Open Quadratic Fermion Systems and Algebras of Affine Transformations

TL;DR

This paper explores the algebraic structure of open quadratic fermion systems, revealing an isomorphism with affine transformations, and introduces a new method for solving their master equations.

Contribution

It establishes an isomorphism between fermionic Liouvillian commutators and affine transformations, providing a novel algebraic framework for analyzing quadratic fermion systems.

Findings

Algebra of fermionic Liouvillians is isomorphic to affine transformations.

New perspective method for solving master equations of quadratic fermion systems.

Brief discussion on algebraic structures for boson and general fermion systems.

Abstract

We study evolution of open quadratic fermion systems in the framework of the quantum Markovian semigroup approach. We show that the algebra concerning commutators of Liouvillians for systems of quadratic interacting fermions of finite number, say $\mathcal{N}$, is isomorphic to that of certain affine transformations on the space of square matrices of size $\mathcal{N}$. By the use of this algebraic structure, we present a perspective method for solving master equations of quadratic fermion systems. Here, we mainly deal with gauge invariant quadratic interacting fermion systems. We briefly mention similar algebraic structures for general quadratic fermion systems and quadratic boson systems. Keywords : open quantum system, Markovian quantum dynamical system, quadratic interacting Fermion, affine transformation, asymptotic behavior, skin effect