A groupoid approach to interacting fermions

TL;DR

This paper develops a mathematical framework using groupoid theory to analyze the algebraic structure of interacting fermions on aperiodic lattices, aiming to facilitate $K$-theoretic methods in quantum many-body physics.

Contribution

It introduces a novel groupoid-based approach to the algebra of interacting fermions on aperiodic sets, connecting physical assumptions with advanced operator algebra techniques.

Findings

The algebra generated by inner-limit derivations can be completed to a groupoid-solvable pro-$C^ullet$-algebra.

Establishes a link between physical assumptions and algebraic structures suitable for $K$-theory.

First step towards applying $K$-theoretic tools to interacting fermion systems on aperiodic lattices.

Abstract

We consider the algebra $\dot\Sigma(\mathcal L)$ generated by the inner-limit derivations over the ${\rm GICAR}$ algebra of a fermion gas populating an aperiodic Delone set $\mathcal L$. Under standard physical assumptions such as finite interaction range, Galilean invariance and continuity with respect to the aperiodic lattice, we demonstrate that the image of $\dot \Sigma(\mathcal L)$ through the Fock representation can be completed to a groupoid-solvable pro-$C^\ast$-algebra. Our result is the first step towards unlocking the $K$-theoretic tools available for separable $C^\ast$-algebra for applications in the context of interacting fermions.