Hidden Grassmann structure in the XXZ model

TL;DR

This paper reveals a hidden Grassmann algebra structure in the critical XXZ model by constructing anti-commutative operators that facilitate expressing vacuum expectation values as exponential quadratic forms.

Contribution

It introduces two anti-commutative operator families acting on a specific operator space, constructed via q-oscillator algebra representations, revealing a Grassmann structure in the XXZ model.

Findings

Operators b(z), c(z) form an anti-commutative family.

Vacuum expectation values are expressed as exponentials of quadratic forms.

The construction parallels Baxter's Q-operators.

Abstract

For the critical XXZ model, we consider the space W of operators which are products of local operators with a disorder operator. We introduce two anti-commutative family of operators b(z), c(z) which act on the space W. 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 W can be expressed in terms of an exponential of a quadratic form of b(z), c(z).