Renormalization-guided inverse blocking for lattice field generation: construction and validation

Abstract

We propose an algorithm for generating lattice field configurations based on the approximate inversion of a renormalization-group blocking transformation. We optimize the blocking transformation using a ``perfect blocking''condition so that the blocked lattice distribution is well approximated by a simple coarse action. The blocking is separated into an invertible smoothing transformation followed by decimation. Machine learning, in the form of a conditional normalizing flow, is used to reconstruct the short-distance degrees of freedom removed by the decimation. A short fine-action rethermalization then removes the residual mismatch. Because the coarse ensemble supplies the long-distance modes, the same blocking transformation and conditional flow can be reused recursively on larger lattices, producing a cascade of configurations from an initial small-volume ensemble. We test the method in two-dimensional $\phi^4$ theory with $\lambda=1$ at criticality and demonstrate stable cascade upscaling from $16^2$ to $2048^2$ lattices on local computational resources. Controlled rethermalization tests show that short-distance mismatches relax rapidly, whereas a deliberately introduced mismatch in the relevant thermal direction relaxes much more slowly. The construction uses ingredients that admit natural extensions to higher-dimensional systems and, ultimately, to gauge and fermionic degrees of freedom.