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Polyakov loop model with exact static quark determinant in the 't Hooft-Veneziano limit: SU(N) case

Sergii Voloshyn

Abstract

We construct an exact solution of the $d$-dimensional $SU(N)$ Polyakov loop model with the exact static quark determinant at finite temperature and non-zero baryon chemical potential in the 't~Hooft--Veneziano limit. In the joint large-$N$, large-$N_f$ limit with fixed ratio $κ= N_f/N$, the mean-field approximation becomes exact, and the core of the Polyakov loop model reduces to a deformed unitary matrix model, which we solve analytically. In particular, we compute the free energy and its derivatives, the expectation values of the Polyakov loop, and the baryon density, and we describe the phase diagram of the model in detail. We show how the $SU(N)$ case differs from the corresponding $U(N)$ model and how the three-phase structure known from one-dimensional QCD at finite density extends to non-zero coupling.

Polyakov loop model with exact static quark determinant in the 't Hooft-Veneziano limit: SU(N) case

Abstract

We construct an exact solution of the -dimensional Polyakov loop model with the exact static quark determinant at finite temperature and non-zero baryon chemical potential in the 't~Hooft--Veneziano limit. In the joint large-, large- limit with fixed ratio , the mean-field approximation becomes exact, and the core of the Polyakov loop model reduces to a deformed unitary matrix model, which we solve analytically. In particular, we compute the free energy and its derivatives, the expectation values of the Polyakov loop, and the baryon density, and we describe the phase diagram of the model in detail. We show how the case differs from the corresponding model and how the three-phase structure known from one-dimensional QCD at finite density extends to non-zero coupling.

Paper Structure

This paper contains 20 sections, 118 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Polyakov loop model and its solution
  3. Particular cases of $SU(N)$ PL model.
  4. Chiral broken phase
  5. Chiral symmetric phase. Phase of "saturation"
  6. Middle phase: Partial restoration of chiral symmetry
  7. Large $\kappa$ expansion
  8. Massless limit: model at $m=0$
  9. Heavy–dense limit ($h_- = 0$)
  10. $b = 0$: the case of $1d$ QCD ($g_+ = g_- = 0$)
  11. $\kappa = 0$ and $\kappa = \infty$ limits
  12. Phase diagram of $SU(N)$ PL model. General case
  13. Top phase transition
  14. Bottom phase transition
  15. Expansion around $u=0$ for $h > h_{\rm cr} (b, \xi)$
  16. ...and 5 more sections

Figures (2)

  • Figure 1: Top: Phase diagram of $SU(N)$ PL model in $\mu$-$b$ coordinates at fixed $m=2$. Left $\kappa$ = 1/2, Center $\kappa$ = 1, Right $\kappa$ = 2, Yellow - exact line, Green - is an asymptotic approximation of exact formula. Bottom: Phase diagram of $SU(N)$ PL model with fixed $\kappa=1$ and different $b$ combined in one picture , $b$=0,1/5,3/5,2/3,3/4,4/5 (related colors are blue, green, red, orange, gray and black)
  • Figure 2: Phase diagram of the PL model with the exact fermion determinant in the $(m,\mu)$ plane at fixed $\kappa = 1$. Top left: $b = 0$; top right: $b = 3/5$; bottom left: $b = 2/3$; bottom right: $b = 4/5$.