transformations similar to those leading to Eq. (3.21) Using exact transformations, we have mapped in the previous Sections the original fermion model, Eqs. (2.1- 2.4), onto the boson model, Eqs. (2.39-2.43), or Eqs. (3.10-3.12c). We have shown how one can treat the latter model diagrammatically making expansion in the inter- action terms and demonstrated in the first two orders in the interaction that the perturbation theory agrees with the conventional perturbation theory for the fermions. In this Section we want to compare the two equiva- lent models from the point of view of their suitability for Monte Carlo simulations. It is well known 22–26 that studying models for fermions one encounters the nega- tive sign problem and the main advantages of the MC get lost. At the same time, the sign in the MC sampling is always positive for bosonic models. Therefore, it is quite natural to investigate the suitability of the boson model, Eqs. (2.39-2.43) or (3.10-3.12c), to MC simulations and clarify the question about the sign problem. Before starting our investigation of the problem we would like to emphasize that we do not attempt dis- cussing here effectiveness of various MC algorithms or compare them with each other. Our goal is to under- stand semi-quantitatively where the problems come from, how one can overcome them and how one can do com- putations in principle. We are aware of the fact that the concrete schemes of the calculations we suggest at the end may be not necessarily the most efficient ones, but writing explicit algorithms is beyond the scope of the present paper. The MC method is used for computation of the average of a quantity A over configurations c in a space of these configurations It is clear that all singularities have disappeared. An- alyzing the expressions above we find that the first sum- mand (3.30a) equals the contribution of the conventional diagram Fig. 4 (c). Thus, we have so far succeeded to show how the boson model reproduces all the conven- tional diagrams without the Hartree bubbles. Finally, we collect all trivial second order contribu- tions arising from the shift of the chemical potential, Eq. (2.14), and the thermodynamic potential, Eq. (2.13). Note that in this procedure, we have to expand also the Fermi functions in the contribution of the first order di- agram, Eq. (3.20). Combining these trivial second order contributions with the contributions (3.30b) and (3.30c) yields the analytical expression for the collection of the Hartree-type diagrams Fig. 4 (d.1) and (d.2). The cal- culation is completely analogous to the one for the first order diagrams but is a little bit more tedious. Thus, the calculations performed in this Section helped us to see that the boson model yields up to the second or- der the same perturbation series as the conventional the- ory. The perturbative calculations in the boson model are somewhat more cumbersome than the standard pertur- bation theory. However, our motivation for constructing the boson model is to develop a new tool for studying both analytically and numerically the low temperature physics, rather than considering lowest order diagrams exactly. The calculations presented in this Section are merely a check of the bosonic approach. We note that the regularization based on the break- ing of translational symmetry is crucial only for a class of diagrams which usually do not play an important role in the framework of analytical investigation of low lying excitations. Therefore, studying, e.g., the non-analytical corrections to the specific heat of an interacting Fermi gas $^{15,21}$, the inherent ambiguity of Eq. (2.32) is rather harmless because the important contributions do not re- quire any additional regularization. According to this approach one chooses a set of M configurations { c$_{i}$ } , i = 1 , 2 , ...M with the probabil- ity Z − $^{1}$p ( c$_{i}$ ). Then, instead of calculating the entire sum in Eq. (4.1), one computes the sample average This approach is very efficient because the time of a nu- merical computation grows with the size of the sample in a power law, which contrasts exponential growth when using other methods. Of course, such an approximation is possible only pro- vided all p ( c ) are not negative. In some cases the proper p ( c ) can be both positive and negative and formally one may not use the approximation (4.2). In this case one 〈S 2 int 〉 (b . 1 − 3 , c . 2) = − N$_{d}$ 2 T ∫ (2 V$_{0}$ − V$_{q}$ )(2 V$_{0}$ − V$_{q}$ ′ ) × n$_{p}$ ( n$_{p}$ − 1)(2 n$_{p}$$_{+}$$_{q}$ − 1)(2 n$_{p}$$_{+}$$_{q}$ ′ − 1)( d p d q d q $^{′}$) = 2 N$_{d}$ ∫ ( d p d q d q $^{′}$) { V$_{q}$V$_{q}$ ′ ∂n$_{p}$ ∂µ n$_{p}$$_{+}$$_{q}$n$_{p}$$_{+}$$_{q}$ ′ (3.30a) + (4 V 2 0 − 2 V$_{0}$V$_{q}$ − 2 V$_{0}$V$_{q}$ ′ ) ∂n$_{p}$ ∂µ n$_{p}$$_{+}$$_{q}$n$_{p}$$_{+}$$_{q}$ ′ (3.30b) − 1 4 (2 V$_{0}$ − V$_{q}$ )(2 V$_{0}$ − V$_{q}$ ′ ) ∂n$_{p}$ ∂µ (2 n$_{p}$$_{+}$$_{q}$ + 2 n$_{p}$$_{+}$$_{q}$ ′ − 1) . (3.30c) 〈 A 〉 = Z − 1 ∑ c A ( c ) p ( c ) , Z = ∑ c p ( c ) (4.1) 〈 A 〉 ≈ 1 M M ∑ i =1 A ( c$_{i}$ ) (4.2) A. Quantum Monte Carlo IV. DISCRETE TIME REPRESENTATION AND MONTE CARLO SIMULATIONS 15