Hidden Grassmann Structure in the XXZ Model

For the critical XXZ model, we consider the space $$\mathcal{W}_{[\alpha]}$$ of operators which are products of local operators with a disorder operator. We introduce two anti-commutative families of operators $${\bf {b}}(\zeta), {\bf {c}}(\zeta)$$ which act on $$\mathcal{W}_{[\alpha]}$$. These operators are constructed as traces over representations of the q-oscillator algebra, in close analogy with Baxter’s Q-operators. We show that the vacuum expectation values of operators in $$\mathcal{W}_{[\alpha]}$$ can be expressed in terms of an exponential of a quadratic form of $${\bf {b}}(\zeta), {\bf {c}}(\zeta)$$.