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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.

重整化群(RG)控制着物理系统的能量尺度依赖性。它是凝聚态和粒子物理学中一个非常重要的概念和仪器。RG的不动点用共形场理论(CFTS)描述。共形对称群(是在时空中作用的群，而不是作用于内部自由度的群)施加了强大的约束，可以用来构建和求解这些理论。共形场理论一般比非共形场理论简单得多，可以用共形微扰理论来近似非共形场理论。共形对称性在两个欧几里得维度上特别强大。在这种情况下，状态空间提供了Virasoro代数的表示--无限维李代数。这种和相关的代数结构是构造和完全求解二维CFT的许多成功的背后原因。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符的线性组合来微扰共形场理论来描述。相应的系数，即耦合常数，随着尺度的变化而变化，这导致了理论空间中的轨迹，也被称为RG流。相应的流线要么出现在具有不同对称性的大质量理论上，例如简并真空，要么出现在其他CFT上。重要的是要掌握流线是否靠近(交叉现象)或接近其他固定点的信息。如果是这种情况，则可以使用适当的共形微扰理论。在凝聚态物理中，对远距离微扰理论的描述相当于确定相图。一般来说，这种相图的整体结构很难确定。作为第一近似值，人们通常依赖于朗道理论的指示，这是一种现象学方法，可以给出相图的大致概念，但当涉及到精确结果时，这种方法并不可靠。或者，人们依赖于手头的理论格子版本中的数值计算，这相当依赖于可用的计算能力，并且也有其他限制。到目前为止，在连续统方法中直接确定来自选定临界点的相的相图的全局结构方面所做的工作很少，也就是使用量子场理论。在本项目中，我们的目标是描述来自描述二维三临界伊辛模型(TIM)的CFT的所有RG流的空间的全局结构。该模型具有丰富的相图，其中显示了一条三相共存的线和三条从原始临界点到临界伊辛模型的临界线。为了绘制相图，我们计划使用一种基于RG界面的新方法--物理表面分离由RG流链接的两个尺度不变理论。