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Progress of Theoretical Physics Supplement

Hidden Grassmannian Structure in the XXZ Model

Tetsuji Miwa

Published Jan 1, 2009• 0 citations
Publisher

Abstract

This review is based on the joint work by Herman Boos, Michio Jimbo, Fedor Smirnov, Yishihiro Takeyama and T. M. We present an exponential formula for the correlation functions of the XXZ model. The formula is given in terms of a kind of monodromy operators in the sense of the quantum inverse scattering method. The opertors satisfy anti-commutation relations, and acting on the space of quasi-local operators as annihilation operators. Construction of creation operators is also given.

DOI
10.1143/ptps.177.99

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This review is based on the joint work by Herman Boos, Michio Jimbo, Fedor Smirnov, Yishihiro Takeyama and T. M. We present an exponential formula for the correlation functions of the XXZ model. The formula is given in terms of a kind of monodromy operators in the sense of the quantum inverse scattering method. The opertors satisfy anti-commutation relations, and acting on the space of quasi-local operators as annihilation operators. Construction of creation operators is also given.
A general approach to the computation of exact correlation functions of lattice quantum integrable models proposed recently is reviewed.It is based on the resolution of the so-called quantum inverse scattering problem, namely, the reconstruction of local operators in terms of the (operator) entries of the quantum monodromy matrix obeying a Yang-Baxter algebra.The application of this method to the XXZ Heisenberg spin-1 2 chain in a magnetic field leads to multiple integral representations of its n-point correlation functions.For zero magnetic field, it gives a proof, in both the massless and massive (anti-ferromagnetic) regimes, of the formulas obtained from the q-deformed KZ equations (massless regime) and the representation theory of the quantum affine algebra U q ( ŝl 2 ) together with the corner transfer matrix approach (massive regime).
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 extend the recent approach of M. Jimbo, K. Miki, T. Miwa, and A. Nakayashiki to derive an integral formula for the N-point correlation functions of arbitrary local operators of the antiferromagnetic spin-1 XXZ model. For this, we realize the quantum affine symmetry algebra Uq(su(2)2)U_q(su(2)_2)Uq​(su(2)2​) of level 2 and its corresponding type I vertex operators in terms of a deformed bosonic field free of a background charge, and a deformed fermionic field. Up to GSO type projections, the Fock space is already irreducible and therefore no BRST projections are involved. This means that no screening charges with their Jackson integrals are required. Consequently, our N-point correlation functions are given in terms of usual classical integrals only, just as those derived by Jimbo et al in the case of the spin-1/2 XXZ model through the Frenkel-Jing bosonization of Uq(su(2)1)U_q(su(2)_1)Uq​(su(2)1​).
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.