Automated analytic solutions to many-electron problems

尽管已经发现了令人眼花缭乱的高能亚原子粒子，但在我们的日常生活中，我们只与四种粒子相互作用：中子、质子、电子和光子（光）。这四种粒子构成了元素周期表上的所有元素，也构成了你生活中所知道和喜爱的一切。电子，以及它们如何与其他三种电子相互作用，是我研究计划的主要兴趣对象。电子是在金属、半导体和绝缘体中形成电流的物体。现代电子设备之所以成为可能，是因为我们有能力理解和控制电子在材料（由中子和质子组成）中如何移动，或者在暴露于光（光子）或施加电压时如何反应。令人惊讶的是，我们对电子在材料中的精确行为的理解非常初级。其原因是电子之间的相互作用与粒子数量呈指数级增长，粒子数量是阿伏伽德罗常数（6.02x10^23）的数量级，这使得精确的计算变得难以进行。这就是众所周知的多电子问题。关键的想法是，我们不能通过简单地建造更大或更快的计算机来克服这个障碍，而是需要创新和新颖的方法来用更少的计算支出来估计解决方案。本研究计划的目标是：1)开发研究多电子问题的新工具；II)将这些工具应用于模型系统；III)将这些工具扩展到学习材料。为了实现这些目标，我们将采用我们小组最近开发的一种方法，使我们能够立即解决相关电子问题的主要部分，这是其他方法的一个关键优势。我们的工具将应用于模型系统，以测试和改进我们的数值算法。首先，我们将研究哈伯德模型，它是相关电子系统的典型模型。在这个项目的过程中，我们将超越该模型的限制，允许量子化学更感兴趣的一般相互作用。这将使我们从纯粹的理论模型系统逐步走向真实的材料计算。作为该计划的一个附加组件，我们打算开发经过良好测试的软件代码，可以解决相关的电子系统，并打算将这些代码作为开源工具发布。这将使其他研究小组从我们的努力中获得最佳收益。所建议的工作将对相互作用系统的详细理解作出重大贡献。充分了解量子系统中自旋自由度和电荷自由度之间的相互作用将允许材料的智能设计，因此将对基础研究计划产生影响，并推动技术进步，从而造福工业，经济和加拿大。

Despite the dizzying array of existing high-energy subatomic particles that have been discovered, in our day-to-day lives we interact with only four: neutrons, protons, electrons and photons (light). These four particles make up all elements on the periodic table and comprise everything you know and love in your life. Electrons, and how they interact with the other three, are the central object of interest in my research program. Electrons are the objects that form current in metals, semi-conductors and insulators. Modern day electronic devices are possible due to our ability to understand and control how electrons move in materials (comprised of neutrons and protons), or react when exposed to light (photons) or an applied voltage. Surprisingly, our understanding of precisely how electrons behave in materials is very rudimentary. The reason for this is that the interactions between electrons scale exponentially with the number of particles, which is on the order of Avogadro's constant (6.02x10^23), making exact calculations intractable.  This is known as the many-electron problem. The key idea is that we cannot overcome this hurdle by simply building bigger or faster computers, but instead require innovative and novel approaches to estimate solutions with less computational expenditure. The goals of this research program are to:I) Develop novel tools for studying the many-electron problem;II) Apply those tools to model systems; III) Extend those tools to study materials.  To accomplish these goals we will employ a method recently developed in our group that allow us to instantaneously solve a dominant fraction of correlated electron problems, a key advantage over other methods. Our tools will be applied to model systems in order to test and improve our numerical algorithms. To begin, we will study the Hubbard model, which is the quintessential model of correlated electron systems. Over the course of this program, we will extend beyond the restrictions of that model to allow more general interactions of interest to quantum chemistry. This will allow us to take incremental steps from purely theoretical model systems, towards real material calculations. As an additional component of the program we intend to develop well-tested software codes that can solve correlated electron systems with the intent of releasing these codes as open-source tools. This will allow other research groups to optimally benefit from our efforts. The proposed work will contribute significantly to the detailed understanding of interacting systems.  A full understanding of the interplay between spin and charge degrees of freedom in quantum systems will allow for the intelligent design of materials and will therefore be of impact to both fundamental research programs and drive forward technological advancement to the benefit of industry, the economy, and Canada.