may use instead, e.g., | p ( c ) | as the probability rewriting Eq. (4.1) as smear it for our dicussion. So, we change the model re- placing the weight W$_{0}$ ( φ ) in Eq. (2.12) by To simplify the discussion we neglect the interaction V (1) in Eq. (2.8) thus restricting ourselves with the on-site repulsion. According to the accepted regularization with the help of the bath we assume that the interaction is equal to V$_{0}$ inside the sphere with the radius R$_{0}$ but vanishes outside this region (c.f. Eq. (2.37)). We are interested in the limit α → 0 but let us take this limit at the end of the calculations. We do not see any reason why this limit should lead to results different from those for the original model. The presence of the derivative in Eq. (4.4) introduces correlations between φ$_{r}$ ( τ ) at different times. We are mostly interested in the low temperature limit when the sign problem is strongest. In this limit, the pair corre- lation function between the fields φ$_{r}$ ( τ ) inside the inter- acting region takes a simple form where s ( c ) = sign[ p ( c )] . One can perform the computations starting with Eq. (4.3) but the average sign s ( c ) becomes in many important cases very small, which leads to exponentially large computation time. The traditional scheme $^{22–25}$of the quantum MC simu- lations for the fermion models is based on the HS trans- formation and using Z [ φ ], Eq. (2.19), as the probability p ( c ) for the MC sampling. The problem arising in this approach is that for some important configurations of φ$_{rσ}$ ( τ ) the functional Z [ φ ] becomes negative. There are some special cases like models with a half-filled band or with attraction between the fermions when Z [ φ ] is neces- sarily positive. However, generically, the MC simulations are inefficient for the fermionic models. In principle, one could propose just to use the bosonized model, Eqs. (3.10-3.12c). However, it contains the superfields Ψ , and one cannot apply the MC method immediately. This diffiuculty can be overcome by decou- pling the interaction with the help of HS transformation. Of course, proceeding in this way we would come back to Eqs. (2.39- 2.43). Now we can argue that the solution A$_{r,r}$ ′ ( z ) of Eq. (2.39) is necessarily real and substituting it into Eq. (2.43) one must obtain positive values of Z [ φ ]. We have explained how one comes to Eqs. (2.39-2.43) us- ing the regularization based on introducing a bath and the small parameter γ . All the transformations leading to (2.39-2.43) were justified and therefore one may enjoy computing using the scheme free of the sign problem. We think that this way of reasoning is correct, but it is definitely somewhat artificial: one introduces super- fields, integrates over the HS fields but then returns to the equations (2.39-2.43). After making these transfor- mations the sign problem desappears but why this has happened is not clear. showing the time correlations on the scale α . In order to proceed with the MC computations, one should discretize the time and basic equations, which can be done in many different ways. The standard pro- cedure is to subdivide the interval (0 , β ) into slices of the length Δ = β/N ≪ 1 and consider discrete times τ$_{l}$ = Δ ( l − 1 / 2). Within this scheme one writes the par- tition function Z in the form where So, it is very instructive to try to understand in a more direct way how the sign problem is overcome, what are the approximations made, etc. Of course, one should discuss these things without using the superfields. This is what the next subsections are devoted to. and the subscript l of φ and c means that these functions are taken at the times τ$_{l}$ . The disadvantage of Eq. (4.6) is that one cannot bring the trace of the fermionic operators c, ¯ c into an useful form without making additional approximations. There- fore, we use a slightly different representation that allows us to calculate this trace exactly and derive the bosonized form. We start the discussion with a slight extension of the original fermionic model, Eqs. (2.1-2.4). The interaction between the electrons V$_{r,r}$ ′ , Eq. (2.3), does not depend on time. As a result, fluctuations of the HS field φ$_{rσ}$ ( τ ) are not correlated in time, Eq. (3.9). The function δ ( τ − τ $^{′}$) describing these correlations does not create difficulties in analytic calculations but it is more convenient to slightly B. Discrete time representation 〈 A 〉 = ∑ c A ( c ) s ( c ) | p ( c ) | ∑ c | p ( c ) (∑ c s ( c ) | p ( c ) | ∑ c | p ( c ) | ) − 1 , (4.3) W$_{α}$ [ φ ] = exp [ − 1 2 V$_{0}$ ∑ r ∫ β 0 [ φ 2 r ( τ )+ α 2 ( ∂φ$_{r}$ ( τ ) ∂τ ) $^{2}$] dτ ] (4.4) 〈 φ$_{r}$ ( τ ) φ$_{r}$ ′ ( τ $^{′}$) 〉 = V$_{0}$δ$_{r,r}$ ′ exp ( − | τ − τ $^{′}$| /α ) 2 α (4.5) Z = lim Δ → 0 ∫ Z $^{′}$[ φ ] W ′ α [ φ ] Dφ ∫ W ′ $_{α}$[ φ ] Dφ , (4.6) Z $^{′}$[ φ$_{r,l}$ ] = Tr$_{¯c}$$_{,}$$_{c}$ [ exp ( − ˆ H$_{0}$/T ) N ∏ l =1 exp ( Δ ∑ r,σ σφ$_{r,l}$ ¯ c$_{r,l}$c$_{r,l}$ )] , W ′ α [ φ ] = exp ( − Δ 2 V$_{0}$ ∑ r N ∑ l =1 ( φ 2 r,l + ( α/ Δ) 2 ( φ$_{r,l}$$_{+1}$$_{/}$$_{2}$ − φ$_{r,l}$$_{−}$$_{1}$$_{/}$$_{2}$ ) 2 )) , 16