1  & 0\\ 0 &  -1 \end{pmatrix}$$ and $\tilde\partial_i$ is a regularized (smeared) derivative \_i f\_=\_\^[-1]{}(-)\_i f\_ \_[i]{}f()=[12 a]{}(f(+a[**e**]{}\_i)-f(-a [**e**]{}\_i)) with $\chi(\xx)$ is a cut-off function defined as the Fourier transform of $\hat\chi(\kk)$, with $\hat\chi(\kk)$ a smooth function which is $\hat\chi(\kk)=0$ for $|\kk|\ge 2^{N+1}$ and $\hat\chi(\kk)=1$ for $|\kk|\le 2^{N}$; $A_i(\xx)$ is a Gaussian random field with short range (but non local) correlation (A\_[i]{}()A\_j())=\_[i,j]{}v(-) and |v(-)|C e\^[-|-|]{}\[alal\] and we will set $\k=1$ for definiteness.

One is mainly interested in the average of the two-point function, from which the density of states can be computed \[&lt;|[1i H- E]{}|0&gt;\]\[s1\] and in the average of the product of two functions \[&lt;|[1i H- E]{}|0&gt;\_[+-]{} &lt;0|[1i H+ E]{}|&gt;\_[+-]{}\]\[s2\] which is related to the conductivity. In the absence of disorder &lt;|[1i H- E]{}|0&gt; =[1L\^2]{}\_e\^[i]{}()

-E & [1a]{}\[ia k\_1+a k\_2\]\
[1a]{}\[ia k\_1-a k\_2\] & -E

\^[-1]{}\[pro\] In the following we will assume that $2^{N}<<{\pi\over a}$ in order to avoid the [*fermion doubling*]{} problem. Indeed at $E=0$ the denominator in the r.h.s. of is vanishing, in the $L\to\io$ limit, not only at $\kk=(0,0)$ but also at $\kk=(0,\pi/a),(\pi/a,0),(\pi/a,\pi/a)$ modulo $2\pi/a$. The condition $2^{N}<<{\pi\over a}$ ensures that the only remaining pole is the one at $\kk=0$, so preventing the fermionic species 