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arXiv:cond-mat/9502072 · Journal of Physics Condensed Matter

Complete solution of the XXZ-model on finite rings; dynamical structure factors at zero temperature

K Fabricius, K -H Mutter, U Low

Published Jul 10, 1995• 2 citations
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Abstract

The finite-size effects of the dynamical structure factors in the XXZ-model are studied in the euclidean time ( tau -) representation. Away from the critical momentum p= pi finite-size effects rum out to be small except for in large- tau limit. The large finite-size effects at the critical momentum p= pi signal the emergence of infrared singularities in the spectral ( omega ) representation of the dynamical structure factors.

DOI
10.1088/0953-8984/7/28/017

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The finite-size effects of the dynamical structure factors in the XXZ-model are studied in the euclidean time ( tau -) representation. Away from the critical momentum p= pi finite-size effects rum out to be small except for in large- tau limit. The large finite-size effects at the critical momentum p= pi signal the emergence of infrared singularities in the spectral ( omega ) representation of the dynamical structure factors.
The transverse and longitudinal structure factors of the spin-1/2 XXZ model are determined from a complete diagonalization of the Hamiltonian on rings with N=4,...,16 sites. At fixed critical momentum p=\ensuremath{\pi}, we observe the critical low-temperature behavior as it is predicted by scaling arguments. Finite-size scaling works well for the transverse structure factors. However, it breaks down for the longitudinal structure factors if the anisotropy is too large (\ensuremath{\gamma}>0.3\ensuremath{\pi}). We also investigate the low-temperature behavior of the uniform susceptibility.
Transition rates and dynamic spin structure factor at zero temperature for the spin-1/2 XXZ chain at critical regime in a magnetic field are numerically evaluated in terms of the exact determinant representations for the form factors and norms of the Bethe eigenstates. We have seen that the transition rates converges toward the constant function with the value 1 in the limit delta -> 0. The observed critical exponent of the singularity at the lower boundary is compared with the one predicted from the comformal field theory. We confirm that they are in good agreement. Further we have discovered that a small peak emerges near the upper boundary in the line shape of S(q,omega) for 0
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.
We study the longitudinal and transverse spin dynamical structure factors of the spin-1/2 XXX chain at finite magnetic field h, focusing in particular on the singularities at excitation energies in the vicinity of the lower thresholds. While the static properties of the model can be studied within a Fermi-liquid like description in terms of pseudoparticles, our derivation of the dynamical properties relies on the introduction of a form of the 'pseudofermion dynamical theory' (PDT) of the 1D Hubbard model suitably modified for the spin-only XXX chain and other models with two pseudoparticle Fermi points. Specifically, we derive the exact momentum and spin-density dependences of the exponents ζ(τ)(k) controlling the singularities for both the longitudinal (τ = l) and transverse (τ = t) dynamical structure factors for the whole momentum range k ∈ ]0,π[, in the thermodynamic limit. This requires the numerical solution of the integral equations that define the phase shifts in these exponents expressions. We discuss the relation to neutron scattering and suggest new experiments on spin-chain compounds using a carefully oriented crystal to test our predictions.