A Scaling Relation, $Z_m$-type Deconfinement Phases and Imaginary Chemical Potentials in Finite Temperature Large-$N$ Gauge Theories

TL;DR

This work demonstrates that Polyakov-loop effective potentials in finite-temperature, large-$N$ gauge theories obey a robust scaling relation with respect to temperature, profoundly constraining the allowed terms and the thermodynamics near confinement. By introducing imaginary chemical potentials or imaginary angular velocities, the authors reveal a landscape of stable $Z_m$-type deconfinement phases, in which Polyakov-loop eigenvalues distribute with $Z_m$ symmetry; these phases obey the same scaling behavior as the underlying effective action. The scaling framework is derived from Schwinger-Dyson equations and is shown to hold nonperturbatively in examples ranging from gauged matrix quantum mechanics to bosonic BFSS, 4D YM on $S^3$, and ${ m N}=4$ SYM on $S^3$, supported by Monte Carlo simulations. The work further discusses the gauge/gravity correspondence implications, suggesting gravity duals for $Z_m$ phases in various theories, and outlines potential connections to real-world QCD and finite-density phenomena via analytic continuation. Overall, the paper reveals a richer phase structure in large-$N$ gauge theories than previously recognized, with broad implications for holography and high-temperature QCD-like physics.

Abstract

We show that the effective potentials for the Polyakov loops in finite temperature SU$(N)$ gauge theories obey a certain scaling relation with respect to temperature in the large-$N$ limit. This scaling relation strongly constrains the possible terms in the Polyakov loop effective potentials. Moreover, by using the effective potentials in the presence of imaginary chemical potentials or imaginary angular velocities in several models, we find that phase transitions to $Z_m$-type deconfinement phases ($Z_m$ phase) occur, where the eigenvalues of the Polyakov loop are distributed $Z_m$ symmetrically. Physical quantities in the $Z_m$ phase obey the scaling properties of the effective potential. The models include Yang-Mills (YM) theories, the bosonic BFSS matrix model and ${\mathcal N}=4$ supersymmetric YM theory on $S^3$. Thus, the phase diagrams of large-$N$ gauge theories with imaginary chemical potentials are very rich and the stable $Z_m$ phase would be ubiquitous. Monte-Carlo calculations also support this. As a related topic, we discuss the phase diagrams of large-$N$ YM theories with real angular velocities in finite volume spaces.

Figures (15)

• Figure 1: Polyakov loops at inverse temperature $\beta$ (left panel) and $3\beta$ (right panel). They are open loop operators in the coordinate space $- \infty < \tau < \infty$. If $u_{n}(\beta)=0$ ($n \notin 3 \mathbf{Z}$), only the same Polyakov loops appear in both the inverse temperature $\beta$ and $3\beta$. Thus, the Polyakov loops $u_{3n}(\beta)$ and $u_{ n}(3\beta)$ should obey the same Schwinger-Dyson equations.
• Figure 2: Three typical eigenvalue distributions $\rho(\alpha)$ of $A_\tau$\ref{['rho']}. In usual gauge theories, the uniform distribution (the left panel) is favored at low temperatures, while the gapped distribution (the right panel) is favored at high temperatures. Depending on the models, the non-uniform distribution (the center panel) appear at middle temperatures near the Hagedorn temperature $T_H$.
• Figure 3: $u_n(T)$ and $F(T)$ in the free matrix quantum mechanics \ref{['action-Free-MQM-omega']} with an imaginary chemical potential $\Omega_I$. The three curves show the results at three different imaginary chemical potentials: $\beta \Omega_I =0$, $2\pi/3$ and $\pi$. There, transitions from the confinement phase to the $Z_1$ phase ($\beta \Omega_I =0$), the $Z_2$ phase ($\beta \Omega_I =\pi$) and the $Z_3$ phase ($\beta \Omega_I =2\pi/3$) occur at $T=T_H$, $2T_H$ and $3T_H$, respectively. $u_n$ satisfy the relation $u_1(T)|_{Z_1} =u_2(T/2)|_{Z_2}= u_3(T/3)|_{Z_3}$. Also, the free energies in the three phases are related by the relation $F(T)|_{Z_1} =F(T/2)|_{Z_2}= F(T/3)|_{Z_3}$. These are the consequence of the scaling relaiton \ref{['F-scaling']}. Note that these plots are obtained from Eqs. \ref{['un-Z1-free-MQM']} and \ref{['F-Z1-free-MQM']}, and they are not accurate away from the phase transition points because of the approximation \ref{['large-D']}. (To improve the approximation \ref{['large-D']}, $D=50$ is used in these plots. We have taken $M=\log D$ so that $T_H=1$ in Eq. \ref{['TH-free-MQM']}).
• Figure 4: Phase diagrams of the free matrix quantum mechanics \ref{['action-Free-MQM-omega']} ($M=1$, $D=2$ and $\tilde{D}=1$) with a real angular velocity $\Omega$ (the left panel) and with an imaginary angular velocity $\Omega_I$ (the right panel). For the real angular velocity, only the conventional confinement phase and the deconfinement phase ($Z_1$ phase) appear, and the transition occurs on the curve $a_1=0$\ref{['a1-free-MQM']}. Also, the system is destabilized when $|\Omega| \ge M=1$. On the other hand, for the imaginary angular velocity, the $Z_m$ phases ($m=1,2,3,4,5,6$ and 7) appear and the transitions to these phases from the confinement phase occur on the curves $a_m=0$\ref{['an-free-MQM']}. Also, we speculate that the transitions between the $Z_m$ phase and the $Z_n$ phase occur on the curves $a_m=a_n$.
• Figure 5: The eigenvalue density functions $\rho(\alpha)$ of the $Z_2$ phase (the left panel) and the $Z_3$ phase (the center panel) in the free matrix quantum mechanics \ref{['action-Free-MQM-omega']}. The right panel is the scatter plot of $\{ e^{ i \alpha_k } \}$ ($k=1,\cdots,N$) for the $Z_3$ phase, and three cuts exist symmetrically.