only when the model correctly predicts the number of defects. The defect count accuracy (i.e., the fraction of cases where the number of defects is correctly predicted) is represented by the color plot. The figure demonstrates that even when the input data capture only a short period of the time evolution of small fluctuations in $\phi$ around $\phi=0$ during the impulse regime, the model can predict both the number and final locations of topological defects with high precision. This is consistent with KZM, where the adiabatic-impulse conjecture suggests that the number of topological defects and the size of domains between them can begin to be established during the impulse regime, before the freeze-out time $+\hat{t}$, i.e., before the order parameter $\phi$ starts to evolve toward the emerging new minima resulting from spontaneous symmetry breaking. Note that RMSE may increase slightly around the red line. This can be attributed to the fact that, at this point, some regions of $\phi$ begin to roll down toward the emerging minima. As a result, the input data can exhibit large fluctuations in $\phi$, ranging from small values near $\phi=0$ to larger values, where $\phi$ starts to roll down. This makes it difficult to analyze the data effectively, especially for the rapid quench timescales $\tau_{Q}$.

Finally, we demonstrate that the predictive abilities of ML also follow the universal behavior predicted by KZM. Fig. 5 shows the validation loss achieved after 60 training epochs for various quench timescales $\tau_{Q}$ (see the upper panels of Fig. 4). It is known that the freeze-out time follows a power law scaling, $\hat{t}\propto\sqrt{\eta\tau_{Q}}$ in the overdamped regime, for the Landau-Ginzburg model represented by Eq. (1) Laguna and Zurek (1997, 1998); Suzuki and Zurek (2024a, b). When using the rescaled time $t/\hat{t}$, the plots for different $\tau_{Q}$ exhibit nearly identical decaying behavior, as shown in the lower panels of Fig. 5. This observation aligns with the general discussion of KZM, where rescaling time by $\hat{t}$ leads to universal dynamics of the order parameter across different quench timescales, consequently resulting in the well-known universal power law for the density of topological defects as a function of the quench timescale Zurek (1985, 1993, 1996); Suzuki and Zurek (2024b). This result shows that the ability of ML to predict the final topological defect configuration remarkably agrees with the universal law based on KZM.

ML has been applied to classify and analyze phases and phase transitions in both classical and quantum systems Carleo et al. (2019); Carrasquilla and Melko (2017); Van Nieuwenburg et al. (2017); Wang (2016); McDermott et al. (2020); Tang et al. (2024); Beach et al. (2018); Rodriguez-Nieva and Scheurer (2019); Zhang et al. (2018); Vargas-Hernández et al. (2018); Miles et al. (2023); Biamonte et al. (2017); Dong et al. (2019), as well as to study nonequilibrium systems Venderley et al. (2018); Seif et al. (2021); Boyd et al. (2022). However, its application to the study of topological defect formation remains largely unexplored. KZM has been widely applied in various fields, ranging from cosmology and condensed matter physics to quantum computing, due to the success in its prediction of a universal power law scaling for the number of topological defects or non-adiabatic excitations generated during second-order phase transitions. Nevertheless, KZM is not solely concerned with the universal power law scaling of defect density, which has been extensively tested and verified across various systems. It also offers insights into the dynamics of the order parameter, as discussed in Suzuki and Zurek (2024b). In particular, the cornerstone of the KZM is the conjecture that the size of the domains which select the same broken symmetry corresponds to the equilibrium correlation length at a moment known as the freeze-out time $+\hat{t}$, when the system’s relaxation rate matches the rate at which the relevant potential is evolving. This insight captures the essence of the non-equilibrium dynamics involved in second-order phase transitions, and it suggests that the final configuration of topological defects is determined during the impulse regime prior to the freeze-out time, and well before the order parameter begins to approach its new equilibrium value in the adiabatic regime. This appears to be indeed the case, as demonstrated in this work. ML was shown to predict the final locations of topological defects from the short-time evolution of the order parameter within the impulse regime before the order parameter begins to move toward one of the emerging minima of the symmetry-breaking potential. Furthermore, it was shown that the capacity of ML to predict also exhibits universal behavior (see rescaling in Fig. 5), as it follows the power law scaling of the freeze-out time implied by KZM: Its ability to correctly anticipate the final topological defect configuration from the input of certain time series, across different quench timescales, exhibits the same behavior when time is rescaled by the freeze-out time.

Information about the final configuration of topological defects is crucial in various physical systems, as these defects often determine the system’s macroscopic behavior and functional properties. The synergy between KZM and ML techniques demonstrated in this work has the potential to advance the prediction of topological defect configurations in second-order phase transitions. It also demonstrates that insights from established physical principles can inspire ML approaches to develop effective and previously unexplored predictive methodologies.

Methods

Preparation of training data

As briefly discussed in the main text, the training data was prepared by numerically solving the Langevin equation in (1+1) dimensions for a real scalar field $\phi$, representing the order parameter evolving under the Landau-Ginzburg potential (Eq. (1)). The time evolution is initiated with $\epsilon=-1.5$ and concluded when $\epsilon$ reaches $1$. The equation is numerically solved using the fourth-order Runge-Kutta method with periodic boundary conditions on a spatial grid of $1024$ points over a length $L=512$, corresponding to $dx=0.5$. By repeating this process, we prepared 3000 samples that show the time evolution of the order parameter during a second-order phase transition, which are then split into a training dataset (80%) and a validation dataset (20%).

RNN model for predicting topological defects configuration

RNN consisting of 256 units and a softsign activation function was used. The network was implemented using a RNN layer followed by a fully connected (Dense) output layer to predict the final configuration of topological defects. The model was trained using the Adam optimizer to minimize the mean squared error (MSE) between the predicted and true $\phi$, using the short-time evolution of $\phi$ during the second-order phase transition (for $-1.5<\epsilon<1$) as input and the corresponding final $\phi$ at $\epsilon=1$ as the target. Training was performed for 60 epochs with a batch size of 10. The results shown in Figs. 4 (lower panel) and 5 represent averages computed over five independent runs of ML model.

Note that the training loss measures the fit of the model to the training data during the learning process, while the validation loss is the measure of the performance of a trained model on unknown data from the validation set. An epoch (i.e., a training cycle) refers to a complete pass through the entire training dataset, during which the model is exposed to every sample and its parameters are updated based on the computed error. The dataset is partitioned into smaller subsets, known as batches, enabling the model to process portions of the training data effectively.