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arXiv · 2505.21826

Breaking the Curse of Dimensionality: Solving Configurational Integrals for Crystalline Solids by Tensor Networks

Duc P. Truong·Benjamin Nebgen·Derek DeSantis·Dimiter N. Petsev·Kim Ø. Rasmussen·Boian S. Alexandrov
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Abstract

Accurately evaluating configurational integrals for dense solids remains a central and difficult challenge in the statistical mechanics of condensed systems. Here, we present a novel tensor network approach that reformulates the high-dimensional configurational integral for identical-particle crystals into a sequence of computationally efficient summations. We represent the integrand as a high-dimensional tensor and apply tensor-train (TT) decomposition together with a custom TT-cross interpolation scheme. This approach avoids the need to explicitly construct the full tensor, which would otherwise be computationally intractable. We introduce tailored rank-1 and rank-2 schemes optimized for sharply peaked Boltzmann probability densities, typical in crystalline solids. When applied to the calculation of internal energy and pressure-temperature curves for crystalline copper (Cu) and argon (Ar), as well as the alpha-to-beta phase transition in tin (Sn), our method accurately reproduces molecular dynamics simulation results using tight-binding, machine learning (HIP-NN), and MEAM potentials, all within seconds of computation time.

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Duc P. Truong, Benjamin Nebgen, Derek DeSantis, Dimiter N. Petsev, Kim Ø. Rasmussen, Boian S. Alexandrov. 2025-05-27. Breaking the Curse of Dimensionality: Solving Configurational Integrals for Crystalline Solids by Tensor Networks. https://arxiv.org/abs/2505.21826

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