2025/03/16 by Kubota, Yosuke · 4 citations
#55P42 #81R15 #Algebraic Topology (math.AT) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Quantum Physics (quant-ph) #Strongly Correlated Electrons (cond-mat.str-el)
paper · doi:10.48550/arxiv.2503.12618
We provide a mathematical realization of a conjecture by Kitaev, on the basis of the operator-algebraic formulation of infinite quantum spin systems. Our main results are threefold. First, we construct an Ω-spectrum IP_* whose homotopy groups are isomorphic to the smooth homotopy group of invertible gapped quantum systems on Euclidean spaces. Second, we develop a model for the homology theory associated with the Ω-spectrum IP_*, describing it in terms of the space of quantum systems placed on an arbitrary subspace of a Euclidean space. This involves introducing the concept of localization flow, a semi-infinite path of quantum systems with decaying interaction range, inspired by Yu's localization C*-algebra in coarse index theory. Third, we incorporate spatial symmetries given by a crystallographic group Γ and define the Ω-spectrum IP_*Γ of Γ-invariant invertible phases. We propose a strategy for computing the homotopy group πn(IPdΓ) that uses the Davis--Lück assembly map and its description by invertible gapped localization flow. In particular, we show that the assembly map is split injective, and hence πn(IPdΓ) contains a computable direct summand.