Using this approach anomalous contributions to the spe- cific heat and magnetic spin susceptibility 16 have been calculated in all dimensions and new logarithmic contri- butions were found. In 1 D , the results obtained are in agreement with those known from renormalization group considerations 19 and exact solutions for spin chains $^{17,18}$. They have also been confirmed in 1 D by the conventional diagrammatic expansion 20 where the first non-vanishing logarithmic contribution was calculated. However, the same diagrammatic method did not exactly reproduce 21 the logarithmic contributions to δC/T d for d > 1, where δC is the anomalous correction to the specific heat. This discrepancy has been attributed to the observation that not all the effects of Fermi surface curvature have been taken into account in Ref. 15. 

One can conclude from this discussion that the prob- lem of the bosonization in d > 1 has not been completely solved yet but such a method is very desirable because it would be a new tool for analytical calculations. Re- markably, fermionic models are shown to be problematic for numerical computations as well. The powerful Monte Carlo (MC) method suffers from the famous fermionic sign problem $^{22–26}$. The MC method implies that the par- tition function can be represented as a sum of terms with positive probabilities and in this case, the computation time is proportional to a power of the size of the sys- tem. This makes the MC method very robust compared to other approaches. However, the fermion determinant is negative for some paths and, actually, its average sign becomes zero in the sampling process. As a result, the computation time grows exponentially with the size of the system or inverse temperature and the advantages of MC get lost. 

We are not able to list here all publications where the sign problem was discussed but (except special cases like electron systems with attraction, systems with repulsion but with half filling and some others) the solution has not been found. Moreover, according to Ref. 26, the sign problem should be NP-hard $^{27}$, which means that its reso- lution is almost impossible. We are not aware about any mathematically rigorous proof of the NP-hardness of the fermionic sign problem but it is anyway clear that it is the main problem for MC simulations. 

This problem does not arise in (non-frustrated) bosonic systems, though. Therefore, it is quite natural to think about a possibility of circumventing the sign problem by mapping the initial fermionic model onto a bosonic one. Of course, it is impossible to convert real fermions into real bosons but one can think of recasting the interacting part of the fermionic action in the language of collective bosonic excitations. This leads us again to the idea of the bosonization. 

In this paper, we suggest a new bosonization scheme that allows one to map in the fermionic model to a model describing bosonic excitations exactly in the sense that the mapping works in any dimension at any tempera- ture and for arbitrary interactions. It can be written in a form of a model of non-interacting bosons in an 

effective Hubbard-Stratonovich (HS) field. The trans- formation from the fermionic system to the bosonic one corresponds to using in quantum mechanics the density matrix instead of the wave functions. We argue that the representation of the model in terms of bosons in the ex- ternal field should be free of the negative sign problem and can be used for MC simulations. 

It should be emphasized, though, that for finite sys- tems used in the MC simulations the new bosonized model is not exactly equivalent to the initial fermionic one. In order to derive the boson model, we need a certain regularization and a bath attached to the sys- tem. This is necessary to avoid some uncertainties in the bosonized expressions. Only after the regularization is carried out we obtain a bosonic model free of the sign problem. Completely exact mapping of the fermionic model onto the bosonic one is impossible because, tech- nically, several electron Green functions with coincid- ing Matsubara frequencies and momenta cannot be con- verted into bosonic excitations. If we did only identi- cal transformations we would not be able to get rid of the sign problem. At the same time, one can see from the derivation that considering a sufficiently large system with the bath any desired precision can be achieved. 

Following an alternative route one can average over the HS field before making any approximation. This goal is achieved using the famous BRST (Becci-Rouet- Stora-Tyutin) transformation based on the introduction of superfields $^{28}$. This allows us to represent the partition function in such a form that the average over the HS field can be done immediately. After the averaging one comes to a model of bosons with quadratic, cubic and quartic interactions. We show how this model can be studied with the help of perturbation theory in the inter- action terms and compare the first orders with the corre- sponding terms of the conventional perturbation theory demonstrating a full agreement. 

The paper is organized as follows: 

In Section II, we formulate the model of interacting fermions, decouple the interaction with the help of the HS transformation and map the model onto the model of bosons in the external field. We introduce a regulariza- tion that allows us to avoid singularities in the equation of motion. 

In Section III, we introduce superfields, average over the HS field and derive the model of interacting bosons. After that we develop a diagrammatic technique for the superfield theory describing the bosonic excitations and demonstrate in the first orders in the interaction its agreement with the conventional perturbation theory. 

Section IV is devoted to the discussion of the bosoniza- tion from the point of view of MC simulations. We dis- cuss in detail how the negative sign contributions origi- nate in the fermionic formulation and why they do not exist in the bosonic regularized model. Properties of the partition functions are considered in the complex plane of the electron-electron interaction, which helps to un- derstand in details what happens in the procedure of the 

