Leveraging recurrence in neural network wavefunctions for large-scale simulations of Heisenberg antiferromagnets on the triangular lattice

TL;DR

The study tackles the sign-structure challenge in simulating the frustrated triangular-lattice Heisenberg antiferromagnet by employing 2D recurrent neural network (RNN) variational wavefunctions, augmented with local basis rotations and variational neural annealing. Through iterative retraining and symmetry-aware optimization, the approach scales to lattices up to $L=30$, enabling reliable finite-size scaling to the thermodynamic limit. The authors estimate a thermodynamic-ground-state energy per spin $E_\infty/N \approx -0.5518$ and a finite sublattice magnetization $M_\infty \approx 0.19$–$0.20$, in good agreement with DMRG and iPEPS results and confirming $120^\circ$ order. Overall, the work demonstrates that RNN wavefunctions, combined with basis transformations and annealing, provide a scalable and effective toolbox for exploring frustrated quantum many-body systems with sign structure.

Abstract

Variational Monte Carlo simulations have been crucial for understanding quantum many-body systems, especially when the Hamiltonian is frustrated and the ground-state wavefunction has a non-trivial sign structure. In this paper, we use recurrent neural network (RNN) wavefunction ansätze to study the triangular-lattice antiferromagnetic Heisenberg model (TLAHM) for lattice sizes up to $30\times30$. In a recent study [M. S. Moss et al. arXiv:2502.17144], the authors demonstrated how RNN wavefunctions can be iteratively retrained in order to obtain variational results for multiple lattice sizes with a reasonable amount of compute. That study, which looked at the sign-free, square-lattice antiferromagnetic Heisenberg model, showed favorable scaling properties, allowing accurate finite-size extrapolations to the thermodynamic limit. In contrast, our present results illustrate in detail the relative difficulty in simulating the sign-problematic TLAHM. We find that the accuracy of our simulations can be significantly improved by transforming the Hamiltonian with a judicious choice of basis rotation. We also show that a similar benefit can be achieved by using variational neural annealing, an alternative optimization technique that minimizes a pseudo free energy. Ultimately, we are able to obtain estimates of the ground-state properties of the TLAHM in the thermodynamic limit that are in close agreement with values in the literature, showing that RNN wavefunctions provide a powerful toolbox for performing finite-size scaling studies for frustrated quantum many-body systems.

Figures (16)

• Figure 1: A 2D RNN wavefunction defined for a triangular lattice with $L = 4$. The bonds of the $4\times4$ triangular lattice are shown in grey to illustrate how the RNN structure maps to the underlying lattice. The autoregressive sequence is defined by the red arrows. Sampling and inference are performed along this path. The information in the network, stored in the hidden vectors, is passed in two directions along the black arrows. Notably, the black arrows follow the nearest-neighbor interactions of a square lattice. The black dotted arrows show how pseudo-periodic boundary connections can be built into the RNN wavefunction. Both the two-dimensional information passing and the pseudo-periodic boundary connections are implemented in a causal way such that the autoregressive sequence is not violated.
• Figure 2: The number of training steps used in the optimization for each system size, as determined by our parameterized training schedule defined in \ref{['eq:schedule']}. We consider three different scales and three rates. The colors and markers are used to indicate these two parameters respectively.
• Figure 3: The energy per spin of the ground state of the TLAHM for $L=6$. We compare results from simulations performed with different local unitary basis transformations applied to the Hamiltonian: no basis transformation, $\mathcal{I}$, the basis transformation given by the MPSR, $\mathcal{U}_\text{sq}$ defined in \ref{['eq:sq_sign_rule']}, and the 120$^{\circ}$ transformation, $\mathcal{U}_\text{tri}$ defined in \ref{['eq:tri_sign_rule']}. Furthermore, we examine how the initial annealing temperature $T_0$ and the scale $s$ from \ref{['eq:schedule']}, which combine to realize different annealing schedules, impacts the accuracy of our simulations. The inset shows a zoomed-in view of the results obtained when $\mathcal{U}_\text{tri}$ is employed. The ground-state energy obtained with exact diagonalization (ED) is shown for reference bernu_exact_1994.
• Figure 4: (a) The variational energies for all system sizes obtained from the RNN wavefunctions optimized according to each of the iterative retraining schedules shown in \ref{['fig:schedules']}. These energies are plotted according to $1/L$ for easier viewing, but they are fit according to the scaling form defined by \ref{['eq:energy_scaling']}. The reference values of the ground-state energy in the thermodynamic limit (TL) are shown for comparison. The dashed line is the value from DMRG simulations using MPS with cylindrical boundary conditions huang_magnetization_2024. The dotted line is the value from variationally-optimized iPEPS hasik_incommensurate_2024. Each of our variational energies is estimated with $10\times10^3$ samples. (b) The variances of the final variational energies shown in (a) plotted as a function of system size $L$.
• Figure 5: Improved estimates of the ground-state energies for finite sizes obtained from extrapolating the energies shown in \ref{['fig:energies_peri']}(a) to their zero-variance limit for each system size $L$. These zero-variance energies, excluding the value for $L=6$, are fit according to the scaling form defined by \ref{['eq:energy_scaling']}. We plot these values as a function of $1/L$ for easier viewing. The reference values of the ground-state energy in the thermodynamic limit (TL) are shown for comparison. The dashed line is the value from DMRG simulations using MPS with cylindrical boundary conditions huang_magnetization_2024. The dotted line is the value from variationally-optimized iPEPS hasik_incommensurate_2024.