Fracton Spin Liquid and Exotic Frustrated Phases in Ising-like Octochlore Magnets

Abstract

For nearly three decades frustrated magnetism research in three dimensions (3D) has centered on the pyrochlore geometry of corner-sharing tetrahedra and the classical spin liquid (CSL) known as spin ice. In this work, we propose that a lattice of corner-sharing octahedra -- appropriately dubbed the octochlore lattice -- may provide a next-generation platform for the study of 3D frustrated magnetism, with realizations in anti-perovskite and certain potassium-fluoride compounds. We study the phase diagram of Ising spins on the octochlore lattice with first- and second-neighbor interactions within each octahedron, which displays a rich variety of frustrated phases, including CSLs with extensive ground state degeneracies, as well as phases with subextensive ground state degeneracies intermediate between spin liquids and long-range order. In addition to a spin ice CSL, we identify a novel fracton CSL with excitations restricted to move along one-dimensional (1D) lines, which is a classical U(1) analog of the celebrated X-cube model, a paradigmatic realization of fracton topological order. The existence of these two CSLs is rationalized as condensation of 1D ferro-spinons bound states from a parent phase with subextensive degeneracy due to frustration of ferromagnetically polarized chains. We also find a spin nematic phase exhibiting two-stage dimensional reduction from cubic to tetragonal (uniaxial) and finally orthorhombic (biaxial) symmetry, driven by strong fluctuations arising from deconfined 1D antiferro-spinons. This work paves the way for the potential realization of fracton CSLs and the exploration of other exotic frustrated states in real materials.

Figures (13)

• Figure 1: Overview of the octochlore Ising system. (a) Octochlore lattice with first ($J_1$) and second ($J_{2a}$ and $J_{2b}$) neighbor interactions. Octahedron centers form a bipartite simple cubic lattice; one "A" and one "B" octahedron are marked. Orthogonal 1D chains are illustrated with red, blue, and green colors. (b,c,d) The Ising moment configurations can be decomposed into multipole contributions---monopole ($A_{1u}$), dipole ($T_{1g}$), and quadrupole ($E_u$) moments---which transform as irreducible representations (irreps) of the octahedral point group symmetry. Note that the $E_u$ irrep is inconsistent with the fixed moment constraint $\vert S_i^z\vert = 1$, as indicated by doubled moment size ($\times\, 2$) in panel (d). (e) Variation of irrep energies from \ref{['eq:irrep_energies']} as a function of $\theta$, with the parameterization $J_1 = J\cos(\theta)$ and $J_{2a} = J\sin(\theta)$. (f) Finite-temperature phase diagram of \ref{['eq:H']} as the function of $\theta$ and temperature, with zero temperature at the outer edge and infinite temperature at the center. Black dots mark thermal phase transition critical temperatures obtained from Monte Carlo simulation \ref{['apx:monte_carlo']}. The radial temperature scale is given by $r(T) = 1-\tanh \sqrt{T/10J}\in[0,1]$, with $T=0$ at the outer edge, increasing towards the center. A linear temperature scale version is provided in \ref{['apx:monte_carlo']}.
• Figure 2: Fragmentation of the $\bm{E_u}$ irrep. There are no spin configurations with fixed-length Ising moments, $S_i^z \in \pm 1$ on all sites, which are purely quadrupolar. Instead, there are two symmetry classes of Ising configurations which mix $E_u$ with one of the other two irreps. An example of each is given on the left side of (a) and (b). On the right, we decompose each as a linear combination of irreducible monopole ($A_{1u}$), dipole ($T_{1g}$), and quadrupole ($E_u$) configurations with variable spin lengths $S_i^z \in \{0,\pm 1, \pm 2\}$, where $S_i^z = 0$ corresponds to the absence of a spin. Beneath each configuration, we have written a polynomial with the corresponding symmetry, where $r^2 = x^2 + y^2 + z^2$.
• Figure 3: 1D spinon interactions in the $\bm{T_{1g}}$ phase. (a) When $J_1=0$, the system decouples into 1D Ising chains. For $J_{2a}> 0$, each chain is ferromagnetic, and the fundamental excitations are positive and negative charged spinons created from a ground state by flipping a straight open string (chain segment) of head-to-tail spins, denoted by the red tube. (b) For small $\vert J_1 \vert$, the energy of the polarized chain states does not change, and the system retains a subextensive ground state degeneracy. (c) The single-spinon energy gap is $2J_{2a}$, independent of $J_1$. (d-g) A finite positive (negative) $J_1$ introduces attraction (repulsion) between spinons of opposite charge.
• Figure 4: Randomly polarized chains in the $\bm{T_{1g}}$ phase. A snapshot of the value the chain ferromagnetic order parameters $\mathcal{D}_c$, \ref{['eq:ferro_chain_OP']}, where each chain is associated with a point on the surface of a cube. This snapshot is from an system with 32 spins per chain that has been slowly annealed from high temperature. In our simulations we always find that the chains adopt an essentially random polarization across the entire $T_{1g}$ frustrated chains phase.
• Figure 5: Spin liquids from 1D spinon condensation. (a) Variation of multi-spinon energies with $J_1/J_{2a}$. Pairs of oppositely charged spinons [\ref{['fig:spinon_interactions']}(f)] condense at the spin ice point $\cramped{J_1/J_{2a}=1}$, while triple-charges bound states [\ref{['fig:spinon_interactions']}(e)] condense at the X-cube point $\cramped{J_1/J_{2a} = -1/2}$, giving rise to two different CSLs. (b) At the spin ice point, a single spinon can turn a corner onto an orthogonal chain without costing any energy; there is no distinction between the colors of spinons, they are free to move in 3D. (c) At the X-cube point, a spinon can only turn a corner by changing its sign and emitting a spinon along the third orthogonal chain, which costs an energy $2J_{2a}$. Thus, individual spinons are confined to move along 1D lines---they are lineons.