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Unconventional Fermi surface in two-dimensional systems of Dirac fermions

At the low energy regime, the decay rate of two-dimensional massless Dirac fermions due to interactions can be written as $\mathrm{Im}Σ(ω) \propto |ω|^{x}$ at zero temperature. We find that the fermion system has: I) no sharp Fermi surface and no well-defined quasiparticle peak for $0<x<1/2$; II) a sharp Fermi surface but no well-defined quasiparticle peak for $1/2\leq x\leq 1$; III) both sharp Fermi surface and well-defined quasiparticle peak for $x>1$. In the presence of long-range gauge/Coulomb interaction or certain massless boson mode, the system exhibits unusual behavior belonging to class II.