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arXiv:1401.4450 · Journal of Statistical Mechanics Theory and Experiment

Computation of dynamical correlation functions of the spin-1 Babujan–Takhtajan chain

Rogier Vlijm, Jean-Sébastien Caux

Published May 16, 2014• 11 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1742-5468/2014/05/p05009

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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.
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.
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 excitations of the Heisenberg antiferromagnetic spin chain in zero field are known as spinons. As pairwise-created fractionalized excitations, spinons are important in the understanding of inelastic neutron scattering experiments in (quasi-)one-dimensional materials. In the present paper, we consider the real space-time dynamics of spinons originating from a local spin flip on the antiferromagnetic ground state of the (an)isotropic Heisenberg spin-1/2 model and the Babujan-Takhtajan spin-1 model. By utilizing algebraic Bethe ansatz methods at finite system size to compute the expectation value of the local magnetization and spin-spin correlations, spinons are visualized as propagating domain walls in the antiferromagnetic spin ordering with anisotropy dependent behavior. The spin-spin correlation after the spin flip displays a light cone, satisfying the Lieb-Robinson bound for the propagation of correlations at the spinon velocity.
Quantum dynamics of strongly correlated systems is a challenging problem. Although the low energy fractional excitations of one-dimensional integrable models are often well understood, exploring quantum dynamics in these systems remains challenging in the gapless regime, especially at intermediate and high energies. Based on the algebraic Bethe ansatz formalism, we study spin dynamics in a representative one-dimensional strongly correlated model, i.e., the antiferromagnetic spin-12\frac{1}{2}21​ XXZ chain with the Ising anisotropy, via the form-factor formulas. Various excitations at different energy scales are identified crucial to the dynamic spin structure factors under the guidance of sum rules. At small magnetic polarizations, gapless excitations dominate the low energy spin dynamics arising from the magnetic-field-induced incommensurability. In contrast, spin dynamics at intermediate and high energies is characterized by the two- and three-string states, which are multiparticle excitations based on the commensurate N\'eel ordered background. Our work is helpful for experimental studies on spin dynamics in both condensed matter and cold atom systems beyond the low energy effective Luttinger liquid theory. Based on an intuitive physical picture, we speculate that the dynamic feature at high energies due to the multiparticle antibound state excitations can be generalized to nonintegrable spin systems.
Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation.
Since a long-time, the quantum integrable systems have remained an area where modern mathematical methods have given an access to interesting results in the study of physical systems. The exact computations, both numerical and asymptotic, of the correlation function is one of the most important subject of the theory of the quantum integrable models. In this context an approach based on the calculation of form factors has been proved to be a more effective one. In this thesis, we develop a new method based on the algebraic Bethe ansatz for the computation of the form-factors in thermodynamic limit. It is both applied to and described in the context of isotropic XXX Heisenberg chain, which is one of the examples of an interesting case of critical models where the Fermi-zone is non-compact. In a particular case of two-spinon form-factors, we obtain an exact result in a closed-form which matches the previous result obtained from an approach based on qqq-vertex operator algebra. This method is then generalised to form-factors in higher spinon sectors where we find a reduced determinant representation for the form-factors, in which a higher-level structure for the form-factors is revealed.