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Systematic investigation of the effects of disorder at the lowest order throughout the BCS-BEC crossover

A systematic investigation of the effects of disorder on the BCS-BEC crossover at the lowest order in the impurity potential is presented for the normal phase above the critical temperature Tc. Starting with the t-matrix approach for the clean system, by which pairing correlations between opposite-spin fermions evolve from the weak-coupling (BCS) to the strong-coupling (BEC) limits by increasing the strength of the attractive inter-particle interaction, all possible diagrammatic processes are considered where the effects of a disordered potential are retained in the self-energy at the lowest order. An accurate numerical investigation is carried out for all these diagrammatic terms, to determine which of them are mostly important throughout the BCS-BEC crossover. Explicit calculations for the values of Tc, the chemical potential, and the Tan's contact are carried out. In addition, the effect of disorder on the single-particle spectral function is analyzed, and a correlation is found between an increase of Tc and a widening of the pseudo-gap energy at Tc on the BCS side of unitarity in the presence of disorder, while on the BEC side of unitarity the presence of disorder favors the collapse of the underlying Fermi surface. The present investigation is meant to orient future studies when the effects of disorder will be considered at higher orders, with the purpose of limiting the proliferation of diagrammatic terms in which interaction and disorder are considered simultaneously.