[Paper Review] Modification of the classical Heisenberg helimagnet by weak uniaxial anisotropy and magnetic field

Igor Zaliznyak, M. E. Zhitomirsky|arXiv (Cornell University)|Jun 13, 2003

Advanced NMR Techniques and Applications3 citations

TL;DR

This paper presents a perturbative microscopic approach to determine the classical ground state of weakly anisotropic Heisenberg helimagnets in an external magnetic field, treating anisotropy as a small perturbation to the isotropic exchange structure. It derives explicit expressions for the spin configuration, uniform magnetization, and magnetic Bragg peak intensities, revealing that a magnetic field stabilizes higher-order Fourier harmonics (e.g., $n\mathbf{Q}$) and induces a magnon gap $\sim H^3$ in antiferromagnetic spirals.

ABSTRACT

Classical ground state of the isotropic Heisenberg spin Hamiltonian on a primitive Bravais lattice is known to be a single-Q planar spin spiral. A uniaxial anisotropy and external magnetic field distort this structure by generating the higher-order Fourier harmonics at wave-vectors nQ in the spatial spin configuration. These features are not captured in the formalism based on the Luttinger-Tisza theorem, where the classical ground state energy is minimized under the ``weak'' condition on the length of the spins. We discuss why the correct solution is lost in that approach and present an alternative microscopic treatment of the problem. It allows to find the classical ground state for general Q for both easy-axis and easy-plane uniaxial second-order anisotropy, and for any orientation of the magnetic field, by treating the effect of anisotropy (but not the field) as a perturbation to the exchange structure. As a result, the classical ground state energy, the uniform magnetization, and the magnetic Bragg peak intensities that are measured in experiment, are calculated.

Motivation & Objective

• To develop a general, microscopic perturbative method for determining the classical ground state of weakly anisotropic Heisenberg helimagnets under magnetic fields, avoiding limitations of the Luttinger-Tisza approach.
• To calculate the spin configuration, uniform magnetization, and magnetic Bragg peak intensities for arbitrary field orientation and both easy-axis and easy-plane anisotropy.
• To provide a tractable framework for subsequent spin-wave theory and fluctuation corrections, enabling comparison with neutron scattering experiments.
• To demonstrate that magnetic fields stabilize higher-order Fourier harmonics (e.g., $n\mathbf{Q}$) in the spin structure, particularly in antiferromagnetic spirals.
• To clarify why the Luttinger-Tisza method fails to capture the correct ground state in the presence of anisotropy and field, and to present an alternative microscopic treatment.

Proposed method

• Treat uniaxial anisotropy (but not the magnetic field) as a perturbation to the isotropic Heisenberg ground state, which is a single-$Q$ planar spiral.
• Use a spin-rotated frame to describe deviations from the isotropic spiral, introducing small angle variations $\delta\theta_i$, $\delta\phi_i$ in spin orientation.
• Expand the energy functional in powers of the anisotropy constant $D$, retaining terms up to first order in $D$ to determine the ground state.
• Derive self-consistent equations for the spin configuration by minimizing the energy under the constraint of fixed spin length, solving order-by-order in $D$.
• Include the Zeeman term from the magnetic field as a linear perturbation, allowing for arbitrary field orientation by retaining $\sin(\mathbf{Q} \cdot \mathbf{r}_i)$ and $\cos(\mathbf{Q} \cdot \mathbf{r}_i)$ components.
• Use the resulting spin configuration to compute the uniform magnetization and magnetic Bragg peak intensities, which are measurable in neutron scattering.

Experimental results

Research questions

• RQ1 Why does the Luttinger-Tisza method fail to capture the correct classical ground state in the presence of weak uniaxial anisotropy and magnetic field?
• RQ2 How do weak uniaxial anisotropy and an external magnetic field modify the classical spin configuration of a Heisenberg helimagnet?
• RQ3 What is the role of higher-order Fourier harmonics ($n\mathbf{Q}$) in the spin structure under anisotropy and field?
• RQ4 How does the magnetic field stabilize a type-B spin structure in antiferromagnetic spirals, and what is the resulting magnon gap?
• RQ5 Can a general analytical framework be developed for the spin reorientation process in helimagnets that avoids the complexity of multi-sublattice models?

Key findings

• The correct classical ground state is lost in the Luttinger-Tisza method due to its weak constraint on spin length, which fails to capture the anisotropy-induced reorientation.
• For an antiferromagnetic spiral with $D>0$ and $\mathbf{H} \perp z$, a spin-flop transition occurs at $H_c$ given by Eq. (32), beyond which the spin structure becomes identical to the one described in Sec. 4.1 with $D>0$.
• A magnetic field stabilizes a type-B spin structure in the sixth order in $H$, leading to a magnon gap $\sim H^3$ in the spin-wave spectrum.
• The uniform magnetization and magnetic Bragg peak intensities are calculated explicitly, providing direct comparison with neutron scattering experiments.
• The method successfully captures the generation of higher-order Fourier harmonics ($n\mathbf{Q}$) in the spin configuration due to the combined effects of anisotropy and field.
• The approach is generalizable to any field orientation by retaining both $\sin(\mathbf{Q} \cdot \mathbf{r}_i)$ and $\cos(\mathbf{Q} \cdot \mathbf{r}_i)$ terms in the spin angle variations.