arXiv:2505.21826 · Physical Review Materials

Breaking the curse of dimensionality: Solving configurational integrals for crystalline solids by tensor networks

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

Published Aug 27, 2025• 2 citations

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 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. This approach circumvents the need to explicitly construct the full tensor. We introduce tailored rank-1 and rank-2 schemes optimized for sharply peaked Boltzmann probability densities, typical for identical-particle crystals. When applied to the calculation of internal energy and pressure-temperature curves for crystalline Cu and Ar at high (GPa) pressures, as well as the alpha-to-beta phase transition diagram of Sn, our method accurately reproduces molecular dynamics simulation results using tight-binding, machine learning, hierarchical interacting particle–neural network, and modified embedded atom method potentials,all within seconds of computation time.

DOI: 10.1103/xrbw-xr49

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