A new numerical approach to strongly correlated quantum physics in 2D.

二维量子物理学在技术上是相关的，从根本上是有趣的;它也是困难的-可用的方法通常限于小系统尺寸或不切实际的近似。主要的问题是必须考虑的自由度的数量，它随系统的大小呈指数增长。解决任何问题的方法，除了最琐碎的问题，都必须选择这些自由度中哪些真正重要，并丢弃其余的。被称为“密度矩阵重整化组”（DMRG）的技术通过以非常有效的方式选择基本自由度，彻底改变了量子系统的数值研究。DMRG已被证明是一个高度准确和强大的工具，用于计算准一维材料的性质：系统可以被描述为一维晶格或“链”的网站。不幸的是，这种方法在两个空间维度及以上的空间维度上遇到了困难--随着系统尺寸的增长，它很快变得低效。量子信息理论的一个结果，即所谓的“面积定律”，表明传统的2D DMRG的失败与1D以上量子纠缠的增强增长有关。系统的不同子区域之间太多的纠缠会导致DMRG逐渐停止。面积定律指出，纠缠与两个子区域之间的边界或“面积”成比例。在1D中，边界只能是一个或两个点，与总系统大小无关。在2D中，该区域实际上将像子区域的周长一样缩放。换句话说，它将随着系统变大而线性增加，直到DMRG方法效率太低而无法使用。有效的数值技术至关重要，因为基于简单近似的分析方法无法解决最有趣的问题，其中量子涨落占主导地位，而在一维中可用的更复杂的精确方法在更高维度中没有类似物。我们可以最小化相关的边界面积，从而最小化纠缠问题。将精确可解子单元的性质与DMRG的能力相结合，导致一个算法，可以有效地执行更大规模的模拟比竞争技术。使用各向异性表示并不禁止我们从应用结果各向同性系统，因为我们通常感兴趣的是“普遍”的数量是独立的微观细节。我打算开发这个算法超越其证明-概念实现为一个通用的工具，用于研究二维量子系统的属性，包括它们的量子信息内容。在这样做的时候，我将把它应用到一个重要的基准问题，与铜氧化物高温超导体（材料导电无电阻）。我还将利用该技术的基本“矩阵乘积态”结构，将其扩展到新兴的非平衡量子问题领域，其中系统通过突然改变其属性之一（例如相互作用的强度）而被“淬火”。

Quantum physics in two dimensions is technologically relevant and fundamentally interesting; it is also difficult - available methods are generally limited to small system sizes or unrealistic approximations. The main problem is the number of degrees of freedom one must consider, which grows exponentially with system size. Methods for solving anything but the most trivial problems must select which of these degrees of freedom really matter and discard the rest. The technique known as the 'density matrix renormalisation group' (DMRG) has revolutionised numerical studies of quantum systems by selecting the essential degrees of freedom in a remarkably efficient manner. DMRG has proven to be a highly accurate and robust tool for calculating properties of materials that are quasi-1D: systems that can be described as a one dimensional lattice or 'chain' of sites. Unfortunately this method stumbles in two spatial dimensions and above - it quickly becomes inefficient as system size grows.A result from quantum information theory, known as the 'area law' shows the failure of conventional 2D DMRG is linked to the enhanced growth of quantum entanglement above 1D. Too much entanglement between different subregions of the system causes DMRG to grind to a halt. The area law states that entanglement scales with the boundary or 'area' between two subregions. In 1D the boundary can only be one or two points, independent of the total system size. In 2D the area will in fact scale like the perimeter of a subregion. In other words, it will increase linearly as the system gets bigger, until the DMRG approach is too inefficient to be useful.Effective numerical techniques are vital because analytic methods based on simple approximations fail for the most interesting problems, where quantum fluctuations are dominant, while more sophisticated exact approaches available in 1D do not have analogues in higher dimensions.By considering the anisotropic 2D case of a coupled array of exactly solvable chains we can minimise the relevant boundary area, thus minimising the entanglement problem. Combining the properties of the exactly solvable subunits with the power of DMRG, leads to an algorithm that can efficiently perform larger scale simulations than competing techniques.Using an anisotropic representation does not prohibit us from the applying the results to isotropic systems as we are generally interested in 'universal' quantities that are independent of such microscopic details.I intend to develop this algorithm beyond its proof-of-concept implementation into a general tool for studying the properties of two dimensional quantum systems, including their quantum information content.In so doing, I will apply it to an important benchmark problem, relevant to the cuprate high temperature superconductors (materials that conduct electricity without resistance). I will also take advantage of the underlying 'matrix product state' structure of the technique to extend it to the emerging field of out-of-equilibrium quantum problems, where a system is 'quenched' by suddenly changing one of its properties (for example the strength of interactions).