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arXiv:cond-mat/9506137 · Physical review. B, Condensed matter

Theory of nuclear magnetic relaxation in Haldane-gap antiferromagnets

Jacob S. Sagi, Ian K. Affleck

Published Apr 1, 1996• 56 citations
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Abstract

A theory of nuclear magnetic resonance is developed for integer-spin, one-dimensional antiferromagnets, which exhibit the Haldane gap. We consider free boson, free fermion, and nonlinear $\ensuremath{\sigma}$-model approaches, all of which give similar results. The detailed anisotropy and magnetic field dependence is calculated and compared with experiment.

DOI
10.1103/physrevb.53.9188

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Journal of Physics Condensed Matter•May 15, 1989•539 citations
A theory of nuclear magnetic resonance is developed for integer-spin, one-dimensional antiferromagnets, which exhibit the Haldane gap. We consider free boson, free fermion, and nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ-model approaches, all of which give similar results. The detailed anisotropy and magnetic field dependence is calculated and compared with experiment.
A comprehensive theory of nuclear magnetic relaxation in S = 1 Haldane gap materials is developed using nonlinear-σ, boson and fermion models. We find that at temperatures much smaller than the lowest gap the dominant contribution to the relaxation rate comes from two magnon processes with T₁⁻¹ ∼ e[sup -Δm/T], where Δm is the smallest gap corresponding to a polarization direction perpendicular to the field direction. As the gap closes, we find that the dominant contribution comes from one magnon processes, and the result depends on the symmetry of the Hamiltonian. Overall the models agree qualitatively, except near the critical regime, where the fermion model is shown to be the best description. We include a thorough discussion of the effects of interchain-couplings, nearest neighbour hyperfine interactions and crystal structure, and introduce a new theory of impurities corresponding to broken chain ends weakly coupled to bulk magnons. The work is then applied to recent measurements on NENP. We find overall fair agreement between available T₁⁻¹ data and our calculations. We finish by suggesting further experimental tests of our conclusions.
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.
In terms of spinless fermions and spin waves, we describe magnetic properties of a spin-1/2 ferromagnetic– antiferromagnetic bond-alternating chain which behaves as a Haldane-gap antiferromagnet. On one hand, we employ the Jordan–Wigner transformation and treat the fermionic Hamiltonian within the Hartree–Fock approximation. On the other hand, we employ the Holstein– Primakoff transformation and modify the conventional spin-wave theory so as to restore the sublattice symmetry. We calculate the excitation gap, the specific heat, the magnetic susceptibility, magnetization curves, and the nuclear spin-lattice relaxation rate with varying bond alternation. These schemes are further applied to a bond-alternating tetramerized chain which behaves as a ferrimagnet. The fermionic language is particularly stressed as a useful tool to investigate one-dimensional spin-gapped antiferromagnets, while the bosonic one works better for ferrimagnets.
In terms of spinless fermions and spin waves, we describe the magnetic properties of a spin-1/2 ferromagnetic-antiferromagnetic bond-alternating chain which behaves as a Haldane-gap antiferromagnet. On the one hand, we employ the Jordan–Wigner transformation and treat the fermionic Hamiltonian within the Hartree–Fock approximation. On the other hand, we employ the Holstein–Primakoff transformation and modify the conventional spin-wave theory so as to restore the sublattice symmetry. We calculate the excitation gap, the specific heat, the magnetic susceptibility, magnetization curves, and the nuclear spin-lattice relaxation rate with varying bond alternation. These schemes are further applied to a bond-alternating tetramerized chain which behaves as a ferrimagnet. The fermionic language is particularly stressed as a useful tool for investigating one-dimensional spin-gapped antiferromagnets, while the bosonic one works better for ferrimagnets.
One-dimensional integer-spin Heisenberg antiferromagnets have disordered ground states and a gap to a triplet magnon near the antiferromagnetic wave vector, k\ensuremath{\approxeq}\ensuremath{\pi}. Near a zero wave vector the lowest energy excitation is a pair of magnons. We calculate the neutron-scattering cross section near k=0, using a Landau-Ginsburg model and exact S-matrix results for the O(3) nonlinear \ensuremath{\sigma} model. The cross section is proportional to k2{\mathit{k}}^{2}k2. As a function of energy, it shows a rounded peak somewhat above the two-magnon threshold. The effects of anisotropy are also considered.