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To investigate finite-volume effects on the chiral symmetry restoration andthe deconfinement transition and some impacts of possible global topologicalbackground for a quantum chromodynamics (QCD) system with $N_f=2$ (two quarkflavors), we apply the Polyakov-loop extended Nambu-Jona-Lasinio model byintroducing a chiral chemical potential $\mu_5$ artificially. The finalnumerical results indicate that the introduced chiral chemical potential doesnot change the critical exponents but shifts the location of critical end point(CEP) significantly; the ratios for the chiral chemical potentials andtemperatures at CEP, $\mu_c/\mu_{5c}$ and $T_c/T_{5c}$, are significantlyaffected by the system size $R$. The behavior is that $T_c$ increases slowlywith $\mu_5$ when $R$ is large and $T_c$ decreases first and then increaseswith $\mu_5$ when $R$ is small. It is also found that for a fixed $\mu_5$,there is a $R_{\text{min}}$, where the critical end point vanishes, and thewhole phase diagram becomes a crossover when $R<R_{\text{min}}$. Therefore, wesuggest that for the heavy-ion collision experiments, which is to study thepossible location of CEP, the finite-volume behavior should be taken intoaccount.

Finite-volume effects on phase transition in the Polyakov-loop extended Nambu-Jona-Lasinio model with a chiral chemical potential