Spin- and charge-density waves in the Hartree-Fock ground state of the two-dimensional Hubbard model

TL;DR

This study investigates the magnetic and charge density wave ground states of the 2D Hubbard model using unrestricted Hartree-Fock theory, revealing the evolution of spin and charge order with doping and interaction strength.

Contribution

It introduces new numerical approaches to minimize finite-size effects and provides a detailed analysis of the SDW phases and their evolution in the Hubbard model.

Findings

Low doping leads to linear spin-density waves with antiferromagnetic order.

Charge order is weaker than spin order, indicating mobile holes.

At higher interactions, holes become localized and phases evolve.

Abstract

The ground states of the two-dimensional repulsive Hubbard model are studied within the unrestricted Hartree-Fock (UHF) theory. Magnetic and charge properties are determined by systematic, large-scale, exact numerical calculations, and quantified as a function of electron doping $h$. In the solution of the self-consistent UHF equations, multiple initial configurations and simulated annealing are used to facilitate convergence to the global minimum. New approaches are employed to minimize finite-size effects in order to reach the thermodynamic limit. At low to moderate interacting strengths and low doping, the UHF ground state is a linear spin-density wave (l-SDW), with antiferromagnetic order and a modulating wave. The wavelength of the modulating wave is $2/h$. Corresponding charge order exists but is substantially weaker than the spin order, hence holes are mobile. As the interaction is increased, the l-SDW states evolves into several different phases, with the holes eventually becoming localized. A simple pairing model is presented with analytic calculations for low interaction strength and small doping, to help understand the numerical results and provide a physical picture for the properties of the SDW ground state. By comparison with recent many-body calculations, it is shown that, for intermediate interactions, the UHF solution provides a good description of the magnetic correlations in the true ground state of the Hubbard model.