Autoregressive Neural Network Extrapolation of Quantum Spin Dynamics Across Time and Space

TL;DR

An autoregressive machine-learning framework is introduced that enables the extrapolation of dynamical spin correlations in both time and space beyond the reach of conventional numerical methods, establishing a new paradigm for studying the dynamics of gapless quantum many-body systems.

Abstract

Understanding the dynamical response of quantum materials is central to revealing their microscopic properties, yet access to long-time and large-scale dynamics remains severely limited by rapidly growing computational costs and entanglement, particularly in gapless systems. Here we introduce an autoregressive machine-learning framework that enables the extrapolation of dynamical spin correlations in both time and space beyond the reach of conventional numerical methods. Trained on time-dependent density matrix renormalization group simulations of the gapless XXZ model, our approach is benchmarked against exact solutions available for this analytically solvable system. Combined with physics-informed spatial extension, multi-layer perceptron model using ReLU activation functions has been shown to be superior than convolutional neural networks and linear regressions for longer time extrapolation. Perturbation study of error accumulation further demonstrates that our autoregressive neural network extrapolations are highly robust to perturbations, suggesting stable and reliable predictions. This work establishes a new paradigm for studying the dynamics of gapless quantum many-body systems, in which machine learning extends and complements the capabilities of state-of-the-art numerical approaches.

Figures (5)

• Figure 1: (a-b) The real and imaginary part of the dynamical spin correlation function $S(r,t)$ for XXZ model with $\Delta = 0.25$. The tDMRG data is shown within the dashed region, while the rest of data beyond the dashed region were produced using the machine-learning autoregressive framework based on a MLP with ReLU activation functions. The blue rectangles denote the input data for the MLP, which outputs the real and imaginary parts of a single point in $S(r,t)$ as shown with the red box. The blue and yellow rows denotes the input and output of the spatial extension. (c) tDMRG data and spatial extension based on the least-square fit to the power-law decay of Luttinger liquid theory [Eq. (\ref{['Eq:LL_Correlation_RealSpace']})] at $t=0$. Finally, the inset of (c) also contains the normalized standard deviations of the parameters used in the model fitting across time $t\in[0,20J^{-1}]$. These parameters correspond to the parameters in Eq. (\ref{['Eq:LL_Correlation_RealSpace']}).
• Figure 2: ML extrapolation of dynamical spin correlation functions for the real part (a)-(c) Re$\left[S(r,t)\right]$ and imaginary part (d)-(e) Im$\left[S(r,t)\right]$ using various NN architectures. The blue dashed line separates the tDMRG data below and the extrapolated region above. While CNN and Linear models perform adequately for short time extrapolation, MLP has the best performance for extrapolation at long time.
• Figure 3: Improved resolution of the dynamical structure factor $S(q,\omega)$ obtained using our autoregressive extrapolation framework for the spin-$\frac{1}{2}$ XXZ chain with $\Delta = 0.25$. (a) $S(q,\omega)$ obtained from tDMRG data. (b) $S(q,\omega)$ obtained from machine learning extrapolated data.
• Figure 4: Perturbative study on error accumulation effect. A set of 1000 randomly perturbed MLP models is generated for extrapolation. (a-b) The average extrapolation value of Re$\left[S(r,t)\right]$ and Im$\left[S(r,t)\right]$, overlapped with test data produced by tDMRG at finite time steps. Evidently, the ML prediction (blue dot) quantitatively matches with the test data (red line). (c) Variance of $S(r,t)$ of the perturbed models over time and space. The gaussian noise is set to $\sigma=10^{-4}$. (d) The maximum variance at each time step with varying magnitudes of gaussian noise decreases with time.
• Figure 5: Calculated dynamical spin correlation function $S(q,\omega)$ using ML autoregressive extrapolation, Bethe ansatz (BA), and Exact Diagonalization (ED). The left column represents $\Delta$ = 0.25 and the right column represents $\Delta$ = 0.75.