Simulating cold interacting atoms on disordered optical lattices

发展相互作用系统中的无序理论是凝聚态物理学中最重要的挑战之一。静态无序会影响材料的导电性，并可以驱动金属-绝缘体转变，虽然人们对非相互作用系统中的无序有很多了解，但对其在相互作用粒子系统中的影响知之甚少。我们的建议是开发一个数值工具，精确地模拟相互作用的冷原子系统的实验探测。理论建模的最新发展，以及冷原子系统提供的非凡的实验控制水平，意味着我们第一次能够计算实验室中产生的实际无序系统的性质。我们将使用密度矩阵重整化群（DMRG）计算（及其更现代的形式，如张量网络状态）来计算被困在晶格中的相互作用粒子的大型系统的基态性质。虽然DMRG方法最初是通过将其嵌入到假设没有无序的有效环境中来逐块构建系统，但该算法现在在同一系统上重复此增长过程多次，直到达到收敛。这样，系统和环境都可以正确地包括无序的影响。这可能会工作已经知道了一段时间，但以前的尝试受到了巨大计算成本的严重限制。随着研究人员在大型矩阵对角化方面的最新算法进展，以及我们高性能计算中心的现有专业知识，我们非常适合执行这些计算，我们相信这可能对相互作用粒子的低维无序系统的研究产生巨大影响。我们将描述不同粒子间相互作用强度和不同无序势强度下系统的性质，以及计算低维相互作用玻色子的基态性质，我们将直接将计算结果与伯明翰冷晶格系统实验组的结果进行比较。由于实验室中研究的系统现在与模型系统非常接近，而且控制得很好，因此我们将能够在理论上研究与实验研究相同的系统。实验数据的解释的重要性的一个问题是实验系统的运动方程的小的时间依赖性变化的影响，这是固有的使用激光图案产生相互作用势。我们将模拟这种势的影响，并设计出将它们的影响与冷原子的固有低温特性分离开来的方法。我们还将能够研究实验密度剖面数据，以寻找接近局域化-离域化过渡的波函数多重分形的证据。来自计算的证据表明，单粒子波函数有非常不寻常的内部相关接近这个本地化过渡。直接访问伯明翰的实验数据，将使我们能够直接寻找这些相关性。

Developing a theory of disorder in interacting systems is one of the most important challenges in condensed matter physics. Static disorder affects the conductivity of materials and can drive a metal-insulator transition and, while a lot is known about disorder in non-interacting systems, far less is known about its effects in systems of interacting particles. Our proposal is to develop a numerical tool to model accurately the systems of interacting cold atoms being probed experimentally. Recent developments in theoretical modeling, together with the extraordinary level of experimental control offered by cold atom systems, mean that for the first time we are in a position to compute properties of the actual disordered systems which are being generated in the laboratory. We will use density matrix renormalization group (DMRG) calculations (and its more modern forms such as tensor network states) to compute the ground state properties of a large system of interacting particles trapped in a lattice. Although the DMRG approach was originally developed to build up a system piece by piece by embedding it in an effective environment which assumed no disorder, the algorithm now repeats this growth process on the same system many times until convergence is reached. This way both the system and the environment can include the effects of disorder correctly. That this might work has been known for a while, but previous attempts were severely restricted by the large computational cost. With recent algorithmic advances in large matrix diagonalisation by the investigators and with the available expertise in our High Performance Computing Centre we are ideally placed to perform these calculations, which we believe could have huge implications for the study of low-dimensional disordered systems of interacting particles. We will characterize the properties of the system for different strengths of interactions between particles and different strengths of disorder potential.As well as computing the ground-state properties of interacting bosons in low dimensions, we will directly compare the results of the calculations with the results from an experimental group in Birmingham working on cold lattice systems. Because the systems studied in the laboratory are now so close to the model systems and so well-controlled, we will be in a position to work on the same systems theoretically as those being studied experimentally. One issue of importance to the interpretation of experimental data is the effect of small time-dependent variations of the equation of motion of the experimental systems, which is intrinsic to the use of laser patterns to generate interaction potentials. We will model the effect of such potentials and devise ways of separating out their effects from the intrinsic low temperature properties of the cold atoms. We will also be able to study experimental density profile data to look for evidence of wavefunction multifractality close to the localization-delocalisation transition. Evidence from computations suggest that the single-particle wavefunctions have very unusual internal correlations close to this localization transition. The direct access to the experimental data from Birmingham, will allow us to look for these correlations directly.