Topological defects and confinement with machine learning: The case of monopoles in compact electrodynamics

We investigate the advantages of machine learning techniques to recognize the dynamics of topological objects in quantum field theories. We consider the compact U(1) gauge theory in three spacetime dimensions as the simplest example of a theory that exhibits confinement and mass gap phenomena generated by monopoles. We train a neural network with a generated set of monopole configurations to distinguish between confinement and deconfinement phases, from which it is possible to determine the deconfinement transition point, and to predict several observables. The model uses a supervised learning approach and treats the monopole configurations as three-dimensional images (holograms). We show that the model can determine the transition temperature with accuracy, which depends on the criteria implemented in the algorithm. More importantly, we train the neural network with configurations from a single lattice size before making predictions for configurations from other lattice sizes, from which a reliable estimation of the critical temperatures is obtained.

Figure 1

The best fits of the expectation value of the Polyakov loop by the function (18).

Figure 2

Neural network model and the typical examples of the monopole configurations as the function of the increasing temperature $\beta$ or, according to Eq. (16), rising temperature $T$.

Figure 3

Training curve (evolution of losses for the training and validation sets with time). The inset: the training curve without regularization.

Figure 4

Learning curve (evolution of losses for the training and validation sets of different sizes). The inset: the learning curve without regularization.

Figure 5

Comparison of the MC and ML distributions of various quantities.

Figure 6

Comparison of absolute errors on the predicted distributions.

Figure 7

The Polyakov loop $L$ in terms of $\beta$ for each configuration for $L_t=4$ and $L_s=16$, with the phase $\varphi$ (red: $p=0$, confined; blue: $p=1$, deconfined).

Figure 8

The monopole density $⟨\rho ⟩$ and the Polyakov loop $⟨L⟩$ in terms of $\beta$. The filled symbols show the actual quantities calculated with the help of the Monte Carlo (MC) technique and the open symbols are the predictions based on the machine learning (ML) of the monopole configurations.

Figure 9

Examples of the distribution of the mean Polyakov loop $L$ in terms of the coupling constant $\beta$ for each configuration, with the phase $\varphi$ (red: $p=0$, confined; blue: $p=1$, deconfined), using $p_c=0.5$.

Figure 10

Examples of $\varphi$ in terms of $\beta$ for each configuration, with the phase $\varphi$ (red: $p=0$, confined; blue: $p=1$, deconfined), using $p_c=0.5$.

Figure 11

(a) Mean $⟨p(\varphi )⟩$ and (b) variance $\mathrm{Var}p(\varphi )$ of the distribution $p(\varphi )$ for different lattice geometries.

Figure 12

Relative errors in terms of $L_t$ and $L_s$ for ${\beta }_c$ computed from $⟨p(\varphi )⟩$.

Figure 13

The critical coupling constant ${\beta }_c$ of compact electrodynamics at various lattices $L_s^2\times L_t$. The full symbols: the Monte Carlo results obtained from the original gauge-field configurations. The open symbols: the prediction of the neural network based on monopole configurations only.