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Journal of Applied Physics

Dynamics of the one-dimensional spin-1 Heisenberg antiferromagnet with exchange and single-site anisotropy

Shu Zhang, Yongmin Yu, V. S. Viswanath, Joachim Stolze, Gerhard H. Muller

Published May 15, 1994• 0 citations
Publisher

Abstract

The T=0 dynamical properties of the one-dimensional s=1 XXZ model with an additional single-site term are investigated by means of the recursion method. The dynamic structure factors Sμμ(q=π,ω), μ=x,z bear the characteristic signatures of several different phase transitions. In the s=1 Heisenberg antiferromagnet, the intrinsic linewidth (at fixed q) of Sμμ(q,ω) is larger for small q than for q near π, in contrast to well-established results for the corresponding s=1/2 model.

DOI
10.1063/1.355519

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arXiv:cond-mat/9712101•Physical review. B, Condensed matter•May 1, 1998•72 citations
The T=0 dynamical properties of the one-dimensional s=1 XXZ model with an additional single-site term are investigated by means of the recursion method. The dynamic structure factors Sμμ(q=π,ω), μ=x,z bear the characteristic signatures of several different phase transitions. In the s=1 Heisenberg antiferromagnet, the intrinsic linewidth (at fixed q) of Sμμ(q,ω) is larger for small q than for q near π, in contrast to well-established results for the corresponding s=1/2 model.
The dynamical properties of the S=1/2 antiferromagnetic XXZ chain are studied by the exact diagonalization and the recursion method of finite systems up to 24 sites. Two types of the exchange interaction are considered: one is the nearest-neighbor type, and the other is the inverse-square one. As the Ising anisotropy becomes larger, there appears a noticeable difference in the transverse component S^{xx}(q,ω) between the two types of the exchange. For the nearest-neighbor type, the peak frequency of S^{xx}(q,ω) for each q approaches the center of the continuum spectrum. On the contrary, the peak frequency for the inverse-square type moves to the upper edge of the continuum, and separates from the continuum for the anisotropy larger than the threshold value. Whether the interaction between domain walls (solitons) is absent or repulsive in the Ising limit leads to this difference in the behavior of S^{xx}(q,ω). In the longitudinal component S^{zz}(q,ω), on the other hand, the feature of the dynamics is scarcely different between the two types. The energy gap and the static properties are also discussed.
The recursion method is applied to the T=0 dynamics of the S=1/2 XXZ model on a linear chain and a square lattice. By means of new calculational techniques for the analysis of the continued-fraction coefficients pertaining to specific dynamical quantities, we obtain reliable information on the type of ordering in the ground state, on the size of gaps in the dynamically relevant excitation spectrum, on the bandwidths of dominant structures in spectral densities, on the exponents of infrared singularities, and on the detailed shape of spectral-weight distributions. We investigate some characteristic properties of the dynamic structure factors ${\mathit{S}}_{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}$(q,\ensuremath{\omega}) and the spin autocorrelation functions ${\mathit{S}}^{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}$(\ensuremath{\omega})=${\mathit{N}}^{\mathrm{\ensuremath{-}}1}$ tsumq{sum}_{\mathbf{q}}sumq​${\mathit{S}}_{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}$(q,\ensuremath{\omega}), specifically their dependence on the uniaxial anisotropy, i.e., on the parameter which controls the type of ordering and the amount of quantum fluctuations in the ground state. We find, for example, that the different degrees of ordering in the planar regime of the one-dimensional and two-dimensional systems (criticality versus antiferromagnetic long-range order) have characteristic signatures in the dynamical properties which are conspicuously displayed in our results.
A combination of world-line quantum Monte Carlo and maximum-entropy methods is used to calculate the dynamic structure factor ${\mathit{S}}_{\mathrm{\ensuremath{\alpha}}\mathrm{\ensuremath{\alpha}}}$(q,\ensuremath{\omega}) for one-dimensional antiferromagnets. A comparison to exact results demonstrates that qualitative features of spectra are reproduced, but are broadened. The broadening is frequency dependent with low-frequency features being reproduced most accurately. Peak positions are well described. Results for the Heisenberg model with S\ensuremath{\le}2 demonstrate the qualitative difference between the dynamics of half-integer and integer spins predicted by Haldane. From the positions of the low-temperature q=\ensuremath{\pi} peaks quantitative estimates are produced for the S=1 and 2 gaps. The q=\ensuremath{\pi} peak position for S=1 increases with temperature in agreement with experiment. When on-site anisotropy is included for S=1 a gap splitting results that resembles that found for NENP [Ni(C2{\mathrm{C}}_{2}C2​H8{\mathrm{H}}_{8}H8​N2{\mathrm{N}}_{2}N2​)2{)}_{2})2​NO2{\mathrm{NO}}_{2}NO2​ClO4{\mathrm{ClO}}_{4}ClO4​].
The recursion method is used to determine the T=0 spin dynamic structure factor Szz(q,ω) in the Luttinger liquid state and in the phase-separated state of the one-dimensional t–J model. As the exchange coupling increases from zero, the dispersions and line shapes of the dominant spin excitations are observed to undergo a major metamorphosis between the free-fermion limit and the onset of phase separation. The familiar two-spinon spectrum of the Heisenberg antiferromagnetic chain emerges gradually in the strongly phase-separated state.
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.