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

TL;DR

This paper introduces a tensor network algorithm to study the Hubbard model on a Bethe lattice, revealing a doping-driven metal-insulator transition and Fermi liquid behavior, providing a new controlled approach for complex lattice phenomena.

Contribution

The paper develops the variational uniform tree state (VUTS) algorithm for tensor networks on tree lattices, enabling controlled studies of phenomena like Fermi liquids and metal-insulator transitions.

Findings

Identified antiferromagnetic insulator and paramagnetic metal phases.

Discovered a first-order doping-driven metal-insulator transition.

Confirmed the metallic state as a Fermi liquid with decreasing quasiparticle weight Z.

Abstract

We numerically solve the Hubbard model on the Bethe lattice with finite coordination number $z=3$, and determine its zero-temperature phase diagram. For this purpose, we introduce and develop the `variational uniform tree state' (VUTS) algorithm, a tensor network algorithm which generalizes the variational uniform matrix product state algorithm to tree tensor networks. Our results reveal an antiferromagnetic insulating phase and a paramagnetic metallic phase, separated by a first-order doping-driven metal-insulator transition. We show that the metallic state is a Fermi liquid with coherent quasiparticle excitations for all values of the interaction strength $U$, and we obtain the finite quasiparticle weight $Z$ from the single-particle 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 tensor-network calculations on tree lattices, and the VUTS algorithm in particular, are a platform for obtaining controlled results for phenomena absent in one dimension, such as Fermi liquids, while avoiding computational difficulties associated with tensor networks in two dimensions. We envision that future studies could observe non-Fermi liquids, interaction-driven metal-insulator transitions, and doped spin liquids using this platform.