Anisotropic Spin Ice on a Breathing Pyrochlore Lattice

TL;DR

This work shows that bond-dependent anisotropy on a breathing pyrochlore spin-ice lattice markedly reduces ground-state degeneracy and can drive dimensional reduction via intermediate symmetries, yielding planes or lines of decoupled degrees of freedom. By introducing independent couplings $\delta_A$ and $\delta_B$ on the two tetrahedral sublattices, the authors map six distinct ground-state limits and analyze thermodynamics with Monte Carlo simulations and a self-consistent Gaussian approximation of the spin structure factor. The resulting phase diagram includes planar ice, omni-plane, line-order, plane-paramagnetic, and ferromagnetic phases, with either finite-temperature transitions or zero-temperature crossovers and characteristic entropy suppressions below the Pauling value. The study also identifies qualitatively distinct spin-structure-factor signatures across phases, including pinch points surviving only in specific planes, and highlights a route to engineer spin-ice states via strain in breathing pyrochlores, with implications for experiments and dipolar-interaction physics.

Abstract

Spin ice systems represent a prime example of constrained spin systems and exhibit rich low-energy physics. In this study, we explore how introducing a tunable anisotropic spin coupling to the conventional Ising spin ice Hamiltonian on the breathing pyrochlore lattice affects the ground state properties of the system. Significant changes are observed in the ground state structure, reflected in the spin structure factor and in a reduction of residual entropy at low temperatures. We theoretically uncover a rich phase diagram by varying the anisotropy and demonstrate how this modification reduces the ground state degeneracy across different phases. Numerical simulations reveal that, at sufficiently low temperatures, the system either undergoes a crossover into a constrained spin ice manifold, characterized by an entropy density that drops below the Pauling entropy of conventional spin ice, or a phase transition into a symmetry-broken state, depending on the perturbations. Additionally, we compute the spin structure factors for the anisotropic model and compare these results to analytical predictions from a large-$N$ expansion, finding good agreement. This work develops the understanding of spin ice in anisotropic limits, which may be experimentally realized by strain, providing, among others, key signatures in entropy and specific heat.

Figures (9)

• Figure 1: Anisotropic breathing pyrochlore. (a) Unit cell of the breathing pyrochlore lattice with the different tetrahedral sublattices shown in two different colors. The strained bonds (thick tubes) lie in the $x,y$-plane of the unit cell. (b) Single tetrahedron with numbering convention of the lattice sites. The strained tetrahedron is shown with the corresponding interaction coefficients. (c) The strain $\delta$ lifts the degeneracy of the six ground states of a single tetrahedron. For $\delta<0$, a single tetrahedron has two ground states, while for $\delta>0$, a single tetrahedron has four ground states. (d) Phase diagram with six different phases depending on the sign of $\delta_{A/B}$. The mid-point, for $\delta_A = \delta_B = 0$, is the conventional isotropic spin ice model.
• Figure 2: Tetrahedra-planes and spin-lines. (a) A plane of tetrahedra of the same kind is shown. Here, $A$ tetrahedra (blue) with equal $z$ coordinate, or an $x,y$-plane, are depicted as an example. The $A$ tetrahedra and the connecting links on the $B$ tetrahedra (lavender) are shown in color. The two-dimensional projection onto the plane is shown in black. Planes can be formed perpendicular to all spatial directions by either $A$ or $B$ tetrahedra. (b) Two spin lines are shown in color. The 03-line connects spins $S_0^z$ and $S_3^z$ in an $x,y$-plane and the 12-line connects spins $S_1^z$ and $S_2^z$.
• Figure 3: Specific heat and entropy density of anisotropic spin ice. (a) While the isotropic case shows only one bump in the specific heat, all anisotropic cases show a second bump at a lower temperature. (b) Zoom in on the second bump in the specific heat. The temperature scale is linear and rescaled by $\delta$. (c) Entropy density for all anisotropic cases. The entropy density of isotropic spin ice approaches the Pauling residual entropy value $\frac{1}{2}\ln{\left(\frac{3}{2}\right)}$, while for all other anisotropic cases, the entropy density is lowered. For planar ice the entropy density approaches the finite value $\frac{3}{8}\ln{\left(\frac{4}{3}\right)}$ and for the line-order case the entropy density approaches $\frac{1}{4L}\ln(2)$. In the symmetry-broken phases the entropy density drops to zero below the transition. All data is obtained for $L_x = L_y = L_z = 4$ face-centered cubic (fcc) unit cells, corresponding to $N_{\text{spin}} = 1024$.
• Figure 4: Normalized spin structure factor in Spin-Flip (SF) channel. Columns show the same system, and rows have the same cut in the Brillouin zone. Columns : 1 Isotropic, 2 Plane Ice, 3 Omni-Plane, 4 Line-Order, 5 Plane-Paramagnetic, 6 Ferromagnetic. Rows: 1 [hhl], 2 [hl0], 3 [h0l]. Every panel shows Monte Carlo (MC) data on the top and self-consistent Gaussian approximation (SCGA) on the bottom. The Monte Carlo data is obtained by sampling in the degenerate ground-state sector for systems with $L_x = L_y = L_z = 10$ fcc unit cells, corresponding to $N_{\text{spin}} = 16000$. The intensities are normalized across a single system between $0$ and $1$ in a.u..
• Figure 5: SCGA spin structure factor in Spin-Flip (SF) channel at different temperatures. While for high temperatures ($T\gg 1$) the signatures for the isotropic (left) and anisotropic (right) systems agree, when the temperature is lowered (top to bottom), the features look significantly different. The line-order regime is evaluated with $\delta_A = \delta_B = 0.05$ and the [hhl] cut in the Brillouin zone is shown. The scale is chosen uniformly across all temperatures and both systems.