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A one-dimensional BDI Dirac-mass extension

Example

Assume AC. In the normalized BDI convention $q = 1, d = 1$ , the strong group is $K O^{0} (p t) = Z$ . The finite matrices below give gapped one-parameter Dirac Hamiltonians $H_{B} (k)$ , $k \in R$ , and integer-labelled difference systems with every invariant $ν \in Z$ . In particular a single negative mass against the positive reference is the generator. An integer invariant is constant along a continuous gapped path; the integer-labelled systems are not asserted to form a continuous path of changing index.

Facts & Assumptions

Given: AC and finite-dimensional complex Hilbert spaces with entrywise conjugation $K$ .

[F1]

The normalized tenfold extension theorem assigns BDI index $q = 1$ , subtracts the one negative Dirac generator, and identifies the resulting index-zero invariant with negative real multiplicity-rank difference (Tenfold real and complex Clifford-extension table).

Verification

technique · direct construction

1.1givenconstructalgebra

Put $e = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix})$ , $γ = (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix})$ and $s = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix})$ . Direct multiplication gives $e^{2} = s^{2} = I$ , $γ^{2} = - I$ and pairwise anticommutation. On $C^{2} \otimes C^{N}$ let $B$ be a real symmetric involution, $M_{B} = s \otimes B$ , and $H_{B} (k) = i k γ \otimes I + M_{B}$ . The kinetic term and mass are Hermitian and anticommute, so $H_{B} (k)^{2} = (k^{2} + 1) I$ . Thus every $k$ has gap at least one.

2.1step 1.1algebra

Take $T = K$ and $C = (e \otimes I) K$ . Both square to $1$ and commute. Complex conjugation changes $i k γ$ to $- i k γ$ and fixes $M_{B}$ , giving $T H_{B} (k) T^{- 1} = H_{B} (- k)$ . Conjugating additionally by $e$ reverses both $γ$ and $s$ , giving $C H_{B} (k) C^{- 1} = - H_{B} (- k)$ . Their product $e \otimes I$ is the self-adjoint chiral involution. These are precisely the momentum-reversing BDI symmetries of the one-dimensional Dirac family.

2.2F1step 1.1algebra

The admissible real symmetric masses anticommuting with both $e \otimes I$ and $γ \otimes I$ are exactly $s \otimes B$ . Indeed anticommutation with $e$ makes the diagonal blocks zero, symmetry makes the other blocks $B, B^{T}$ , and anticommutation with $γ$ forces $B = B^{T}$ . Squaring the mass gives $B^{2} = I$ . Therefore the reduced multiplicity problem is the actual real index-zero mass problem of [F1], with invariant $ν = \dim E_{-} (B) - \dim E_{-} (B_{0})$ . Rank is constant along an involution homotopy because the continuous idempotent $(I - B) / 2$ has integer-valued continuous trace. Fixed-rank involutions form a connected real Grassmannian, so this invariant also distinguishes exactly the stable components.

3.1F1step 1.1step 2.2constructalgebra∎

For $ν \geq 0$ take $N = ν$ , $B = - I_{N}$ , $B_{0} = + I_{N}$ ; for $ν < 0$ take $N = - ν$ , $B = + I_{N}$ , $B_{0} = - I_{N}$ . Use the zero-dimensional empty system at $ν = 0$ . The formula of step 2.2 gives invariant exactly $ν$ in both cases. For $N = 1$ , the negative mass against the positive reference gives $ν = 1$ , which [F1] sends to the unit of $K O^{0} (p t)$ . Direct sum adds the integers and reversing the difference negates them. The entire continuous $k$ family in step 1.1 has the same invariant; no gap-preserving continuous integer jump is claimed.

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Sources

Alexei Kitaev, Periodic table for topological insulators and superconductors (2009) (standard reference, not scraped)
M. F. Atiyah, R. Bott, and A. Shapiro, Clifford Modules, Topology 3, Supplement 1 (1964) (standard reference, not scraped)
M. F. Atiyah, K-Theory (lectures, notes by D. W. Anderson, 1967), with K-Theory and Reality reprint (standard reference, not scraped)