Learning topological defects formation with neural networks in a quantum phase transition

TL;DR

This work uses neural-network quantum states, specifically a Restricted Boltzmann Machine (RBM), to study nonequilibrium quantum dynamics in a one-dimensional transverse-field quantum Ising model (TFQIM) during a linear quench across its quantum critical point. By applying time-dependent variational Monte Carlo to the RBM ansatz, the authors track the time evolution of the energy, the formation of topological defects (kinks), and the full statistics of kink numbers, testing the Kibble-Zurek mechanism (KZM) and exploring universalities beyond KZM. They find that the excitation energy scales as a power law in the quench rate, consistent with the kink-number scaling, and that the first three cumulants of the kink-pair distribution follow universal power laws with exponents close to, but slightly shifted from, the KZM prediction due to finite-size effects. Additionally, kink-kink correlations computed from the neural-network states agree with analytic predictions, validating the method's accuracy for critical dynamics in quantum many-body systems.

Abstract

Neural networks possess formidable representational power, rendering them invaluable in solving complex quantum many-body systems. While they excel at analyzing static solutions, nonequilibrium processes, including critical dynamics during a quantum phase transition, pose a greater challenge for neural networks. To address this, we utilize neural networks and machine learning algorithms to investigate the time evolutions, universal statistics, and correlations of topological defects in a one-dimensional transverse-field quantum Ising model. Specifically, our analysis involves computing the energy of the system during a quantum phase transition following a linear quench of the transverse magnetic field strength. The excitation energies satisfy a power-law relation to the quench rate, indicating a proportional relationship between the excitation energy and the kink numbers. Moreover, we establish a universal power-law relationship between the first three cumulants of the kink numbers and the quench rate, indicating a binomial distribution of the kinks. Finally, the normalized kink-kink correlations are also investigated and it is found that the numerical values are consistent with the analytic formula.

Figures (7)

• Figure 1: Sketchy map of the structure of the RBM. RBM has a hidden layer with $h_i$$(i=1,\cdots M)$ as hidden neurons and a visible layer with $s_j$$(j=1,\cdots N)$ as visible neurons. The lines linking the hidden points and visible points represent the interactions. There is no intralayer interactions in the hidden layer or visible layer themselves.
• Figure 2: (Left)Time evolution of the energy expectation value $E(t)/J$ per lattice point with respect to the reduced time $t/\tau_Q$. The solid lines represent the evolution of energy for three different quench rates from machine learning methods, while the black dashed lines are the comparing results from the methods in dziarmaga2005dynamics. The inset plot exhibits the relative errors for both methods, from which we see that they match each other very well. (Right) The excitation energy density $\Delta E$ with respect to $\tau_Q$ at the end of the quench. The fitting line has a power-law scaling as $\Delta E\approx 0.245\times\tau_Q^{-0.56}$, which indicates that the excitation energy is proportional to the kink numbers.
• Figure 3: Double logarithmic plots for the cumulants $\kappa_q$ ($q=1,2,3$) of the kink pair number distributions with respect to quench rate $\tau_Q$. The circles, triangles and squares are the data from the neural network methods, while the solid lines are from the analytic method introduced in dziarmaga2005dynamics. The dashed line is the reference line with theoretical power law $\tau_Q^{-0.5}$. The error bars represent the standard errors.
• Figure 4: The normalized kink-kink correlations $C_r^{KK}$ against the normalized distances $r/\hat{\xi}$. The green line is from the analytic formula in Eq.\ref{['analytic']} with $\tau_Q=5$. The error bars in the numerical data represent the standard error.
• Figure 5: The first three cumulants against the quench rate for various sites numbers $N$. The dashed lines are the theoretical predictions with the power-law scaling $\tau_Q^{-0.5}$. The numerical data are from the neural network methods.