[Paper Review] $q$ deformed formulation of Hamiltonian SU(3) Yang-Mills theory

Tomoya Hayata, Yoshimasa Hidaka|arXiv (Cornell University)|Jun 21, 2023

Black Holes and Theoretical Physics4 citations

TL;DR

This paper proposes a $q$-deformed Hamiltonian formulation of $(2+1)$-dimensional SU(3) Yang-Mills theory using Wilson line networks, realizing the gauge symmetry as the quantum group $\mathrm{SU}(3)_k$. By employing a mean-field computation based on infinite projected entangled pair states (iPEPS), the method reproduces key features of the SU(3) theory—such as string tension and Casimir scaling—with good quantitative agreement to Monte Carlo simulations at large $k$, demonstrating the potential of tensor networks for non-Abelian gauge theories.

ABSTRACT

We study $\mathrm{SU}(3)$ Yang-Mills theory in $(2+1)$ dimensions based on networks of Wilson lines. With the help of the $q$ deformation, networks respect the (discretized) $\mathrm{SU}(3)$ gauge symmetry as a quantum group, i.e., $\mathrm{SU}(3)_k$, and may enable implementations of $\mathrm{SU}(3)$ Yang-Mills theory in quantum and classical algorithms by referring to those of the stringnet model. As a demonstration, we perform a mean-field computation of the groundstate of $\mathrm{SU}(3)_k$ Yang-Mills theory, which is in good agreement with the conventional Monte Carlo simulation by taking sufficiently large $k$. The variational ansatz of the mean-field computation can be represented by the tensor networks called infinite projected entangled pair states. The success of the mean-field computation indicates that the essential features of Yang-Mills theory are well described by tensor networks, so that they may be useful in numerical simulations of Yang-Mills theory.

Motivation & Objective

• To develop a regularized Hamiltonian formulation of SU(3) Yang-Mills theory in (2+1)D that preserves gauge invariance and enables numerical simulation.
• To generalize the $q$-deformation approach from SU(2) to SU(3) by constructing a quantum group $\mathrm{SU}(3)_k$ to approximate the continuous gauge group.
• To demonstrate the viability of tensor network methods—specifically iPEPS—for simulating non-Abelian gauge theories by performing a mean-field computation of the ground state.
• To compare the results of the $\mathrm{SU}(3)_k$ mean-field computation with conventional Monte Carlo simulations of the full SU(3) theory to validate the approach.
• To assess the convergence of observables like string tension and Casimir scaling in the $k \to \infty$ limit, establishing the required $k$ for physical accuracy.

Proposed method

• Construct a Kogut-Susskind Hamiltonian on a square lattice using networks of Wilson lines, with gauge invariance enforced via $\mathrm{SU}(3)_k$ quantum group symmetry.
• Implement $q$-deformation of the SU(3) group to discretize the continuous gauge group, mapping infinite-dimensional Hilbert spaces to finite-dimensional ones via $k$-dependent representations.
• Use a variational ansatz based on infinite projected entangled pair states (iPEPS) to represent the ground state wavefunction, enabling efficient computation of expectation values.
• Compute expectation values of Wilson loops using graph-based contraction techniques, with the string tension derived from large-area loops.
• Perform mean-field computations by optimizing variational parameters under the iPEPS ansatz, with observables evaluated via tensor network contractions.
• Compare results—especially string tension and Casimir scaling—between the $\mathrm{SU}(3)_k$ mean-field approach and SU(3) Monte Carlo simulations at various $k$ and $1/g^2$ values.

Experimental results

Research questions

• RQ1 Can the $q$-deformed $\mathrm{SU}(3)_k$ quantum group formulation preserve the essential non-Abelian structure of SU(3) Yang-Mills theory while enabling finite-dimensional simulations?
• RQ2 To what extent does the iPEPS-based mean-field computation of $\mathrm{SU}(3)_k$ Yang-Mills theory reproduce the string tension and Casimir scaling observed in full SU(3) Monte Carlo simulations?
• RQ3 What is the required value of $k$ for the $\mathrm{SU}(3)_k$ theory to accurately approximate the $k \to \infty$ limit and recover physical observables of the original SU(3) theory?
• RQ4 How do finite-size effects and correlations in Wilson loops affect the accuracy of the mean-field iPEPS ansatz, and can they be improved via more sophisticated tensor network ansätze?
• RQ5 Can tensor network methods like iPEPS be extended to simulate real-time dynamics and glueball masses in non-Abelian gauge theories beyond the mean-field level?

Key findings

• The mean-field computation of $\mathrm{SU}(3)_k$ Yang-Mills theory using iPEPS shows good quantitative agreement with Monte Carlo simulations of the full SU(3) theory, particularly at large $k$.
• The string tension of the fundamental Wilson loop in the $\mathrm{SU}(3)_k$ mean-field computation matches the Monte Carlo data for $k=50$, though small discrepancies remain due to mean-field approximations.
• Casimir scaling $\sigma_a \propto C_2(a)$ holds in the intermediate coupling regime ($1/g^2 \sim 3-30$) for $k=50$, but breaks down in the strong coupling and topological phases.
• The number of irreducible representations in the $k$-deformed theory scales as $N_v = k^2/2 + 3k/2 + 1$, reaching $N_v = 1326$ at $k=50$, indicating a significant but finite Hilbert space dimension.
• The success of the iPEPS ansatz suggests that tensor networks can capture essential features of Yang-Mills theory, supporting their use in future simulations of QCD.
• The method enables computation of Wilson loop expectation values and string tensions via graph-based tensor contractions, providing a scalable framework for non-Abelian gauge theories.