Renormalisation Group Interfaces in Tricritical Ising Model

重整化群（RG）控制着物理系统的能量尺度依赖性。它是凝聚态物理和粒子物理中一个非常重要的概念和工具。RG的不动点由共形场论（CFTs）描述。共形对称群（作用于时空而不是内部自由度的群）设置了强大的约束，可以用来构建和解决这些理论。共形场论一般比非共形场论简单得多，可以用共形微扰理论来近似非共形场论。共形对称在两个欧几里德维度中特别强大。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。这一点和相关的代数结构是构造和完全求解二维CFTs的许多成功的背后。在固定点附近，共形对称性被破坏。相应的理论可以通过用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，随着尺度而变化，这导致理论空间中的轨迹，也称为RG流。相应的流线要么在具有不同对称性质的大质量理论（例如简并真空）处结束，要么在其他CFTs处结束。重要的是要有关于流线是否经过附近（交叉现象）或接近其他固定点的信息。在凝聚态物理学中，对大距离扰动理论的描述相当于确定一个相图。这种相图的全局结构一般很难确定。作为第一近似，人们通常依赖于朗道理论的指示，朗道理论是一种唯象方法，可以给出相图的粗略概念，但在精确结果方面不是可靠的方法。或者，人们依赖于手头理论的格子版本中的数值，这非常依赖于可用的计算能力，并且还有其他限制。很少已经做了迄今为止，在确定全球结构的相图的相位从选定的临界点直接在连续的方法，即工作与量子场theors.In本项目中，我们的目标是描述的全球结构的空间的所有RG流起源于CFT描述的二维三临界伊辛模型（TIM）。这个模型有一个丰富的相图，展示了一条线，其中三相共存和三个关键线导致从原来的临界点的关键伊辛模型。为了绘制相图，我们计划使用一种基于RG接口的新方法-物理表面分离由RG流连接的两个尺度不变理论。

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.