Cold atoms, free fermions and the Kardar-Parisi-Zhang equation

As the simplest example of a system of noninteracting cold atoms, I’ll discuss the example of N free fermions trapped in a harmonic well. We will see that even without interactions, this is an interesting many-body system with nontrivial quantum fluctuations arising purely from the Pauli exclusion principle. In 1d and at T = 0, the quantum fluctuations of the positions
of the fermions can be exactly mapped to the eigenvalues of a Gaussian Hermitian random matrix. A lot of nice exact results for the fermions can be obtained using this correspondence.In particular, the position of the rightmost fermion in 1 − d, T = 0, is described by the celebrated Tracy-Widom distribution for the top eigenvalue of a random matrix. I’ll then consider how these results can be generalized to finite temperature. Remarkably, at finite T, the position of the rightmost fermion has the same distribution as the height at finite time of the (1+1)-dimensional interfaces described by the continuum Kardar-Parisi-Zhang equation. If time permits, I’ll also discuss the generalizations to higher dimensions.