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arXiv:1109.2607 · Physical Review B

Single-ion anisotropy in Haldane chains and the form factor of the O(3) nonlinear sigma model

Shunsuke C. Furuya, Takafumi J. Suzuki, Shintaro Takayoshi, Yoshitaka Maeda, Masaki Oshikawa

Published Nov 21, 2011• 8 citations
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Abstract

We consider spin-1 Haldane chains with single-ion anisotropy, which exists in known Haldane chain materials. We develop a perturbation theory in terms of anisotropy, where the magnon-magnon interaction is important even in the low-temperature limit. The exact two-particle form factor in the $O(3)$ nonlinear sigma model leads to quantitative predictions on several dynamical properties, including the dynamical structure factor and electron-spin-resonance frequency shift. These agree very well with numerical results, and with experimental data on the Haldane chain material Ni(C${}_{5}$H${}_{14}$N${}_{2}$)${}_{2}$N${}_{3}$(PF${}_{6}$).

DOI
10.1103/physrevb.84.180410

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We consider spin-1 Haldane chains with single-ion anisotropy, which exists in known Haldane chain materials. We develop a perturbation theory in terms of anisotropy, where the magnon-magnon interaction is important even in the low-temperature limit. The exact two-particle form factor in the O(3)O(3)O(3) nonlinear sigma model leads to quantitative predictions on several dynamical properties, including the dynamical structure factor and electron-spin-resonance frequency shift. These agree very well with numerical results, and with experimental data on the Haldane chain material Ni(C5{}_{5}5​H14{}_{14}14​N2{}_{2}2​)2{}_{2}2​N3{}_{3}3​(PF6{}_{6}6​).
We discuss electron spin resonance (ESR) shifts in spin-1 Heisenberg antiferromagnetic chains with a weak single-ion anisotropy, based on several effective field theories: the O(3) nonlinear sigma model (NLSM) in the Haldane phase, free-fermion theories around the lower and the upper critical fields. In the O(3) NLSM, the single-ion anisotropy corresponds to a composite operator which creates two magnons at the same time and position. Therefore, even inside a parameter range where free magnon approximation is valid for thermodynamics, we have to take interactions among magnons into account in order to include the single-ion anisotropy as a perturbation. Although the O(3) NLSM is only valid in the Haldane phase, an appropriate translation of Faddeev-Zamolodchikov operators of the O(3) NLSM to fermion operators enables one to treat ESR shifts near the lower critical field in a similar manner to discussions in the Haldane phase. Our theory gives quantitative agreements with a numerical evaluation using quantum Monte Carlo simulation, and also with recent ESR experimental results on a spin-1 chain compound Ni(C5H14N2)2N3(PF6)\mathrm{Ni}{({\mathrm{C}}_{5}{\mathrm{H}}_{14}{\mathrm{N}}_{2})}_{2}{\mathrm{N}}_{3}({\mathrm{PF}}_{6})Ni(C5​H14​N2​)2​N3​(PF6​).
We study both the static and dynamic properties of gapped, one-dimensional, Heisenberg, antiferromagnetic, spin chains at finite temperature through an analysis of the O(3) nonlinear sigma model. Exploiting the integrability of this theory, we are able to compute an exact low-temperature expansion of the finite temperature correlators. We do so using a truncated ``form-factor'' expansion and so provide evidence that this technique can be successfully extended to finite temperature. As a direct test, we compute the static zero-field susceptibility and obtain an exact match to the susceptibility derived from the low-temperature expansion of the exact free energy. We also study transport properties, computing both the spin conductance and the NMR-relaxation rate, 1/T1.{1/T}_{1}.1/T1​. We find these quantities to show ballistic behavior. In particular, the computed spin conductance exhibits a nonzero Drude weight at finite temperature and zero applied field. The physics thus described differs from the spin diffusion reported by Takigawa et al. [Phys. Rev. Lett. 76, 2173 (1996)] from experiments on the Haldane gap material, AgVP2S6.{\mathrm{AgVP}}_{2}{S}_{6}.AgVP2​S6​.
We use a quantum O(N) nonlinear \ensuremath{\sigma} model to describe one-dimensional quantum Heisenberg antiferromagnets with integer spin S. A large-N analytical treatment of this effective theory leads to definite predictions for thermodynamic quantities in the low-temperature regime. We obtain the Haldane gap as a function of the temperature and note a trend similar to that found in the compound NENP. The specific heat and magnetic susceptibility are also obtained and comparison with exact numerical results for the S=1 spin chain allows us to determine the domain of validity of our approximation. By a calculation at subleading 1/N order we compute the quasiparticle lifetime. Results are given on-shell, for the width observed at the quasiparticle pole as seen in neutron-scattering experiments, and also off-shell, for the NMR relaxation rate ${\mathit{T}}_{1}^{\mathrm{\ensuremath{-}}1}$. The phenomenological interacting boson model is compared with our approach. We discuss the effects of anisotropy and compare our results with present experiments on NENP.
The structure function S(k) for the s=1, Haldane-gap antiferromagnetic chain is measured accurately using the recent density-matrix renormalization-group method, with a chain length of 100. Excellent agreement with the nonlinear-\ensuremath{\sigma}-model prediction is obtained, both at k\ensuremath{\approxeq}\ensuremath{\pi}, where a single-magnon process dominates, and at k\ensuremath{\approxeq}0, where a two-magnon process dominates. We repeat our calculation with crystal-field anisotropy chosen to model 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​), obtaining good agreement with both field-theory predictions and recent experiments. Correlation lengths, gaps, and velocities are determined for both polarizations.
Improving matrix-product state techniques based on the purification of the density matrix, we are able to accurately calculate the finite-temperature dynamic response of the infinite spin-1 XXZ\text{XXZ}XXZ chain with single-ion anisotropy in the Haldane, large-DDD, and antiferromagnetic phases. Distinct thermally activated scattering processes make a significant contribution to the spectral weight in all cases. In the Haldane phase, intraband magnon scattering is prominent, and the on-site anisotropy causes the magnon to split into singlet and doublet branches. In the large-DDD phase response, the intraband signal is separated from an exciton-antiexciton continuum. In the antiferromagnetic phase, holons are the lowest-lying excitations, with a gap that closes at the transition to the Haldane state. At finite temperatures, scattering between domain-wall excitations becomes especially important and strongly enhances the spectral weight for momentum transfer $\ensuremath{\pi}$.
We investigate the magnetic field behaviour of an antiferromagnetic Heisenberg spin-1 chain with the most general single-ion anisotropy. We discuss the regime in which the magnetic field is below the transition value. The splitting of the Haldane triplet is obtained as a function of a field applied in an arbitrary orientation by means of a Lanczos exact diagonalization of chains of up to 16 spins. Our results are well summarized in terms of a first-order perturbation theory. We explain various level crossings that occur by the existence of discrete symmetries. A discussion is given of the electron-spin-resonance and neutron-scattering experiments on the compound Ni(C 2 H 8 N 2 ) 2 NO 2 ClO 4 (NENP).