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

Abstract

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 = 1 2 Heisenberg model to calculate and numerically evaluate transition rates pertaining to dynamic spin structure factors. For real solutions z1, . . . , zr 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.

Evaluation of Dynamic Spin Structure Factor for the Spin-1/2 XXZ Chain in a Magnetic Field

“…In the XXX case ∆ = 1, it has double singularities at the lower and upper boundary. 16 Almost the same line shapes are observed for 0.7 ≤ ∆ < 1. When we further decrease the value of ∆, a small peak near the upper boundary emerges for ∆ ≤ 0.5, and grows as ∆ is further decreasing.…”

Quasiparticles governing the zero-temperature dynamics of the one-dimensional spin-1/2 Heisenberg antiferromagnet in a magnetic field

“…͑b͒ Integrated intensity S zz (q) ͑inset͒ and relative P2 contribution ͑main plot͒ for N ϭ12,16,20,24,28,32. ͑c͒ Integrated intensity S DD (q) ͑inset͒ and relative P2 contribution ͑main plot͒ for Nϭ12, 16,20,24,28. The lines in ͑b͒ connect the Nϭ32 data points and the lines in ͑c͒ the Nϭ28 data points.…”

“…The solid line represents a two-parameter fit, a 1/ Ϫ2 ϩb, of the data points representing the lowest excitation for N ϭ12, 16,20,24,28. The transition rate data at higher frequencies appear to approach zero sufficiently rapidly to overcome the divergent trend of the density of states to produce a monotonically decreasing spectral-weight distribution with a cusp singularity at the upper continuum boundary.…”

“…The use of the determinantal expressions of Bethe ansatz transition rates for the calculation of dynamic structure factors will be reported elsewhere. 20 In designing an efficient algorithm, we must heed the fact that in the calculation of a single matrix element, the sum P S r is evaluated many times, once for every coefficient a(n 1 , . .…”

The two-spinon transverse structure factor of the gapped Heisenberg antiferromagnetic chain

“…Our purpose will be to compute the contribution to this quantity coming from all two-spinon intermediate states, using two separate methods relying on integrability. The first method applies to finite lattices, and makes use of determinant expressions for spin operator form factors derived within the Algebraic Bethe Ansatz [12,13] and used to compute structure factors of Heisenberg chains for general fields and anisotropies, both for two-particle [14,15,16] and general multiparticle states [17,18]. The second method starts from an algebraic analysis of the infinite chain in zero field [19], and uses the quantum group symmetry of the model to express states and form factors directly in the thermodynamic limit.…”