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 ${\mathcal{E}}_{\infty }$-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 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.

Minor revisions from v1 and a new appendix with background on operads, E infinity spaces and simplicial sets, as well as additional functional analytic background in section 2

References in corpus (31)

• Quantum Spin Hall Effect in Graphene
• Topological Insulators in Three Dimensions
• Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect
• Topological invariants of time-reversal-invariant band structures
• Periodic table for topological insulators and superconductors
• Entanglement spectrum of a topological phase in one dimension
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Topological phases of fermions in one dimension
• Classifying quantum phases using Matrix Product States and PEPS
• Spectral Gap and Exponential Decay of Correlations
• Braiding statistics approach to symmetry-protected topological phases
• Topological Phases of One-Dimensional Fermions: An Entanglement Point of View
• Lieb-Robinson Bounds and the Exponential Clustering Theorem
• Classification of Topological Insulators and Superconductors
• Reflection positivity and invertible topological phases
• Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology
• Locality in Quantum and Markov Dynamics on Lattices and Networks
• Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems
• Symmetry Protected Topological phases and Generalized Cohomology
• Higher Anomalies, Higher Symmetries, and Cobordisms I: Classification of Higher-Symmetry-Protected Topological States and Their Boundary Fermionic/Bosonic Anomalies via a Generalized Cobordism Theory
• Higher-dimensional generalizations of the Berry curvature
• Higher Anomalies, Higher Symmetries, and Cobordisms III: QCD Matter Phases Anew
• Local Noether theorem for quantum lattice systems and topological invariants of gapped states
• On adiabatic cycles of quantum spin systems
• A classification of phases of bosonic quantum lattice systems in one dimension
• Gamma Spaces and Information
• Classification and Construction of Topological Phases of Quantum Matter
• Unital operads, monoids, monads, and bar constructions

Cited by in corpus (9)

• Multi wavefunction overlap and multi entropy for topological ground states in (2+1) dimensions
• Higher Berry Curvature from the Wave Function I: Schmidt Decomposition and Matrix Product States
• Charting the space of ground states with tensor networks
• Higher Berry Curvature from the Wave function II: Locally Parameterized States Beyond One Dimension
• Higher Berry curvature from matrix product states
• Anomalous continuous symmetries and quantum topology of Goldstone modes
• Charge pumps, boundary modes, and the necessity of unnecessary criticality
• A Classifying Space for Phases of Matrix Product States
• Equivariant Parameter Families of Spin Chains: A Discrete MPS Formulation