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David Thouless coauthored a theory connecting the Hall conductivity of a two-dimensional electron gas to a topological invariant. In a uniform magnetic field and periodic potential, the Kubo formula can be written so that quantization becomes explicit when the Fermi energy lies in a gap. The integer character of the response explains why continuous changes of microscopic parameters need not continuously change the Hall plateau. A transition between distinct plateaus requires leaving the conditions that protect that quantization.^[ref]

A physicist who researched superfluidity and superconductivity in thin systems.^[ref]

The Kubo formula expresses a linear response through an equilibrium correlation function. For a perturbation $H^{\prime }(t)=-f(t)B$ and observable $A$, one convention gives $\delta ⟨A(t)⟩={\int }_{-\infty }^t{\chi }_{AB}(t-t^{\prime })f(t^{\prime })d{t}^{\prime }$, withThe step function enforces causality, and the commutator records the quantum response.

The sign depends on the declared perturbing-Hamiltonian convention. Applying the framework to currents or polarization relates transport coefficients to fluctuations in an unperturbed system, forming a foundation of linear response theory and the fluctuation-dissipation theorem.^[ref]