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.

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