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

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>n</a:mi> <a:mi>th</a:mi> </a:mrow> </a:math> -nearest-neighbor hopping, where the <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mn>2</b:mn> <b:mi>n</b:mi> </b:mrow> </b:math> rescaled couplings generated in the renormalization procedure precisely match the QTT bond dimension of the one-particle Green's function.