content_image/436828.jpg content_image/436829.jpg content_image/436830.jpg content_image/436831.jpg content_image/436832.jpg content_image/436826.jpg content_image/436827.jpg content_image/436834.jpg content_image/436833.jpg
a system with a small number sites N$_{d}$ (quantum dot), al- though properties of the systems with and without leads can be different. total 

The form of the interaction V r,r ′ , Eq. (2.37), formally violates the translational invariance and the matrix ele- ments 

<img src='content_image/436828.jpg'>

are finite for q$_{0}$ ̸ = 0. 

Of course, the matrix element with q$_{0}$ = 0 remains finite and, for such momenta, the two Green functions in Fig. 1 (b) still have the same momenta and frequen- cies. However, we can neglect them because they give a small contribution provided the number of the contribut- ing momenta q$_{0}$ is large. The latter is achieved for a large number of the sites in the bath. 

We emphasize, that neglecting the contribution of graphs containing several electron Green functions with coinciding momenta and frequencies is possible because their contribution is not singular. One can estimate the relative error of this approximation as being proportional to N$_{d}$/N total d , where N total d is the total number of sites in the entire system including the bath. 

Following this idea we could simply exclude in the bosonization approach all propagators with ω = 0 , q = 0 “by hand”. However, this is possible only when doing a perturbation theory and such an approach is not suffi- cient for non-perturbative and numerical studies. 

The bosonic modes with ω = 0 , q = 0 correspond to solutions of the homogeneous equation in Eq. (2.32). In order to discard these modes by a regular procedure we regularize Eq. (2.32) slightly changing it. This is not a trivial task because this slight modification should guar- antee the complete absence of the solutions of the homo- geneous equation corresponding to Eq. (2.32). 

We regularize the bosonized model replacing Eq. (2.32) by a system of two equations 

<img src='content_image/436829.jpg'>

where a = 1 , 2, H r,r ′ is a 2 × 2 matrix 

<img src='content_image/436830.jpg'>

and 

The functions A$_{a}$$_{;}$$_{r,r}$ ′ ( z ) satisfy the bosonic boundary conditions 

<img src='content_image/436831.jpg'>

<img src='content_image/436832.jpg'>

and 2 × 2 Pauli matrices are used 

<img src='content_image/436826.jpg'>

The real parameter γ should be put to zero, γ → 0, at the end of calculations. 

With this modification we write the function Z [ φ ] in the form 

<img src='content_image/436827.jpg'>

The exponent in Eq. (2.43) is an even function of γ . It is not difficult to see that putting γ = 0 in Eqs. (2.39- 2.43) one returns to Eqs. (2.31, 2.32). However, keeping in Eqs. (2.39-2.43) the parameter γ finite allows us to discard all the solutions of the homogeneous equation in Eq. (2.32). 

This can be understood rather easily because the op- erator H is hermitian and, therefore, its eigenvalues are real. Moreover, as will be shown in the next subsection, they appear in pairs: if an eigenvalue E exists, then the eigenvalue − E exists, too. If, when changing parameters of the operator H , an eigenvalue E with its counterpart − E turned to zero at some point, this would mean that after crossing this point they become imaginary. How- ever, the latter is forbidden by the hermiticity of the operator H . A general proof of the absence of the solu- tions of the homogeneous equation in Eq. (2.39) or, in other words, absence of zero eigenvalues E of the opera- tor H r,r ′ ( τ ), Eq. (2.40), is given in Appendix A. The absence of zero eigenvalues makes the operator 

H invertible. Since all the parameters of Eqs. (2.39- 2.40) are real, the solutions A$_{a}$$_{;}$$_{r,r}$ ′ ( z ) are real, too. This means that the exponent in Eq. (2.43) is real and Z [ φ ] is real and positive. This property of the Z [ φ ] can be very important for numerical computations using the MC method. 

Using the regularization with the parameter γ , Eqs. (2.39-2.43), we can prove a stronger than Eq. (2.29) relation for A$_{r,r}$ ′ ( z ), 

Eq. (2.44) is fulfilled automatically for any solution of Eqs. (2.39-2.40). This property can be proven putting in Eqs. (2.39, 2.40) r = r ′ and summing over r . Then, one obtains the equations 

where I ′ $_{a}$( z ) = ∑ r A ′ a ; r,r ( z ), I ′′ a ( z ) = ∑ r A ′′ a ; r,r ( z ). Only the trivial solution I ′ $_{a}$( z ) = I ′′ a ( z ) = 0 satifies the boundary condition, Eq. (2.41), leading to Eq. (2.44). 

<img src='content_image/436834.jpg'>

<img src='content_image/436833.jpg'>

