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

二维的量子物理在技术上是相关的，而且从根本上说是有趣的；它也很困难--可用的方法通常限于小系统规模或不切实际的近似。主要的问题是人们必须考虑的自由度的数量，它随着系统的大小呈指数增长。除了最微不足道的问题之外，解决任何问题的方法都必须选择这些自由度中真正重要的自由度，并丢弃其余的自由度。这种被称为“密度矩阵重整化群”(DMRG)的技术以一种非常有效的方式选择了基本自由度，从而彻底改变了量子系统的数值研究。DMRG已被证明是一种计算准1D材料性质的高精度和强大的工具：可以被描述为一维晶格或位置链的系统。不幸的是，这种方法在两个空间维度和更高的维度上步履蹒跚--随着系统规模的增加，它很快就会变得低效。量子信息论的一个结果，即所谓的面积定律，表明传统的2D DMRG的失败与一维以上量子纠缠的加速增长有关。系统不同子区域之间的太多纠缠导致DMRG陷入停顿。面积定律指出，纠缠度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，与系统的总大小无关。在2D中，该区域的规模实际上就像一个分区域的周长。换言之，它将随着系统的变大而线性增加，直到DMRG方法太过低效而无法使用。有效的数值技术是至关重要的，因为基于简单近似的分析方法无法处理最有趣的问题，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的维度上没有类似的方法。通过考虑由精确可解的链组成的耦合阵列的各向异性2D情况，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，产生了一种算法，它可以比竞争技术有效地执行更大规模的模拟。使用各向异性表示并不妨碍我们将结果应用于各向同性系统，因为我们通常对与此类微观细节无关的万有量感兴趣。我打算将该算法从概念验证实现发展为一个通用工具，用于研究二维量子系统的性质，包括它们的量子信息含量。在这样做的过程中，我将把它应用于一个重要的基准问题，与铜酸盐高温超导体(没有电阻导电的材料)相关。我还将利用这项技术的潜在“矩阵积态”结构，将其扩展到新兴的非平衡量子问题领域，在这个领域中，系统通过突然改变其一个属性(例如，相互作用的强度)来“猝灭”。

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).