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bosonization. We derive formulas directly suitable for numerical simulations and give an example of numerical convergence in a case of static fields. 

We discuss the obtained results in Section V. 

The main results of this paper have already been pre- sented in a shorter publication $^{29}$. 

## II. FERMIONIC MODEL AND BOSONIZATION 

## A. Reduction to fermions in the auxiliary field 

Our method of the bosonization is quite general and is applicable to a broad class of models. Therefore we start our considerations with a general model of interacting fermions on a lattice of an arbitrary dimension. We write the Hamiltonian ˆ H of the system as follows 

where ˆ H$_{0}$ is the bare part describing non-interacting elec- trons, 

with a chemical potential µ , a hopping amplitude t$_{r,r}$ ′ between the sites r and r ′ and the spin σ = +1 , − 1. The term ˆ H$_{int}$ describes the electron-electron interac- 

tion, 

The partition function Z can obtained from ˆ H the Hamiltonian ˆ H using the standard relation 

where T is temperature. 

In order to proceed with the mapping of the fermion model onto a boson on, we will decouple the interaction term ˆ H$_{int}$ by integration over an auxiliary HS field. In principle, the decoupling can be performed in different ways and we should choose one of them. The choice can be different depending on the problem studied. In this work we are mostly interested in systems with an on-site repulsion. We also keep in mind a possibility of using the method for MC computations. The latter desire enforces us to use real HS fields φ$_{r}$ depending only on one coordinate r . 

At the same time, it would be advantageous to keep an arbitrary sign of the off-diagonal interaction V$_{r,r}$ ′ . The on-site attraction is less interesting for us because MC schemes can be free of sign in this case 25 anyway. The bosonization procedure can be carried out also in this case with minimal changes but we do not do this here. 

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In order to carry out the HS transformation with the real fields φ$_{r}$ we split the matrix V$_{r,r}$ ′ into its diagonal V$_{r,r}$ and off-diagonal ˜ V$_{r,r}$ ′ parts, 

Assuming that ˜ V$_{r,r}$ ′ is bounded from above, we can add a sufficiently large constant ¯ V to − ˜ V$_{r,r}$ ′ , ensuring the matrix 

to be positive definite. (Its Fourier transform V (1) q should be positive for all momenta q .) Introducing the coupling constant 

assuming it is site independent, and using the fermionic commutation relations we rewrite the interaction term ˆ H$_{int}$ as 

while the chemical potential µ should be replaced by ˜ µ = µ V$_{0}$ + ¯ V / 2. 

− ( ) Due a somewhat arbitrary choice of the constant ¯ V the coupling constants V$_{0}$ and V (1) are also not uniquely defined. However, this does not create any problems. 

As the interaction term ˆ H$_{int}$ does not commute with ˆ H$_{0}$ the HS transformation can be performed first sub- dividing the interval (0 , β ) into segments of the length Δ = β/N with N ≫ 1 and then integrating over the auxiliary field for each τ$_{l}$ = Δ ( l − 1 / 2) , l = 1 , 2 ...N . In this Section we are interested in analytical calcula- 

tions and therefore we write all formulas in the continu- ous limit (Δ → 0) with respect to the “imaginary time” τ . Transformations for the discrete time (finite Δ) will be performed in Section IV . 

We rewrite the partition function Z , Eq. (2.4), as 

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where T$_{τ}$ is the time ordering operator (the time grows from the right to the left) and ˆ H$_{int}$ ( τ ) is the operator in the interaction representation, 

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