Factorization of the finite temperature correlation functions of the XXZ chain in a magnetic field

We present a conjecture for the density matrix of a finite segment of the XXZ chain coupled to a heat bath and to a constant longitudinal magnetic field. It states that the inhomogeneous density matrix, conceived as a map which associates with every local operator its thermal expectation value, can be written as the trace of the exponential of an operator constructed from weighted traces of the elements of certain monodromy matrices related to $U_q (\hat{\mathfrak{sl}}_2)$ and only two transcendental functions pertaining to the one-point function and the neighbour correlators, respectively. Our conjecture implies that all static correlation functions of the XXZ chain are polynomials in these two functions and their derivatives with coefficients of purely algebraic origin.Comment: 35 page

Computation of static Heisenberg-chain correlators: Control over length and temperature dependence

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 $\omega$ 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 $\omega$ 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 of the leading order terms for up to eight lattice sites.Comment: 5 pages, 3 figures, v2: text slightly shortened, typos in eqns. (16), (17) corrected, Fig. 1 replaced, v3: typo in eqn. (11) correcte

Short-distance thermal correlations in the XXZ chain

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 $- 1 < \Delta < 1$. 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.Comment: 25 pages, 5 colored figure