Colloids interacting with periodic substrates such as those created with optical traps are an ideal system in which to study various types of phase transitions such as commensurate to incommensurate states and melting behaviors, and they can also be used to create new types of ordering that can be mapped to spin systems. Here we numerically demonstrate how magnetic colloids interacting with an array of elongated two-state traps can be used to realize square artificial spin ice. By tuning the magnetic field, it is possible to precisely control the interaction strength between the colloids, making it possible to observe a transition from a disordered state to an ordered state that obeys the two-in/two-out ice rules. We also examine the dynamics of excitations of the ground state, including pairs of monopoles, and show that the monopoles have emergent attractive interactions. The strength of the interaction can be modified by the magnetic field, permitting the monopole velocity to be tuned.
Optical traps have been extensively employed to create tailored colloidal crystalline structures where the crystals
can have long range order. Here we discuss how colloidal particles on periodic substrates can be used to
understand how frustration can produce partially ordered states. We demonstrate how to create artificial spin
ice systems using colloidal particles and describe variations on this system that include geometries in which a
random loop model can be realized. We also discuss how frustration effects can be used to control grain boundary
formation by creating energetic defects in the ground state ordering of these systems.