Phases of the $q$-deformed $\mathrm{SU}(N)$ Yang-Mills theory at large $N$

TL;DR

This paper studies the phase structure of large-$N$ $q$-deformed $ ext{SU}(N)$ Yang-Mills theory in 2+1 dimensions using a lattice Hamiltonian approach, revealing the robustness of topological order across different regimes.

Contribution

It introduces a variational mean-field analysis of the large-$N$ phase diagram of $q$-deformed $ ext{SU}(N)$ Yang-Mills theory, connecting confinement and topological order.

Findings

Topological order persists at large $N$ under certain scalings.

The phase structure depends on the 't Hooft coupling and the ratio $k/N$.

The continuum limit may involve more complex behavior than expected.

Abstract

We investigate the $(2+1)$-dimensional $q$-deformed $\mathrm{SU}(N)_k$ 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 $\lambda_\mathrm{tH}=g^2N$ 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.