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Short-distance thermal correlations in the massive XXZ chain

Frank Göhmann, Andreas Klümper, Fachbereich C Physik

Published Jan 1, 2009• 28 citations
OpenAlex

Abstract

We explore short-distance static correlation functions in the infinite XXZ chain using previously derived formulae which represent the correlation functions in factorized form. We compute two-point functions ranging over 2, 3 and 4 lattice sites as functions of the temperature and the magnetic field in the massive regime ∆> 1, extending our previous results to the full parameter plane of the antiferromagnetic chain ( ∆> −1 and arbitrary field h). The factorized formulae are numerically efficient and allow for taking the isotropic limit ( ∆ = 1) and the Ising limit ( ∆ = ∞). At the critical field separating the fully polarized phase from the Néel phase, the Ising chain possesses exponentially many ground states. The residual entropy is lifted by quantum fluctuations for large but finite ∆ inducing unexpected crossover phenomena in the correlations. 1

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We explore short-distance static correlation functions in the infinite XXZ chain using previously derived formulae which represent the correlation functions in factorized form. We compute two-point functions ranging over 2, 3 and 4 lattice sites as functions of the temperature and the magnetic field in the massive regime ∆> 1, extending our previous results to the full parameter plane of the antiferromagnetic chain ( ∆> −1 and arbitrary field h). The factorized formulae are numerically efficient and allow for taking the isotropic limit ( ∆ = 1) and the Ising limit ( ∆ = ∞). At the critical field separating the fully polarized phase from the Néel phase, the Ising chain possesses exponentially many ground states. The residual entropy is lifted by quantum fluctuations for large but finite ∆ inducing unexpected crossover phenomena in the correlations. 1
Recent studies have revealed much of the mathematical structure of the static correlation functions of the XXZ chain. Here we use the results of those studies in order to work out explicit examples of short-distance correlation functions in the infinite chain. We compute two-point functions ranging over two, three and four lattice sites as functions of the temperature and the magnetic field for various anisotropies in the massless regime −1<Δ<1. It turns out that the new formulae are numerically efficient and allow us to obtain the correlation functions over the full parameter range with arbitrary precision.
Recent studies have revealed much of the mathematical structure of the static correlation functions of the XXZ chain. Here we use the results of those studies in order to work out explicit examples of short-distance correlation functions in the infinite chain. We compute two-point functions ranging over 2, 3 and 4 lattice sites as functions of the temperature and the magnetic field for various anisotropies in the massless regime −1 < ∆ < 1. It turns out that the new formulae are numerically efficient and allow us to obtain the correlations functions over the full parameter range with arbitrary precision.
Abstract We use the form factors of the quantum transfer matrix in the zero-temperature limit in order to study the two-point ground-state correlation functions of the XXZ chain in the antiferromagnetic massive regime. We obtain novel form factor series representations of the correlation functions which differ from those derived either from the q -vertex-operator approach or from the algebraic Bethe Ansatz approach to the usual transfer matrix. We advocate that our novel representations are numerically more efficient and allow for a straightforward calculation of the large-distance asymptotic behaviour of the two-point functions. Keeping control over the temperature corrections to the two-point functions we see that these are of order T ∞ in the whole antiferromagnetic massive regime. The isotropic limit of our result yields a novel form factor series representation for the two-point correlation functions of the XXX chain at zero magnetic field.
We communicate results on correlation functions for the spin-1/2 Heisenberg chain in two particularly important cases: (a) for the infinite chain at arbitrary finite temperature T, and (b) for finite chains of arbitrary length L in the ground state. In both cases we present explicit formulas expressing the short-range correlators in a range of up to seven lattice sites in terms of a single function ω encoding the dependence of the correlators on T (L). These formulas allow us to obtain accurate numerical values for the correlators and derived quantities like the entanglement entropy. By calculating the low T (large L) asymptotics of ω we show that the asymptotics of the static correlation functions at any finite distance are T(2) (1/L(2)) terms. We obtain exact and explicit formulas for the coefficients for up to eight sites.
The XXZ quantum spin chain has a triple point in its ground state h –1/Δ phase diagram. This first order critical point is located at the joint end point of the two second order phase transition lines marking the transition from the gapless phase to the fully polarized phase and to the Néel ordered phase, respectively. We explore the magnetization and the short-range correlation functions in its vicinity using the exact solution of the model. In the critical regime above the triple point we observe a strong variation of all physical quantities on a low energy scale of order 1/Δ induced by the transverse quantum fluctuations. We interpret this phenomenon starting from a strong coupling perturbation theory around the highly degenerate ground state of the Ising chain at the triple point. From the perturbation theory we identify the relevant scaling of the magnetic field and of the temperature. Applying the scaling to the exact solutions we obtain explicit formulae for the magnetization and short-range correlation functions at low temperatures.
We consider the longitudinal dynamical two-point function of the XXZ quantum spin chain in the antiferromagnetic massive regime. It has a series representation based on the form factors of the quantum transfer matrix of the model. The nth summand of the series is a multiple integral accounting for all n-particle–n-hole excitations of the quantum transfer matrix. In previous works, the expressions for the form factor amplitudes appearing under the integrals were either again represented as multiple integrals or in terms of Fredholm determinants. Here, we obtain a representation which reduces, in the zero-temperature limit, essentially to a product of two determinants of finite matrices whose entries are known special functions. This will facilitate the further analysis of the correlation function.