Sum rules for four-spinon dynamic structure factor in XXX model

Abstract

In the context of the antiferromagnetic spin 1/2 Heisenberg quantum spin chain (XXX model), we estimate the contribution of the exact four-spinon dynamic structure factor S 4 by calculating a number of sum rules the total dynamic structure factor S is known to satisfy exactly. These sum rules are: the static susceptibility, the integrated intensity, the total integrated intensity, the first frequency moment and the nearest-neighbor correlation function. We find that the contribution of S 4 is between 1% and 2.5%, depending on the sum rule, whereas the contribution of the exact two-spinon dynamic structure factor S 2 is between 70% and 75%. This is consistent with the expected scattering weight of states from outside the spin-wave continuum. The calculations are numerical and Monte Carlo based. Good statistics are obtained.

The four-spinon dynamical structure factor of the Heisenberg chain

“…We have corrected two inaccuracies in [47]: first, the correct normalization is presented here (compare (20) with formula (5.10) there), and most importantly, we have explicitly written that the momentum δ function fixes k only modulo 2π. This has the crucial consequence that two sectors must be considered when solving the dynamical constraints of momentum and energy conservation for 4-spinon intermediate states (this was overlooked in [48,49,50], leading in particular to an incorrect description of the four-spinon continuum). Bearing in mind that the spinon momenta p i are by definition constrained to the interval [−π, 0], we define sectors 0 and 1 as 0 : k + p 1 + p 2 + p 3 + p 4 = 0, 1 : k + 2π + p 1 + p 2 + p 3 + p 4 = 0 (21) with in both cases the energy constraint explicitly written as ω + π 2 (sin p 1 + sin p 2 + sin p 3 + sin p 4 ) = 0.…”

“…Further analytical work [48] yielded little progress, and in fact (as we will show below) incorrectly identified the boundaries of the four-spinon continuum. To this day, nobody has been able to extract curves from these expressions, and previous attempts [49,50] have not yielded acceptable results due to the inappropriateness of the chosen method and the incorrect continuum used.…”

Spin dynamics of the antiferromagnetic spin-12chain at finite magnetic fields and intermediate temperatures

“…28,33,91,92 For the present work sum rules can be applied to assess the quality of the analytic continuation. We will focus on the sum rules for the static structure factor S ␣␤ ͑q͒ and the static susceptibility ␣␤ ͑q͒ which are obtained by integral transforming the dynamical structure factor, 93…”

“…In principle, also BA allows to determine dynamical correlation functions, however calculating the corresponding matrix elements is highly non-trivial and progress has been made only recently. By now analytic expressions for dynamical spin correlation functions are available for the two- 27,28,29 and the fourspinon sector 30,31,32,33 at ∆ = 1, h = 0 and T = 0. In addition, determinant approaches 34,35 allow for numerical treatment of two- 36,37,38 and many-spinon 39,40 states of the XXZ chain in finite magnetic fields, at T = 0.…”