Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Drinfeld Center as Quantum State Monodromy over Bloch Hamiltonians around Defects

Hisham Sati, Urs Schreiber

Abstract

The Drinfeld center fusion category $\mathcal{Z}(\mathrm{Vec}_G)$ famously models anyons in certain lattice models. Here we demonstrate how its fusion rules may also describe topological order in fractional topological insulator materials, in the vicinity of point defects in the Brillouin zone. Concretely, we prove that $\mathcal{Z}(\mathrm{Vec}_G)$ reflects, locally over a punctured disk in the Brillouin zone, the monodromy (topological order) of gapped quantum states over the parameter space of Bloch Hamiltonians whose classifying space has fundamental group $G$.

Drinfeld Center as Quantum State Monodromy over Bloch Hamiltonians around Defects

Abstract

The Drinfeld center fusion category famously models anyons in certain lattice models. Here we demonstrate how its fusion rules may also describe topological order in fractional topological insulator materials, in the vicinity of point defects in the Brillouin zone. Concretely, we prove that reflects, locally over a punctured disk in the Brillouin zone, the monodromy (topological order) of gapped quantum states over the parameter space of Bloch Hamiltonians whose classifying space has fundamental group .
Paper Structure (19 sections, 12 theorems, 68 equations)

This paper contains 19 sections, 12 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Topological Order and Parameter Monodromy
  3. Classifying Spaces of Bloch Hamiltonians
  4. Homotopy of spaces of Bloch Hamiltonians
  5. Local Bloch Hamiltonian Monodromy
  6. Local Parameter Monodromy via the Drinfeld Center Z(G)
  7. Simple objects of Z(G) as Local Parameter Monodromy
  8. Fusion in Z(G) as Fusion of Local Parameter Monodromy
  9. Conclusion
  10. Background
  11. Homotopy
  12. Groupoids
  13. Groupoid Representations
  14. Inertia Groupoids
  15. Induced Representations
  16. ...and 4 more sections

Key Result

Lemma 3.1

[lemma]ConjugacyClassesAsFundamentalGroupOfFreeLoopSpace The local topological phases form the set of conjugacy classes of elements of $G$:

Theorems & Definitions (26)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • Proposition 3.4
  • proof
  • ...and 16 more