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Journal of Physics A Mathematical and General

Monte Carlo study of four-spinon dynamic structure function in antiferromagnetic Heisenberg model

B. Si-Lakhal, Abdessamad Abada

Published Dec 15, 2003• 4 citations
Publisher

Abstract

Using Monte Carlo integration methods, we describe the behaviour of the exact four-spinon dynamic structure function S 4 in the antiferromagnetic spin Heisenberg quantum spin chain as a function of the neutron energy ω and momentum transfer k . We also determine the four-spinon continuum, the extent of the region in the ( k , ω) plane outside which S 4 is identically zero. In each case, the behaviour of S 4 is shown to be consistent with the four-spinon continuum and compared to that of the exact two-spinon dynamic structure function S 2 . Overall shape similarity is noted.

DOI
10.1088/0305-4470/37/2/017

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Using Monte Carlo integration methods, we describe the behaviour of the exact four-spinon dynamic structure function S 4 in the antiferromagnetic spin Heisenberg quantum spin chain as a function of the neutron energy ω and momentum transfer k . We also determine the four-spinon continuum, the extent of the region in the ( k , ω) plane outside which S 4 is identically zero. In each case, the behaviour of S 4 is shown to be consistent with the four-spinon continuum and compared to that of the exact two-spinon dynamic structure function S 2 . Overall shape similarity is noted.
Using Monte Carlo integration methods, we describe the behavior of the exact four-spinon dynamic structure function S4S_{4}S4​ in the antiferromagnetic spin 1/2 Heisenberg quantum spin chain as a function of the neutron energy ωωω and momentum transfer kkk. We also determine the four-spinon continuum, the extent of the region in the (k,ω)(k,ω)(k,ω) plane outside which S4S_{4}S4​ is identically zero. In each case, the behavior of S4S_{4}S4​ is shown to be consistent with the four-spinon continuum and compared to the one of the exact two-spinon dynamic structure function S2S_{2}S2​. Overall shape similarity is noted.
We discuss some properties of the exact four-spinon dynamical correlation function S4S_4S4​ in the antiferromagnetic spin 1/2 XXXXXXXXX-model, the expression of which we derived recently. We show that the region in which it is not identically zero is different from and larger than the spin-wave continuum. We describe its behavior as a function of the neutron momentum transfer kkk for fixed values of the neutron energy ωωω and compare it to the one corresponding to the exact two-spinon dynamical correlation function S2S_2S2​. We show that the overall shapes are quite similar, even though the expression of S4S_4S4​ is much more involved than that of S2S_2S2​. We finish with concluding remarks.
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.
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.
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.
We consider the transverse dynamical structure factor of the anisotropic Heisenberg spin-1/2 chain ( XXZ model) in the gapped antiferromagnetic regime (Δ>1). Specializing to the case of zero field, we use two independent approaches based on integrability (one valid for finite size, the other for the infinite lattice) to obtain the exact two-spinon part of this correlator. We discuss in particular its asymmetry with respect to the π/2-momentum line, its overall anisotropy dependence, and its contribution to sum rules.