Confronting the conformal bootstrap with Monte Carlo simulations of lattice models

临界现象主要与连续相变有关。重正化群理论是上个世纪70年代发展起来的一种现代的临界现象理论框架。在第二阶相变附近，热力学量发散，遵循幂律。这些幂称为临界指数。RG理论的一个重要预测是相变属于普适类。在一个普适类中，临界指数假设完全相同的值。一个普适类是由一些定性的特征，如系统的维数，相互作用的范围和序参量的对称性。理论计算主要基于朗道-金斯伯格理论（场论方法）的扩展或基于晶格模型，例如伊辛模型。晶格模型可以通过平均场理论、级数展开或蒙特卡罗模拟来研究。临界指数的实验结果大多不如理论计算的准确。近年来，共形引导方法在理论方面取得了很大的进展。特别是对于三维伊辛模型的普适性类，临界指数的计算具有前所未有的精度。此外，计算了表征临界点处三点相关函数行为的结构常数。由于共形引导方法从一个相当抽象的不动点的特征出发，因此需要检查结果是否与其他方法所得到的结果一致。事实上，临界指数的估计与以前的Monte Carlo模拟的最准确的结果相吻合。特别是，在过去的两年中，精确估计的临界指数的三维XY和海森堡普适性类得到了使用共形引导方法。这些证实了我在当前项目中获得的准确的蒙特卡罗结果。现在我打算集中讨论基本对称性的扰动。在晶体的情况下，自然会出现具有立方对称性的微扰。几十年来，人们一直在争论这样的微扰是否与以O（3）对称性为特征的三维海森堡普适类有关。最近的结果证明，这确实是使用共形引导方法的情况下，这意味着立方不动点支配的物理。本提案的主要目标是准确描述这一固定点。此外，我喜欢研究所谓的危险无关扰动。虽然这样的扰动不会改变高温阶段的长程物理特性，并且直接在临界温度下，但低温阶段的行为从根本上改变。

Critical phenomena are mainly related with continuous phase transitions. The renormalization group (RG) theory developed in the seventies of the last century is the modern theoretical framework of critical phenomena. In the neighbourhood of a second order phase transition, thermodynamic quantities diverge, following power laws. The powers are called critical exponents. An important prediction of RG theory is that phase transitions fall into universality classes. Within a universality class, critical exponents assume exactly the same values. A universality class is characterized by a few qualitative features such as the dimension of the system, the range of the interaction and the symmetry properties of the order parameter. Theoretic calculations are mostly based on extensions of the Landau- Ginsburg theory (field theoretic methods) or on lattice models, such as the Ising model. Lattice models can be studied for example by using mean-field theory, series expansions or Monte Carlo simulations. Experimental results for critical exponents are mostly less accurate than those obtained from theoretical calculations. Recently there has been great progress on the theoretical side by using the so called conformal bootstrap method. In particular for the universality class of the three-dimensional Ising model, critical exponents were computed with unprecedented accuracy. In addition, structure constants, that characterize the behavior of three-point correlation functions at the critical point were computed. Since the conformal bootstrap method starts from a rather abstract characterization of fixed points, it is desirable to check whether the results agree with those obtained by other methods. Indeed the estimates of critical exponents coincide with the most accurate results of previous Monte Carlo simulations. In particular, in the last two years precise estimates for the critical exponents of the three-dimensional XY and Heisenberg universality classes were obtained by using the conformal bootstrap method. These confirm the accurate Monte Carlo results I obtained within the current project. Now I intend to focus on perturbations of the underlying symmetry. In the case of crystals a perturbation with a cubic symmetry arises naturally. It had been debated for decades, whether such a perturbation is relevant for the three-dimensional Heisenberg universality class, characterized by O(3) symmetry. Recent results obtained by using the conformal bootstrap method prove that this is indeed the case, implying that the cubic fixed point governs the physics. The accurate characterization of this fixed point is the main objective of the current proposal. Furthermore I like to study so called dangerously irrelevant perturbations. While such perturbations do not alter the long range physics in the high temperature phase and directly at the critical temperature, the behavior in the low temperature phase is fundamentally changed.