Ab-initio variational wave functions for the time-dependent many-electron Schrödinger equation

TL;DR

This work introduces a variational approach for fermionic time-dependent wave functions, surpassing mean-field approximations by accurately capturing many-body correlations, and extends this approach to real-time evolution, providing improved accuracy over traditional methods.

Abstract

Understanding the real-time evolution of many-electron quantum systems is essential for studying dynamical properties in condensed matter, quantum chemistry, and complex materials, yet it poses a significant theoretical and computational challenge. Our work introduces a variational approach for fermionic time-dependent wave functions, surpassing mean-field approximations by accurately capturing many-body correlations. Therefore, we employ time-dependent Jastrow factors and backflow transformations, which are enhanced through neural networks parameterizations. To compute the optimal time-dependent parameters, we utilize the time-dependent variational Monte Carlo technique and a new method based on Taylor-root expansions of the propagator, enhancing the accuracy of our simulations. The approach is demonstrated in three distinct systems. In all cases, we show clear signatures of many-body correlations in the dynamics. The results showcase the ability of our variational approach to accurately capture the time evolution, providing insight into the quantum dynamics of interacting electronic systems, beyond the capabilities of mean-field.

Figures (11)

• Figure 1: Schematic representation of different approaches to capture correlations. Single-determinant approaches with single-particle orbitals are mostly applicable where strong correlations can be neglected. Correlations can be built in using a polynomial number of determinants, a multiplicative time-dependent Jastrow factor that depends on all the electron positions, or in the most powerful case by using a determinant with higher-dimensional many-body time-dependent orbitals using backflow transformations.
• Figure 2: Monopole $Q$ for the harmonic interaction model with $30$ particles, subject to a quench of the harmonic confinement and a time-dependent interaction strength. (top panel) We show the predictions with tVMC using the ($i$) time-dependent constants Ansatz in green, and ($ii$) the neural quantum state in orange, both introduced in the main text. We compare to the exact solution in blue. The curves are overlapping and therefore hardly distinguishable. (bottom panel) The relative error $\Delta Q$ on the predicted monopole, averaged over a rolling window of $\Delta t=0.2$ to reduce the effect of statistical noise.
• Figure 3: Time-dependent dipole moment of $H_{2}$ in an intense, time-dependent laser field modeled with an NQS and tVMC. We show the effect of capturing correlations with a slater determinant with time-dependent neural backflow transformation and a Jastrow representing the cusp condition (S+C+BF), as well as results obtained in the STO-3g basis in second quantization (RBM). We compare to predictions from TDHF and ED.
• Figure 4: The integrated $R^2(t)$ error in Eq. \ref{['eq:R2']} (top panel) and pair correlation $G^{(2)}(t)$ in Eq. \ref{['eq:integrated_G2']} (bottom panel) as a function of time for a fully polarized quantum dot with $N=6$ and subject to a quench $\kappa(t) = 1 \to 2$ at $t=0$. We compare with predictions using tVMC with a single Slater determinant (S), a Slater-Jastrow model (S+J), and a Slater-Jastrow-Backflow (S+J+BF) model.
• Figure 5: The integrated $R^2$ for the same quantum dot quenched experiment as in Fig. \ref{['fig:quantumdot_monopole_1_2']}, using the tre-tVMC approach with given order $K$. Between brackets, we indicate the number of electrons and the time step: $(N, \delta t\times 10^2)$. We compare the results using the basis expanded Slater-Jastrow model (S+J) with $K=2$ TRE to a fully neural-network-based wave function ansatz (NQS).