Quantum phase transitions and collapse of the Mott gap in the $d=1+ϵ$ dimensional half-filled Hubbard model

We study the low-energy asymptotics of the half-filled Hubbard model with a circular Fermi surface in $d=1+ϵ$ continuous dimensions, based on the one-loop renormalization-group (RG) method. Peculiarity of the $d=1+ϵ$ dimensions is incorporated through the mathematica structure of the elementary particle-partcile (PP) and particle-hole (PH) loops: infrared logarithmic singularity of the PH loop is smeared for $ϵ>0$. The RG flows indicate that a quantum phase transition (QPT) from a metallic phase to the Mott insulator phase occurs at a finite on-site Coulomb repulsion $U$ for $ϵ>0$. We also discuss effects of randomness.