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Homotopical Foundations of Parametrized Quantum Spin Systems

Agnes Beaudry, Michael Hermele, Juan Moreno, Markus Pflaum, Marvin Qi, Daniel Spiegel

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.

Homotopical Foundations of Parametrized Quantum Spin Systems

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 -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.
Paper Structure (23 sections, 57 theorems, 350 equations)

This paper contains 23 sections, 57 theorems, 350 equations.

Table of Contents

  1. Introduction
  2. Functorial properties of quasi-local algebras and their states
  3. Quasi-local algebras
  4. States and pure states
  5. Quantum state types
  6. Formal properties of quantum state types
  7. Cellularizing quantum state types
  8. The classifying spaces for quantum systems and phases
  9. Stacking structure
  10. Parametrized phases and group completion
  11. Loop-spectra and May's Recognition Principle
  12. Berry phase and free fermions
  13. Gapped Quantum Systems in spacetime dimension $0+1$
  14. K-theory and free fermions
  15. The fundamental group of the pure state space
  16. ...and 8 more sections

Key Result

Theorem 1

There is a topologically enriched functor which maps $\mathcal{H}$ to $\mathscr{P}_\Gamma(\mathcal{H})$ and with the property that for a linear isometric embedding $f\colon \mathcal{H} \longrightarrow \mathcal{K}$, is a closed embedding. Furthermore, the map which takes states to their tensor product gives a natural transformation making $\mathscr{P}_\Gamma$ into a lax monoidal functor.

Theorems & Definitions (174)

  • Example 1.2
  • Example 1.3
  • Example 1.4: qpump
  • Example 1.5
  • Theorem : \ref{['thm:uniquantstatetype']}
  • Example 1.6
  • Definition : \ref{['defn:quantumstatetype']}
  • Theorem : \ref{['thm:jmathHspace']}
  • Theorem : \ref{['thm:jquotientiso']}
  • Theorem : \ref{['thm:locisgroup']}
  • ...and 164 more