Quantum Spherical Models for Dirty Phase Transitions

We construct large-$n$ (spherical) limits for a series of interesting quantum phase transitions in disordered systems, including quantum ferromagnets and spin glasses, superconducting thin films with and without an external magnetic field, the dirty boson problem, the fractional quantum Hall effect, and Nelson's model of flux lines in high-temperature superconductors. The spherical limit always produces a random matrix inversion problem with self-consistency conditions, which then must be solved numerically. We present preliminary results for the dirty boson problem in two spatial dimensions.