Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fragile Topological Phases and Topological Order of 2D Crystalline Chern Insulators

Hisham Sati, Urs Schreiber

Abstract

We apply methods of equivariant homotopy theory, which may not previously have found due attention in condensed matter physics, to classify first the fragile/unstable topological phases of 2D crystalline Chern insulator materials, and second the possible topological order of their fractional cousins. We highlight that the phases are given by the equivariant 2-Cohomotopy of the Brillouin torus of crystal momenta (with respect to wallpaper point group actions) -- which, despite the attention devoted to crystalline Chern insulators, seems not to have been considered before. Arguing then that any topological order must be reflected in the adiabatic monodromy of gapped quantum ground states over the covariantized space of these band topologies, we compute the latter in examples where this group is non-abelian, showing that any potential FQAH anyons must be localized in momentum space. We close with an outlook on the relevance for the search for topological quantum computing hardware. Mathematical details are spelled out in a supplement.

Fragile Topological Phases and Topological Order of 2D Crystalline Chern Insulators

Abstract

We apply methods of equivariant homotopy theory, which may not previously have found due attention in condensed matter physics, to classify first the fragile/unstable topological phases of 2D crystalline Chern insulator materials, and second the possible topological order of their fractional cousins. We highlight that the phases are given by the equivariant 2-Cohomotopy of the Brillouin torus of crystal momenta (with respect to wallpaper point group actions) -- which, despite the attention devoted to crystalline Chern insulators, seems not to have been considered before. Arguing then that any topological order must be reflected in the adiabatic monodromy of gapped quantum ground states over the covariantized space of these band topologies, we compute the latter in examples where this group is non-abelian, showing that any potential FQAH anyons must be localized in momentum space. We close with an outlook on the relevance for the search for topological quantum computing hardware. Mathematical details are spelled out in a supplement.
Paper Structure (19 sections, 1 theorem, 51 equations, 8 figures)

This paper contains 19 sections, 1 theorem, 51 equations, 8 figures.

Table of Contents

  1. Agenda
  2. Fragile Crystalline Chern Phases
  3. Topological Order of FCIs
  4. Covariantization under Diffeos
  5. Introduction and Concepts
  6. Symmetry and Equivariance
  7. Topological Order and Monodromy
  8. FQAH Systems.
  9. General Topological Order.
  10. Covariance under Diffeomorphisms
  11. Methods and Results
  12. Generalities
  13. Topological phases
  14. No Symmetry
  15. p3 Symmetry
  16. ...and 4 more sections

Key Result

Proposition 3.3.2

The unitary irrep $\mathbf{2}$ of $\mathrm{Sym}(3)$ is generated by a couple of quantum gates(cf. NielsenChuang2010) known as the Pauli Z-gate$Z$, and the rotation gate$R_y(8\pi/3)$, where the latter corresponds to the cyclic permutation of defect anyons (cf. Fig. CyclicPermutation).

Figures (8)

  • Figure 1: Under adiabatic tuning of classical parameters $p$ along paths $\gamma$ in parameter space, the ground states of a gapped quantum system $\mathcal{H}$ undergo unitary transformations $U_\gamma$. For topological states, these $U_\gamma$ depend only on the homotopy class of $\gamma$ (relative endpoints), thus making the Hilbert spaces $\mathcal{H}_p$ constitute a local system or flat bundle over parameter space, hence equivalently a linear representation of the fundamental group of closed paths (loops) $\ell$ at any base point $p_0$:
  • Figure 2: There is a duality between the FQH effect and its anomalous FQAH version, under which the ordinary space of positions inside the 2D material is exchanged for the "reciprocal" space of crystal momenta, while the external magnetic flux density is exchanged for the Berry curvature of the Bloch bands over this momentum space.
  • Figure 3: The quantum adiabatic theorem implies that gapped quantum ground states depending "topologically" (homotopically) on the configurations of points in a surface, transform under unitary braid representations as the configurations trace out loops in parameter space, forming braids of worldlines.
  • Figure 4: The non-trivial group commutator in the algebra of anyon quantum observables on a torus in terms of the anyon braiding phase$\zeta$.
  • Figure 5: The action on the 2-sphere of the non-trivial cyclic groups $\mathbb{Z}_{ / {n} }$, $n \geq 1$, is by rotation around a fixed axis and hence fixes precisely an antipodal pair of points, to be denoted "$\mathrm{n}$" and "$\mathrm{s}$".
  • ...and 3 more figures

Theorems & Definitions (7)

  • Definition 2.2.1: General topological order
  • Example 2.2.2: Traditional anyon braiding
  • Example 2.2.3: Topological order over the torus
  • Example 2.2.4: The situation for FQAH systems
  • Example 2.3.1
  • Definition 3.3.1
  • Proposition 3.3.2: SS25-FQH