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

TL;DR

This paper explores the geometric structure of quantum critical submanifolds using information theory and differential geometry, revealing how relevant, marginal, and irrelevant operators influence the quantum metric's behavior at critical points.

Contribution

It establishes a geometric framework linking the singular behavior of the quantum metric to the nature of operators in the renormalization group, with explicit examples in XY and Haldane models.

Findings

Normal directions correspond to singular metric behavior.

Tangent directions have vanishing metric contributions.

The framework applies to well-known quantum models.

Abstract

We analyze the thermodynamical limit of the quantum metric along critical submanifolds of theory space. Building upon various results previously known in the literature, we relate its singular behavior to normal directions, which are naturally associated with relevant operators in the renormalization group sense. We formulate these results in the language of information theory and differential geometry. 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 behavior, while for the tangent ones it vanishes -- these directions lie in the kernel of the metric.