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Renormalization group on tensor networks

Shinichiro Akiyama

Abstract

We review recent developments in tensor network approaches, focusing on renormalization group methods. Since they are free from the negative sign and complex action problems, there is growing interest in their application to lattice field theories, particularly with a view toward future studies of quantum chromodynamics (QCD) at finite temperature and density. They are also of broad interest in quantum field theory, with recent advances in approaches that allow one to directly investigate universal aspects of critical behavior by making use of theoretical insights from conformal field theory. We highlight several recently explored topics that are expected to play important roles in forthcoming tensor-network studies of QCD.

Renormalization group on tensor networks

Abstract

We review recent developments in tensor network approaches, focusing on renormalization group methods. Since they are free from the negative sign and complex action problems, there is growing interest in their application to lattice field theories, particularly with a view toward future studies of quantum chromodynamics (QCD) at finite temperature and density. They are also of broad interest in quantum field theory, with recent advances in approaches that allow one to directly investigate universal aspects of critical behavior by making use of theoretical insights from conformal field theory. We highlight several recently explored topics that are expected to play important roles in forthcoming tensor-network studies of QCD.
Paper Structure (8 sections, 3 figures)

This paper contains 8 sections, 3 figures.

Table of Contents

  1. Introduction
  2. Basic formulation of TRG
  3. Grassmann tensor networks
  4. Applications to two-color QCD and strong-coupling QCD
  5. Partition-function ratios
  6. CFT data, spectroscopy, and entanglement entropy
  7. Synergistic and hybrid computational approaches
  8. Summary and outlook

Figures (3)

  • Figure 1: Thermodynamic free energy density as a function of $\theta$ for the massive Schwinger model with staggered fermions, adapted from Ref. Kanno:2025hgp.
  • Figure 2: TRG results for (3+1)D two-color QCD in the strong-coupling limit, adapted from Ref. Sugimoto:2025vui. (Left) Quark number density $\langle n \rangle$, chiral condensate $\langle \bar{\chi}\chi \rangle$, and diquark condensate $\langle \chi\chi \rangle$ as functions of the chemical potential $\mu$. The phase transition points are compared with the MF prediction Nishida:2003uj. (Right) The diquark condensate scales as $(\mu-\mu_c)^{\beta_m}$ with $\beta_m=0.514(27)$, consistent with the MF value $\beta_m=1/2$.
  • Figure 3: GW ratio for the (1+1)D $U(1)$ gauge-Higgs model at $\theta=\pi$ as a function of the Higgs mass $M$. The admissibility condition for the $U(1)$ gauge fields is imposed Akiyama:2024qer. The phase transition is associated with the spontaneous breaking of $\mathbb{Z}_{2}$ charge conjugation symmetry. The value at the scale-invariant point, where data from different volumes intersect, is consistent with the 2D Ising CFT Morita:2025hsv.