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

TL;DR

This paper introduces a tensor network method that efficiently computes high-dimensional configurational integrals in crystalline solids, enabling rapid and accurate thermodynamic property calculations.

Contribution

It develops a novel tensor-train based approach with specialized schemes for sharply peaked densities, improving computational efficiency in solid-state statistical mechanics.

Findings

Accurately reproduces MD simulation results for Cu, Ar, and Sn phase transition.

Achieves rapid computation within seconds for complex integrals.

Handles sharply peaked Boltzmann densities effectively.

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.