Dimer models on quasicrystals

用多米诺骨牌铺棋盘有多少种方法？这是经典二聚体模型的一个例子。移走一张多米诺骨牌，露出它覆盖的两个方块：一个黑，一个白。这些可以被认为是一个粒子-反粒子对，进一步的多米诺骨牌重新排列允许粒子在棋盘上移动。这个简单的模型对强相互作用系统的理论有着深远的影响。这个项目将考虑准晶体上的二聚体模型：可以通过高维晶体构造成切片的非周期切片。多米诺骨牌（二聚体）可以放置在瓷砖的边缘，这样每个顶点都连接到一个二聚体吗？或者必须有一个有限密度的颗粒状缺陷而不与二聚体相连？研究准晶体的优点是，它们足够随机，可以得到非平凡的结果，但又足够有序，可以用分析方法证明这些结果。在之前的工作[物理Rev X 10, 011005(2020)]中，我们发现了彭罗斯瓦片中这种模型的最小缺陷密度，彭罗斯瓦片是由罗杰·彭罗斯爵士（2020年诺贝尔物理学奖获得者）教授发现的原始准晶体。这个项目有广泛的可能路线，取决于兴趣。这些包括扩展到量子二聚体模型，如Rokhsar-Kivelson模型（最初引入模型高温超导，该模型具有其他奇异状态，如量子自旋液体）；障碍的影响;约束和质量缺口（克莱研究所的千禧年奖问题之一）；相关的模型，如环二聚体模型和哈密顿循环。这个项目可以采取数值方向（使用经典和量子蒙特卡罗技术和机器学习），或分析方向，寻求精确的结果。与卡迪夫和其他地方的实验小组合作是有空间的，例如在介观磁阵列中实现类似的模型。该项目将是与全球理论家和数字学家合作开展的一系列工作的核心，包括与牛津大学和康奈尔大学的小组密切合作。

How many ways are there of tiling a chess board with dominoes? This is an example of a classical dimer model. Removing one domino reveals the two squares it covered: one black, one white. These can be thought of as a particle-antiparticle pair, with further domino re-arrangements allowing the particles to move around on the board. This simple model turns out to have deep implications for the theory of strongly-interacting systems.This project will consider dimer models on quasicrystals: aperiodic tilings which can be constructed as slices through higher-dimensional crystals. Can dominoes (dimers) be placed on the tiling's edges such that every vertex connects to one dimer? Or must there be a finite density of particle-like defects not connected to dimers? The advantage of working with quasicrystals is that they are random enough to get non-trivial results, but ordered enough that these results can be proven analytically. In previous work [Phys Rev X 10, 011005 (2020)] we found the minimum density of defects for such models in Penrose tilings, the original quasicrystals discovered by Prof. Sir Roger Penrose (recipient of the 2020 Nobel Prize for Physics).This project has a broad range of possible routes depending on interest. These include extending to quantum dimer models such as the Rokhsar-Kivelson model (originally introduced to model high-temperature superconductivity, this model features other exotic states such as quantum spin liquids); disorder effects; confinement and the mass gap (one of the Clay Institute's Millennium Prize Problems); related models such as loop-dimer models and Hamiltonian cycles. This project can be taken in a numerical direction (using classical and quantum Monte Carlo techniques and machine learning), or an analytical direction, seeking exact results. There is scope to work with experimental groups in Cardiff and elsewhere, for example implementing similar models in mesoscopic magnetic arrays.The project will be central to a range of work being carried out in collaboration with theorists and numericists around the globe, including close collaboration with groups in Oxford and Cornell.