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Entanglement across scales: Quantics tensor trains as a natural framework for renormalization

Stefan Rohshap1,*, Jheng-Wei Li2, Alena Lorenz1,3, Serap Hasil1, Karsten Held1, Anna Kauch1, and Markus Wallerberger1

  • *Contact author: stefan.rohshap@tuwien.ac.at
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Phys. Rev. Research 7, 043313 – Published 18 December, 2025

DOI: https://doi.org/10.1103/qlrs-6f8t

Abstract

Understanding entanglement remains one of the most intriguing problems in physics. While particle and site entanglement have been studied extensively, the investigation of length- or energy-scale entanglement—quantifying the information exchange between different length scales—has received far less attention. Here, we identify the quantics tensor train (QTT) technique, a matrix product state-inspired approach for overcoming computational bottlenecks in resource-intensive numerical calculations, as a renormalization group method by analytically expressing an exact cyclic reduction-based real-space renormalization scheme in QTT language, which serves as a natural formalism for the method. In doing so, we precisely match the QTT bond dimension—a measure of length-scale entanglement—to the number of rescaled couplings generated in each coarse-graining renormalization step. While QTTs have so far been applied almost exclusively to numerical problems in physics, our analytical calculations demonstrate that they are also powerful tools for mitigating computational costs in semianalytical treatments. We present our results for the one-dimensional tight-binding model with nth-nearest-neighbor hopping, where the 2n rescaled couplings generated in the renormalization procedure precisely match the QTT bond dimension of the one-particle Green's function.

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Phys. Rev. Research 7, 043313– Published 18 December, 2025
  • Received 25 July 2025
  • Accepted 13 November 2025
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