arXiv:cond-mat/0205430 · Europhysics Letters (EPL)

Transition rates via Bethe ansatz for the spin-½ Heisenberg chain

Daniel Biegel, Michael Karbach, Gerhard H. Muller

Published Sep 1, 2002• 42 citations

Abstract

We use the exact determinantal representation derived by Kitanine, Maillet, and Terras for matrix elements of local spin operators between Bethe wave functions of the one-dimensional s = ½ Heisenberg model to calculate and numerically evaluate transition rates pertaining to dynamic spin structure factors. For real solutions z 1 ,..., z r of the Bethe ansatz equations, the size of the determinants is of order r × r . We present applications to the zero-temperature spin fluctuations parallel and perpendicular to an external magnetic field.

DOI: 10.1209/epl/i2002-00125-0

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arXiv:cond-mat/0205430 · Europhysics Letters (EPL)

Transition rates via Bethe ansatz for the spin-½ Heisenberg chain

Daniel Biegel, Michael Karbach, Gerhard H. Muller

Published Sep 1, 2002• 42 citations

arXiv HTML PDF

Abstract

DOI: 10.1209/epl/i2002-00125-0

Related papers

Determinantendarstellung von Übergangsmatrixelementen für das Spin-½ XXZ-Modell

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The one-dimensional spin-½ Heisenberg model was solved, in principle, by H. Bethe in 1931 using an ad hoc trial wavefunction, now famously known as the Bethe ansatz. Decades went by before the power and scope of this method of exact analysis became widely known for applications to spectrum and thermodynamics of a select class of completely integrable model systems. Until recently, the most vexing exception to immense progress in the further development of the Bethe ansatz has been the absence of a practical method to use the exactly known and readily available Bethe wave functions for the explicit calculation of transition matrix elements. The knowledge of such transition rates is of paramount importance for an understanding of dynamic correlation functions in relation to the underlying quasiparticles and for the interpretation of experimental probes of quantum fluctuations in quasi-one-dimensional magnetic compounds. It was most remarkable, therefore, when Kitanine, Maillet, and Terras succeeded in reducing matrix elements between Bethe wave functions for local spin operators to determinantal expressions. Here these expressions are used to calculate dynamic spin structure factors of the Heisenberg antiferromagnet (XXX model) and the one-dimensional spin-½ XXZ model in the planar regime (|Δ| < 1), both with periodic boundary conditions and an external magnetic field: Determinantal representations are derived for the zero-temperature spin fluctuations parallel and perpendicular to the external field. The singular nature of the Bethe ansatz equations for the case Δ = 0 (XX model) is discussed and the XX limit is performed in the determinantal expressions for the transition matrix elements. As an application, lineshapes for dynamic structure factors relevant for experiments are calculated. The predominant excitations are identified within the framework of quasiparticles excited from a corresponding vacuum.The one-dimensional spin-½ Heisenberg model was solved, in principle, by H. Bethe in 1931 using an ad hoc trial wavefunction, now famously known as the Bethe ansatz. Decades went by before the power and scope of this method of exact analysis became widely known for applications to spectrum and thermodynamics of a select class of completely integrable model systems. Until recently, the most vexing exception to immense progress in the further development of the Bethe ansatz has been the absence of a practical method to use the exactly known and readily available Bethe wave functions for the explicit calculation of transition matrix elements. The knowledge of such transition rates is of paramount importance for an understanding of dynamic correlation functions in relation to the underlying quasiparticles and for the interpretation of experimental probes of quantum fluctuations in quasi-one-dimensional magnetic compounds. It was most remarkable, therefore, when Kitanine, Maillet, and Terras succeeded in reducing matrix elements between Bethe wave functions for local spin operators to determinantal expressions. Here these expressions are used to calculate dynamic spin structure factors of the Heisenberg antiferromagnet (XXX model) and the one-dimensional spin-½ XXZ model in the planar regime (|Δ| < 1), both with periodic boundary conditions and an external magnetic field: Determinantal representations are derived for the zero-temperature spin fluctuations parallel and perpendicular to the external field. The singular nature of the Bethe ansatz equations for the case Δ = 0 (XX model) is discussed and the XX limit is performed in the determinantal expressions for the transition matrix elements. As an application, lineshapes for dynamic structure factors relevant for experiments are calculated. The predominant excitations are identified within the framework of quasiparticles excited from a corresponding vacuum.

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As part of a study that investigates the dynamics of the

s=\frac{1}{2}

\mathrm{XXZ}

model in the planar regime $|\ensuremath{\Delta}|<1,$ we discuss the singular nature of the Bethe ansatz equations for the case $\ensuremath{\Delta}=0$

(XX

model). We identify the general structure of the Bethe ansatz solutions for the entire

\mathrm{XX}

spectrum, which include states with real and complex magnon momenta. We discuss the relation between the spinon or magnon quasiparticles (Bethe ansatz) and the lattice fermions (Jordan-Wigner representation). We present determinantal expressions for transition rates of spin-fluctuation operators between Bethe wave functions and reduce them to product expressions. We apply the formulas to two-spinon transition rates for chains with up to

N=4096

sites.

The spin fluctuations parallel to the external magnetic field in the ground state of the one-dimensional (1D)

s=\frac{1}{2}

Heisenberg antiferromagnet are dominated by a two-parameter set of collective excitations. In a cyclic chain of N sites and magnetization

0<{M}_{z}<N/2,

the ground state, which contains

{2M}_{z}

spinons, is reconfigured as the physical vacuum for a different species of quasiparticles, identifiable in the framework of the coordinate Bethe ansatz by characteristic configurations of Bethe quantum numbers. The dynamically dominant excitations are found to be scattering states of two such quasiparticles. For $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty},$ these collective excitations form a continuum in $(q,\ensuremath{\omega})$ space with an incommensurate soft mode. Their matrix elements in the dynamic spin structure factor ${S}_{\mathrm{zz}}(q,\ensuremath{\omega})$ are calculated directly from the Bethe wave functions for finite N. The resulting line- shape predictions for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ complement the exact results previously derived via algebraic analysis for the exact two-spinon part of ${S}_{\mathrm{zz}}(q,\ensuremath{\omega})$ in the zero-field limit. They are relevant for neutron-scattering experiments on quasi-1D antiferromagnetic compounds in a strong magnetic field.