The Hubbard model on the Bethe lattice via variational uniform tree states: metal-insulator transition and a Fermi liquid

We numerically solve the Hubbard model on the Bethe lattice with finite coordination number z=3, and determine its T=0 phase diagram. For this purpose, we introduce and develop the `variational uniform tree state' (VUTS) algorithm, a tensor network (TN) algorithm which generalizes the variational uniform matrix product state algorithm to tree TNs. Our results reveal an AFM insulating phase and a PM metallic phase, separated by a first-order doping-driven metal-insulator transition (MIT). We show that the metallic state is a Fermi liquid (FL) with coherent quasiparticle (qp) excitations for all values of the interaction strength U, and we obtain the finite qp weight Z from the occupation function of a generalized "momentum" variable. We find that Z decreases with increasing U, ultimately saturating to a non-zero, doping-dependent value. Our work demonstrates that TN calculations on tree lattices, and the VUTS algorithm in particular, are a platform for obtaining controlled results for phenomena absent in 1D, such as FLs, while avoiding computational difficulties associated with TNs in 2D. We envision that future studies could observe non-FLs, interaction-driven MIT, and doped spin liquids using this platform.