The density matrix of a system at finite temperature is give by $$\langle\psi_1|\rho|\psi_2\rangle=\frac{1}{Z}\langle\psi_1|e^{-\beta H}|\psi_2\rangle, $$ where $Z$ is a normalization constant. We can then express this matrix element as the Euclidean path integral $$ \frac{1}{Z}\langle\psi_1|e^{-\beta H}|\psi_2\rangle= \int_{\psi(t_E=0)=\psi_1}^{\psi(t_E=\beta)=\psi_2}\mathcal D\psi~e^{i \int_0^\beta dt_E~\int d^3 x ~\mathcal L_E[\phi]} ,$$ where $t_E$ is the Euclidean time and $\mathcal L_E$ is the Euclidean Lagrangian.
Question: If I want to describe a system at finite temperature with chemical potential $\mu$ corresponding to some $U(1)$ conserved charge $Q$ so that the density matrix is now given by $$\langle\psi_1|\rho|\psi_2\rangle=\frac{1}{Z}\langle\psi_1|e^{-\beta (H-\mu Q)}|\psi_2\rangle, $$ is there a path-integral representation for this matrix element?
