Designing rules behind renormalization group(RG), and invertible RG

Hong-Ye Hu
​March 19, 2019

Renormalization group(RG) plays a central role in the study of quantum field theory and many-body physics, and its influence has been spread to other fields outside physics. The RG transformation progressively coarse-grains the field configuration to extract relevant features. The coarse-graining rules or RG schemes are generally system-dependent and requires human design. Take the real-space RG of Ising model for example. One should coarse-grain the uniform spin component for a ferromagnetic model, but the staggered spin component for an anti-ferromagnetic model, and the RG rule can be more complicated if the Ising couplings are random. When it comes to the momentum-space RG, the rule becomes to renormalize the low-energy degrees of freedom by decimating the high-energy degrees of freedom. What is the general designing principle behind all these RG schemes? Can a machine learns to design the optimal RG scheme based on the field theory action?

Fig.1: Relation between (a) RG and (b) generative model.