Approaching the Thermodynamic Limit with Neural-Network Quantum States

• URL: http://arxiv.org/abs/2602.02665v1
• Date: Mon, 02 Feb 2026 19:00:05 GMT
• Title: Approaching the Thermodynamic Limit with Neural-Network Quantum States
• Authors: Luciano Loris Viteritti, Riccardo Rende, Subir Sachdev, Giuseppe Carleo,
• Abstract summary: We introduce the Spatial Attention mechanism, a minimal and physically interpretable inductive bias for Neural-Network Quantum States.<n>This bias stabilizes large-scale optimization and enables access to thermodynamic-limit physics.<n>We show that our approach achieves a state-of-the-art energies for a $20times20$ square lattice, outperforming Residual Convolutional Neural Networks.
• Score: 0.6999740786886536
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Accessing the thermodynamic-limit properties of strongly correlated quantum matter requires simulations on very large lattices, a regime that remains challenging for numerical methods, especially in frustrated two-dimensional systems. We introduce the Spatial Attention mechanism, a minimal and physically interpretable inductive bias for Neural-Network Quantum States, implemented as a single learned length scale within the Transformer architecture. This bias stabilizes large-scale optimization and enables access to thermodynamic-limit physics through highly accurate simulations on unprecedented system sizes within the Variational Monte Carlo framework. Applied to the spin-$\tfrac12$ triangular-lattice Heisenberg antiferromagnet, our approach achieves state-of-the-art results on clusters of up to $42\times42$ sites. The ability to simulate such large systems allows controlled finite-size scaling of energies and order parameters, enabling the extraction of experimentally relevant quantities such as spin-wave velocities and uniform susceptibilities. In turn, we find extrapolated thermodynamic limit energies systematically better than those obtained with tensor-network approaches such as iPEPS. The resulting magnetization is strongly renormalized, $M_0=0.148(1)$ (about $30\%$ of the classical value), revealing that less accurate variational states systematically overestimate magnetic order. Analysis of the optimized wave function further suggests an intrinsically non-local sign structure, indicating that the sign problem cannot be removed by local basis transformations. We finally demonstrate the generality of the method by obtaining state-of-the-art energies for a $J_1$-$J_2$ Heisenberg model on a $20\times20$ square lattice, outperforming Residual Convolutional Neural Networks.

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