Magnetic field-induced degenerate ground state in the classical antiferromagnetic XX model on the icosahedron

TL;DR

This study analyzes the ground state of the classical antiferromagnetic XX model on the icosahedron in an external magnetic field. Using numerical energy minimization of $12$ planar classical spins with the Hamiltonian $H=J \sum_{\langle ij\rangle}(s_i^x s_j^x+s_i^y s_j^y) - h \sum_i s_i^x$, the authors reveal two magnetization discontinuities and a broad degenerate ground-state window near $h=J$. They show that the degeneracy arises from interpentagon coupling that creates a triangular interaction unit, allowing the ten spins to form two zero-magnetization pentagons while two spins align with the field; the resulting degeneracy persists up to the saturated field $h_{sat}=(5+\sqrt{5})J$. The results illuminate how frustration and connectivity in finite molecular magnets produce degenerate manifolds and abrupt magnetization responses, contributing to the understanding of classical spin liquids in highly symmetric frustrated systems.

Abstract

The ground state of the classical antiferromagnetic XX model in a magnetic field is calculated for spins mounted on the vertices of the icosahedron. The magnetization is characterized by two discontinuities as a function of the external field. For a wide field range above the first discontinuity the ground state is degenerate, with two spins related by spatial inversion aligned with the field and the rest forming two magnetization units in the form of pentagons. It is shown that the degeneracy originates from the coupling of the two pentagons, which introduces the triangle, associated with ground-state degeneracy, as an interaction unit in the icosahedron. The magnetization discontinuities are shown to evolve first from the coupling of isolated triangles and then from the coupling of the two spins related by spatial inversion.

Figures (16)

• Figure 1: A projection of the icosahedron on a plane. The circles are classical spins of unit magnitude and each interacts with its five nearest neighbors with strength $J$, as shown by the connecting lines.
• Figure 2: Magnetization per spin $\frac{M^x}{N}$ along the field as a function of the magnetic field over its saturation value $\frac{h}{h_{sat}}$ in the ground state of Hamiltonian (\ref{['eqn:model']}) for the icosahedron. The (red) solid arrows point at the locations of the magnetization discontinuities.
• Figure 3: Polar-angle configuration of the ground state of Hamiltonian (\ref{['eqn:model']}) below the first magnetization discontinuity for the icosahedron. Circles with the same pattern (and color) correspond to polar angles adding up to $2\pi$.
• Figure 4: Polar angles $\phi_i$, $i=1,\dots,N$ as a function of the magnetic field over its saturation value $\frac{h}{h_{sat}}$ in the ground state of Hamiltonian (\ref{['eqn:model']}) for the icosahedron. The (red) solid arrows point at the locations of the magnetization discontinuities.
• Figure 5: The arrows show the spin configuration of the ground state of Hamiltonian (\ref{['eqn:model']}) for $h=J$ for the icosahedron. The solid lines highlight the two pentagons, each of which consists of nearest neighbors of a spin pointing along the magnetic field direction $\hat{x}$, shown in black. The spins of the two corresponding pentagons are highlighted in red and green. The dashed lines connect spins belonging to different pentagons. The dotted lines connect the two spins aligned with the field with their nearest neighbors. These two spins are related by spatial inversion. The ground-state energy is invariant under simultaneous rotation of all pentagon spins around the field by an angle ranging continuously from 0 to $\frac{\pi}{5}$.