  The Mott insulator is characterized by having small deviations around the
(integer) average particle density n, with pairs with n-1 and n+1 particles
forming bound states. In one dimension, the effect is captured by a non-zero
value of a non-local "string" of parities, which instead vanishes in the
superfluid phase where density fluctuations are large. Here, we investigate the
interaction induced transition from the superfluid to the Mott insulator, in
the paradigmatic Bose Hubbard model at n=1. By means of quantum Monte Carlo
simulations and finite size scaling analysis on LxM ladders, we explore the
behavior of "brane" parity operators for L going to infinity from one dimension
(i.e., M=1) to two dimensions (i.e., M going to infinity). We confirm the
conjecture that, adopting a standard definition, their average value decays to
zero in two dimensions also in the insulating phase, evaluating the scaling
factor of the "perimeter law" [S.P. Rath et al., Ann. Phys. (N.Y.) 334, 256
(2013)]. Upon introducing a further phase in the brane parity, we show that its
expectation value becomes non-zero in the insulator, while still vanishing at
the transition to the superfluid phase. These quantities are directly
accessible to experimental measures, thus providing an insightful signature of
the Mott insulator.
