全组态相互作用量子蒙特卡罗方法及其在Hubbard模型中的应用研究

The full configuration interaction quantum Monte Carlo is a newly developed method by combining characteristics of the full configuration interaction (FCI) (which is also known as the exact diagonalization for physicists) method and the projector quantum Monte Carlo. FCIQMC can yield FCI quality results while requiring a substantially smaller memory than FCI calculation using the same basis. A key advantage of FCIQMC is that the nodal structure of the ground-state wave function emerges spontaneously by sampling the discrete space of Slater determinants. Although this method can't solve the minus-sign problem completely, the dynamic evolution of walkers (or psips) in the discrete determinant space does not require the fixed node approximation. What's more, it can increase the meeting and cancelling probability of the opposite sign walkers. Therefore, one can achieve a stable signal to noise ratio for any large but finite basis if the walker population is sufficiently large. On the basis of FCIQMC, the recently developed initiator approximation (i-FCIQMC), the semi-stochastic method (S-FCIQMC) and CASSCF-FCIQMC can improve the efficiency and accuracy remarkably. In this project, we will apply these methods to the widely used Hubbard model, for which the focus will be placed on the near half-filling and medium strength coupling region of two-dimensional Hubbard model, where many calculations can’t consistent with each other. And we will explore some studies, such as whether the ground state is ferromagnetic under thermodynamic limit in the Nagaoka's theory. The results of this project will improve the knowledge about Hubbard model, and provide important references for the development of FCIQMC and its application to solid in the future.

全组态相互作用量子蒙特卡罗(FCIQMC)是把全组态相互作用(FCI)与投影量子蒙特卡罗相结合发展起来的一类新方法。FCIQMC能给出与FCI相同精度的结果，但在相同基组下FCIQMC模拟所需内存比FCI低很多数量级。FCIQMC中基态波函数的节点结构是在Slater行列式离散空间进行抽样的过程中自动出现的。该方法虽不能从根本上消除负符号问题，但具有很高的正负粒子对湮灭的概率，从而能够很好地克服负符号问题。本项目将利用基于FCIQMC新提出的i-FCIQMC、S-FCIQMC和CASSCF-FCIQMC等方法来研究富有争议的近半满、中间耦合强度下的二维Hubbard模型的基态性质。重点针对Nagaoka理论中“系统的基态在热力学极限下是否铁磁性”等问题进行探索性研究。该项目的研究成果将有助于提高人们对Hubbard模型的理解和认识，同时也为FCIQMC的未来发展和在固体中的应用提供参考。

基于实空间下的Hubbard模型在全组态相互作用量子蒙特卡罗方法(FCIQMC)下具有弱“负符号问题”的优势,我们和马克斯普朗克固体物理研究所的Alavi课题组共同发展和完善了总自旋可选的全组态相互作用量子蒙特卡罗(spin-adapted FCIQMC)和约化密度矩阵下的全组态相互作用量子蒙特卡罗(RDM-FCIQMC)这两种方法。在此基础上，我们重点研究了Hubbard模型中Nagaoka铁磁性及其相变。首先，当正方格子体系中只存在1个空穴时，基态的总自旋S随着关联强度U的增大而从最小值增加到最大值，由此我们可以确定Nagaoka铁磁性的临界关联强度Uc(其正比于格子大小)。当U小于Uc时，除16个格子以外总自旋S始终保持最小值。而当U等于Uc时，所有体系的总自旋能量非常接近，这意味着体系处于强烈的自旋振荡状态。其次，当正方格子体系中存在多于1个空穴时spin-adapted FCIQMC的计算结果表明体系的基态始终对应于最小的总自旋S。由于顺磁、反铁磁和部分铁磁畴所对应的总自旋S均为最小值，我们为了弄清楚多空穴下体系的具体状态而研究了电子自旋的空间分布图像。RDM-FCIQMC所给出的自旋-自旋关联函数的计算结果表明：当U大于或等于Uc时，18、20和24格子的双空穴计算结果均表明存在空间局部区域的铁磁畴状态(Sz同号，向上或者向下)。同时24格子下的三空穴计算结果也表明存在铁磁畴现象。由此可见，这一新的铁磁畴相发生在空穴密度足够小而关联强度足够大的区域。最后，本项目的研究成果表明把量子化学领域中的全组态相互作用量子蒙特卡罗方法用于研究凝聚态物理中的Nagaoka铁磁性及其相变是非常成功的。该项目不仅加深了人们对近半满区域的Hubbard模型铁磁性相变的理解和掌握，同时也将进一步推动量子化学和凝聚态物理所共同关注的强关联电子体系中磁性的研究工作。