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 transition 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 $\mu \neq 0$. The universality class of the second order phase transition at $\mu\neq 0$ 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 also second order and can be understood as a mean field transition.