content_image/391557.jpg content_image/391556.jpg
<img src='content_image/391557.jpg'>

plot in Fig. 7 (b) this quantity as a function of the in- verse total number of the sites N total d in the system. As previously, the HS is non-zero on two sites only and all the parameters are the same. 

Fig. 7 (b) shows that the deviation log ( Z$_{f}$/Z$_{b}$ ) de- cays with the size of the sample inversely proportional to the total number of the sites N total d . This dependence is natural because the regularization with the parameter γ cuts out N total d of ( N total d ) 2 states in the system. This ex- ample of the convergence completes the analytical proof of Appendix B, and shows that the bath is necessary in order to equate Z$_{f}$ and Z$_{b}$ . It shows as well that for any finite size system, Z$_{f}$ and Z$_{b}$ are different, Z$_{b}$ converging asymptotically towards Z$_{f}$ within an infinite bath. 

The numerical procedure carried out here can be ex- tended in a straightforward way to the case of the HS field ˜ φ$_{r}$ ( τ ) varying in time, which corresponds to the case of many slices N . According to our analytical con- sideration there should not be principal problems with this general case and the function Z$_{b}$ [ ˜ φ ] must remain real and positive. This is demonstrated in Fig. 8 

For the calculation of the bosonic function Z$_{b}$ [ ˜ φ ], the integral over u in Eq. (4.63) has been calculated analyt- 

<img src='content_image/391556.jpg'>

ically (see Eq. (4.65)). The resulting matrix entering log( Z$_{b}$ [ ˜ φ ] /Z$_{0}$ ) was evaluated using numerical diagonal- ization. This quantity remains real to a high precision [Imaginary parts due to numerical inaccuracy are of or- der 10 − $^{13}$] showing the positivity of Z$_{b}$ [ ˜ φ ], whereas the imaginary part of log Z$_{f}$ [ ˜ φ ] jumps frequently between 0 and π showing the sign fluctuations of Z$_{f}$ [ ˜ φ ]. It is inter- esting to note that log Z$_{f}$ is spread over a broader region of the real axis than log Z$_{b}$ [ ˜ φ ]. 

The form of the distribution of log Z$_{b}$ [ ˜ φ ], Fig. 8, shows that the sign problem is avoided in the bosonized rep- resentation. Moreover, the distribution is narrower than that of log Z$_{f}$ [ ˜ φ ] and we hope that the computational scheme may be efficient. 

We have mapped models of interacting fermions onto models describing collective bosonic excitations. This was performed by decoupling the interaction between the fermions by a Hubbard-Stratonovich transformation and considering non-interacting fermions in the fluctuating HS field. The standard step of tracing out the fermions, which led to the fermionic determinant, was followed by converting this determinant into an integral containing in the integrand the solution of a linear differential equation. The latter step resembles writing quasiclassical equations 

## V. DISCUSSION 

