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

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