 ],
  \end{equation}
which satisfies the usual Wetterich equation without extra terms because the background field
$\phi^0$ is scale-independent. 
For finite $\Lambda$, the vertex expansion of  $\tilde{\Gamma}_{\Lambda} [ \varphi ]$
is then of the form
 \begin{eqnarray}
 \tilde{\Gamma}_{\Lambda} [ \varphi ] & = & \tilde{\Gamma}^{(0)}_{\Lambda} +
 \tilde{\Gamma}^{(1)}_{\Lambda} \varphi_{Q=0} 
 + \frac{1}{2} \int_Q \tilde{\Gamma}^{(2)}_{\Lambda} ( Q ) \varphi_{-Q} \varphi_Q
 \nonumber
 \\
 & + &
 \sum_{ n =3}^{\infty} \frac{1}{n!} \int_{Q_1} \cdots \int_{Q_n}
 \delta ( Q_1 + \ldots + Q_n ) 
 \nonumber
 \\
 & & \hspace{10mm} \times \tilde{\Gamma}_\Lambda^{(n)} ( Q_1, \ldots, Q_n )
 \varphi_{Q_1} \cdots \varphi_{Q_n}.
 \hspace{7mm}
 \label{eq:tildeGammaexpansion}
 \end{eqnarray}
By construction, at the initial scale 
 \begin{equation}
\tilde{\Gamma}_{\Lambda_0} [ \varphi ] =  S_{\rm eff} [ \phi^0 + \varphi],
 \end{equation}
where the
proper initial value should be determined by the
boundary condition
 \begin{equation}
 \lim_{\Lambda \rightarrow 0} \tilde{\Gamma}^{(1)}_{\Lambda} =0.
 \label{eq:boundarycondition}
 \end{equation}
The initial vertex 
$\tilde{\Gamma}^{(1)}_{\Lambda_0}$ is then given by
 \begin{equation}
 \tilde{\Gamma}^{(1)}_{\Lambda_0} =    \omega_0^2 \phi^0 + \gamma_0 \int_K \tilde{G}_0 ( K ).
 \label{eq:onepoint1}
 \end{equation}
Here 
 \begin{equation}
\tilde{G}_0 ( K ) =
 \frac{1}{i \omega - \epsilon_{\bm{k}} + \mu -   \gamma_0 \phi^0 }
 \end{equation}
is the fermion propagator with shifted chemical potential, where the shift
$- \gamma_0 \phi^0=   \gamma_0^2 \rho / \omega_0^2$
 takes all 
tadpole contributions to the electronic self-energy into account.
The initial conditions for all  other shifted vertices 
$\tilde{\Gamma}^{(n)}_{\Lambda_0} ( Q_1, \ldots, Q_n )$ with $ n \geq 2$ can be obtained from the
corresponding unshifted expressions 
$\Gamma^{(n) }_{\Lambda_0} ( Q_1, \ldots, Q_n )$ by replacing $G_0 ( K ) \rightarrow 
\tilde{G}_0 ( K )$ in all fermion loops. In particular,
the two-point vertex is initially given by
 \begin{equation}
 \tilde{\Gamma}_{\Lambda_0}^{(2)} ( Q ) = \bar{\omega}^2 + \omega_0^2 +\gamma_0^2 \int_K \tilde{G}_0 ( K ) 
 \tilde{G}_0 ( K + Q ).
 \end{equation}

To obtain the exact flow equation for the free energy (in units of temperature)
we set $\phi = \phi^0$ in the Wetterich equation (\ref {eq:Wetterich1}) 
and obtain
 \begin{equation}
 \partial_{\Lambda} \tilde{\Gamma}^{(0)}_{\Lambda} = \frac{1}{2} \int_Q \tilde{D}_{\Lambda} ( Q )
 \partial_{\Lambda} R_{\Lambda} ( Q ),
 \label{eq:zeropointflow}
 \end{equation}
where
 \begin{equation}
 \tilde{D}_{\Lambda} ( Q ) = \frac{1}{ \tilde{\Gamma}^{(2)}_\Lambda ( Q ) + R_{\Lambda} ( Q ) }
 \label{eq:phonreg}
 \end{equation}
is the regularized phonon propagator.
Next, we substitute the expansion (\ref{eq:tildeGammaexpansion}) 
into Eq.~(\ref{eq:Wetterich1}) and compare the coefficients on both sides to 
obtain
 \begin{equation}
  \partial_{\Lambda} \tilde{\Gamma}^{(1)}_{\Lambda}  =
   \frac{1}{2} \int_Q \dot{\tilde{D}}_{\Lambda} ( Q ) \tilde{\Gamma}^{(3)}_{\Lambda} ( - Q, Q, 0),
 \label{eq:onepointflow}
 \end{equation}
where the single-scale propagator
is defined by
 \begin{equation}
 \dot{ \tilde{D}}_{\Lambda} ( Q ) = - \tilde{D}_{\Lambda}^2 ( Q ) \partial_{\Lambda} R_{\Lambda} ( Q).
 \end{equation}
Similarly, we find that  the two-point vertex satisfies the exact flow equation
 \begin{eqnarray}
 & & \partial_{\Lambda} \tilde{\Gamma}^{(2)}_{\Lambda} ( Q )  =  
 \frac{1}{2} \int_{ Q^{\prime} } \dot{\tilde{D}}_{\Lambda} ( Q^{\prime} ) 
 \tilde{\Gamma}^{(4)}_{\Lambda} ( - Q^{\prime}, Q^{\prime}, -Q, Q )
 \nonumber
 \\
 &  & - \frac{1}{2} \int_{ Q^{\prime}} [  {\tilde{D}}_{\Lambda} ( Q^{\prime} ) 
 \tilde{D}_{\Lambda} ( Q^{\prime} + Q )]^{\bullet}
  \tilde{\Gamma}^{(3)}_{\Lambda} ( - Q,   Q+ Q^{\prime},-  Q^{\prime} )
 \nonumber
 \\
 & & \hspace{7mm} \times 
\tilde{\Gamma}^{(3)}_{\Lambda} ( Q^{\prime},  -  Q^{\prime}  - Q, Q ),
 \label{eq:twopointflow}\\
\nonumber
 \end{eqnarray}
where we have introduced the product notation for single-scale propagators,
 \begin{eqnarray}
 & & [  {\tilde{D}}_{\Lambda} ( Q^{\prime} ) \tilde{D}_{\Lambda}
 (  Q^{\prime} + Q )]^{\bullet}
 \nonumber
 \\
 & = &   \dot{\tilde{D}}_{\Lambda} ( Q^{\prime} ) \tilde{D}_{\Lambda}
 ( Q^{\prime} + Q ) +  {\tilde{D}}_{\Lambda} ( Q^{\prime} ) \dot{\tilde{D}}_{\Lambda}
 (  Q^{\prime} + Q ). \hspace{7mm}
 \label{eq:productrule}
 \end{eqnarray}
Finally, we also need the flow of the three-point vertex which is given by
 \begin{widetext}
 \begin{eqnarray}
& &  \partial_{\Lambda} \tilde{\Gamma}^{(3)}_{\Lambda} ( Q_1, Q_2, Q_3 )
  =  
  \frac{1}{2} \int_{ Q} \dot{\tilde{D}}_{\Lambda} ( Q ) 
 \tilde{\Gamma}^{(5)}_{\Lambda} ( - Q, Q, Q_1, Q_2, Q_3 )
 \nonumber
 \\
 &  & -  \frac{1}{2} \int_Q     \left\{   [  {\tilde{D}}_{\Lambda} ( Q ) 
 \tilde{D}_{\Lambda} ( Q + Q_1 )]^{\bullet}  
 \tilde{\Gamma}^{(3)}_{\Lambda} ( Q_1, Q , - Q_1 - Q )
  \tilde{\Gamma}^{(4)}_{\Lambda} ( -  Q , Q_1 + Q, Q_2, Q_3)
+ ( Q_1 \leftrightarrow Q_2) + ( Q_1 \leftrightarrow Q_3) 
 \right\}
 \nonumber
 \\
 & & + \int_Q   [  {\tilde{D}}_{\Lambda} ( Q ) 
 \tilde{D}_{\Lambda} ( Q + Q_1 )  \tilde{D}_{\Lambda} ( Q - Q_2 )  ]^{\bullet}  
 \tilde{\Gamma}^{(3)}_{\Lambda} ( Q_1, Q, - Q_1 - Q )
 \tilde{\Gamma}^{(3)}_{\Lambda} ( Q_2, - Q, - Q_1 + Q )
\tilde{\Gamma}^{(3)}_{\Lambda} ( Q_3,  Q_1 + Q, -Q + Q_2 ).
 \nonumber
 \\
 & &
 \label{eq:threepointflow}
\end{eqnarray}
 \end{widetext}
Diagrammatic representations of the exact FRG flow equations 
(\ref{eq:zeropointflow}), (\ref{eq:onepointflow}), (\ref{