Phase diagram of two-color lattice QCD in the chiral limit

We study thermodynamics of strongly coupled lattice QCD with two colors of massless staggered fermions as a function of the baryon chemical potential $\mu$ in $3+1$ dimensions using a new cluster algorithm. We find evidence that the model undergoes a weak first order phase transition at $\mu =0$ which becomes second order at a finite $\mu$. Symmetry considerations suggest that the universality class of these phase transitions should be governed by a $O(N)\times O(2)$ field theory with collinear order, with $N=3$ at $\mu =0$ and $N=2$ at $\mu \ne 0$. The universality class of the second order phase transition at $\mu \ne 0$ appears to be governed by the decoupled $XY$ fixed point present in the $O(2)\times O(2)$ field theory. Finally we show that the quantum ($T=0$) phase transition as a function of $\mu$ is a second order mean field transition.

Figure 1

An example of a dimer-baryonloop configuration.

Figure 2

The inset shows the data for $Y_C$ and $Y_B$ as a function of $L$. The lines shown are fits to Eq. (28) which yields $F_{\pi }^2=0.0965(5)$ and $F_B^2=0.0839(6)$. The main figure shows the plot of ${\chi }_C$ versus $L$ for $T=2.918$. The solid line is a fit to Eq. (29) with $F_{\pi }^2$ and $F_B^2$ fixed to the values quoted above. The fit yields $\Sigma =0.3364(7)$ and $a=2.6(1)$ with a ${\chi }^2/\mathrm{DOF}=0.5$.

Figure 3

This figure shows the plot of ${\chi }_C$ versus $L$ for different values of $T$ across the phase transition. The solid lines for $T\le 2.9275$ are fits to Eq. (29) while those for $T=2.9285$ and $T=2.9292$ are fits to Eq. (31). We find $\tilde{a}=1650(200)$, 1150(50), $\tilde{b}=0.030(5)$, 0.027(4), $\tilde{c}=2.5(5)$, 2.0(4) and $\Delta F=2\times {10}^{-6}$, $4\times {10}^{-6}$ for the two temperatures. All the fits have ${\chi }^2/\mathrm{DOF}$ less than 1.

Figure 4

This figure shows the plot of $M_B$, $F_{\pi }^2$ and $F_B^2$ as a function of $T$. The solid lines are fits to the data as discussed in the text.

Figure 5

This figure shows ${\chi }_B$ as a function of $L$ at $\mu =0.3$ at three different temperatures close to the critical temperature. The inset shows that $Y_B$ goes to a constant as a function of $L$ at $T=2.85$. The solid line is a plot of Eq. (33) with $\Delta =0.117$ and $a=2.4(3)$. Clearly, ${\chi }_B$ grows as $L^3$ at $T=2.85$ but begins to saturate at $T=2.87$ indicating that there is a transition between these two temperatures.

Figure 6

(Left) Plot of $Y_{C}L$ versus $T$ close to $T_c$ for various values of $L$. The solid lines are plots of $Y_{C}L=1.128-0.161(T-2.85946)L^{1/0.60}$. The inset shows ${\chi }_B$ as a function of $L$ at $T=2.8594$ and the solid line is a plot of $0.306L^{1.958}$. (Right) Plot of $M_B$, $F_{\pi }^2$, ${\Delta }^2$ versus $T$ close to $T_c$. The solid lines show the plots of Eq. (34) with $a_0=0.70$, $c_0=0.79$, $d_0=0.252$, $\nu =0.610$, $\beta =0.311$ and $T_c=2.85946$.

Figure 7

Same as Fig. 6 but now showing the solid lines show $O(2)$ scaling including corrections. (Left) The solid lines represent $Y_{C}L=2.321-1.26L^{-0.0218}-0.299(T-2.8590)L^{1/0.6715}$. The inset shows ${\chi }_B$ as a function of $L$ at $T=2.8590$ and the solid line is a plot of $0.302L^{1.962}$. (Right) The solid lines are plots to Eqs. (35) with $a_0=4.8(5)$, $a_0^{\prime }=-0.8(1)$, $c_0=4.6(4)$, $c_0^{\prime }=-0.83(8)$, $d_0=2.1(1)$, $d_0^{\prime }=-0.89(5)$, $\nu =0.6715$, $\beta =0.3485$, $\omega =0.0218$.

Figure 8

Results at $T=1.0$ and $\mu =0.01$. Note that ${\chi }_B$ grows with the volume while ${\chi }_C$ saturates. Both $Y_C$ and $Y_B$ grow linearly with $L$. The lines are drawn to guide the eye.

Figure 9

The figure shows our results for $\Delta$, ${\xi }^{-1}$ and $E_B$ as a function of $\mu$. For $\mu <{\mu }_c$ the dotted line is the mean field result for $\Delta$ given in Eq. (37) and the solid line contains the one-loop corrections. For $\mu >{\mu }_c$ the dashed line is the plot of $3.549\sqrt{\mu -{\mu }_c}$ and the solid line is the plot of $2(\mu -{\mu }_c)$. We have used ${\mu }_c=0.90922$ here. The inset shows the baryon density $n_B$ as a function of $\mu$.

Figure 10

The phase diagram of two-color lattice QCD with staggered fermions at strong couplings.

Figure 11

The baryon density as a function of $\mu$ at $T=2.859$, $L_t=4$, $L=12$. Note for these parameters ${\mu }_c=0.3$.