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

TL;DR

The paper investigates the phase structure of the $(2+1)$-dimensional $q$-deformed $ ext{SU}(N)_k$ Yang–Mills theory in the large-$N$ limit, focusing on the competition between confinement and topological order when the Hilbert-space truncation by $k$ and the coupling $g$ are varied. A variational mean-field approach bridges the strong- and weak-coupling vacua, with the phase boundary diagnosed from the Hessian around the topological ground state and encoded in fusion rules, quantum dimensions, and Casimirs; the analysis is organized in terms of the 't Hooft coupling $λ_ ext{tH}$ and the ratio $k/N$. The key finding is a universal collapse of the critical lines onto a curve in the $(1/λ_ ext{tH}, k/N)$ plane for $N\ge 3$, implying a well-defined large-$N continuum limit within mean-field and indicating that the topological phase persists under suitable scalings of $g$ and $k$. The study also clarifies the abelian nature of the $k=1$ regime, discusses the lack of level-rank duality in the phase structure due to the Casimir choice, and outlines avenues beyond mean-field toward a fuller understanding of confinement and continuum limits in large-$N$ gauge theories, with implications for quantum simulations of gauge theories.

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 $λ_\mathrm{tH}=g^2N$ 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.

Figures (5)

• Figure 1: Phase structure in terms of the inverse square of YM coupling $1/g^2$ and the cutoff $k$.
• Figure 2: Phase structure in terms of the inverse 't Hooft coupling $1/\lambda_\mathrm{tH}$ and $k/N$.
• Figure 3: The critical coupling $g_\mathrm{c}^2$ at $k = 1$ for various $N = 2,~\cdots,~50$. The solid line represents a guideline $g_\mathrm{c}^2(N)|_{k=1} = \frac{2}{3}\qty(1-\cos\qty(\frac{2\pi}{N}))$.
• Figure 4: The critical coupling $g_\mathrm{c}^2$ at $k = 1,2,3$ for various $N = 2,~\cdots,~20$. The dotted and dashed lines represent the fitted results, while the solid line is a guideline $g_\mathrm{c}^2(N)|_{k=1} = \frac{2}{3}\qty(1-\cos\qty(\frac{2\pi}{N}))$.
• Figure 5: Minimum of the inverse critical coupling for $N = 2,~\cdots,~10$. The dashed line represents the fitted result of a linear function.