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Progress of Theoretical and Experimental Physics

Tensor Renormalization Group Study of the 3D SU(2) and SU(3) Gauge Theories with the Reduced Tensor Network Formulation

Atis Yosprakob, Kouichi Okunishi

Published Feb 17, 2025• 3 citations
Publisher

Abstract

Abstract We perform a tensor renormalization group simulation of non-Abelian gauge theory in three dimensions using a formulation based on the “armillary sphere”. In this formulation, matrix indices are completely traced out, eliminating the degeneracy in the singular-value spectrum of the initial tensor. We demonstrate the usefulness of this technique by computing the average plaquette at zero temperature and the Polyakov-loop susceptibility at finite temperatures for the 2+1D SU(2) and SU(3) gauge theories. The deconfinement transition is identified for both gauge groups, with the SU(2) case being consistent with previous Monte Carlo results.

DOI
10.1093/ptep/ptaf028

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Abstract We perform a tensor renormalization group simulation of non-Abelian gauge theory in three dimensions using a formulation based on the “armillary sphere”. In this formulation, matrix indices are completely traced out, eliminating the degeneracy in the singular-value spectrum of the initial tensor. We demonstrate the usefulness of this technique by computing the average plaquette at zero temperature and the Polyakov-loop susceptibility at finite temperatures for the 2+1D SU(2) and SU(3) gauge theories. The deconfinement transition is identified for both gauge groups, with the SU(2) case being consistent with previous Monte Carlo results.
We perform a tensor renormalization group simulation of non-Abelian gauge theory in three dimensions using a formulation based on the `armillary sphere.' In this formulation, matrix indices are completely traced out, eliminating the degeneracy in the singular value spectrum of the initial tensor. We demonstrate the usefulness of this technique by computing the average plaquette at zero temperature and the Polyakov loop susceptibility at finite temperatures for 2+1D SU(2) and SU(3) gauge theories. The deconfinement transition is identified for both gauge groups, with the SU(2) case being consistent with previous Monte Carlo results.
Abstract Formulating non-Abelian gauge theories as a tensor network is known to be challenging due to the internal degrees of freedom that result in the degeneracy in the singular value spectrum. In two dimensions, it is straightforward to “trace out” these degrees of freedom with the use of character expansion, giving a reduced tensor network where the degeneracy associated with the internal symmetry is eliminated. In this work, we show that such an index loop also exists in higher dimensions in the form of a closed tensor network that we call the “armillary sphere”. This allows us to completely eliminate the matrix indices and reduce the overall size of the tensors in the same way as is possible in two dimensions. This formulation allows us to include significantly more representations with the same tensor size, thus making it possible to reach a greater level of numerical accuracy in the tensor renormalization group computations.
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.
Abstract We propose a method to represent the path integral over gauge fields as a tensor network. We introduce a trial action with variational parameters and generate gauge field configurations with the weight defined by the trial action. We construct initial tensors with indices labelling these gauge field configurations. We perform the tensor renormalization group (TRG) with the initial tensors and optimize the variational parameters. As a first step to the TRG study of non-Abelian gauge theory in more than two dimensions, we apply this method to three-dimensional pure SU(2) gauge theory. Our result for the free energy agrees with the analytical results in the weak and strong coupling regimes.
We study the SU(2)SU(2)SU(2) gauge-Higgs model in two Euclidean dimensions using the tensor renormalization group (TRG) approach. We derive a tensor formulation for this model in the unitary gauge and compare the expectation values of different observables between TRG and Monte Carlo simulations finding excellent agreement between the two methods. In practice we find the TRG method to be far superior to Monte Carlo simulation for calculations of the Polyakov loop correlation function which is used to extract the static quark potential.
Tensor-network methods enable probing dynamics of strongly interacting quantum many-body systems, including gauge theories, via Hamiltonian simulation, hence bypassing sign problems. They also have the potential to inform efficient quantum-simulation algorithms of the same theories. We develop and benchmark a matrix-product-state ansatz for the SU(2) lattice gauge theory using the loop-string-hadron formulation. This formulation has been demonstrated to be advantageous in Hamiltonian simulation of non-Abelian gauge theories. It is applicable to both SU(2) and SU(3) gauge groups, to periodic and open boundary conditions, and to 1+1 and higher dimensions. In this work, we report on progress in computing static and dynamical observables in a SU(2) gauge theory in (1+1)D, pushing the boundary of existing studies.