Self-similarity on 4d cubic lattice

TL;DR

This paper extends the concept of algebraic self-similarity from 3d to 4d cubic lattices, revealing a recursive structure involving matrices over _2 and their block spin decompositions.

Contribution

It demonstrates a 4d analogue of algebraic self-similarity on cubic lattices, showing how block matrices over _2 exhibit recursive decompositions with elegant formulations.

Findings

Existence of 4d self-similarity analogous to 3d case

Decomposition of 16 copies of a matrix into four block spin summands

Additional block spins appear with different matrices when entries are in _2

Abstract

A phenomenon of "algebraic self-similarity" on 3d cubic lattice, providing what can be called an algebraic analogue of Kadanoff--Wilson theory, is shown to possess a 4d version as well. Namely, if there is a $4\times 4$ matrix $A$ whose entries are indeterminates over the field $\mathbb F_2$, then the $2\times 2\times 2\times 2$ block made of sixteen copies of $A$ reveals the existence of four direct "block spin" summands corresponding to the same matrix $A$. Moreover, these summands can be written out in quite an elegant way. Somewhat strikingly, if the entries of $A$ are just zeros and ones -- elements of $\mathbb F_2$ -- then there are examples where two more "block spins" split out, and this time with different $A$'s.