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International Symposium on Lattice Field Theory (LATTICE)

Recent progress in the tensor renormalization group

Daisuke Kadoh

Published May 25, 2022• 6 citations
Publisher

Abstract

The tensor renormalization group is a very attractive approach for numerically calculating lattice field theory. It does not suffer from the sign problem, and it is a method that enables calculation with an extremely large volume lattices and direct calculation of free energy. I report on recent developments in this field and clarify the problems to be solved in the future.

DOI
10.22323/1.396.0633

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The tensor renormalization group is a very attractive approach for numerically calculating lattice field theory. It does not suffer from the sign problem, and it is a method that enables calculation with an extremely large volume lattices and direct calculation of free energy. I report on recent developments in this field and clarify the problems to be solved in the future.
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.
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.
The tensor renormalization group is a promising complementary approach to traditional Monte Carlo methods for lattice systems, as it is inherently free from the sign problem. We discuss recent developments crucial for its application to lattice QCD: the multi-layer construction for multi-flavor gauge theory and the armillary sphere formulation for non-Abelian gauge theory. These techniques are important for reducing the size of the initial tensor and for eliminating non-local entanglement structures within the tensor network. We present selected numerical results and discuss potential generalizations to lattice QCD.
We review the development of numerical renormalization group methods,including Wilson's numerical renormalization group,White's density matrix renormalization group,and several recent rapidly developing tensor network state-based algorithms.Among these,particular emphasis is laid on the linearized tensor renormalization group method,which was recently proposed to accurately calculate the thermodynamic properties of quantum many-body lattice systems,and which may have broad applications in both one-and two-dimensional quantum systems.
One goal in understanding quantum chromodynamics (QCD) includes solving how quarks and gluons combine to form the hadrons and nuclei seen in nature. With lattice QCD, progress has been made regarding the calculation of masses and couplings. However, the real-time evolution and the critical behavior at finite density of strong particles in colliders, stars, or after the big bang remain a challenging problem despite their potential to detect the existence of new physics. The tensor methods for lattice field theories provide a route to handle strongly correlated systems across different subfields using renormalization group methods or quantum computing.
Tensor network study of two dimensional lattice ϕ4\phi^{4}ϕ4 theory
The tensor renormalization group attracts great attention as a new numerical method that is free of the sign problem. In addition to this striking feature, it also has an attractive aspect as a coarse-graining of space-time; the computational cost scales logarithmically with the space-time volume. This fact allows us to aggressively approach the thermodynamic limit. While taking this advantage, we study the critical coupling of the two dimensional ϕ4\phi^{4}ϕ4 theory on large and fine lattices. We present the numerical results along with the extrapolation procedure to the continuum limit and compare them with the previous ones by Monte Carlo simulations.