Partial Deconfinement

TL;DR

The paper proposes partial deconfinement as an intermediate phase in gauge theories where an SU(M) subsector is deconfined within SU(N), potentially connecting confined and fully deconfined regimes. It motivates this mechanism through holographic intuition (small black holes) and an ant-trail analogy, and supports it with quantitative tests across 4d N=4 SYM on S^3, other 4d theories, 4d pure YM on flat space, matrix quantum mechanics, and 2d maximal SYM. In each case, the Polyakov loop phase distribution exhibits features consistent with a mixed confined/deconfined state and a Gross-Witten-Wadia transition governing the partial-to-full deconfinement boundary. The results suggest a universal picture potentially relevant to real-world QCD and finite-density transitions, with implications for understanding holographic phase structure and related gravity duals.

Abstract

We argue that the confined and deconfined phases in gauge theories are connected by a partially deconfined phase (i.e. SU(M) in SU(N), where M<N, is deconfined), which can be stable or unstable depending on the details of the theory. When this phase is unstable, it is the gauge theory counterpart of the small black hole phase in the dual string theory. Partial deconfinement is closely related to the Gross-Witten-Wadia transition, and is likely to be relevant to the QCD phase transition. The mechanism of partial deconfinement is related to a generic property of a class of systems. As an instructive example, we demonstrate the similarity between the Yang-Mills theory/string theory and a mathematical model of the collective behavior of ants [Beekman et al., Proceedings of the National Academy of Sciences, 2001]. By identifying the D-brane, open string and black hole with the ant, pheromone and ant trail, the dynamics of two systems closely resemble with each other, and qualitatively the same phase structures are obtained.

Figures (7)

• Figure 1: [Left] The phase structure of 4D ${\cal N}=4$ SYM on S$^3$. The orange and red dashed lines denote the Hagedorn string phase and small black hole phase, respectively, which are well-defined in the microcanonical theory. In the canonical treatment, the small black hole phase is the unstable saddle which is responsible for strong hysteresis. The green line marks the transition temperature in the canonical ensemble, $T_c$. Above $T_c$ the large black hole phase is favored, while below $T_c$ the graviton gas phase is favored. [Center] A more common example of a first order transition, for example, between ice and liquid water. Small perturbations can destabilize the metastable states. At the critical temperature, there can be a mixture of two phases, in which ice and liquid water coexist. [Right] Superheated ice (red point) and supercooled liquid water (blue point) are not stable even in the microcanonical ensemble. There is an instability toward the mixture of the ice and liquid water.
• Figure 2: $x\equiv\frac{N_{\rm trail}}{N}$ versus $\tilde{p}=Np$ in the ant trail model \ref{['eq:ant-equation']}. $\alpha=\frac{1}{N}$, $\tilde{s}\equiv\frac{s}{N}=0.1$ (left), $1.0$ (center) and $5.0$ (right), $N=10^5$.
• Figure 3: Cartoon pictures of $T$ vs $N_{\rm BH}$ in gauge theory. The blue and red lines are confined and completely deconfined phases, where $N_{\rm BH}\sim O(N^0)$ and $N_{\rm BH}=N$, respectively. Orange lines indicate the partially deconfined phase.
• Figure 4: Cartoon pictures of the Polyakov loop $P$ as a function of temperature. The blue, orange and red lines represent the confined, partially deconfined and completely deconfined phases. Similar curves are obtained by using $E/N^2$ as the vertical axis as well.
• Figure 5: The distribution of Polyakov line phases. [Left] $\mu=3.0$, where the partially deconfinement phase is stable. We have plotted the distribution at $N=32$, the number of lattice points $L=18$, and temperature $T=1.15$, which is in the middle of the partially deconfined phase. The fit is $\rho(\theta)=\frac{1}{2\pi}\left( 1+\frac{2}{\kappa}\cos\theta \right)$ with $\kappa=2.87$. [Right] $\mu=5.0$, where the partially deconfinement phase is unstable and the transition is of first order. We have plotted the distribution at $N=128$, $L=16$, $T=1.54$, which is very close to $T_2$. The fit is $\rho(\theta)=\frac{2}{\pi\kappa}\cos\frac{\theta}{2} \sqrt{\frac{\kappa}{2}-\sin^2\frac{\theta}{2}}$ with $\kappa=1.88$.