Department of Physics, College of William and Mary, Williamsburg, VA 23187, USA.
J Phys Condens Matter. 2011 Dec 21;23(50):505601. doi: 10.1088/0953-8984/23/50/505601. Epub 2011 Nov 30.
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 evolve 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.
在无限制哈特利-福克（UHF）理论的框架内，研究了二维排斥哈伯德模型的基态。通过系统的、大规模的、精确数值计算，确定了磁和电荷性质，并作为电子掺杂量 h 的函数进行量化。在自洽 UHF 方程的求解中，使用多个初始配置和模拟退火来促进收敛到全局最小值。采用新方法来最小化有限尺寸效应，以达到热力学极限。在低到中等相互作用强度和低掺杂的情况下，UHF 基态是线性自旋密度波（l-SDW），具有反铁磁序和调制波。调制波的波长为 2/h。存在相应的电荷序，但远弱于自旋序，因此空穴是可移动的。随着相互作用的增强，l-SDW 态演变成几个不同的相，空穴最终被局域化。提出了一个简单的配对模型，用于低相互作用强度和小掺杂的解析计算，以帮助理解数值结果，并为 SDW 基态的性质提供物理图像。通过与最近的多体计算进行比较，表明对于中等相互作用，UHF 解为哈伯德模型的真实基态中的磁关联提供了很好的描述。
