Rigorous Construction of Luttinger Liquids Through Ward Identities

Abstract

There are up to now two different ways to prove the key property on which our Luttinger liquid rigorous construction rests, the vanishing of the leading part of the Beta function. The first one was developed in the last years and is based in an essential way on the exact Mattis-Lieb solution of the Luttinger model. More recently, we found a new proof, based on the Ward identities obtained by a chiral local gauge transformation, applied to a Luttinger model with ultraviolet and infrared cutoffs. This is an old approach in the physical literature, but its implementation in an RG scheme is not trivial at all, because the ultraviolet and infrared cutoffs destroy local gauge invariance and produce “correction terms” with respect to the formal Ward identities. We discover however a new set of identities, called “Correction Identities”, relating the corrections to the Schwinger functions. By combining Ward and Correction identities with a Dyson equation, the vanishing of the Beta function follows, so that the infrared cutoff can be removed. As a byproduct, even the ultraviolet cutoff can be removed, after a suitable ultraviolet renormalization, so that a Quantum Field Theory corresponding to the Thirring model is constructed, showing the phenomenon of Chiral anomaly.