Eight-dimensional quantum Hall effect and "Octonions".

We construct a generalization of the quantum Hall effect where particles move in an eight-dimensional space under an SO(8) gauge field. The underlying mathematics of this particle liquid is that of the last normed division algebra, the octonions. Two fundamentally different liquids with distinct configuration spaces can be constructed, depending on whether the particles carry spinor or vector SO(8) quantum numbers. One of the liquids lives on a 20-dimensional manifold with an internal component of SO(7) holonomy, whereas the second liquid lives on a 14-dimensional manifold with an internal component of G2 holonomy.

Magnetic instability in strongly correlated superconductors.

Recently, a new phenomenological Hamiltonian has been proposed to describe the superconducting cuprates. This so-called Gossamer Hamiltonian is an apt model for a superconductor with strong on-site Coulomb repulsion between the electrons. It is shown that at half-filling the Gossamer superconductor with strong repulsion is unstable toward an antiferromagnetic insulator. The superconducting state undergoes a quantum phase transition to an antiferromagnetic insulator as one increases the on-site Coulomb repulsion. Near the transition the Gossamer superconductor becomes spectroscopically indistinguishable from the insulator.