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

二维强相关量子物理的新数值方法。

• 批准号: EP/L010623/1
• 负责人: Andrew James
• 金额: $ 41.78万
• 依托单位: University College London
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2014
• 资助国家: 英国
• 起止时间: 2014 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FL010623%2F1
• 关键词: new; numerical; approach; strongly; correlated

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).

二维量子物理学在技术上是相关的，从根本上是有趣的；它也很困难-可用的方法通常限于小系统大小或不切实际的近似值。主要问题是必须考虑的自由度的数量，它随着系统大小呈指数增长。除了最琐碎的问题之外，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那个，而放弃其余的。这种被称为“密度矩阵重整化群”（DMRG）的技术通过以一种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已被证明是一种高度精确和强大的工具，用于计算准一维材料的性质：可以被描述为一维晶格或“链”的系统。不幸的是，这种方法在两个或以上的空间维度上出错了——随着系统大小的增长，它很快变得低效。量子信息理论的一个结果，被称为“面积定律”，表明传统二维DMRG的失败与一维以上量子纠缠的增强增长有关。系统的不同子区域之间过多的纠缠会导致DMRG逐渐停止。面积定律指出，缠结与两个子区域之间的边界或“面积”有关。在一维中，边界只能是一个或两个点，与系统的总尺寸无关。在二维图像中，这个区域实际上相当于一个子区域的周长。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量子涨落占主导地位的最有趣的问题，而一维中可用的更复杂的精确方法在高维中没有类似的方法。通过考虑精确可解链耦合阵列的各向异性二维情况，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特性与DMRG的功能相结合，导致一种算法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，因为我们通常对独立于这些微观细节的“普遍”量感兴趣。我打算把这个算法发展成一个通用的工具，用于研究二维量子系统的特性，包括它们的量子信息内容。在此过程中，我将把它应用于一个重要的基准问题，与铜高温超导体（无电阻导电的材料）有关。我还将利用该技术的基础“矩阵积态”结构，将其扩展到非平衡量子问题的新兴领域，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“淬灭”。

项目成果

Signatures of rare states and thermalization in a theory with confinement

• DOI: 10.1103/physrevb.99.195108
• 发表时间: 2019-05-06
• 期刊: PHYSICAL REVIEW B
• 影响因子: 3.7
• 作者: Robinson, Neil J.;James, Andrew J. A.;Konik, Robert M.
• 通讯作者: Konik, Robert M.

Non-perturbative methodologies for low-dimensional strongly-correlated systems: From non-abelian bosonization to truncated spectrum methods

低维强相关系统的非微扰方法：从非阿贝尔玻色子化到截断谱方法

• DOI: 10.48550/arxiv.1703.08421
• 发表时间: 2017
• 作者: James A

Nonthermal States Arising from Confinement in One and Two Dimensions

• DOI: 10.1103/physrevlett.122.130603
• 发表时间: 2019-04-05
• 期刊: PHYSICAL REVIEW LETTERS
• 影响因子: 8.6
• 作者: James, Andrew J. A.;Konik, Robert M.;Robinson, Neil J.
• 通讯作者: Robinson, Neil J.

Quantum quenches in two spatial dimensions using chain array matrix product states

使用链阵列矩阵积态在两个空间维度进行量子淬灭

• DOI: 10.1103/physrevb.92.161111
• 发表时间: 2015
• 期刊: Physical Review B
• 影响因子: 3.7
• 作者: James A

Itinerant effects and enhanced magnetic interactions in Bi-based multilayer cuprates

• DOI: 10.1103/physrevb.90.220506
• 发表时间: 2014-12-04
• 期刊: PHYSICAL REVIEW B
• 影响因子: 3.7
• 作者: Dean, M. P. M.;James, A. J. A.;Hill, J. P.
• 通讯作者: Hill, J. P.