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Splitting obstructions and $\mathbb{Z}_2$ invariants in time-reversal symmetric topological insulators

Alessandro Ferreri, Domenico Monaco, Gabriele Peluso

Abstract

The Fu-Kane-Mele $\mathbb{Z}_2$ index characterizes two-dimensional time-reversal symmetric topological phases of matter. We shed some light on some features of this index by investigating projection-valued maps endowed with a fermionic time-reversal symmetry. Our main contributions are threefold. First, we establish a decomposition theorem, proving that any such projection-valued map admits a splitting into two projection-valued maps that are related to each other via time-reversal symmetry. Second, we provide a complete homotopy classification theorem for these maps, thereby clarifying their topological structure. Third, by means of the previous analysis, we connect the Fu-Kane-Mele index to the Chern number of one of the factors in the previously-mentioned decomposition, which in turn allows to exhibit how the $\mathbb{Z}_2$-valued topological obstruction to constructing a periodic and smooth Bloch frame for the projection-valued map, measured by the Fu-Kane-Mele index, can be concentrated in a single pseudo-periodic Kramers pair.

Splitting obstructions and $\mathbb{Z}_2$ invariants in time-reversal symmetric topological insulators

Abstract

The Fu-Kane-Mele index characterizes two-dimensional time-reversal symmetric topological phases of matter. We shed some light on some features of this index by investigating projection-valued maps endowed with a fermionic time-reversal symmetry. Our main contributions are threefold. First, we establish a decomposition theorem, proving that any such projection-valued map admits a splitting into two projection-valued maps that are related to each other via time-reversal symmetry. Second, we provide a complete homotopy classification theorem for these maps, thereby clarifying their topological structure. Third, by means of the previous analysis, we connect the Fu-Kane-Mele index to the Chern number of one of the factors in the previously-mentioned decomposition, which in turn allows to exhibit how the -valued topological obstruction to constructing a periodic and smooth Bloch frame for the projection-valued map, measured by the Fu-Kane-Mele index, can be concentrated in a single pseudo-periodic Kramers pair.

Paper Structure

This paper contains 15 sections, 23 theorems, 171 equations.

Table of Contents

  1. From (symmetric) periodic Hamiltonians to (symmetric) projection-valued maps
  2. Continuous parallel transport
  3. Review: frames in absence of time-reversal symmetry
  4. Chern number through the matching matrices
  5. Pseudo-periodic frames
  6. From the symmetric matching matrices to the $\mathbb{Z}_2$-valued invariant
  7. Review: time-reversal symmetric matching matrices
  8. The $\mathbb{Z}_2$-valued invariant
  9. Results
  10. Symmetric splitting and pseudo-periodic symmetric frame
  11. Addivity and stability of the $\mathbb{Z}_2$ invariant
  12. From pseudo-periodic frames to periodic Parseval frames
  13. Emerging spin--type simmetry
  14. Completeness of the invariant
  15. Tools from algebraic geometry

Key Result

Proposition 1.1

The operator $\mathcal{U}_{\mathrm{BFZ}}$ defined above extends uniquely to a unitary operator

Theorems & Definitions (56)

  • Proposition 1.1
  • Proposition 1.2: Riesz formula
  • proof
  • Remark 1.3
  • Definition 1.4: Time-reversal symmetric projection-valued map
  • Definition 1.5: Symmetric periodic (Bloch) frame
  • Definition 1.6: Pseudo–periodic symmetric frame
  • Definition 1.7: Symmetric splitting
  • Definition 1.8: Equivalence relations for time-reversal symmetric projection-valued maps
  • Remark 1.9: Role of Murray--von Neumann, unitary and homotopy equivalences
  • ...and 46 more