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We investigate the finite-size scaling of the boundary quantum geometrictensor (QGT) numerically close to the Anderson localization transition in thepresence of small external magnetic fields. The QGT exhibits universal scalingand reveals the crossover between the orthogonal and unitary critical states inweak random magnetic fields. The flow of the QGT near the critical pointsdetermines the critical exponents. Critical distributions of the QGT areuniversal and exhibit a remarkable isotropy even in a homogeneous magneticfield. We predict universal and isotropic Hall conductance fluctuations at themetal-insulator transition in an external magnetic field.

Universal Scaling Theory of the Boundary Geometric Tensor in Disordered Metals