Confronting the conformal bootstrap with Monte Carlo simulations of lattice models

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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.