Fractional Conductances: S-matrix vs. Kubo Formalism

This lecture explores the conditions for coexistence of various perturbations in strongly interacting one-dimensional systems with N channels. Learn about how the neutrality requirement restricts interactions beyond forward-scattering quadratic terms in the Lagrangian, requiring each term in the Hamiltonian to conserve particle numbers. Discover how perturbations must represent new fields while preserving the Lagrangian form to become relevant and open a gap. Examine Haldane's constraint on coexisting perturbations and understand how conductance (in e²/h units) can be expressed as the difference between initial conductance of all N channels and conductance eliminated by compatible relevant perturbations. Compare two approaches to calculating conductance—scattering matrix and Kubo formula—and explore the proof of conductance independence from wire interactions in the presence of relevant perturbations. The presentation demonstrates how various combinations of relevant perturbations create different possible fractional conductances.