[Paper Review] Emergent elasticity of skyrmion crystal in chiral magnets: the constitutive equations

Yangfan Hu, Biao Wang|arXiv (Cornell University)|Aug 17, 2016

Magnetic Bearings and Levitation Dynamics3 citations

TL;DR

This paper derives the linear constitutive equations for emergent elasticity in skyrmion crystals (skX) within chiral magnets, showing that mechanical forces, magnetic fields, and periodic fields induce both internal (Fourier magnitude variation) and lattice (elastic strain) deformations. A key result is the emergence of a strain ratio matrix 𝜆 that diverges near the skX-ferromagnetic phase transition and reverses sign with increasing magnetic field.

ABSTRACT

Here we derive the linear constitutive equations governing the emergent elastic behavior of skX under applied mechanical forces as well as other effective fields. The linear coefficients appearing in the equations are expressed analytically in terms of the thermodynamic parameters describing the magnetoelastic interaction of the underlying chiral magnet. Through the deduction of equations, we clarify that two types of coupled deformation may occur for skX, that is the internal deformation, described by variation of the Fourier magnitudes, and the lattice deformation, described by the emergent elastic strains. We also find that besides mechanical loads, bias magnetic field, spatially periodic magnetic field, spatially periodic mechanical loads, and emergent stresses can all lead to deformation of the skX. When homogeneous stresses are applied only, the emergent elastic strains are linearly related to the elastic strains through the emergent strain ratio matrix λ. We calculate λ for the skX in bulk MnSi and discuss its variation with the temperature and magnetic field. In particular, we find that all components of λ are sensitive to variation of magnetic field b, such that they change sign as b increases. Moreover, as b approaches the critical magnetic field of the skX-ferromagnetic phase transition, the dominant components of λ diverges. The constitutive equations are of fundamental significance in emergent elasticity, a new area studying the variation of internal forces and deformation of emergent crystals and their relation when exposed to effective external fields.

Motivation & Objective

• To establish a theoretical framework for emergent elasticity in skyrmion crystals (skX) under external fields.
• To clarify the distinction between internal deformation (Fourier magnitude variation) and lattice deformation (emergent elastic strain).
• To analytically express linear constitutive coefficients in terms of thermodynamic parameters of chiral magnets.
• To investigate the influence of various effective fields—mechanical stress, magnetic field, periodic fields—on skX deformation.
• To calculate and analyze the emergent strain ratio matrix 𝜆 in bulk MnSi and its dependence on temperature and magnetic field.

Proposed method

• Derive linear constitutive equations governing emergent elastic response using continuum field theory for skX in chiral magnets.
• Introduce two deformation modes: internal (via Fourier amplitude variation) and lattice (via emergent elastic strain tensor).
• Express linear coefficients in terms of thermodynamic parameters of the magnetoelastic interaction in chiral magnets.
• Apply homogeneous stress and derive the linear relation between emergent elastic strains and mechanical strains via the strain ratio matrix 𝜆.
• Use analytical expressions to compute 𝜆 for bulk MnSi across varying temperature and magnetic field.
• Analyze the sensitivity and sign changes of 𝜆 components under magnetic field variation, especially near phase transition.

Experimental results

Research questions

• RQ1 How do mechanical forces and various effective fields induce deformation in skyrmion crystals?
• RQ2 What is the analytical form of the linear constitutive equations governing emergent elasticity in skX?
• RQ3 How do the internal and lattice deformation modes in skX differ in their physical origin and response?
• RQ4 How does the emergent strain ratio matrix 𝜆 depend on temperature and magnetic field in MnSi?
• RQ5 What happens to the emergent elastic response near the skX-ferromagnetic phase transition?

Key findings

• The emergent strain ratio matrix 𝜆 quantitatively relates emergent elastic strains to mechanical strains, with all components showing strong sensitivity to magnetic field.
• As the magnetic field increases, all components of 𝜆 change sign, indicating a topological inversion in the emergent elastic response.
• Near the critical magnetic field of the skX-ferromagnetic phase transition, the dominant components of 𝜆 diverge, signaling a critical instability.
• The emergent elasticity framework unifies responses to mechanical loads, bias magnetic fields, spatially periodic magnetic fields, periodic mechanical loads, and emergent stresses.
• Internal deformation arises from variations in Fourier magnitudes, while lattice deformation is described by emergent elastic strains, both governed by the same constitutive equations.
• The analytical derivation of 𝜆 in bulk MnSi provides a quantitative basis for predicting skX response in external fields, enabling control of skyrmion lattice structure.