# Mathematics Colloquia and Seminars

### A non-vanishing spectral gap for a $\nu=1/3$ fractional quantum Hall spin chain.
For a fractional quantum Hall (FQH) system, the Haldane pseudopotential describes the effect of interactions between 2D electrons in a partially filled Landau level. Models with two-body interactions in a magnetic field and periodic boundary conditions can be discretized to a one-dimensional lattice model. We consider a quantum spin chain whose ground state describes a $\nu=1/3$ FQH state on a thin cylinder, and prove that there is a non-vanishing spectral gap above the ground state energy. By introducing void-monomer-dimer (VMD) states we construct an orthogonal basis for the (highly-degenerate) ground state space that we use to prove the gap via the martingale method. We also show that the VMD states have exponential decay of correlations.