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We analyze the phase structure of $d=4$ $\mathcal{N}=2$ large $N$ SYM theorywith flavor on $S^1\times S^3$ by using geometric entropy as an orderparameter. We introduce chemical potential conjugate to global U(1) symmetryand find the third order phase transition at finite density by using thegeometric entropy as the order parameter. We also find that the geometricentropy at the finite density has an interesting behavior at low temperatureand for large $N_f$.

Geometric entropy and third order phase transition in d=4 N=2 SYM with flavor