Renormalisation of Fermionic Cellular Automata

TL;DR

This work introduces an exact, tile-based renormalisation scheme for fermionic cellular automata, enabling coarse-grained FCA that remain local and unitary. By pairing coarse-graining with a finite-dimensional renormalisation equation, it reduces renormalisability to a concrete algebraic condition and fully characterises the flow for 1D spinless, nearest-neighbour FCA, identifying fixed points corresponding to shifts and a qubit-like fixed point. The framework leverages the Wrapping Lemma to move between infinite and finite lattices, preserving tractable analysis and enabling exact multicellular, multistep to single-step descriptions. The findings illuminate how renormalisation acts within FCA space, with implications for efficient quantum simulation, circuit design, and the exploration of dynamical topological features in discrete fermionic systems.

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.

Figures (4)

• Figure 1: The coarse-grained lattice $\mathbb{L'}$ of $\mathbb{L}$ is given by the identification $\Lambda_{\boldsymbol{\mathrm x}}\rightarrow \boldsymbol{\mathrm x}=(x,y)$.
• Figure 2: Representation of the maps $\mathcal{E} ^\dag, \mathcal{E}$ between the algebra $\mathsf{A}(\mathbb{L})$ on the lattice $\mathbb{L}$ and its coarse-grained counterpart $\mathsf{A}'(\mathbb{L}')\cong \mathsf{A}^\Pi(\mathbb{L})$ on the coarse-grained lattice $\mathbb{L}'$.
• Figure 3: The renormalisation equation \ref{['eq:cg']} amounts to ask the commutativity of the above diagram. If the FCA $\mathcal{S}$ is a renormalisation of the FCA $\mathcal{T}$, then evolving the coarse-grained algebra by one step by $\mathcal{S}_w$ ($\mathcal{S}$) and subsequently embedding this algebra into the original one via $\mathcal{E}$ is the same than first embedding the coarse-grained algebra in the original one and then evolving for $N$ steps by $\mathcal{T}_w$ ($\mathcal{T}$). In this way, a single step of $\mathcal{S}$ reproduces the action of $\mathcal{T}^N$ when restricted to the chosen degrees of freedom. The renormalisation equation is expressed on the wrapped FCA for technical convenience, but it applies to the unwrapped automata (see Appendix \ref{['sec:appendiceThm1']}).
• Figure 4: Trivial examples of $(2,\Pi_{\Lambda_0})$-renormalisation over a one-dimensional lattice $\mathbb Z$. Here we have $\Lambda_x=\{2x,2x+1\}$. (a) Renormalisation of cell-wise transformations. Two steps of gates $\mathcal{U}$ over two cells are mapped in a single step of new cell-wise gates $\mathcal{U}'$. (b) Renormalisation of a right shift. After two steps, all the content of $\Lambda_x$ is moved to $\Lambda_{x+1}$. The renormalised evolution is a right shift $\mathsf A'_x\mapsto \mathsf A'_{x+1}$ over the coarse-grained lattice $\mathbb{L}'$.

Theorems & Definitions (102)

• Definition 1: Shift
• Definition 2
• Proposition 1
• Definition 3: Fermionic Gate
• Remark 1
• Remark 2
• Definition 4: Wrapping
• Definition 5: Regular wrapping
• Lemma 1: Wrapping Lemma
• Definition 6