The Loring--Schulz-Baldes Spectral Localizer Revisited

TL;DR

The paper presents an elementary spectral-flow proof of the Loring–Schulz-Baldes spectral localizer result, unifying the 1D chiral and 2D IQHE cases and recasting the localizer within a bulk-edge framework. By introducing a bounded, infinite-volume localizer and a quantitative gap analysis, the authors derive explicit conditions under which the finite-volume signature recovers the Zak phase or Chern number without extrapolating to infinite volume. Central to the approach are three ingredients: locality bounds on commutators with the position operator, stable spectral flow, and a decoupling argument that isolates finite-volume contributions. The bulk-edge interpretation links the spectral flow to edge indices of a domain-wall Hamiltonian, providing a physically transparent explanation for the invariants and enabling explicit Chern-number calculations via a flattened operator formalism. Overall, the work sharpens the computational toolkit for disordered topological insulators, yielding robust finite-volume diagnostics with explicit parameter control and suggesting natural extensions to mobility gaps, boundary conditions, and higher symmetry classes.

Abstract

The spectral localizer, introduced by Loring in 2015 and Loring and Schulz-Baldes in 2017, is a method to compute the (infinite volume) topological invariant of a quantum Hamiltonian on $\ZZ^d$, as the signature of the (finite) localizer matrix. We present a direct and elementary spectral-theoretic proof treating the $d=1$ and $d=2$ cases on an almost equal footing. Moreover, we re-interpret the localizer as a higher-dimensional topological insulator via the bulk-edge correspondence.

Figures (3)

• Figure 1: A depiction of the function $f_{10}$ on $x\in[-20,20)\cap\mathbb{Z}$ in case $d=1$. Note $0\notin\operatorname{im} f_{10}$.
• Figure 2: (Left) The shaded area indicates where the spectrum of $Q_t$ must lie for $t\in[0,1]$. The only crossing of $0$ can occur at $t=\tfrac{1}{2}$. Black lines indicate the two eigenspaces we know explicitly, namely $E_{-1}(Q_0)\cap E_{1}(Q_1)$ and $E_{1}(Q_0)\cap E_{-1}(Q_1)$. (Right) The two paths for computing the spectral flow of $H_{t,s}$ defined by \ref{['eq:Hst-def']}.
• Figure 3: The domain-wall system in mixed real-space for $x$ momentum space for $\kappa$.

Theorems & Definitions (30)

• Theorem 1.1: Loring--Schulz-Baldes
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Proposition 2.4
• Lemma 2.5: Spectral flow may be flattened
• proof
• Lemma 2.6: An index theorem for the flattened spectral flow
• Remark 2.7
• proof : Proof of \ref{['lem:spectral-flow-Q']}