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have opposite signs for certain functions ˜ φ , we present ar- guments that after integration over ˜ φ the partition func- tion Z$_{b}$ obtained in the bosonic model has to be a good approximation to the partition function Z$_{f}$ of the orig- inal fermion model provided both the models are taken with the bath and the regularizing parameter γ is small. The absence of the sign oscillations in Z$_{b}$ [ ˜ φ ] should lead to smaller variations of the modulus of this quantity and, hence, to a better convergence in the MC simulations. 

The logics of all these manipulations can be illustrated calculating two different integrals 

<img src='content_image/389366.jpg'>

Calculating the integral I$_{1}$ analytically, one can reduce it by turning the contours of the integration to the integral I$_{2}$ and show exactly that I$_{1}$ = I$_{2}$ for any a . 

At the same time, we can discretize the variable x , replace the integrals by sums and calculate these sums by the MC method. Then, one can see that difficulties in the calculation of these sums are quite different. The integrand of I$_{2}$ is positive and the sum converges very fast. In contrast, the integrand of I$_{1}$ oscillates faster and faster with growing x and one cannot use it as the weight in the MC sampling. Of course, one could replace the integrand by its modulus and use the latter as the weight but then one would calculate the average sign of cos ax 2 as in Eq. (4.3), which is not a very pleasant task. 

Our scheme of the replacement of the fermionic deter- minant Z$_{f}$ [ ˜ φ ] by the bosonic functional Z$_{b}$ [ ˜ φ ] resembles the analytical reduction of the integral I$_{1}$ to the integral I$_{2}$ . This transformation can easily be carried out in the continuous limit by using the powerful theory of complex variables but making the same for finite length of slices is difficult and cannot be done exactly. It is clear that the discrete versions of the integrals I$_{1}$ and I$_{2}$ are different and the only justification for a replacement of I$_{1}$ by I$_{2}$ is that they must be equal to each other in the continuous limit. 

A nice example relating the occurrence of the sign problem to the representation frame of a physical prob- lem is discussed on p. 103 of Ref. 24. In this example, the model of a single spin in an external magnetic field is discussed. Although the corresponding Hamiltonian H = − h · σ can be solved in a simple way, tackling the problem using the path integral approach can either lead to sums containing both positive and negative terms [choosing h = ( h$_{x}$, h$_{y}$, 0) to lie in the xy -place], or lead to purely positive matrix elements [after rotating h into the xz -plane, h = ( h$_{x}$, 0 , h$_{z}$ )]. While the result of the former choice resembles the fermionic sign problem, this sign problem is apparently “solved” by switching to the other reference frame. 

This illuminating example shows that in some cases the sign problem can be overcome by a simple analytical transformation. Our approach for solving the fermionic sign problem looks similar: we make a rotation in a “gen- eralized space” of bosons and fermions from the “fermion 

plane” to the “boson plane”. Actually, it is not surpris- ing that one does not encounter the sign problem in a bosonic model, but the fact that there can be an exact mapping between a fermion and boson model has not been anticipated previously. 

The sign problem is a long standing problem of quan- tum MC computations for fermionic systems. There have been many attempts to overcome this problem writing sophisticated algorithms but it has not gener- ally been solved until now. It has even been asserted 26 that the sign problem belonged to the class of NP-hard problems $^{27}$, which would imply that chances to solve it are very low. However, we do not think that the argu- ments presented in Ref. 26 can be considered as a rig- orous proof. Even if our bosonization scheme will really work for the fermion models, we do not know yet how to apply it for any NP problem. 

The functional Z$_{b}$ [ ˜ φ ] can be computed for any ˜ φ$_{r}$ ( τ ) using directly Eq. (4.63). One can perform integration over u either numerically or analytically. The numerical integration over u looks more convenient because one can easily carry out the MC updating. Then, one can use the standard MC scheme for integration over the field ˜ φ$_{r}$ ( τ ). One can also use the “Ising spin” auxiliary field 23 instead of the Gaussian field φ , which can make the computation faster. 

In order to make an independent check and get a feel- ing of how one could compute using Eq. (4.63) we have calculated the function Z$_{b}$ [ ˜ φ ] for a static field φ$_{r}$ integrat- ing over u both analytically and numerically and found a good agreement with the corresponding function Z$_{f}$ [ ˜ φ ] of the original fermionic model. We do not expect es- sential difficulties in extending the computation to time- dependent HS fields ˜ φ$_{r}$ ( τ ) and hope that our bosoniza- tion scheme will be checked numerically in the nearest future. 

We thank Transregio 12 of DFG, the French ANR for financial support and the Aspen Center for Physics where part of this work was completed. We acknowledge useful discussions with Y. Alhassid, A. Bulgac, D. Galanakis, M. Jarrell, S. Kettemann, M.Yu. Kharitonov, O. Parcol- let, M. Troyer, Ph. Werner and S-X. Yang. 

## Appendix A: Absence of zero eigenvalues in the regularized model 

Here, we demonstrate that the operator H r,r ′ ( τ ), Eq. (2.40), does not have zero eigenvalues E $^{K}$. This property is guaranteed by the presence of the regular- izer γ . If we put γ = 0 in Eq. (2.40), zero eigenvalues E exist. 

Putting γ = 0 we come to equations (2.46, 2.47). As the operators in the L.H.S. are not hermitian, the eigen- 

## Acknowledgements 

