Hidden Grassmann Structure in the XXZ Model II: Creation Operators

Abstract

Abstract. In this article we unveil a new structure in the space of operators of the XXZ chain. For each α we consider the space W α of all quasi-local operators, which are products of the disorder field q α P 0 j=−∞ σ 3 j with arbitrary local operators. In analogy with CFT the disorder operator itself is considered as primary field. In our previous paper, we have introduced the annhilation operators b(ζ), c(ζ) which mutually anti-commute and kill the "primary field". Here we construct the creation counterpart b * (ζ), c * (ζ) and prove the canonical anti-commutation relations with the annihilation operators. We conjecture that the creation operators mutually anti-commute, thereby upgrading the Grassmann structure to the fermionic structure. The bosonic operator t * (ζ) is the generating function of the adjoint action by local integrals of motion, and commutes entirely with the fermionic creation and annihilation operators. Operators b * (ζ), c * (ζ), t * (ζ) create quasi-local operators starting from the primary field. We show that the ground state averages of quasi-local operators created in this way are given by determinants.

Classical tau-function for quantum spin chains

“…Among other things, this result means that quantum integrable models have a hidden free fermionic (Grassmann) structure in the auxiliary ("horizontal") space. It would be very interesting to relate it to the hidden Grassmann structure uncovered in [71][72][73][74] in the quantum ("vertical") space of quantum integrable models. Another context where τ -functions of classical integrable hierarchies enter the theory of quantum integrable models and associated 2D lattice models of statistical mechanics is calculation of scalar products and partition functions with domain wall boundary conditions [75][76][77].…”

Quantum transfer-matrices for the sausage model

“…Based on the results of these papers, we derived non-linear integral equations for determining the vacuum eigenvalues of the chiral transfermatrix which work both for the cigar and the parafermionic regimes. We believe that this might be a good starting point for applying the powerful fermionic methods [92][93][94][95][96] to the sausage/O(3) NLSM.…”

Problems with Using Separated Variables for Computing Expectation Values for Higher Ranks

“…Then one can try to understand the algebraic structure behind the expressions for these expectation values. In the case whose classical limit corresponds to hyperelliptic spectral curves, the algebra A qua allows fermionic structure [9,10], and the expectation values are expressed in terms of a quantum deformation of the canonical second kind differential ω(P 1 , P 2 ). This is explained by the fact that in the quantum case one can use the method of separation of variables [12,13] for computing the expectation values, which provides the direct analogue of (1.1).…”