$q$ deformed formulation of Hamiltonian SU(3) Yang-Mills theory

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.