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In this article we study the two-dimensional (2D) ultracold Fermi gas withweak impurity in the framework of mean-field theory where the impurity isintroduced through Gaussian fluctuations. We have investigated the role of theimpurity by studying the experimentally accessible quantities such ascondensate fraction and equation of state of the ultracold systems. Ouranalysis reveals that, at the crossover the disorder enhances superfluidity,which we attribute to the unique nature of the unitary region and to thedimensional effect.

Effect of Weak Disorder on the BCS-BEC crossover in a two-dimensional Fermi Gas