Concept of contact spectrum and its applications in atomic quantum Hall states

A unique feature of ultracold atoms is the separation of length scales, \(r_0\ll k_F^{-1}\), where \(k_F\) and \(r_0\) are the Fermi momentum characterizing the average particle distance and the range of interaction between atoms respectively. For \(s\)-wave scattering, Shina Tan discovered that such diluteness leads to universal relations, all of which are governed by contact, among a wide range of thermodynamic quantities. Here, we show that the concept of contact can be generalized to an arbitrary partial-wave scattering. Contact of all partial-wave scatterings form a contact spectrum, which establishes universal thermodynamic relations with notable differences from those in the presence of \(s\)-wave scattering alone. Moreover, such a contact spectrum has an interesting connection with a special bipartite entanglement spectrum of atomic quantum Hall states, and enables an intrinsic probe of these highly correlated states using two-body short-ranged correlations.