In this paper, we present a homotopical framework for studying invertible gapped phases of matter from the point of view of infinite spin lattice systems, using the framework of algebraic quantum mechanics. We define the notion of quantum state types. These are certain lax-monoidal functors from the category of finite-dimensional Hilbert spaces to the category of topological spaces. The universal example takes a finite-dimensional Hilbert space ${\mathscr{H}}$ to the pure state space of the quasi-local algebra of the quantum spin system with Hilbert space ${\mathscr{H}}$ at each site of a specified lattice. The lax-monoidal structure encodes the tensor product of states, which corresponds to stacking for quantum systems. We then explain how to formally extract parametrized phases of matter from quantum state types, and how they naturally give rise to ${\mathcal{ℰ}}_{\infty }$-spaces for an operad we call the “multiplicative” linear isometry operad. We define the notion of invertible quantum state types and explain how the passage to phases for these is related to group completion. We also explain how invertible quantum state types give rise to loop-spectra. Our motivation is to provide a framework for constructing Kitaev’s loop-spectrum of bosonic invertible gapped phases of matter. Finally, as a first step toward understanding the homotopy types of the loop-spectra associated to invertible quantum state types, we prove that the pure state space of any UHF algebra is simply connected.

在本文中，我们提出了一个同伦框架，利用代数量子力学的框架，从无限自旋晶格系统的角度研究物质的可逆带隙相。我们定义量子态类型的概念。这些是从有限维希尔伯特空间范畴到拓扑空间范畴的某些松散幺半函子。通用示例采用有限维希尔伯特空间${\mathscr{H}}$到带有希尔伯特空间的量子自旋系统的准局域代数的纯态空间${\mathscr{H}}$在指定晶格的每个位置。松散幺半结构对状态的张量积进行编码，这对应于量子系统的堆叠。然后，我们解释如何从量子态类型中正式提取物质的参数化相，以及它们如何自然地产生${\mathcal{ℰ}}_{\mathrm{无穷大}}$- 操作数的空间，我们称为“乘法”线性等距操作数。我们定义了可逆量子态类型的概念，并解释了这些阶段的过渡如何与群完成相关。我们还解释了可逆量子态类型如何产生环谱。我们的动机是提供一个框架来构建基塔耶夫的玻色可逆带隙物质相环谱。最后，作为理解与可逆量子态类型相关的环谱同伦类型的第一步，我们证明任何 UHF 代数的纯态空间都是单连通的。