Vortex duality: Observing the dual nature using order propagators

In condensed matter physics, Kramers-Wannier duality implies that the state disordered by quantum fluctuations or temperature actually corresponds to an ordered state formed from the topological excitations of the “original” ordered state. At first sight it might appear to be impossible to observe this dual order using means associated with the original order. Although true for Ising models, we demonstrate in this paper that this is not a general statement by considering the well-known vortex duality, in particular in the quantum interpretation in 2+1D where it is associated with the quantum phase transition from a superfluid to a Bose Mott insulator. Here, the disordered Mott insulating state is at the same time a dual superconductor corresponding to a Bose condensate of vortices. We present a simple formalism making it possible to compute the velocity propagator associated with the superfluid in terms of the degrees of freedom of the dual theory. The Mott insulator is characterized by a doublet of massive modes, and we demonstrate that one of these modes is nothing else than the longitudinal photon (gauged second sound) of the dual superconductor. For increasing momenta, the system rediscovers the original order, and the effect on the velocity correlator is that the longitudinal photon loses its pole strength. The quantum critical regime as probed by the velocity correlator is most interesting. We demonstrate that at infinite wavelength the continua of critical modes associated with second sound and the dual longitudinal photon are indistinguishable. However, at finite momenta they behave differently, tracking the weight reshuffling found in the quasiparticle spectrum of the disorder state closely.