Frustrated magnets are a surprisingly rich place to explore the interplay between `classical' orders (or the tendency of the magnetic moment on each site in a crystal to align in a particular direction) and quantum fluctuations. There are two main kinds of questions that we usually ask about these systems: first, what is the static (classical) ordering pattern? And second, are there systems in which quantum fluctuations dominate and there is no classical order at all, even at zero temperature?
Complex classical ordering patterns arise in geometrically frustrated magnets, where each spin wants to be anti-parallel to its neighbors, but the geometric arrangement of the sites on which the spins sit makes this impossible. (Here's a nice picture of this from an article by Leon Balents). In collaboration with Prof. Shivaji Sondhi, I studied the low-temperature order of classical spins on a class of geometrically frustrated lattice which give the least dense known stable packing arrangement of spheres.
Classical antiferromagnetism on Torquato-Stillinger packings
Phys. Rev. B 78, 024407 (2008)
In some situations, the frustration may be so extreme that quantum fluctuations play an essential role in determining the low-temperature order. In this case the ordering is not one of local spin moments, but rather involves ordering in correlations between spins on multiple different crystal sites. This opens the possibility for `spin liquid' orders that (unlike ordering of local spin moments) do not break the translation symmetry of the original lattice. Shivaji Sondhi and I proposed such a spin-liquid order on the highly frustrated pyrochlore lattice.
Monopole flux state on the pyrochlore lattice
Phys. Rev. B 79, 144432 (2009)
This particular spin liquid has some interesting algebraic structure, which is related to the relationship between the geometry of the pyrochlore lattice and that of the fundamental representation of SU(4). R. Shankar, Shivaji Sondhi, and I found that there are a family of lattices whose geometries are related to representations of other Lie groups, where similar spin liquid states can in principle exist.
Flux Hamiltonians, Lie algebras, and root lattices with minuscule decorations
Annals of Physics 324, 267 (2009)