content_image/426299.jpg content_image/426298.jpg content_image/426297.jpg content_image/426296.jpg
Eq. (4.47), the function Z$_{b}$ [ ˜ φ ] is given after discretization 

by 

<img src='content_image/426299.jpg'>

In Eq. (4.63) the symbol T r stands for the trace of the 2 × 2 matrices introduced in the regularization, Eq. (2.40). The expression entering the square brackets under T r is a 2 × 2 matrix in this “regularization space” and an N 2 d × N 2 d N × N matrix with the space { r, r $^{′}$} , { r$_{1}$, r ′ 1 } and time τ, τ$_{1}$ matrix elements. This matrix is obtained from the matrix G $^{0}$σ ˜ ΦΛ$_{1}$ using the conventional rules of matrix operations. The only peculiarity is that the coordinates enter in pairs { r, r $^{′}$} and { r$_{1}$, r ′ 1 } . Matrix elements can be written explicitly as 

<img src='content_image/426298.jpg'>

Eq. (4.63) can directly be used for computations. The integral over u cannot lead to singularities, which is guar- anteed by the absence of the zero eigenvalues of the op- erator in the L.H.S. of Eq. (4.62) written for an arbitrary u . This property is valid for an arbitrarily large ˜ Φ. 

In principle, one could easily perform the integration over u already analytically before doing numerics, which gives in the continuous time limit 

<img src='content_image/426297.jpg'>

The integrand in Eq. (4.65) is free of singularities as well. At small ˜ φ$_{r}$ ( τ ) one can expand the logarithm and take the first order of the expansion. This leads us again to the RPA formula, Eq. (3.14). However, Eq. (4.63) looks more convenient for MC computations because it allows one to peform updating more easily. 

Note that in the present paper, we decoupled the in- teraction by the Gaussian integration over ˜ φ$_{r}$ ( τ ). For the on-site interaction, one can also use the “Ising spin” auxiliary field s$_{r,l}$ $^{23}$. All the steps of the derivation of the functional Z$_{b}$ [ ˜ φ ] can be repeated with this field. The final expression, Eq. (4.44), should be replaced by the following sum over all configurations of s$_{r,l}$ 

where N$_{d}$ is the number of the sites in the system. The functional Z$_{b}$ [ ˜ φ ] for the Ising spin field has the same form 

<img src='content_image/426296.jpg'>

as previously with the field ˜ φ$_{r}$ ( τ ) = ¯ λs$_{r,l}$/ Δ , cosh ¯ λ = exp ( V$_{0}$ Δ / 2), for ( l − 1) Δ ≤ τ < l Δ. Using the “Ising spin” auxiliary field can further improve the efficiency of the computational schemes. 

As we have discussed previously the final formulas, Eqs. (4.62, 4.47) or Eq. (4.63), are different from the for- mula of the original fermionic model, Eqs. (4.13-4.16). These deviations are due to the introduced bath and states effectively discarded by introducing the parame- ter γ . So, generally speaking there is no reason to expect that the functions Z$_{f}$ [ ˜ φ ] and Z$_{b}$ [ ˜ φ ] are close to each other. As we have argued, they should be considerably different for, e.g., piece-wise functions with big jumps. The agree- ment between the partition functions Z is possible only after integration over ˜ φ$_{r}$ and only in the limit of a large number of the sites N total d in the system. 

However, what one can expect is the agreement be- tween the values of Z$_{f}$ [ ˜ φ ] and Z$_{b}$ [ ˜ φ ] for static ˜ φ (one slice in time) for large N$_{d}$ . Although we have demonstrated this agreement analytically in Appendix B, it is instruc- tive to check this property numerically using Eqs. (4.62, 4.47) or Eq. (4.63). Such a test can also give a feeling of how well the numerical procedure based on using these equations can work. 

We present in Fig. 7 an example of the convergence be- tween Z$_{f}$ and Z$_{b}$ in a bath for the static case of one time slice. We consider a two sites Hubbard model embed- ded in a bath of N total d − 2 sites with periodic boundary conditions. The HS fields φ$_{r}$ are not equal to zero on the sites r = { 1 , 2 } but are equal to zero elsewhere, such that only the sites 1 and 2 are interacting. Both analyt- ical and numerical integration over u in Eq. (4.63) are suitable and we use both of them. Integrating over u in Eq. (4.63) analytically makes the computation faster but we have found a good agreement between the results obtained using both the approaches. 

In Fig. 7 (a) the convergence between log ( Z$_{f}$ /Z$_{0}$ ) (up- per curve) and log ( Z$_{b}$/Z$_{0}$ ) (lower curve) is demonstrated in the static case, as the number of the sites in the bath is increased. Although the difference between these quanti- ties is fairly small, the complete convergence is not very fast. Therefore, in order to understand the dependence of the deviation log ( Z$_{f}$/Z$_{b}$ ) on the size of the system we 

## 3. A numerical test 

