Renormalisation of Fermionic Cellular Automata

TL;DR

This paper introduces an exact renormalisation scheme for fermionic cellular automata, enabling analysis of their large-scale behavior and fixed points through a systematic grouping and subspace selection method.

Contribution

It provides a necessary and sufficient condition for renormalisability and characterises the renormalisation flow for specific fermionic automata models.

Findings

Identified all fixed points of the renormalisation flow.

Derived conditions for automaton renormalisability.

Mapped the renormalisation flow for two-cell tiles and two time steps.

Abstract

We present an exact renormalisation scheme for fermionic cellular automata on hypercubic lattices. By grouping neighbouring cells into tiles and selecting subspaces within them, multiple evolution steps on the original system correspond to a single step of an effective automaton acting on the subspaces. We derive a necessary and sufficient condition for renormalisability and fully characterise the renormalisation flow for two-cell tiles and two time steps of nearest-neighbour fermionic automata on a chain of spinless modes, identifying all fixed points.