Stable homotopy theory of invertible gapped quantum spin systems I: Kitaev's $Ω$-spectrum

• URL: http://arxiv.org/abs/2503.12618v1
• Date: Sun, 16 Mar 2025 19:21:06 GMT
• Title: Stable homotopy theory of invertible gapped quantum spin systems I: Kitaev's $Ω$-spectrum
• Authors: Yosuke Kubota,
• Abstract summary: We provide a realization of a proposal by Kitaev, on the basis of the operator-algebraic formulation of infinite quantum spin systems.<n>We develop a model for the homology theory associated with the $Omega$-spectrum $mathitIP_*$, describing it in terms of the space of quantum systems placed on an arbitrary subspace of a Euclidean space.<n>We incorporate spatial symmetries given by a crystallographic group $Gamma $ and define the $Omega$-spectrum $mathitIP_*Gamma$ of $
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We provide a mathematical realization of a proposal by Kitaev, on the basis of the operator-algebraic formulation of infinite quantum spin systems. Our main results are threefold. First, we construct an $\Omega$-spectrum $\mathit{IP}_*$ 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 $\Omega$-spectrum $\mathit{IP}_*$, 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 $\Gamma $ and define the $\Omega$-spectrum $\mathit{IP}_*^\Gamma$ of $\Gamma$-invariant invertible phases. We propose a strategy for computing the homotopy group $\pi_n(\mathit{IP}_d^\Gamma )$ that uses the Davis--L\"{u}ck assembly map and its description by invertible gapped localization flow. In particular, we show that the assembly map is split injective, and hence $\pi_n(\mathit{IP}_d^\Gamma)$ contains a computable direct summand.

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