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Experimental realizations of topologically ordered states of matter, such asfractional quantum Hall states, with cold atoms are now within reach. Inparticular, optical lattices provide a promising platform for the realizationand characterization of such states, where novel detection schemes enable anunprecedented microscopic understanding. Here we show that the central chargecan be directly measured in current cold atom experiments using the numberentropy as a proxy for the entanglement entropy. We perform density-matrixrenormalization-group simulations of Hubbard-interacting bosons on coupledchains subject to a magnetic field with $\alpha=\frac{1}{4}$ flux quanta perplaquette. Tuning the inter-chain hopping, we find a transition from a trivialquasi-one dimensional phase to the topologically ordered Laughlin state atmagnetic filling factor $\nu=\frac{1}{2}$ for systems of three or more chains.We resolve the transition using the central charge, on-site correlations,momentum distributions and the many-body Chern number. Additionally, we proposea scheme to experimentally estimate the central charge from Fock basissnapshots. The model studied here is experimentally realizable with existingcold atom techniques and the proposed observables pave the way for thedetection and classification of a larger class of interacting topologicalstates of matter.

Snapshot-based detection of $\frac{1}{2}$-Laughlin states: coupled chains and central charge