Z s f and Z s b in the complex plane of s from real negative s < 0 to s = 1. In order to analytically continue Z s $_{f}$, one can use Eqs. (4.33, 4.34) giving a finite sum of exponen- tials of s and the analytical continuation is performed just putting s = 1 everywhere. Then, one obtains im- mediately all standard formulas for the fermionic system with repulsion and Z$_{f}$ [ ˜ φ ] derived from these formulas can be both positive and negative depending on ˜ φ$_{r}$ ( τ ), which leads to the sign problem. Having constructed the function Z s $_{b}$[ ˜ φ ] for real negative s, Eqs. (4.36-4.38), one should integrate over ˜ φ$_{r}$ ( τ ) using Eq. (4.7) and obtain the partition function Z s $_{b}$. After this function has been found for all real s < 0 one can find uniquely an analytical continuation ˜ Z s b for complex s coinciding with Z s b for real s < 0. This means that one can represent ˜ Z s b in a form of a convergent series in powers of the complex s in a certain region of complex s including the negative real half-axis. Below are some arguments for the analyticity of Z $_{b}$. As we have seen, the difference between the terms of this expansion and those for the fermionic partition function Z s f arises from a different treatment of two or more elec- tronic Green functions at coinciding Matsubara frequen- cies and momenta. This difference should be small in the limit of a large bath and small parameter γ and therefore the Taylor expansion of the function Z s b should converge for all complex s in the same way as the expansion for Z s f does. In Section III we have checked explicitly the agreement between the partition functions Z$_{f}$ and Z$_{b}$ in the first two orders of the expansion in the interaction. The calculation can be repeated without changes for arbi- trary complex s including real s < 0. Thus, the function Z s b is analytic for all complex s except the infinite point and the physical partition function Z$_{b}$ can be obtained putting s = 1 in this series. This result allows us to ex- pect that the function Z$_{b}$ is a good approximation for Z$_{f}$ . other for real negative s and would remain always close for other complex s including s = 1 if they were an- alytical in the complex plane of s for all ˜ φ$_{r}$ ( τ ), which would imply a convergent series in s . If this is not so one can expect that the functional Z s $_{b}$[ ˜ φ ] does not well ap- proximate the functional Z s $_{f}$[ ˜ φ ] for certain ˜ φ ( τ ) in some regions of complex s including s = 1. In this case, the functional Z$_{b}$ [ ˜ φ ] used for our explicit calculations can be considerably different from the analytical continuation of Z s $_{b}$[ ˜ φ ] from real s < 0 to s = 1 and, hence, from Z s $_{f}$[ ˜ φ ]. In particular, the functionals Z$_{b}$ [ ˜ φ ] and Z s $_{f}$[ ˜ φ ] may have different signs for certain functions ˜ φ ( τ ) when Z$_{f}$ [ ˜ φ ] be- comes negative. It is not easy to consider the analyticity properties of the functionals Z s $_{f}$[ ˜ φ ] and Z s $_{b}$[ ˜ φ ] analytically contin- ued from Eq. (4.35) and Eqs. (4.36-4.38), respectively. It seems that Z s $_{f}$[ ˜ φ ] is still analytical everywhere in the complex plane of s because the fermionic determinant is a finite product of exponentials and therefore can be expanded in series in integer powers of s . However, the same cannot be said about Z s $_{b}$[ ˜ φ ] , Eqs. (4.36-4.38), and we think that this quantity cannot be analytical every- where in the complex plane of s for any ˜ φ$_{r}$ ( τ ). A simple way to see this is to try to repeat the regular- ization of Appendix A for complex s . The only difference with respect to s = 1 in the derivation is that the eigen- value in Eqs. (2.47) written for complex M$_{r,r}$ ′ is not just λ K $^{∗}$, where λ K is is taken from Eq. (2.46), but generally another value ˜ λ K $^{∗}$. It coincides with λ K ∗ only for real M$_{r,r}$ ′ corresponding to positive real s . As a result, one should replace in Eqs. (A3, A5) the value λ ∗ by ˜ λ $^{∗}$. Then, one cannot exclude that for some functions ˜ φ$_{r}$ ( τ ) and some complex s the eigenvalue E turns to zero. The subsequent integration over u in Eq. (4.36) would lead to cuts in the complex plane of s. ˜ In this situation, the functional Z$_{b}$ [ φ ] obtained by just putting s = 1 in Eqs. (4.36-4.38) can differ from the analytical continuation from real s < 0 to s = 1. Then, the functionals Z$_{f}$ [ ˜ φ ] and Z$_{b}$ [ ˜ φ ] can essentially be differ- ent from each other for certain functions ˜ φ$_{r}$ ( τ ) and can even have different signs. At the same time, the physical partition functions Z$_{f}$ and Z$_{b}$ can still be close to each other because the functions Z s f and Z s b are analytical ev- erywhere in the complex plane. This discussion may help the reader to understand why replacing the not necessarily positive fermionic determi- nant Z$_{f}$ [ ˜ φ ] by the always positive functional Z$_{b}$ [ ˜ φ ] we still expect the correct result for the physical partition function Z obtained after integration over all functions ˜ φ$_{r}$ ( τ ). The arguments of this subsection are strongly based on the assumption that the system is finite, which allowed us to speak about the analyticity of the partition func- tion Z s in the interaction constant sV . This analyticity can be lost at low temperatures and real s < 0 in the limit of an infinite sample when the system becomes a superconductor but this does not seem to create prob- The analyticity in the the complex plane is evident for the partition function Z $^{s}$, Eq. (4.33), and our arguments justifying the approximation done in the bosonization procedure have been based on this property. However, the analyticity of Z s f and Z s b does not automatically im- ply the analyticity of the functions Z s $_{f}$[ ˜ φ ] and Z s $_{b}$[ ˜ φ ] for any given ˜ φ$_{r}$ ( τ ) and one should check this property us- ing explicit formulas. These functionals are close to each Having put s = 1 everywhere in the series for Z s b we can represent it again in a form of an expansion of the integral over ˜ φ with Z$_{b}$ [ ˜ φ ], Eq. (2.43), thus coming back to the bosonization formulas obtained in the previous sections. This means, that replacing the original fermionic model with the repulsion by the bosonic one, we expect a precise agreement between the results obtained by integration over ˜ φ of the functional Z$_{f}$ [ ˜ φ ] and Z$_{b}$ [ ˜ φ ] in the limit γ → 0 and N$_{d}$ → ∞ . Although both the functionals can equally be used for analytical calculations, the use of Z$_{b}$ [ ˜ φ ] may be preferable for the MC method because it is always positive in contrast to Z$_{f}$ [ ˜ φ ]. 21