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Physical Review X

Symmetry Classification of Many-Body Lindbladians: Tenfold Way and Beyond

Lucas Sá, Pedro Ribeiro, Tomaž Prosen

Published Aug 16, 2023• 58 citations
Publisher

Abstract

A framework for the classification of universal properties of realistic, chaotic, dissipative quantum systems offers building blocks of dynamic dissipative evolution, with a potential impact on the fabrication of complex quantum structures.
DOI
10.1103/physrevx.13.031019

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The tenfold way is a classification scheme for the building blocks of free fermion systems. More precisely, it classifies the isomorphism classes of spaces of equivariant free Hamiltonians in irreducible fermion systems with symmetries. This classification scheme naturally leads to the K-theoretical classification of topological phases of matter, known as the periodic table of topological insulators and superconductors. Topological K-theory is represented by spectra KUKUKU and KOKOKO, and in this article we present realizations of these spectra in terms of time evolution operators of irreducible free fermion systems with symmetries, with explicit formulas for the structural suspension maps. We introduce a geometric definition of the space of weakly interacting time evolution operators, as the complement of the cut locus of the subspace of free operators. Our main result is that spectra KUwiKU^{wi}KUwi and KOwiKO^{wi}KOwi of weakly interacting time evolution operators deformation retract to KUKUKU and KOKOKO. We thus have a stable homotopy theoretical proof that the tenfold way is stable to weak interactions.