Abstract

Quantum Hall scanning images are often read as maps of a local potential, temperature, or viscosity, whereas a probe in fact records a finite-resolution functional of a transport operator. We formulate that functional explicitly, combining Landau-level projection, a particle-number Ward identity, magnetization-subtracted thermoelectric transport, a hydrodynamic Stokes–Ohm inversion, and finite-tip Fisher information. Two results follow in complementary transport regimes. In the strong-field, sharp-Landau-level regime, the defect-induced thermoelectric and electrical Hall contrasts of a smooth scalar defect obey δαxy^tr/δσxy = (Ec−µ)/(eT). At the retained long-wavelength order, the orbital form factor, defect geometry, and common tip kernel cancel after the heat-magnetization current is removed, so the zero of the thermoelectric contrast is pinned by energy weighting at Ec = µ rather than by defect shape. In the hydrodynamic regime, the measurable q2 tensor amplitudes mix Hall, longitudinal, transverse, boundary, electrothermal, and kinetic channels, so a Hall-odd image is not by itself a Hall-viscosity measurement. For a representative graphene geometry, a Schur-complement fit against the stated nuisance library yields a conditional one-standard-deviation sensitivity δdH ≃68 nm2 at SNR₀ = 200, with boundary slip the limiting nuisance. The framework turns visual interpretation of quantum Hall nanoscopy into a quantitative observability test for electrical, thermoelectric, and viscous response channels.

Abstract

Quantum Hall scanning images are often read as maps of a local potential, temperature, or viscosity, whereas a probe in fact records a finite-resolution functional of a transport operator. We formulate that functional explicitly, combining Landau-level projection, a particle-number Ward identity, magnetization-subtracted thermoelectric transport, a hydrodynamic Stokes–Ohm inversion, and finite-tip Fisher information. Two results follow in complementary transport regimes. In the strong-field, sharp-Landau-level regime, the defect-induced thermoelectric and electrical Hall contrasts of a smooth scalar defect obey δαxy^tr/δσxy = (Ec−µ)/(eT). At the retained long-wavelength order, the orbital form factor, defect geometry, and common tip kernel cancel after the heat-magnetization current is removed, so the zero of the thermoelectric contrast is pinned by energy weighting at Ec = µ rather than by defect shape. In the hydrodynamic regime, the measurable q2 tensor amplitudes mix Hall, longitudinal, transverse, boundary, electrothermal, and kinetic channels, so a Hall-odd image is not by itself a Hall-viscosity measurement. For a representative graphene geometry, a Schur-complement fit against the stated nuisance library yields a conditional one-standard-deviation sensitivity δdH ≃68 nm2 at SNR₀ = 200, with boundary slip the limiting nuisance. The framework turns visual interpretation of quantum Hall nanoscopy into a quantitative observability test for electrical, thermoelectric, and viscous response channels.

Abstract

Quantum Hall scanning images are often read as maps of a local potential, temperature, or viscosity, whereas a probe in fact records a finite-resolution functional of a transport operator. We formulate that functional explicitly, combining Landau-level projection, a particle-number Ward identity, magnetization-subtracted thermoelectric transport, a hydrodynamic Stokes–Ohm inversion, and finite-tip Fisher information. Two results follow in complementary transport regimes. In the strong-field, sharp-Landau-level regime, the defect-induced thermoelectric and electrical Hall contrasts of a smooth scalar defect obey δαxy^tr/δσxy = (Ec−µ)/(eT). At the retained long-wavelength order, the orbital form factor, defect geometry, and common tip kernel cancel after the heat-magnetization current is removed, so the zero of the thermoelectric contrast is pinned by energy weighting at Ec = µ rather than by defect shape. In the hydrodynamic regime, the measurable q2 tensor amplitudes mix Hall, longitudinal, transverse, boundary, electrothermal, and kinetic channels, so a Hall-odd image is not by itself a Hall-viscosity measurement. For a representative graphene geometry, a Schur-complement fit against the stated nuisance library yields a conditional one-standard-deviation sensitivity δdH ≃68 nm2 at SNR₀ = 200, with boundary slip the limiting nuisance. The framework turns visual interpretation of quantum Hall nanoscopy into a quantitative observability test for electrical, thermoelectric, and viscous response channels.

Abstract

Quantum Hall scanning images are often read as maps of a local potential, temperature, or viscosity, whereas a probe in fact records a finite-resolution functional of a transport operator. We formulate that functional explicitly, combining Landau-level projection, a particle-number Ward identity, magnetization-subtracted thermoelectric transport, a hydrodynamic Stokes–Ohm inversion, and finite-tip Fisher information. Two results follow in complementary transport regimes. In the strong-field, sharp-Landau-level regime, the defect-induced thermoelectric and electrical Hall contrasts of a smooth scalar defect obey δαxy^tr/δσxy = (Ec−µ)/(eT). At the retained long-wavelength order, the orbital form factor, defect geometry, and common tip kernel cancel after the heat-magnetization current is removed, so the zero of the thermoelectric contrast is pinned by energy weighting at Ec = µ rather than by defect shape. In the hydrodynamic regime, the measurable q2 tensor amplitudes mix Hall, longitudinal, transverse, boundary, electrothermal, and kinetic channels, so a Hall-odd image is not by itself a Hall-viscosity measurement. For a representative graphene geometry, a Schur-complement fit against the stated nuisance library yields a conditional one-standard-deviation sensitivity δdH ≃68 nm2 at SNR₀ = 200, with boundary slip the limiting nuisance. The framework turns visual interpretation of quantum Hall nanoscopy into a quantitative observability test for electrical, thermoelectric, and viscous response channels.