(i) Magnetic Frustration from Geometry and SpinOrbit Coupling (Itamar Kimchi) :
Many magnetic systems display "frustration", competing interactions that lead to multiple ground states at the classical level. The system is then poised to realize a variety of phases, including, it is believed, exotic spin liquid states that possess an unusual "topological order" arising from the expulsion of certain topological defect configurations. The different flavors of spin liquid states were studied in, for the triangular and Kagome magnets. Symmetries are realized in a subtle way for these states, which severely constrains them. Finding all possible phases consistent with these constraints led us to new spinliquid states (see figure) with enhanced stability.
Thermal and quantum fluctuations, which are generally associated with their disordering effects, can paradoxically select ordered states in frustrated magnets ("order by disorder"). These have been studied in the context of two spin 1/2 frustrated magnets on a distorted kagome (volborthite lattice and on the hyperkagome lattice [6] (NaIrO), a new three dimensional frustrated lattice.
Unified theory of spiral magnetism in the 3D harmonic honeycomb iridates Li2IrO3: arxiv:14083640 (With Itamar Kimchi, Radu Coldea, James Analytis) We show that the Kitaev interaction stabilizes the counterrotating spiral. By studying a minimal model of zig zag chains. This is argued to be the key to understanding of the remarkably similar magnetic ordering found in two different structures of Li2IrO3.

[4]Spin Liquid States on the Triangular and Kagome Lattices: A Projective Symmetry Group Analysis
Fa Wang and A. Vishwanath.
Phys. Rev. B, 74, 174423 (2006) 
[5] Quantum and Classical Spins on the Spatially Distorted Kagome Lattice: Applications to Volborthite
Fa Wang, A. Vishwanath and Y. B. Kim
condmat/07040933. Submitted to Phys. Rev.B. 
[6] Topological spin liquid on the hyperkagome lattice of Na4Ir3O8
M. J. Lawler, H.Y. Kee, Y. B. Kim and A. Vishwanath
condmat/07050990. Submitted to Phys. Rev. Lett.
(ii) Unconventional Quantum Criticality: In [1] we studied the physics of 2D quantum magnets when certain spacetime defects (hedgehog defects  see figure) are absent. This lack of "topological disorder" completely changes the physics of the model, in particular, even when the spins are fluctuating, the system possesses a hidden order that manifests itself in the form of an emergent photon excitation (light!) and excitations with fractional quantum numbers. Also, continuous phase transitions in these models turn out to be very different in character; and in one situation even turns out to be selfdual.
Interestingly, these deconfined critical points may actually occur quite naturally in certain quantum magnets [2] where quantum interference effects help to a suppress hedgehog defects. Moreover, they can control `Landau Forbidden' continuous transition  that is, transitions between two states of different symmetry that according to Landau's theory of phase transitions (and common sense) would be continuous without special fine tuning.
We are currently engaged in searching for microscopic models where such phenomena arise, and looking to generalize these results to three dimensions.

[1] Emergent Photons and New Transitions in the O(3) Sigma Model with Hedgehog Suppression
O. I. Motrunich, A. Vishwanath
Phys. Rev. B 96, 207202 (2004) 
[2] Deconfined quantum critical points
T. Senthil, Ashvin Vishwanath, Leon Balents, Subir Sachdev, M. P. A. Fisher
Science 303,1490 (2004).
TALK: Emerging Photons and Decaying Electrons
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TALK: Entanglement and Quantum Matter