Phases of the q-deformed SU(N) Yang-Mills theory at large N

We investigate the (2+1)-dimensional q-deformed SU(N)_k Yang-Mills theory in the lattice Hamiltonian formalism, which is characterized by three parameters: the number of colors N, the coupling constant g, and the level k. By treating these as tunable parameters, we explore how key properties of the theory, such as confinement and topological order, emerge in different regimes. Employing a variational mean-field analysis that interpolates between the strong- and weak-coupling regimes, we determine the large-N phase structure in terms of the 't Hooft coupling λ_tH=g^2N and the ratio k/N. We find that the topologically ordered phase remains robust at large N under appropriate scalings of these parameters. This result indicates that the continuum limit of large-N gauge theory may be more intricate than naively expected, and motivates studies beyond the mean-field theory, both to achieve a further understanding of confinement in gauge theories and to guide quantum simulations of large-N gauge theories.