the CuO$_2$ plane in HTSC: $$\label{hamtot}\hat H=\tau \sum_{{\bf r},{\bf a}_1,{\bf a}_2,\sigma ,\sigma
^{\prime }}c_{{\bf r}+{\bf a}_1,\sigma }^{\dagger }c_{{\bf r}+{\bf a}
_2,\sigma ^{\prime }}(\frac 12\delta _{\sigma \sigma ^{\prime }}+2\vec s
_{\sigma \sigma ^{\prime }}\vec S_{{\bf r}})+\frac J2\sum_{{\bf r},{\bf g}}
\vec S_{{\bf r}}\vec S_{{\bf r}+{\bf g}},$$ ${\bf a}_1,{\bf a}_2=\pm \frac 12{\bf g}_x,\pm \frac 12{\bf g}_y,{\bf g}=\pm
{\bf g}_x,\pm {\bf g}_y.$

Here and below ${\bf g}_{x,y}$ are basic vectors of a copper square lattice ( $|{\bf g}|\equiv 1$), ${\bf r}+{\bf a}$ are four vectors of O sites nearest to the Cu site ${\bf r}$, the operator $c_\sigma ^{\dagger }$ creates a hole with the spin index $\sigma =\pm 1$ at the O site, $\vec s
_{\sigma \sigma ^{\prime }}=\frac 12\vec \sigma _{\sigma \sigma ^{\prime }}$ , operator ${\bf S}$ represents the localized spin on the copper site. As mentioned above $\tau $ is the integral of oxygen hole hoppings that takes into account the coupling of the hole motion with copper spin subsystem. $J$ is the constant of nearest neighbor AFM exchange between the copper spins.

It is well known that the most prominent feature of the Hamiltonian (\[hamtot\]) is that the low-energy physics of hole excitations is described by the Bloch sums 