Dreaming up scale invariance via inverse renormalization group

Area of Science:

• Statistical Physics
• Machine Learning
• Computational Physics

Background:

• The renormalization group (RG) is crucial for understanding critical phenomena and scale invariance.
• Inverting the RG coarse-graining procedure to generate microscopic configurations from coarse-grained states is theoretically challenging.

Purpose of the Study:

• To investigate the capability of minimal neural networks to invert the RG coarse-graining procedure.
• To explore the generation of critical configurations in the two-dimensional Ising model using machine learning.

Main Methods:

• Utilizing minimal neural networks with few trainable parameters to learn probability distributions.
• Applying probabilistic approaches to reconstruct scale-invariant distributions without microscopic data.
• Performing real-space renormalization group analysis on generated configurations.

Main Results:

• Neural networks, even with three parameters, can generate critical configurations for the Ising model.
• Generated configurations reproduce key scaling behaviors (magnetic susceptibility, heat capacity, Binder ratios).
• Models capture scale invariance and nontrivial RG transformation eigenvalues, despite imperfect inversion.

Conclusions:

• Minimal neural networks can effectively learn and reproduce the RG-relevant structure of critical distributions.
• Simple, local generative rules are sufficient to encode universality in critical phenomena.
• This approach offers potential for efficient generative models of statistical physics ensembles.