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Acta Polytechnica

Correlation Functions for Lattice Integrable Models

Fedor Aleksandrovich Smirnov

Published Jan 2, 2008• 0 citations
Publisher

Abstract

In this lectures I consider the problem of calculating the correlation functions for XXZ spin chain. First, I explain in details the free fermion case. Then I show that for generic coupling constant the fermionic operators acting on the space of quasi-local fields can be introduced. In the basis generated by these fermionic operators the correlation functions are given by determinants as in the free fermion case.

DOI
10.14311/959

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In this lectures I consider the problem of calculating the correlation functions for XXZ spin chain. First, I explain in details the free fermion case. Then I show that for generic coupling constant the fermionic operators acting on the space of quasi-local fields can be introduced. In the basis generated by these fermionic operators the correlation functions are given by determinants as in the free fermion case.
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.
Abstract We give a descriptive review of the fermionic basis approach to the theory of correlation functions of the XXZ quantum spin chain. The emphasis is on explicit formulae for short-range correlation functions which will be presented in a way that allows for their direct implementation on a computer. Within the fermionic basis approach a huge class of stationary reduced density matrices, compatible with the integrable structure of the model, assumes a factorized form. This means that all expectation values of local operators and all two-point functions, in particular, can be represented as multivariate polynomials in only two functions ρ and ω and their derivatives with coefficients that are rational in the deformation parameter q of the model. These coefficients are of ‘algebraic origin’. They do not depend on the choice of the density matrix, which only impacts the form of ρ and ω . As an example we work out in detail the case of the grand canonical ensemble at temperature T and magnetic field h for q in the critical regime. We compare our exact results for the fourth-neighbour two-point functions with asymptotic formulae for h , T = 0 and for finite h and T .
ntegrable models in statistical mechanics and quantum field theory constitute a rich research field at the crossroads of modern mathematics and theoretical physics. An important issue to understand is the space of local operators in the system and, ultimately, their correlation functions and form factors. This book is the first published monograph on this subject. It treats integrable lattice models, notably the six-vertex model and the XXZ Heisenberg spin chain. A pair of fermions is introduced and used to create a basis of the space of local operators, leading to the result that all correlation functions at finite distances are expressible in terms of two transcendental functions with rational coefficients. Step-by-step explanations are given for all materials necessary for this construction, ranging from algebraic Bethe ansatz, representations of quantum groups, and the Bazhanov-Lukyanov-Zamolodchikov construction in conformal field theory to Riemann surfaces and their Jacobians. Several examples and applications are given along with numerical results.
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​).