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

TL;DR

This paper introduces the use of quantics tensor trains (QTT) as a natural framework for renormalization in physics, linking entanglement across scales with computational efficiency in numerical and semi-analytical models.

Contribution

It analytically demonstrates that QTT can serve as a renormalization group method, connecting length scale entanglement with QTT bond dimension in a novel way.

Findings

QTT bond dimension matches the number of rescaled couplings in renormalization.

QTT effectively reduces computational costs in semi-analytical physics calculations.

Application to a 1D tight-binding model shows precise correspondence between couplings and QTT bond dimension.

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 semi-analytical treatments. We present our results for the one-dimensional tight-binding model with n-th-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.