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We develop a theory for quantum phases and quantum multicriticality inbilayer graphene in the presence of an explicit energy gap in thenon-interacting spectrum by extending previous renormalization group (RG)analyses of electron-electron interactions in gapless bilayer graphene atfinite temperature to include the effect of an electric field appliedperpendicular to the sample. We determine the possible outcomes of theresulting RG equations, represented by "fixed rays" along which ratios of thecoupling constants remain constant and map out the leading instabilities of thesystem for an interaction of the form of a Coulomb interaction that is screenedby two parallel conducting plates placed equidistant from the electron. We findthat some of the fixed rays on the "target plane" found in the zero-field caseare no longer valid fixed rays, but that all four of the isolated rays arestill valid. We also find five additional fixed rays that are not present inthe zero-field case. We then construct maps of the leading instability (orinstabilities) of the system for the screened Coulomb-like interaction as afunction of the overall interaction strength and interaction range for fourvalues of the applied electric field. We find that the pattern of leadinginstabilities is the same as that found in the zero-field case, namely that thesystem is unstable to a layer antiferromagnetic state for short-rangedinteractions, to a nematic state for long-ranged interactions, and to both forintermediate-ranged interactions. However, if the interaction becomes toolong-ranged or too weak, then the system will exhibit no instabilities. Theranges at which the nematic instability first appears, the antiferromagneticinstability disappears, and the nematic instability disappears all decreasewith increasing applied electric field.

Quantum multicriticality of bilayer graphene in the presence of a tunable energy gap