  To investigate finite-volume effects on the chiral symmetry restoration and
the deconfinement transition and some impacts of possible global topological
background for a quantum chromodynamics (QCD) system with $N_f=2$ (two quark
flavors), we apply the Polyakov-loop extended Nambu-Jona-Lasinio model by
introducing a chiral chemical potential $\mu_5$ artificially. The final
numerical results indicate that the introduced chiral chemical potential does
not change the critical exponents but shifts the location of critical end point
(CEP) significantly; the ratios for the chiral chemical potentials and
temperatures at CEP, $\mu_c/\mu_{5c}$ and $T_c/T_{5c}$, are significantly
affected by the system size $R$. The behavior is that $T_c$ increases slowly
with $\mu_5$ when $R$ is large and $T_c$ decreases first and then increases
with $\mu_5$ 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 the
whole phase diagram becomes a crossover when $R<R_{\text{min}}$. Therefore, we
suggest that for the heavy-ion collision experiments, which is to study the
possible location of CEP, the finite-volume behavior should be taken into
account.
