Ground states of a family of frustrated spin models for quasicrystals and their approximants

TL;DR

We address how quasiperiodicity influences magnetic order in quasicrystals and their approximants by studying a family of classical cluster-spin models on square-triangle tilings with intra-cluster coupling $J_1$ and inter-cluster coupling $J_2$. The periodic lattices yield a spectrum of ordered states, from ferromagnetic and antiferromagnetic to coplanar spirals, with some exact solutions via bond–triangle decompositions, while the 6-fold quasicrystal reveals two novel long-range ordered phases: a quasiperiodic three-color cluster antiferromagnet and a mixed infinite-cluster with fluctuating islands. The results provide intuitive, analytically tractable pictures that guide numerical studies and help interpret experimental trends in RKKY-driven magnetism of quasicrystal alloys and their approximants, including the nuanced relationship between Curie-Weiss temperature and the type of low-temperature order. Overall, the work demonstrates that quasiperiodicity can stabilize unique magnetic textures and highlights future directions in extending to quantum effects, longer-range interactions, and spin-glass phenomena.

Abstract

Many new families of quasicrystal-forming magnetic alloys have been synthesized and studied in recent years. For small changes of composition, the alloys can go from quasiperiodic to periodic (approximant crystals) while conserving most of the local atomic environments. Experiments show that many of the periodic approximants order at low temperatures, with clear signatures of ferromagnetic or antiferromagnetic transitions, and also in some cases undergo non-equilibrium spin glass transitions. In contrast, the quasicrystals are mostly found to be spin glasses. Systematically studying these alloys could help elucidate the role played by quasiperiodicity in (de)stabilizing long range magnetic order. In this work, we study cluster spin models with the aim of understanding the mechanisms behind various types of long range magnetic ordering in approximants and quasicrystals. These models embody key features of real systems, and to some extent are analytically tractable, both for periodic and quasiperiodic cases. For the quasicrystal, we describe two novel magnetic phases with quasiperiodic ordering. Our results should serve to motivate further studies with detailed numerical explorations of this family of models.

Figures (18)

• Figure 1: Five examples of spatial arrangements of hexagonal clusters of 6 spins and inter-cluster bonds. The figures show close-ups of local environments in a) Square array b) Hexagonal array c) Square staggered array d) Sigma lattice array and e) 6-fold quasiperiodic tiling. Intra-cluster bonds are shown in red, inter cluster bonds in blue. The underlying square-triangle lattice is shown in grey.
• Figure 2: a) A ground state for the $hex-\triangle$ model in the $+-$ sector showing spin orientations. The orientations for spins on sublattice $\mathcal{S}_1$ are indicated by circles of darker shades, colored red (spins aligned with an axis of reference), blue (at an angle of $2\pi/3$) and green (at an angle of $4\pi/3$). The orientations for spins on sublattice $\mathcal{S}_2$ are indicated by the circles of lighter shades, directed along $-\theta$ (red), $2\pi/3-\theta$ (blue) and $4\pi/3-\theta$ (green). b) Geometrical representation of constraint satisfied by spins in the ground state of $H_{tri}$ defined in Eq.5, for $r\le 2$.
• Figure 3: Three out of the four collinear states of the $hex-\square$ lattice (a trivial ferromagnetic phase is not shown). a) The $+-$ quadrant cluster-antiferromagnet phase c-AFM1, b) the $-+$ quadrant cluster-antiferromagnet phase c-AFM2 and c) the $--$ quadrant cluster-ferromagnet phase c-FM2.
• Figure 4: (Left) The spiral state is illustrated here by showing the variation of angle of orientation for one of the spins of the unit cell along the diagonal direction. The state is periodic along the direction perpendicular to the spiral. The angle $\phi$ is defined in the text. (Right) Spin configuration in the ground state for $J_1=J_2$, when $\phi=\pi/6$. In this case the magnetic ground state is commensurate with the lattice. The colors red, green and blue correspond to the spin orientations such that ${\bf{S}}_{r}+{\bf{S}}_{b}+{\bf{S}}_{g}=0$. A black cross on a red circle indicates that the spin points in the opposite direction $-{\bf{S}}_{r}$ and similarly for the other colors.
• Figure 5: Phase diagram of the $hex-\square$ lattice in the $J_1-J_2$ plane. Ferromagnetic phases are shown in blue and gray-blue, antiferromagnetic phases are shown in red and red-gray. The structures of these phases, which are collinear, are shown in Figs.\ref{['fig:collinear']}. Spiral phases can be incommensurate, or incommensurate but always periodic along one of the diagonals of the lattice. The dashed line represents the line $T_{cw}=0$. The blue line is a path of fixed negative $T_{cw}<0$ in the $J_1-J_2$ plane, and it is seen to intersect four different phases, depending on the values of the couplings.