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In a progress toward searching for the QCD critical point, we study thefinite density phase transition of $N_f = 4$ and 2 lattice QCD at finitetemperature with the canonical ensemble approach. We develop a winding numberexpansion method to accurately project out the particle number from the fermiondeterminant which greatly extends the applicable range of baryon number sectorsto make the study feasible. Our lattice simulation was carried out with theclover fermions and improved gauge action. For a given temperature, wecalculate the baryon chemical potential from the canonical approach to look forthe mixed phase as a signal for the first order phase transition. In the caseof $N_f=4$, we observe an "S-shape" structure in the chemical potential-densityplane due to the surface tension of the mixed phase in a finite volume which isa signal for the first order phase transition. We use the Maxwell constructionto determine the phase boundaries for three temperatures below $T_c$. Theintersecting point of the two extrapolated boundaries turns out to be at theexpected first order transition point at $T_c$ with $\mu = 0$. This serves as acheck for our method of identifying the critical point. We also studied the$N_f =2$ case, but do not see a signal of the mixed phase for temperature aslow as 0.83 $T_c$.

Finite density phase transition of QCD with $N_f=4$ and $N_f=2$ using canonical ensemble method