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arXiv:2301.06069

Open Quadratic Fermion Systems and Algebras of Affine Transformations

Tamura, Hiroshi

Published Jan 15, 2023• 0 citations
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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

DOI
10.48550/arxiv.2301.06069

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We study the steady states of translation-invariant open quantum many-body systems governed by Lindblad master equations, where the Hamiltonian is quadratic in the ladder operators, and the Lindblad operators are either linear or quadratic and Hermitian. These systems are called quasifree and quadratic, respectively. We find that steady states of one-dimensional systems with finite-range interactions necessarily have exponentially decaying Green's functions. For the quasifree case without quadratic Lindblad operators, we show that fermionic systems with finite-range interactions are noncritical for any number of spatial dimensions and provide bounds on the correlation lengths. Quasifree bosonic systems can be critical in D>1 dimensions. Last, we address the question of phase transitions in quadratic systems and find that, without symmetry constraints beyond invariance under single-particle basis and particle-hole transformations, all gapped Liouvillians belong to the same phase.