Numerical Studies of Phase Transitions in Disorderd Systems

这笔拨款支持无序系统相变的理论研究。这项工作主要是数值方面的。这项研究将集中在两个领域。第一个是研究自旋玻璃态的性质，它发生在具有挫折感和无序性的系统中。尽管这些术语指的是一类磁性系统，但在自旋玻璃中发展出来的概念具有更广泛的适用性，并在其他科学领域找到了用途，如蛋白质折叠和优化问题。只有有限的解析结果，所以数值模拟起到了重要作用。过去的工作已经相当确凿地证明，确实发生了有限的温度转变。然而，在相变温度以下的自旋玻璃态的性质是相当有争议的。最近，Pi的小组开始了一系列的研究，通过研究当施加不同类型的微扰时基态如何变化来阐明自旋玻璃态的性质。由于受挫，确定基态不是一件容易的事，需要相当复杂的遗传算法。这些计算提供了一张自旋玻璃的图片，它是所提出的理论之一的液滴图片的修改版本，但也包含了另一种理论的重要组成部分，即复制对称破缺。在这一资助过程中，PI将更详细地研究这种情况，并且最重要的是，检查它是否也与有限温度蒙特卡罗模拟相一致，所述有限温度蒙特卡罗模拟在相当小的尺寸上也是一致的，但是使用交换蒙特卡罗方法将温度降低到比以前低得多的温度，这大大减少了传统蒙特卡罗中在低温下发生的低温下的减速。由于自旋玻璃的数值结果只能在低温下的相当小的系统上进行，因为由于相空间的多谷性质(其仅被交换蒙特卡罗方法部分地减少)，在对数值数据的解释中，一个主要问题是对比例的修正的大小。因此，PI还打算系统地研究不同模型下这些修正的大小是如何变化的，以确定是否存在修正最小的最佳模型。最后，将研究的第二个领域涉及具有无序的量子相变。与有限温度下的经典相变相比，人们对这些相变的理解较差。这是因为在经典跃迁中成功使用的许多技术在量子情况下不起作用，也许是因为非微扰(Griffiths-McCoy)效应很重要。因此，数值工作再次变得非常重要。将开展几项研究。%这笔赠款支持无序系统相变的理论研究。这项工作主要是数值方面的。这项研究将集中在两个领域。第一个是研究自旋玻璃态的性质，它发生在具有挫折感和无序性的系统中。尽管这些术语指的是一类磁性系统，但在自旋玻璃中发展出来的概念具有更广泛的适用性，并在其他科学领域找到了用途，如蛋白质折叠和优化问题。只有有限的解析结果，所以数值模拟起到了重要的作用。第二个需要研究的领域是无序量子相变。与有限温度下的经典相变相比，人们对这些相变的理解较差。这是因为在经典跃迁中成功使用的许多技术在量子情况下不起作用，也许是因为非微扰(Griffiths-McCoy)效应很重要。因此，数值工作再次变得非常重要。将进行几项研究。*

0086287YoungThis grant supports theoretical research on phase transitions in disordered systems. The work is primarily numerical. The research will focus on two areas. The first is to investigate the nature of the spin glass state, which occurs in systems with frustration and disorder. Although the terminology refers to a class of magnetic systems, the concepts developed in spin glasses have a much wider applicability and have found use in other areas of science such as protein folding and optimization problems. Only limited results have been obtained analytically, so numerical simulations have played an important role. Past work has established fairly conclusively that a finite temperature transition does occur. The nature of the spin glass state below the transition temperature is, however, fairly controversial.Recently the PI's group has embarked on a series of studies to elucidate the nature of the spin glass state by investigating how the ground state changes when various types of perturbations are applied. Because of frustration, determination of the ground state is non-trivial and required a fairly sophisticated genetic algorithm. These calculations suggest a picture of the spin glass which is a modified version of one of the proposed theories, the droplet picture, but also contains an important ingredient of the alternative theory, replica symmetry breaking. During the course of this grant, the PI will investigate this scenario in more detail, and, most importantly, check that it is also consistent with finite temperature Monte Carlo simulations on rather small sizes but taken down to temperatures much lower than before using the exchange Monte Carlo method, which considerably reduces the slowing down that occurs at low temperature in conventional Monte Carlo.Since numerical results on spin glasses can only be done on rather small systems at low temperatures, because of long relaxation times due to the many-valley nature of the phase space (which is only partially reduced by the exchange Monte Carlo method), a major concern in the interpretation of numerical data is the size of the corrections to scaling. The PI therefore also intends to look systematically at how the size of these corrections varies for different models, to see if there is an optimal model for which the corrections are smallest.Finally, the second area that will be studied concerns quantum phase transitions with disorder. These are poorly understood compared with classical phase transitions at finite temperature. This is because many of the techniques used successfully in classical transitions don't work in the quantum case, perhaps because non-perturbative (Griffiths-McCoy) effects are important. Hence again, numerical work has been very important. Several studies will be carried out.%%%This grant supports theoretical research on phase transitions in disordered systems. The work is primarily numerical. The research will focus on two areas. The first is to investigate the nature of the spin glass state, which occurs in systems with frustration and disorder. Although the terminology refers to a class of magnetic systems, the concepts developed in spin glasses have a much wider applicability and have found use in other areas of science such as protein folding and optimization problems. Only limited results have been obtained analytically, so numerical simulations have played an important role.The second area that will be studied concerns quantum phase transitions with disorder. These are poorly understood compared with classical phase transitions at finite temperature. This is because many of the techniques used successfully in classical transitions don't work in the quantum case, perhaps because non-perturbative (Griffiths-McCoy) effects are important. Hence again, numerical work has been very important. Several studies will be carried out.***