  Applied magnetic fields are an important tuning parameter for artificial spin
ice (ASI) systems, as they can drive phase transitions between different
magnetic ground states, or tune through regimes with high populations of
emergent magnetic excitations (e.g., monopole-like quasiparticles). Here, using
simulations supported by experiments, we investigate the thermodynamic
properties and magnetic phases of square and quadrupolar ASI as a function of
applied in-plane magnetic fields. Monte Carlo simulations are used to generate
field-dependent maps of the magnetization, the magnetic specific heat, the
thermodynamic magnetization fluctuations, and the magnetic order parameters,
all under equilibrium conditions. These maps reveal the diversity of magnetic
orderings and the phase transitions that occur in different regions of the
phase diagrams of these ASIs, and are experimentally supported by
magneto-optical measurements of the equilibrium "magnetization noise" in
thermally-active ASIs.
