Large Deviations and Limit Shapes for Emptiness Formation Probabilities

Emptiness formation probability (EFP) is one of the simplest non-trivial many-body correlation functions in a quantum spin chain, and can probe properties of a ground state inaccessible to regular two point functions, i.e topological order. The expectation value of this quantity can be mapped onto the partition function of a system of repulsive directed polymers on the cylinder, and analyzed through the hydrodynamic 'instanton' framework to reproduce the asymptotic scaling behaviour. We find a third-order phase transition in the EFP for free fermions at finite temperature and derive the geometric transition in the shape of the corresponding instanton; which is an example of the limit shape phenomenon first observed in dimer tiling problems.