Failure of Mean Field Theory at Large $N$

We study strongly coupled lattice QCD with $N$ colors of staggered fermions in $3+1$ dimensions. While mean field theory describes the low temperature behavior of this theory at large $N$, it fails in the scaling region close to the finite temperature second order chiral phase transition. The universal critical region close to the phase transition belongs to the 3D $XY$ universality class even when $N$ becomes large. This is in contrast to Gross-Neveu models where the critical region shrinks as $N$ (the number of flavors) increases and mean field theory is expected to describe the phase transition exactly in the limit of infinite $N$. Our work demonstrates that infrared fluctuations can be important close to second order phase transitions even when $N$ is strictly infinite.

Figure 1

Plot of $\Sigma$ vs $T$ for various values of $N$. The errors in the Monte Carlo data are less than the size of the symbols. The solid line is the mean field result given in Eq. (5). The dashed line represents the next-to-leading order (NLO) result for $N=48$, which includes the $1/N$ corrections. The lower inset shows $T_c$ as a function of $N$, with the solid line representing the fit $T_c=1.5525N+3.126-0.88/N$.

Figure 2

Plot of $\Sigma$ vs $T/T_c$ for various values of $N$. The solid lines represent the fit to $A(1-T/T_c{)}^{0.3485}$ which is expected from 3D $XY$ model. The values of $A$ for different $N$ are given in the text.

Figure 3

Plot of $F^2$ vs $T/T_c$ for $N=48$, 24, and 12. The data for the various $N$ may not be clearly seen since they are superimposed on top of each other. The lower inset shows the same graph for a larger range of $T/T_c$ for $N=48$.