Understanding magnetism at the shortest length and time scales requires the quantum-mechanical approach. As the Hilbert space of this quantum many-body problem grows exponentially, no exact solution is available, and numerical approaches are challenging as well. Even for one of the simplest models, the antiferromagnetic Heisenberg model on a square lattice, no numerically exact solutions are available. Recently, new variational approaches inspired by machine learning have emerged, which go beyond some of the limitations of existing methods. These Neural Quantum States (NQS) ansätze have proven to be a powerful tool to accurately represent the many-body wave function for a wide class of physical systems . Extensive research has been done to explore the linear regime of magnetic dynamics using NQS . However, the nonlinear regime has not been explored at all.
The dynamics of NQS models are governed by the time-dependent variational principle (TDVP) equation of motion. However, TDVP is known to suffer from numerical instabilities for non-linear driving, or otherwise induced dynamical complexity. While the precise origin of instabilities is not well understood, they are related to the overdetermination of the parameter basis. So far, these instabilities have been addressed with regularization, but this proved to be useful only in the linear response regime .
In order to identify the cause of numerical instabilities, we study the Heisenberg antiferromagnet on a 2x2 square lattice, using the Restricted Boltzmann Machine (RBM) neural network ansatz. The system is driven out of equilibrium by a constant perturbation of the exchange interaction in one of the lattice dimensions. We investigate the numerical accuracy and possible breakdown due to instabilities, as a function of the perturbation strength. The advantage of this simple system is the availability of the exact diagonalization (ED), which we use as a benchmark. The dynamical properties of this system are monitored through the observables such as energy and spin-spin correlation function. We compare the exact dynamics with those obtained by three different numerical procedures designed to treat the instabilities. Interestingly, distinct from previous investigations for larger lattice models and in the presence of noise, we identify a specific value of perturbation strength in which RBM solutions break down, regardless of the stabilization procedure.
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 D. Hofmann, G. Fabiani, J. H. Mentink, G. Carleo, M. A. Sentef: Role of stochastic noise and generalization error in the time propagation of neural-network quantum states, SciPost Physics 12, 165 (2022), DOI: 10.21468/SciPostPhys.12.5.165