The authors have a long experience in developing novel methods providing a better understandin of the physics contained in the simple quantum Hamiltonians that may describe generic properties of material with strong electronic correlations (Hubbard model for instance). Here they report on the application of the cellular DMFT (DMFT) method to the attractive Hubbard model (AHM) to gain new insights into the smooth crossover between BCS superconductivity and Bose-Einstein condensation (BEC). CDMFT is an improvement of DMFT that should suppress the effect of infinite dimensions on which DMFT is based.

There has been recently an application of DMFT to the BCS-BEC crossover problem which yielded some surprizes (Capone, Keller). The most interesting point (see below why) is a possible first-order phase transition at T=0 (and densitie n different from half-filling) in the normal (metastable) state between a Fermi liquid (weak coupling, U<Uc) and a insolating liquid of bosonic bound pairs for U>Uc. This transition is expected to also affect the finite temperature properties of the model, especially above Tc. Now if we recall that the AHM is one of the simplest model that is able to describe the problematic pseudogap phase of the high Tc superconductore above Tc, then we easily realize that a better understanding of the normal phase of the AHM may also be relevant in this context.

The main result of the paper is that the first-order phase transition obtained with DMFT does not survive in finite dimensions (here D=2). It is replaced by a smooth transition showing however all the expected spectral properties: a pseudogap for small U and a large gap for large Us. I regret that the authors do not discuss more deeply the origins of these properties. Indeed, their study essentially reveals the how the strong coupling gap (~ U for large Uc) becomes a pseudogap in the weak coupling and eventually vanishes in the limit U->0, yielding the metal phase. It is however accepted (Mermin-Wagner theorem) that, at least in D=2, an additional effect affects the spectral properties: the enhanced superconducting fluctuations. The interplay between these two factors contributes to the difficulties in understanding the high-Tc. Adding the effect of fluctuations to CDMFT may certainly open new perspectives in separating the "correlation contribution" (as described here, also present for D>2) from the "order parameter flucutation contribution" that plays a dominant role in D=2 only. Analyzing the pseudogap experiments under this persepective may provide a robust framework because it does not rely on any assumption about the actual mechanism leading to the pairing of the electrons (or holes).