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6 HIDDEN GRASSMANN STRUCTURE IN THE XXZ MODEL

H. Boos, M. Jimbo, T. Miwa, Fedor Aleksandrovich Smirnov, Y. Takeyama

Published Jun 30, 2006• 79 citations
OpenAlex

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).

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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).
In the recent study of correlation functions for the infinite XXZ spin chain, a new pair of anti-commuting operators b(z),c(z)b(z), c(z)b(z),c(z) was introduced. They act on the space of quasi-local operators, which are local operators multiplied by the disorder operator. For the inhomogeneous chain with the spectral parameters ξkξ_{k}ξk​, these operators have simple poles at z2=ξk2z^2=ξ_{k}^2z2=ξk2​. The residues are denoted by bk,ckb_{k}, c_{k}bk​,ck​. At q=iq=iq=i, we show that the operators bk,ckb_{k}, c_{k}bk​,ck​ are cubic monomials in free fermions. In other words, the action of these operators is very simple in the fermion basis. We give an explicit construction of these fermions. Then, we show that the existence of the fermionic basis is a consequence of the Grassmann relation, the equivariance with respect to the action of the symmetric group and the reduction property, which are all valid for the operators bk,ckb_{k}, c_{k}bk​,ck​ in the case of generic qqq.
We address the problem of computing temperature correlation functions of the XXZ chain, within the approach developed in our previous works. In this paper we calculate the expected values of a fermionic basis of quasi-local operators, in the infinite volume limit while keeping the Matsubara (or Trotter) direction finite. The result is expressed in terms of two basic quantities: a ratio $ρ(\z)$ of transfer matrix eigenvalues, and a nearest neighbour correlator $ω(\z,ξ)$. We explain that the latter is interpreted as the canonical second kind differential in the theory of deformed Abelian integrals.
We address the problem of computing temperature correlation functions of the XXZ chain, within the approach developed in our previous works. In this paper we calculate the expected values of a fermionic basis of quasi-local operators, in the infinite volume limit while keeping the Matsubara (or Trotter) direction finite. The result is expressed in terms of two basic quantities: a ratio ρ(ζ) of transfer matrix eigenvalues and a nearest neighbour correlator ω(ζ, ξ). We explain that the latter is interpreted as the canonical second kind differential in the theory of deformed Abelian integrals.
In this paper, we construct a Q -operator as a trace of a representation of the universal R -matrix of U q over an infinite-dimensional auxiliary space. This auxiliary space is a four-parameter generalization of the q -oscillator representations used previously. We derive generalized T – Q relations in which three of these parameters shift. After a suitable restriction of parameters, we give an explicit expression for the Q -operator of the 6-vertex model and show the connection with Baxter's expression for the central block of his corresponding operator.