From Integer Quantized Hall Plateaus to Z2 Topological Surfaces: Quantum Capacitance as a Probe of the Boundary Density of States

The integer quantum Hall effect (IQHE), Z2 topological insulators (TIs), and quantum capacitance are generally treated as separate subjects, yet they share a common thread: bulk topology gives rise to a protected boundary state, and quantum capacitance is one of the few electrostatic methods able to detect the density of states (DOS). This paper examines that connection, restricted to single-particle band topology; the fractional QHE is a many-body phase outside this scope, treated separately. The IQHE showed that quantized boundary transport has a topological (Chern-number) origin, a result later extended to zero field in Z2 insulators with helical edge and Dirac surface states. Quantum capacitance has become a key technique for mapping the Landau-level bulk DOS underlying the QHE and the boundary Dirac DOS of topological surfaces. Self-consistent screening theory complicates this: the bulk divides into compressible and incompressible regions that shift with the magnetic field, and local impedance measurements read as quantum capacitance show the bulk is not uniformly incompressible even while the plateau remains quantized. This paper argues that quantum capacitance is one of the clearest windows into the boundary DOS that topology imposes—while remaining insensitive to the invariant itself—and reveals the local, field-dependent character of the ‘insulating bulk’ assumption behind the QHE–TI analogy.