Data supporting "Constructing Emergent U(1) Symmetries in the Gamma-Prime (Γ′) model"

<p>Frustrated magnets can elude the paradigm of conventional symmetry breaking and instead exhibit signatures<br>of emergent symmetries at low temperatures. Such symmetries arise from &ldquo;accidental&rdquo; degeneracies within the<br>ground state manifold and have been explored in a number of disparate models, in both two and three dimen-<br>sions. Here we report the systematic construction of a family of classical spin models that, for a wide variety<br>of lattice geometries with triangular motifs in one, two and three spatial dimensions, such as the kagome or hy-<br>perkagome lattices, exhibit an emergent, continuous U(1) symmetry. This is particularly surprising because the<br>underlying Hamiltonian actually has very little symmetry &mdash; a bond-directional, off-diagonal exchange model<br>inspired by the microscopics of spin-orbit entangled materials (the &Gamma;&prime;-model). The construction thus allows<br>for a systematic study of the interplay between the emergent continuous U(1) symmetry and the underlying<br>discrete Hamiltonian symmetries in different lattices across different spatial dimensions. We discuss the impact<br>of thermal and quantum fluctuations in lifting the accidental ground state degeneracy via the thermal and quan-<br>tum order-by-disorder mechanisms, and how spatial dimensionality and lattice symmetries play a crucial role<br>in shaping the physics of the model. Complementary Monte Carlo simulations, for representative one-, two-,<br>and three-dimensional lattice geometries, provide a complete account of the thermodynamics and confirm our<br>analytical expectations.</p>