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arXiv:cond-mat/9606068 · Physical review. B, Condensed matter

Two-spinon dynamic structure factor of the one-dimensional s= Heisenberg antiferromagnet

Michael Karbach, Gerhard H. Muller, A. Hamid Bougourzi, Andreas Fledderjohann, K.-H. Mütter

Published May 1, 1997• 165 citations
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Abstract

The exact expression derived by Bougourzi, Couture, and Kacir for the two-spinon contribution to the dynamic spin structure factor S(q,\ensuremath{\omega}) of the one-dimensional s=1/2 Heisenberg antiferromagnet at T=0 is evaluated for direct comparison with finite-chain transition rates (N\ensuremath{\leqslant}28) and an approximate analytical result previously inferred from finite-N data, sum rules, and Bethe ansatz calculations. The two-spinon excitations account for 72.89% of the total intensity in S(q,\ensuremath{\omega}). The singularity structure of the exact result is determined analytically and its spectral-weight distribution evaluated numerically over the entire range of the two-spinon continuum. The leading singularities of the frequency-dependent spin autocorrelation function, static spin structure factor, and q dependent susceptibility are determined via sum rules. The impact of the non-two-spinon excitations on the integrated intensity, the susceptibility, the frequency moments, and the Euclidian time representation of S(q,\ensuremath{\omega}) is studied on the basis of finite-size data.

DOI
10.1103/physrevb.55.12510

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The exact expression derived by Bougourzi, Couture, and Kacir for the two-spinon contribution to the dynamic spin structure factor S(q,\ensuremath{\omega}) of the one-dimensional s=1/2 Heisenberg antiferromagnet at T=0 is evaluated for direct comparison with finite-chain transition rates (N\ensuremath{\leqslant}28) and an approximate analytical result previously inferred from finite-N data, sum rules, and Bethe ansatz calculations. The two-spinon excitations account for 72.89% of the total intensity in S(q,\ensuremath{\omega}). The singularity structure of the exact result is determined analytically and its spectral-weight distribution evaluated numerically over the entire range of the two-spinon continuum. The leading singularities of the frequency-dependent spin autocorrelation function, static spin structure factor, and q dependent susceptibility are determined via sum rules. The impact of the non-two-spinon excitations on the integrated intensity, the susceptibility, the frequency moments, and the Euclidian time representation of S(q,\ensuremath{\omega}) is studied on the basis of finite-size data.
Significance of the non-2-spinon part of S(q,ω)S(q,ω)S(q,ω) in the one-dimensional S=1/2S=1/2S=1/2 Heisenberg antiferromagnet at zero temperature
The impact of the non-2-spinon excitations of the one-dimensional S=1/2S=1/2S=1/2 Heisenberg antiferromagnet on the integrated intensity, the susceptibility, the frequency moments, and the Euclidian time representation of the T=0T=0T=0 dynamic spin structure factor S(q,ω)S(q,ω)S(q,ω) is studied on the basis of finite-size data for chains with up to N=28N=28N=28 sites.
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.
We revisit our initial derivations of the exact 2-spinon S2S_2S2​ and 4-spinon S4S_4S4​ dynamical structure factors (DSF). First we show that the latter derivations had normalization factors that are twice and quadriple the correct ones respectively. This means that S2S_2S2​ contributes not 72% as was previously thought but 36% to the total DSF. We also calculated the contribution of S4S_4S4​ to be between 18% and 20% and not 27% as was calculated by Caux and Hagemans. In fact we show that the latter reference had also the normalization factor twice the correct value and had it done the numeric integrations correctly it would have obtained a contribution between 36% and 40% for S4S_4S4​. Furthermore, we prove that its claim that our initial derivation of S4S_4S4​ was also incorrect in its dependency on the spectral parameters is incorrect because fixing the momentun transfer kkk up to 2π2π2π as the latter reference did to justify its claim only amounts to mutltiplying the overall factor by 2 because as we will prove in this paper S4S_4S4​ is periodic in kkk with period 2π2π2π. Also in this paper we derive SnS_nSn​ for general nnn by following a different approach compared to our initial derivation of S4S_4S4​. Although for S4S_4S4​ both the new derivation and the initial one lead to equivalent formulas that are expressed differently, the new form presented in this paper is much more elegant and compact and also reveals new hidden and nontrivial symmetries which substantially simplify the numeric evaluation of S4S_4S4​ and its sum rules. Moreover based on the results of this paper we propose a simple approximation to the total DSF of the Heisenberg model. Finally we comment on how our work might resolve the discrepancy between the exact theoretic results and experiemntal data as reported by Zaliznyak et al.
The authors have evaluated an expression obtained previously for the dynamical spin correlation function of a linear Heisenberg antiferromagnet. The authors discuss the results in relation to both recent neutron scattering experiments and recent computations for finite chains with S= 1 / 2 . It is concluded that contributions to the dynamical structure factor from many accumulating low-frequency excitations provide the high-frequency strength required from sum rule arguments for small wavevectors.
The exact two-spinon part of the dynamic spin structure factor ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ for the one-dimensional s=1/2,s=1/2,s=1/2, XXZ\mathrm{XXZ}XXZ model at T=0T=0T=0 in the antiferromagnetically ordered phase is calculated using recent advances in the algebraic analysis based on (infinite-dimensional) quantum group symmetries of this model and the related vertex models. The two-spinon excitations form a two-parameter continuum consisting of two partly overlapping sheets in $(Q,\ensuremath{\omega})$ space. The spectral threshold has a smooth maximum at the Brillouin zone boundary $(Q=\ensuremath{\pi}/2)$ and a smooth minimum with a gap at the zone center (Q=0).(Q=0).(Q=0). The two-spinon density of states has square-root divergences at the lower and upper continuum boundaries. For the two-spinon transition rates, the two regimes $0<~Q<{Q}_{\ensuremath{\kappa}}$ (near the zone center) and ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2$ (near the zone boundary) must be distinguished, where ${Q}_{\ensuremath{\kappa}}\ensuremath{\rightarrow}0$ in the Heisenberg limit and ${Q}_{\ensuremath{\kappa}}\ensuremath{\rightarrow}\ensuremath{\pi}/2$ in the Ising limit. In the regime ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2,$ the two-spinon transition rates relevant for ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ are finite at the lower boundary of each sheet, decrease monotonically with increasing \ensuremath{\omega}, and approach zero linearly at the upper boundary. The resulting two-spinon part of ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ is then square-root divergent at the spectral threshold and vanishes in a square-root cusp at the upper boundary. In the regime $0<{Q}_{\ensuremath{\kappa}}<~\ensuremath{\pi}/2,$ in contrast, the two-spinon transition rates have a smooth maximum inside the continuum and vanish linearly at either boundary. In the associated two-spinon line shapes of ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega}),$ the linear cusps at the continuum boundaries are replaced by square-root cusps. Existing perturbation studies have been unable to capture the physics of the regime ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2.$ However, their line-shape predictions for the regime $0<~Q<{Q}_{\ensuremath{\kappa}}$ are in good agreement with the exact results if the anisotropy is very strong. For weak anisotropies, the exact line shapes are more asymmetric.
The T=0 dynamics of the one-dimensional s= 1 / 2 planar antiferromagnet is studied by an approach which consists of exact analytic calculations in the Bethe formalism and numerical finite-chain calculations on rings up to 10 spins. The method makes use of well known critical exponents for the correlation functions and of exact sum rules. The authors obtain approximate analytic expressions for both the out-of-plane and the in-plane dynamic structure factors, and for related quantities such as integrated intensities, susceptibilities and autocorrelation functions. The results are discussed in relation to possible experiments on quasi-1D magnetic compounds at low T. The calculations make clear that the T=0 dynamic structure factors are dominated by two-parameter continua of excitations rather than by single branches of spin-waves as predicted by classical spin-wave theory. By varying the planar anisotropy the autocorrelation functions display interesting features in their longtime asymptotic behaviour, such as a crossover from a uniform power-law decay to an oscillatory decay and a crossover between oscillatory decays with different frequencies. They conjecture a possibility of approaching the classical limit s= infinity starting from the quantum limit s= 1 / 2 . This provides a qualitative, and in some aspects even quantitative, understanding of the dynamical behaviour of s> 1 / 2 systems in terms of a quantum approach.