Theoretical studies of quantum magnets and strongly correlated metals

这个研究计划的目标是发现和研究理论框架，这些框架捕捉到量子材料的一般潜在和统一的物理原理，如受挫磁铁和强关联金属。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，一对自旋具有特定的相对取向，从而使它们的相互作用能最小。当没有同时最小化所有自旋对之间的相互作用能量的自旋排列时，系统会受挫。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相连网络的顶点上。在焦绿石晶体中观察到了许多不同的现象，包括普通铁磁性(自旋彼此对齐的状态)、反铁磁性(自旋改变方向)、自旋冰(自旋符合冰规则的系统：每个四面体有两个自旋指向和两个自旋指向)、量子自旋-冰(具有大的量子波动的自旋-冰态)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，而在具有高底层对称性的晶体中，完全通用的模型将使用群论(对称性)的方法来推导；这些相同的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(为具有少量自旋的系统产生精确解)和经典自旋模拟。强相关金属是电子之间存在强烈相互作用的导体，产生了丰富的相图，其中包括金属到绝缘体的转变、超导或磁序。电子关联可以用Hubbard相互作用来模拟：一种强烈的、持续的排斥力，它阻止一个以上的传导电子占据每个原子位置。尽管概念上很简单，但这个模型带来了困难的计算挑战，即使是在二维情况下也是如此。我们在MUN的团队最近取得的进展，将分析和计算方法结合在一种称为算法松原积分的过程中，极大地减少了计算时间，达到了微扰理论中六阶精确结果的触手可及的程度。这项建议旨在完成这项任务。除了二维以外，哈伯德模型还被应用于许多系统，特别是那些含有过渡元素或稀土的系统。其中包括金属稀土焦绿石，如Pr2Ir2O7；这项工作的长期目标是将我们的数值技术应用到这个系统中。

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.