Information geometry of quantum critical submanifolds: relevant, marginal and irrelevant operators

Abstract

We analyze the thermodynamical limit of the quantum metric along critical submanifolds of theory space. We relate its singular behaviour to normal directions, which are naturally associated to relevant operators in the renormalization group sense. We exemplify our theory through the paradigmatic examples of the XY and Haldane models, where the normal directions to the critical submanifolds are seen to be precisely those along which the metric has singular behaviour, while for the tangent ones it vanishes-these directions lie in the kernel of the metric.