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arXiv:cond-mat/0502365 · Physical Review Letters

Computation of Dynamical Correlation Functions of Heisenberg Chains in a Magnetic Field

Jean-Sébastien Caux, Jean Michel Maillet

Published Aug 8, 2005• 224 citations
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Abstract

We compute the momentum- and frequency-dependent longitudinal spin structure factor for the spin-1/2 XXZ Heisenberg spin chain in a magnetic field, using exact determinant representations for form factors on the lattice. Multiparticle (i.e., multispinon) contributions are computed numerically throughout the Brillouin zone, yielding saturation of the sum rule to high precision.

DOI
10.1103/physrevlett.95.077201

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We compute the momentum- and frequency-dependent longitudinal spin structure factor for the spin-1/2 XXZ Heisenberg spin chain in a magnetic field, using exact determinant representations for form factors on the lattice. Multiparticle (i.e., multispinon) contributions are computed numerically throughout the Brillouin zone, yielding saturation of the sum rule to high precision.
We compute all dynamical spin–spin correlation functions for the spin-1/2 XXZ anisotropic Heisenberg model in the gapless antiferromagnetic regime at zero temperature, using numerical sums of exact determinant representations for form factors of spin operators on the lattice. Contributions from intermediate states containing many particles and string (bound) states are included. We present modified determinant representations for the form factors valid in the general case with string solutions to the Bethe equations. Our results are such that the available sum rules are saturated to high precision. We Fourier transform our results back to real space, allowing us in particular to make a comparison with known exact formulae for equal-time correlation functions for small separations in zero field, and with predictions for the zero-field asymptotics from conformal field theory.
Considering the adjacent spin operators SjzSj+1zS^z_jS^z_{j+1}Sjz​Sj+1z​ and Sj−Sj+1−S^-_jS^-_{j+1}Sj−​Sj+1−​ in the S=1/2 Heisenberg chain, we give a determinant representation of their form factors. The dynamical structure factors of the respective operators are computed over the whole Brillouin zone in several magnetic fields and the resulting signal is analyzed in terms of excitation types. Among other results, we find that the SjzSj+1zS^z_jS^z_{j+1}Sjz​Sj+1z​ dynamical structure factor carries a large weight of the 4-spinon excitations which are distinguishable from the 2-spinon signal because they are located outside the 2-spinon spectrum.
Inelastic neutron scattering experiments are commonly used to unveil how excitations on Heisenberg spin models play a role in dynamical correlations functions. For a certain class of materials, like CsCoCl3_33​ or CsCoBr3_33​ salts, it turns out that their magnetic properties are fairly well approximated by quasi-one dimensional XXZ models, which enjoy the property of quantum integrability. In these instances, one can in principle use their underlying algebraic structure to describe very precisely how excitations, the so-called spinons, participate in dynamical correlations functions. Even though the available theories (either algebraic Bethe ansatz or quantum group approach) provide all the needed physical quantities such as form factors, complete set of eigenstates and spectrum, it is typically a rather daunting task, however, to obtain sufficiently simple analytical expressions for computing Dynamical Structure Factors (DSFs), valuable, e.g., for parameters estimation based on experimental data. This is particularly the case for the longitudinal DSF of the XXZ model, which has eluded a formal mathematical treatment thus far. Using the quantum group approach, we present here an exact and simple expression of the 2-spinon longitudinal DSF and show our results to be consistent with the expected sum rules and the isotropic and Ising antiferromagnet limiting cases.
We compute the exact four-spinon contribution to the zero-temperature dynamical structure factor of the spin-1/2 Heisenberg isotropic antiferromagnet in zero magnetic field, directly in the thermodynamic limit. We make use of the expressions for matrix elements of local spin operators obtained by Jimbo and Miwa using the quantum affine symmetry of the model, and of their adaptation to the isotropic case by Abada, Bougourzi and Si-Lakhal (correcting some overall factors). The four-spinon contribution to the first frequency moment sum rule at fixed momentum is calculated. This shows, as expected, that most of the remaining correlation weight above the known two-spinon part is carried by four-spinon states. Our results therefore provide an extremely accurate description of the exact structure factor.
The dynamical structure factor of the Babujan–Takhtajan antiferromagnetic spin-1 chain is computed numerically at zero temperature and zero magnetic field, using the higher spin generalization of an algebraic Bethe ansatz-based method previously used for spin-1/2 integrable chains. This method, which consists in the explicit construction of eigenstates and the summation of the Lehmann representation of the correlator, is particularly challenging to implement here in view of the presence of strongly deviated string solutions to the Bethe equations. We show that a careful treatment of these deviations makes it possible to obtain perfect saturation of sum rules for small system sizes, and extremely good saturation for large system sizes where the dynamical structure factor is computed by including all two-spinon and four-spinon contributions. The real-space spin–spin correlation, obtained by Fourier transforming our results, displays asymptotics fitting predictions from conformal field theory.
The longitudinal spin structure factor for the XXZ-chain at small wave vector q is obtained using Bethe ansatz, field theory methods, and the density matrix renormalization group. It consists of a peak with a peculiar, non-Lorentzian shape and a high-frequency tail. We show that the width of the peak is proportional to q2 for finite magnetic field compared to q3 for a zero field. For the tail we derive an analytic formula without any adjustable parameters and demonstrate that the integrability of the model directly affects the line shape.