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arXiv:cond-mat/0502569 · Journal of Physics A Mathematical and General

High temperature expansion of the emptiness formation probability for the isotropic Heisenberg chain

Zengo Tsuboi, Masahiro Shiroishi

Published May 3, 2005• 21 citations
arXivHTMLPDF

Abstract

Recently, Göhmann, Klümper and Seel have derived novel integral formulae for the correlation functions of the spin-1/2 Heisenberg chain at finite temperature. We have found that the high temperature expansion (HTE) technique can be effectively applied to evaluate these integral formulae. Actually, as for the emptiness formation probability P ( n ) of the isotropic Heisenberg chain, we have found a general formula of the HTE for P ( n ) with arbitrary up to O (( J / T ) 4 ). If we fix a magnetic field to a certain value, we can calculate the HTE to much higher order. For example, the order up to O (( J / T ) 42 ) has been achieved in the case of P (3) when h = 0. We have compared these HTE results with the data by quantum Monte Carlo simulations. They exhibit excellent agreement.

DOI
10.1088/0305-4470/38/20/l05

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Recently, Göhmann, Klümper and Seel have derived novel integral formulae for the correlation functions of the spin-1/2 Heisenberg chain at finite temperature. We have found that the high temperature expansion (HTE) technique can be effectively applied to evaluate these integral formulae. Actually, as for the emptiness formation probability P ( n ) of the isotropic Heisenberg chain, we have found a general formula of the HTE for P ( n ) with arbitrary up to O (( J / T ) 4 ). If we fix a magnetic field to a certain value, we can calculate the HTE to much higher order. For example, the order up to O (( J / T ) 42 ) has been achieved in the case of P (3) when h = 0. We have compared these HTE results with the data by quantum Monte Carlo simulations. They exhibit excellent agreement.
We review some recent advances in the computation of exact correlation functions of the XXZ-½ Heisenberg chain. We first give a general introduction to our method which is based on the algebraic Bethe ansatz and the resolution of the quantum inverse scattering problem, leading in particular to multiple integral representations for the correlation functions. Then we describe recently obtained compact formulas for the spin-spin correlation functions of the XXZ-½ Heisenberg chain. We outline how this leads to several explicit results including the known two point functions in the limit of free fermions, the so-called emptiness formation probability at anisotropy Δ=1/2 and its large distance asymptotic behaviour in the massless phase of the model.
We study the spin-1/2 Heisenberg XXX antiferromagnet for which the spectrum of the Hamiltonian was found by Bethe in 1931. We study the probability of the formation of ferromagnetic string in the antiferromagnetic ground state, which we call emptiness formation probability P ( n ). This is the most fundamental correlation function. We prove that, for short strings, it can be expressed in terms of the Riemann zeta function with odd arguments, logarithm ln 2 and rational coefficients. This adds yet another link between statistical mechanics and number theory. We have obtained an analytical formula for P (5) for the first time. We have also calculated P ( n ) numerically by the density matrix renormalization group. The results agree quite well with the analytical results. Furthermore, we study the asymptotic behaviour of P ( n ) at finite temperature by quantum Monte Carlo simulation. This also agrees with our previous analytical results.
The method for calculating the isotropic exchange interactions in the paramagnetic phase is proposed. It is based on the mapping of the high-temperature expansion of the spin-spin correlation function calculated for the Heisenberg model onto the Hubbard Hamiltonian one. The resulting expression for the exchange interaction has a compact and transparent formulation. The quality of the calculated exchange interactions is estimated by comparing the eigenvalue spectra of the Heisenberg model and low-energy magnetic part of the Hubbard model. By the example of quantum rings at half-filling with different hopping setups, we analyze contributions from different parts of the Hubbard model spectrum to the resulting exchange interaction. The developed method can be applied for simulating magnetic couplings in correlated materials and surface nanostructures.
By combining conventional finite-temperature many-body perturbation theory with cluster expansions, we develop a systematic method to carry out high order arbitrary temperature perturbative calculations on a computer. The method is well suited to studying the thermodynamic properties of quantum disordered and quantum critical phases at finite temperatures. As an application, we calculate the magnetic susceptibility, internal energy, and specific heat of the bilayer Heisenberg model. It is shown that for a wide range of coupling constants these expansions show excellent convergence at all temperatures. Comparing the direct series (without extrapolations) for the bulk susceptibility to quantum Monte Carlo simulations we find an almost perfect agreement between the two methods even at the quantum critical coupling separating the dimerized and antiferromagnetic phases. The convergence fails only at very low temperatures, which are also difficult to reach by quantum Monte Carlo simulations.
We propose a new correlator in one-dimensional quantum spin chains, the s -emptiness formation probability ( s -EFP). This is a generalization of the emptiness formation probability (EFP), which is the probability that the first n spins of the chain are all aligned downwards. In the s -EFP we let the spins in question be separated by s sites. The usual EFP corresponds to the special case when s = 1. Taking s > 1 allows us to quantify non-local correlations. We express the s -EFP for the anisotropic XY model in a transverse magnetic field, a system with both critical and non-critical regimes, in terms of a Toeplitz determinant. For the isotropic XY model we find that the magnetic field induces an interesting length scale.