Theoretical studies of quantum magnets and strongly correlated metals

这项研究计划的目标是发现和研究理论框架，这些框架可以捕获量子材料（如受挫磁铁和强相关金属）的一般潜在和统一的物理原理。几何受挫磁体的特征是其磁性离子亚晶格的几何形状：磁矩（“自旋”）位于三角形或四面体的顶点。一般来说，一对自旋有一个特定的相对方向，使它们的相互作用能最小。当一个系统没有同时使所有自旋对之间的相互作用能量最小化的自旋排列时，系统就会受挫。一个著名的例子是稀土焦绿石晶体，其中磁性稀土离子位于四面体连接网络的顶点上。在焦绿石晶体中已经观察到许多不同的现象，包括普通铁磁性（一种自旋相互排列的状态），反铁磁性（自旋交替方向），自旋冰（自旋服从冰规则的系统：两个自旋指向每个四面体，两个自旋指向每个四面体），量子自旋冰（自旋冰状态具有大量子波动）和自旋液体（具有宏观量子纠缠的状态）。拟议的研究将开发和使用一套计算和分析方法。所有方法的基础都是模型，在具有高度底层对称性的晶体中，将使用群论（对称）方法推导出完全一般的模型；这些方法可以大大降低模型的复杂性。计算方法包括精确对角化（对具有少量自旋的系统产生精确解）和经典自旋模拟。强相关金属是一种导体，其中电子之间存在强相互作用，产生丰富的相图，包括金属到绝缘体的转变，超导性或磁顺序。电子的相关性可以用哈伯德相互作用来建模：一种强烈的、持续的斥力，可以抑制一个以上的传导电子占据每个原子位置。虽然在概念上很简单，但这个模型在计算上提出了困难的挑战，即使在二维中也是如此。我们在联合国的小组最近取得的进展，将分析和计算方法结合在一个称为“算法松原积分”的程序中，大大减少了计算时间，使微扰理论的精确结果达到六阶。本提案旨在完成这一任务。在二维之外，哈伯德模型已经应用于许多系统，特别是那些有过渡元素或稀土的系统。其中有金属稀土焦绿石，如Pr2Ir2O7；这项工作的长期目标是将我们的数值技术应用于这个系统。

The goal of this research program is to discover and study theoretical frameworks which capture the general underlying and unifying physical principles of quantum materials such as frustrated magnets and strongly correlated metals. Geometrically frustrated magnets are characterized by the geometry of their magnetic ion sub-lattice: the magnetic moments ("spins") are located at the vertices of triangles or tetrahedra. Generally, a pair of spins has a particular relative orientation which minimizes their interaction energy. A system is frustrated when there is no arrangement of the spins that simultaneously minimizes the interaction energy between all pairs of spins. A famous example is the rare-earth pyrochlore crystals, where the magnetic rare earth ions are located on the vertices of a connected network of tetrahedra. Many different phenomena have been observed in pyrochlore crystals, including ordinary ferromagnetism (a state where the spins are all aligned with each other), anti-ferromagnetism (in which the spins alternate their orientation), spin ice (systems with spins obeying the ice-rule: two spins pointing into and two spins pointing out of each tetrahedron), quantum spin-ice (spin-ice states with large quantum fluctuations) and spin-liquid (a state that has macroscropic quantum entanglement). The proposed research will develop and use a suite of computational and analytical methods. The foundation for all approaches is the model, and in crystals with a high underlying symmetry, completely general models will be derived using the methods of group theory (symmetry); these same methods can greatly reduce the complexity of the models. Computational methods include exact diagonalization (which yields exact solutions for systems with a small number of spins) and classical spin simulations. Strongly correlated metals are conductors in which there are strong interactions between electrons, yielding a rich phase diagram that includes metal-to-insulator transitions, superconductivity or magnetic order. The electronic correlations can be modeled by the Hubbard interaction: a strong, constant repulsion that inhibits more than one conduction electron from occupying each atomic site. Although conceptually simple, this model presents difficult computational challenges, even in two-dimenstions. Recent advances by our group at MUN, which combine analytic and computational methods in a procedure called "Algorithmic Matsubara Integration" have considerably decreased computing times to the point where exact results up to sixth order in perturbation theory are within reach. This proposal aims to fulfill this task. Beyond two-dimensions, the Hubbard model has been applied to many systems, especially those with transition elements or rare earths. Among these are the metallic rare-earth pyrochlores such as Pr2Ir2O7; the longer term goal of this work is to apply our numerical techniques to this system.