Renormalisation Group Interfaces in Tricritical Ising Model

三临界 Ising 模型中的重正化群接口

基本信息

批准号：
EP/W010283/1
负责人：
Anatoly Konechny
金额：
$ 10.18万
依托单位：
Heriot-Watt University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FW010283%2F1
关键词：
Renormalisation Group Interfaces Tricritical Ising

项目摘要

The renormalisation group (RG) governs the energy scale dependence of a physical system. It is a very important concept and instrument in condensed matter and particle physics. Fixed points of the RG are described by conformal field theories (CFTs). The conformal symmetry group (being a group acting in space-time rather than on internal degrees of freedom) places powerful constraints which can be used to construct and solve such theories. Conformal field theories are generically much simpler than the non-conformal ones and can be used to approximate the non-conformal theories by means of conformal perturbation theory. Conformal symmetry is particularly powerful in two Euclidean dimensions. In this case the state spaces furnish representations of the Virasoro algebra - an infinite-dimensional Lie algebra. This and the associated algebraic structures are behind a lot of the success in constructing and fully solving two-dimensional CFTs.Near a fixed point the conformal symmetry is broken. The corresponding theories can be described by perturbing the conformal field theory by a linear combination of relevant operators. The corresponding coefficients, i.e. the coupling constants, change with scale which results in a trajectory in theory space also known as an RG flow. The corresponding flow lines end up either at massive theories which have different symmetry properties, e.g. degenerate vacua, or at other CFTs. It is important to have information on whether the flow line passes near (crossover phenomenon) or approaches other fixed points. If that is the case the appropriate conformal perturbation theory can be used.In condensed matter physics the description of the perturbed theories at large distances amounts to determining a phase diagram. The global structure of such phase diagrams is in general very hard to determine. As a first approximation one usually relies on indications from Landau theory which is a phenomenological approach that can give a rough idea of the phase diagram but is not a reliable method when it comes to precision results. Alternatively one relies on numerics done in the lattice versions of the theory at hand which are quite dependent on computing power available and have other limitations as well. Little has been done to date in terms of determining the global structure of phase diagrams for phases originating from chosen critical points directly in the continuum approach, that is, working with quantum field theories.In the present project we aim to describe the global structure of the space of all RG flows originating from the CFT describing the two-dimensional tricritical Ising model (TIM). This model has a rich phase diagram exhibiting a line where three phases coexist and three critical lines leading from the original critical point to the critical Ising model. To chart the phase diagram we plan to use a novel approach based on RG interfaces -- physical surfaces separating two scale invariant theories linked by an RG flow.

重正化群 (RG) 控制物理系统的能量尺度依赖性。它是凝聚态物理和粒子物理中非常重要的概念和工具。 RG 的不动点由共形场论 (CFT) 描述。共形对称群（是作用于时空而不是内部自由度的群）施加了强大的约束，可用于构建和解决此类理论。共形场论一般比非共形场论简单得多，并且可以通过共形微扰理论来近似非共形理论。共形对称性在两个欧几里得维度中特别强大。在这种情况下，状态空间提供了 Virasoro 代数（无限维李代数）的表示。这个和相关的代数结构是构造和完全求解二维 CFT 的许多成功的原因。在固定点附近，共形对称性被打破。相应的理论可以通过相关算子的线性组合扰动共形场论来描述。相应的系数，即耦合常数，随尺度变化，从而产生理论空间中的轨迹，也称为 RG 流。相应的流线最终要么处于具有不同对称性的大规模理论，例如简并真空，或其他 CFT。了解流线是否经过（交叉现象）或接近其他固定点的信息非常重要。如果是这种情况，则可以使用适当的共形微扰理论。在凝聚态物理学中，远距离微扰理论的描述相当于确定相图。这种相图的整体结构通常很难确定。作为第一个近似，通常依赖于朗道理论的指示，朗道理论是一种现象学方法，可以给出相图的粗略想法，但在获得精确结果时并不是一种可靠的方法。或者，我们依赖于现有理论的点阵版本中完成的数值，这些数值很大程度上取决于可用的计算能力，并且还具有其他限制。迄今为止，在确定直接源自连续介质方法（即使用量子场论）中选定的临界点的相图的全局结构方面，几乎没有做任何工作。在本项目中，我们的目标是描述源自描述二维三临界伊辛模型 (TIM) 的 CFT 的所有 RG 流的空间。该模型具有丰富的相图，显示了三相共存的线以及从原始临界点通向临界伊辛模型的三条临界线。为了绘制相图，我们计划使用一种基于 RG 接口的新颖方法 - 物理表面分隔由 RG 流链接的两个尺度不变理论。