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Mean-field theories for many-body systems have been highly desired. For example, Bogoliubov theory of weakly interacting bosons has been a great success in describing Bose-Einstein condensation (BEC) and superfluids in ultra-cold atoms. Extending Bogliubov theory to finite temperature or stronger-coupling regimes, however, can be difficult and most available theories violate at least one of the criteria that a consistent theory for interacting bosons should satisfy. Although the large-N expansion has been applied to this problem, it was widely believed that the leading-order large-N theory is not consistent with Bogoliubov theory and thus should be rejected based on experimental evidence. In this talk I will show that the large-N theory indeed has a Bogoliubov-like dispersion if the correct vacuum is constructed. Moreover, by including the anomalous density in a consistent fashion, one can construct a mean-field theory that satisfies all the criteria. In hindsight, this should be the natural way because the anomalous density has been included in Bogoliubov theory. Our mean-field theories are more transparent if written in the path-integral formalism, which I will present in this talk. Our theoretical framework can be applied to other problems including fermions with tunable attractive interactions and two-component repulsive Bose gases.
Reference: Chien et al., PRA 86, 023634 (2012); PRL 105, 240402 (2010); PRA 83, 053622 (2011), PRA 83, 053637 (2011).