$q$-deformed Grassmann field and the two-dimensional Ising model

A. I. Bugrij ; V. N. Shadura

Teoretičeskaâ i matematičeskaâ fizika, Tome 103 (1995) no. 3, pp. 388-412

Résumé

We construct an exact representation of the Ising partition function in the form of the $SL_q(2,R)$-invariant functional integral for the lattice-free $q$-fermion field theory ($q=-1$). It is shown that the $q$-fermionization allows one to rewrite the partition function of the eight-vertex model in an external field through a functional integral with four-fermion interaction. To construct these representations, we define a lattice $(l,q,s)$-deformed Grassmann bispinor field and extend the Berezin integration rules to this field. At $q=-1$, $l=s=1$ we obtain the lattice $q$-fermion field which allows us to fermionize the two-dimensional Ising model. We show that the Gaussian integral over $(q,s)$-Grassmann variables is expressed through the $(q,s)$-deformed Pfaffian which is equal to square root of the determinant of some matrix at $q=\pm 1$, $s=\pm 1$.