  Using holographic idea, we study the entanglement of purification in a field
theory with a critical point in intermediate and low temperature regime. This
theory includes temperature $T$ as well as chemical potential $\mu$. In the
intermediate regime, due to chemical potential, we observe that new terms
proportional to temperature square appear in the final result of entanglement
of purification or equivalently, apart from $T^0$ and $T^4$ terms in the case
of $\mu=0$, it contains the terms proportional to $T^2$. Our results also
indicate that the entanglement of purification, i.e. the correlation between
subsystems, can decrease or increase depending on the value of $\frac{\mu}{T}$
when the other parameters are kept fixed. However, in the low temperature
limit, the correlation always decreases, comparing to the $\mu=0$ case,
independent of the value of $\frac{\mu}{T}$ when the other parameters do not
alter. The existence of a critical point in the theory changes the behavior of
entanglement of purification in such a way that the entanglement of
purification experiences a maximum or minimum. Moreover, near the critical
point, we analytically show that the critical exponent is equal to 0.5 in both
regimes and also the term proportional to $T^2$ changes sign and becomes
negative in the intermediate regime.
