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

Yosuke Kubota

TL;DR

This work provides a rigorous operator-algebraic realization of Kitaev's program by constructing an invertible gapped phase spectrum $\mathit{IP}_d$ whose smooth, equivariant realizations encode the topology of interacting quantum spin systems. It develops a sheaf-theoretic framework on the category of manifolds to realize $IP_d$ as an $\Omega$-spectrum via Kitaev’s pump, a Hastings-based adiabatic transport, and localization flows that connect bulk and boundary phenomena. A coarse geometric formulation yields a generalized IP-homology theory $IP_{loc}$ with a split-injective assembly map in the crystalline setting, enabling computable invariants and partial validations of the generalized cohomology hypothesis. The paper also derives concrete low- and mid-degree structure, clarifying the relationships to symmetry-protected topological phases, and establishing a path to rational (Chern–Dold) descriptions of these interacting phases. Overall, it furnishes a robust mathematical bridge between stable homotopy theory, coarse index theory, and the classification of spatially symmetric invertible quantum phases.

Abstract

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 $\mathit{IP}_*$ 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 $Ω$-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 $Γ$ and define the $Ω$-spectrum $\mathit{IP}_*^Γ$ of $Γ$-invariant invertible phases. We propose a strategy for computing the homotopy group $π_n(\mathit{IP}_d^Γ)$ 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 $π_n(\mathit{IP}_d^Γ)$ contains a computable direct summand.

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

TL;DR

This work provides a rigorous operator-algebraic realization of Kitaev's program by constructing an invertible gapped phase spectrum whose smooth, equivariant realizations encode the topology of interacting quantum spin systems. It develops a sheaf-theoretic framework on the category of manifolds to realize as an -spectrum via Kitaev’s pump, a Hastings-based adiabatic transport, and localization flows that connect bulk and boundary phenomena. A coarse geometric formulation yields a generalized IP-homology theory with a split-injective assembly map in the crystalline setting, enabling computable invariants and partial validations of the generalized cohomology hypothesis. The paper also derives concrete low- and mid-degree structure, clarifying the relationships to symmetry-protected topological phases, and establishing a path to rational (Chern–Dold) descriptions of these interacting phases. Overall, it furnishes a robust mathematical bridge between stable homotopy theory, coarse index theory, and the classification of spatially symmetric invertible quantum phases.

Abstract

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 -spectrum 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 -spectrum , 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 and define the -spectrum of -invariant invertible phases. We propose a strategy for computing the homotopy group 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 contains a computable direct summand.

Paper Structure

This paper contains 56 sections, 122 theorems, 446 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Functional-analytic realization of Kitaev's conjecture
  3. (1) Analysis of the gapped interpolation
  4. (2) Difference of continuous and smooth homotopies
  5. (3) Uniformity of the internal degrees of freedom
  6. Computations of homotopy groups and SPT phases
  7. Models of generalized homology theory
  8. Invertible phases with spatial symmetry and the assembly map
  9. Construction of the paper
  10. Analysis of quantum spin systems
  11. Coarse geometry of a lattice of the Euclidean space
  12. Quantum spin system
  13. Smooth family of Hamiltonians and the adiabatic theorem
  14. Adiabatic interpolation
  15. The ground state of invertible gapped Hamiltonians
  16. ...and 41 more sections

Key Result

Theorem 1

There is an $\Omega$-spectrum $\{ \mathit{IP}_d, \kappa_d\}_{d \in \mathbb{Z}_{\geq 0}}$ such that the group $\mathrm{IP}^d(\mathscr{M})\coloneqq [\mathscr{M}, \mathit{IP}_d]$ is isomorphic to the set of smooth homotopy classes of smooth families of invertible gapped uniformly almost local (IG UAL)

Figures (7)

  • Figure 4.1: The picture of $\kappa_d(\boldsymbol{\mathsf{H}})$ (here $\mathsf{H}|_t$ is an abbreviation for $\mathrm{ev}_t\mathsf{H}$).
  • Figure 4.5: The picture of $\kappa_d^R(\boldsymbol{\mathsf{H}})$ and $\kappa_d^R(\boldsymbol{\mathsf{H}}) \circ \check{\kappa}_d^R(\boldsymbol{\mathsf{H}})$.
  • Figure 2.66: Example of linearly coarsely transverse pair $Z_L$, $Z_R$.
  • Figure 4.10: The case of $0 \leq t \leq 1/2$.
  • Figure 4.17: The homotopy \ref{['eqn:pump.constant.homotopy']}. The shaded squares represent the homotopy $\overline{\mathsf{H}}$. The middle white square stands for $\mathsf{H}$.
  • ...and 2 more figures

Theorems & Definitions (327)

  • Theorem 1: \ref{['cor:spectrum', 'cor:fermionic.equivariant.spectrum']}
  • Theorem 2: \ref{['thm:IP.bivariant', 'thm:Atiyah.duality']}
  • Theorem 3: \ref{['thm:covering', 'thm:BCI']}
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.8
  • Remark 2.9
  • ...and 317 more