Homotopical Foundations of Parametrized Quantum Spin Systems

TL;DR

This paper develops a homotopical framework for classifying invertible gapped phases of quantum spin systems using algebraic quantum mechanics, connecting quantum states to topological and operadic structures.

Contribution

It introduces quantum state types as functors to topological spaces, linking phases of matter to $ ext{E}_ ext{infty}$-spaces and loop-spectra, advancing the mathematical understanding of topological phases.

Findings

Pure state space of any UHF algebra is simply connected.

Invertible quantum state types relate to group completion and loop-spectra.

Framework for constructing Kitaev's loop-spectrum of phases.

Abstract

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 to the pure state space of the quasi-local algebra of the quantum spin system with this Hilbert space 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 $\mathscr{E}_\infty$-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 towards 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.