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Multiple-band nature of electronic energy bands leads to novel physicaleffects in solids. In this paper, we clarify physical properties of a Fermisystem with a pair of electron and hole Fermi surfaces (FSs), whose coupling ismediated by a critical U(1) boson field. By using a one-loop renormalizationgroup analysis, we show that when the boson field undergoes a quantum phasetransition with broken U(1) symmetry, the multiple-band Fermi system shows anon-Fermi liquid (non-FL) behaviour in its thermodynamic and magneticproperties. At a quantum critical point (QCP), Fermi velocities of the two FSsare renormalized into a same critical velocity as a boson velocity, and thefermion's density of states (DOS) shows a pseudo-gap behaviour with alogarithmic energy dependence at the QCP.

Fate of a multiple-band Fermi liquid that is coupled with critical $\phi^4$ bosons