Here we just provide a quick summary.

We defined the two particle object by:

\begin{eqnarray} \chi_{i_0,i_1,i_2,i_3}(\nu_1,\nu_2,\Omega) = \int_0^\beta e^{i\nu_1\tau_e^1+i(\nu_2-\Omega)\tau_e^2-\nu_2\tau_s^2-i(\nu_1-\Omega)\tau_s^1 } \langle T_\tau \psi_{i_0}^\dagger(\tau_e^1)\psi_{i_1}^\dagger(\tau_e^2)\psi_{i_2}(\tau_s^2)\psi_{i_3}(\tau_s^1)\rangle \end{eqnarray}Here index $i$ denotes the bath index, which we block diagonalize. (In general, we have index $i$ to denote the block, and internal index $b$ to denote the entry in the block, i.e., $i = (i,b)$.)

The two contributions computed by the CTQMC are:

\begin{eqnarray} \chi_{i_0,i_1,i_2,i_3}(\nu_1,\nu_2,\Omega) =\delta_{i_1 i_2}\delta_{i_0 i_3}\int_0^\beta e^{i\nu_1\tau_e^1+i(\nu_2-\Omega)\tau_e^2-\nu_2\tau_s^2-i(\nu_1-\Omega)\tau_s^1 } \langle T_\tau \psi_{i_0}^\dagger(\tau_e^1)\psi_{i_1}^\dagger(\tau_e^2)\psi_{i_1}(\tau_s^2)\psi_{i_0}(\tau_s^1)\rangle \\ +\delta_{i_0 i_2}\delta_{i_1 i_3}\int_0^\beta e^{i\nu_1\tau_e^1+i(\nu_2-\Omega)\tau_e^2-\nu_2\tau_s^2-i(\nu_1-\Omega)\tau_s^1 } \langle T_\tau \psi_{i_0}^\dagger(\tau_e^1)\psi_{i_1}^\dagger(\tau_e^2)\psi_{i_0}(\tau_s^2)\psi_{i_1}(\tau_s^1)\rangle \end{eqnarray}Here $i$ is not just a single bath, bath can mean a matrix of possibilities.

First, we can define the two frequency object $M_{i}(\nu_1,\nu_2)$, which is (when SVD sampling is turned off) computed directly in Matsubara frequency space

\begin{eqnarray} M_{i}(\nu_1,\nu_2) = \frac{1}{\beta}\sum_{\tau_s,\tau_e}e^{i\nu_1\tau_e}M_{i}(\tau_e,\tau_s)e^{-i\nu_2\tau_s} \end{eqnarray}This is the same object as needed for measuring the Green's function, except that Green's function corresponds to the part in which $\nu_1=\nu_2$. Hence, measuring this quantity does not cost much overhead. We here used index $i$ to denote the block of baths in which this $M$ is calculated. Note that the baths, which are not coupled, form independent matrices $M_i$.

Once we compute this $M$ object, we can construct the two parts of the two particle vertex by (derivation requires one to take the second derivative of the partition function with respect to the hybridization $\frac{\delta^2 logZ}{\delta\Delta_{i_0 i_3}\delta\Delta_{i_1 i_2}}$ -- see Hyowon's thesis above):

\begin{eqnarray} V_H[i_0,i_1][\nu_1,\nu_2;\Omega] = M_{i_0}(\nu_1,\nu_1-\Omega)\; M_{i_1}(\nu_2-\Omega,\nu_2)\\ V_F[i_0,i_1][\nu_1,\nu_2;\Omega] = M_{i_0}(\nu_1,\nu_2) \; M_{i_1}(\nu_2-\Omega,\nu_1-\Omega) \end{eqnarray}The two particle Green's function then can be constructed by:

\begin{eqnarray} \chi_{i_0,i_1,i_2,i_3}(\nu_1,\nu_2,\Omega) = V_H[i_0,i_1][\nu_1,\nu_2;\Omega]\delta_{i_0,i_3}\delta_{i_1,i_2} - V_F[i_0,i_1][\nu_1,\nu_2;\Omega]\delta_{i_1,i_3}\delta_{i_0,i_2} \end{eqnarray}This does not require much time, as $M(\nu_1,\nu_2)$ is already computed and stored.

These two objects ($V_H$ and $V_F$) are printed into *tvertex.dat file.*

Looking at the definition of $V_F$, we see that if sufficient number of Matsubara frequencies are available, $V_F$ can be computed from $V_H$ in the following way:

\begin{eqnarray} V_F[i_0,i_1][\nu_1,\nu_2;\Omega] = V_H[i_0,i_1][\nu_1,\nu_1-\Omega;\nu_1-\nu_2] \end{eqnarray}*svd_lmax>0*) is used (see paper by Shinaoka Hiroshi).