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Theory of nuclear magnetic relaxation in Haldane gap materials: an illustration of the use of (1+1)-dimensional field theory techniques

Jacob S. Sagi

Published Jan 1, 2009• 0 citations

Abstract

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.

DOI: 10.14288/1.0085111

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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.

We consider quasi-one-dimensional spin-1 Heisenberg chains with crystal field anisotropy in a uniform magnetic field. We determine the dynamical structure factor in various limits and obtain a fairly complete qualitative picture of how it changes with the applied field. In particular, we discuss how the width of the higher energy single magnon modes depends on the field. We consider the effects of a weak interchain coupling. We discuss the relevance of our results for recent neutron scattering experiments on the quasi-1D Haldane-gap compound NDMAP.

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(

{\mathrm{C}}_{2}

{\mathrm{H}}_{8}

{\mathrm{N}}_{2}

{)}_{2}

{\mathrm{NO}}_{2}

(

{\mathrm{ClO}}_{4}

), obtaining good agreement with both field-theory predictions and recent experiments. Correlation lengths, gaps, and velocities are determined for both polarizations.

We consider the isotropic

S=1

Heisenberg chain with a finite Haldane gap $\mathrm{\ensuremath{\Delta}}$ and use state-of-the-art numerical techniques to investigate its dynamical properties at finite temperature, focusing on the nuclear spin-lattice relaxation rate

1/{T}_{1}

measured in nuclear magnetic resonance (NMR) experiments, for instance. In particular, we analyze the contributions from modes with momenta close to $q\ensuremath{\approx}0$ and $q\ensuremath{\approx}\ensuremath{\pi}$ as a function of temperature. At high-temperature we observe spin diffusion, while at low-temperature we argue that a simple activated behavior $1/{T}_{1}\ensuremath{\propto}exp(\ensuremath{-}\mathrm{\ensuremath{\Delta}}/T)$ can be observed only at temperatures much smaller than the gap $\mathrm{\ensuremath{\Delta}}$.

The low-temperature properties of spin-1/2 one-dimensional (1D) Heisenberg antiferromagnetic (HAF) chains which have relatively small exchange couplings between the spins can be tuned using laboratory-scale magnetic fields. Magnetization measurements, made as a function of temperature, provide phase diagrams for these systems and establish the quantum critical point (QCP). The evolution of the spin dynamics behavior with temperature and applied field in the quantum critical (QC) region, near the QCP, is of particular interest and has been experimentally investigated in a number of 1D HAFs using neutron scattering and nuclear magnetic resonance as the preferred techniques. In the QC phase both quantum and thermal spin fluctuations are present. As a result of extended spin correlations in the chains, magnon excitations are important at finite temperatures. An expression for the NMR spin-lattice relaxation rate

1/{T}_{1}

of probe nuclei in the QC phase of 1D HAFs is obtained by considering Raman scattering processes which induce nuclear spin flips. The relaxation rate expression, which involves the temperature and the chemical potential, predicts scaling behavior of

1/{T}_{1}

consistent with recent experimental findings for quasi-1D HAF systems. A simple relationship between

1/{T}_{1}

and the deviation of the magnetization from saturation (${M}_{S}\ensuremath{-}M$) is predicted for the QC region.