Coarse-Graining and Renormalization without Locality

The renormalization group (RG) describes how to systematically coarse-grain physical models and calculate their scaling behavior near a critical point. This approach has been immensely successful for systems where interactions are local, with a high degree of symmetry that effectively reduces the number of parameters that can contribute to the behavior in the ultraviolet limit. Here, we introduce a scheme inspired by the recent work on coarse-graining neural dynamics [1], which is capable of detecting infra-red behavior directly from experimental data without explicit reference to locality or symmetry. Specifically, our approach selects maximally correlated pairs of system elements, taking the correlations themselves as a proxy for local interaction, and compresses their activity using the Information Bottleneck method, while preserving the information that the compressed variables contain about the next-closest scale. Repeatedly applying such transformations recovers the renormalization group flow for the coupling strengths and variation of nearest-neighbor correlations with length scale for data taken from a 2D Ising system on the square lattice, showing the viability of this data-driven approach.

[1] L. Meshulam, et al. Phys. Rev. Lett. 123, 178103 (2019).