Confronting the conformal bootstrap with Monte Carlo simulations of lattice models

• Applicant: Dr. Martin Herbert Hasenbusch
• Subject Area: Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics
• Term: since 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 411753772

Continuous phase transitions are primary examples of critical phenomena. The renormalization group (RG) theory developed in the seventies of the last century is their modern theoretical framework. In the neighbourhood of a second order phase transition, thermodynamic quantities diverge, following power laws. The powers are called critical exponents. RG predicts that continuous 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. Recently there has been great theoretical progress by using the so-called conformal bootstrap (CB) method. In particular for the universality class of the three-dimensional Ising model, critical exponents and structure constants were computed with unprecedented accuracy. Highly accurate results for the XY and Heisenberg universality classes were obtained recently. Since the CB method starts from a rather abstract characterisation of RG fixed points, it is desirable to check whether the results agree with those obtained by other methods. In the current project we have improved the accuracy for the XY and Heisenberg universality classes by using Monte Carlo simulations of improved lattice models to a level similar to that of the CB. The XY universality class is of particular interest, since the lambda-transition of Helium shares this universality class. Accurate experimental results are available for this transition. There is a small, but significant difference between experiment and theory, where CB and Monte Carlo simulations of improved lattice models give consistent and accurate results. In the most recent period of the project I studied a cubic perturbation of the Heisenberg fixed point and in particular the cubic fixed point. This problem had been studied since the early days of the RG. Basic questions have been settled only recently. A cubic perturbation of the O(3) internal symmetry might be caused by the crystal lattice. The cubic symmetry might also be underlying in structural transitions as for example of perovskite. In the case of O(3) symmetry, it the turns out that the O(3) invariant fixed point is unstable against a cubic perturbation. However, the RG flow is very slow and the cubic fixed point is close to the Heisenberg one. It follows that critical exponents take very similar values for the two universality classes. Group theory suggests that for spin l=2 than for l=0 a larger effect of the cubic perturbation is expected. I intend to compare with recent CB results.

• DFG Programme: Research Grants