Non-local order in Mott insulators, duality and Wilson loops

• Number statistics of cold atoms in optical lattices show non-local correlations.
• These correlations are measurable via single site resolution imaging.
• Incompressible phases exhibit an area law in particle number fluctuations.
• This leads to long-range parity order of Mott-insulators in one dimension.
• Parity order in 2d is connected with a Wilson-loop in a lattice gauge theory.

It is shown that the Mott insulating and superfluid phases of bosons in an optical lattice may be distinguished by a non-local ‘parity order parameter’ which is directly accessible via single site resolution imaging. In one dimension, the lattice Bose model is dual to a classical interface roughening problem. We use known exact results from the latter to prove that the parity order parameter exhibits long range order in the Mott insulating phase, consistent with recent experiments by Endres et al. [M. Endres, M. Cheneau, T. Fukuhara, C. Weitenberg, P. Schauß, C. Gross, L. Mazza, M.C. Bañuls, L. Pollet, I. Bloch, et al., Science 334 (2011) 200]. In two spatial dimensions, the parity order parameter can be expressed in terms of an equal time Wilson loop of a non-trivial U(1)U(1) gauge theory in 2+12+1 dimensions which exhibits a transition between a Coulomb and a confining phase. The negative logarithm of the parity order parameter obeys a perimeter law in the Mott insulator and is enhanced by a logarithmic factor in the superfluid.