Condensed Matter Seminar Series

Chris Hooley

University of St. Andrews

Classifying and characterising multicritical points using the numerical conformal bootstrap

A physical system at a second-order phase transition (a “critical point”) shows scale-invariant spatial correlations, which are usually described by a field theory with conformal symmetry, i.e. a conformal field theory (“CFT”).  The exponents of the power laws in those scale-invariant correlation functions (“scaling dimensions”) are directly related to the exponents that describe the divergence of physical observables, e.g. susceptibilities, as the critical point is approached (“critical exponents”).  These exponents typically depend only on the spatial dimensionality of the system and the group describing any global symmetries that it respects: this is universality.  Conventionally, one determines the critical exponents for a given universality class either through a renormalisation group analysis or by simulating (through e.g. Monte Carlo methods) a specific model.  Both of these methods are somewhat indirect, i.e. they study the approach to the critical point, not the critical point itself.
An alternative strategy is to work directly at the critical point, exploiting the conformal symmetry to constrain the possible values of the scaling dimensions, which then in turn places bounds on the critical exponents.  This approach is referred to as the “conformal bootstrap”.  In two-dimensional CFTs it can often be carried out analytically; in three-dimensional CFTs it was traditionally regarded as hopeless – until fairly recently, when the realisation that it can be turned into a problem in semidefinite programming allowed rapid progress [1].
In this talk, I will present the basics of this numerical conformal bootstrap technique, in a way designed to require no prior knowledge beyond basic quantum mechanics and statistical physics, and I will advertise some of its successes, e.g. world-record precision in the prediction of the critical exponents of the 3D Ising model.  I will then present some of our recent work on three-dimensional CFTs with product-group symmetries [2], and discuss its relevance to the nature of multicritical points in systems with competing ordered phases.
[1] D. Poland, S. Rychkov, and A. Vichi, Rev. Mod. Phys. 91, 015002 (2019).
[2] M. T. Dowens and CAH, arXiv:2004.14978 (2020); accepted for publication in J. High Energy Phys..