Luttinger liquids from a microscopic perspective

Luttinger liquid theory is a powerful and widely applicable framework for modelling one-dimensional many-body quantum systems. Within this framework, one supposes that the macroscopic behaviour of such systems is entirely determined by two phenomenological parameters, g ₂ and g ₄. While there exists an intuitive and seemingly sensible physical interpretation of these parameters in terms of the scattering of the system's constituent particles, g ₂ and g ₄ are traditionally either fixed by experiment or calculated using heavy-duty numerical techniques, rather than inferred using scattering theory, and for this reason the interpretation remains untested. By applying Luttinger liquid theory in a simple setting, we show that a widely-held and repeatedly-stated belief, namely that the intrabranch terms appearing in Luttinger's model originate from microscopic intrabranch interactions, is a misconception. We begin with the microscopic model of an interacting one-dimensional, spin-polarized Fermi gas, which we systematically transform into a Luttinger model by introducing an effective interaction, linearizing the dispersion, and renormalizing. By this method, we are able to show that the usual interpretation of g ₄ as a measure of intrabranch scattering implies that it must vanish in the dilute limit. Since this runs contrary to conservation of particle number, we conclude that g ₄ cannot be related to intrabranch scattering. Rather, we show that g ₄ interactions must be included in the effective model in order to compensate for the deleterious effect that introducing an effective interaction has upon the model's energetics. We explicitly calculate an approximation to this correction for our simple system, and find that it agrees with the value of g ₄ found in the literature. We therefore propose a new fermionic Hamiltonian which agrees with the traditional model after bosonisation, but which better reflects the underlying microscopic physics.