We study the filling-controlled metal-insulator transition in the two-dimensional Hubbard model near half-filling with the use of zero temperature quantum Monte Carlo methods. In the metallic phase, the compressibility behaves as
κ∝∣μ−μc∣−0.58±0.08 where
μc is the critical chemical potential. In the insulating phase, the localization length follows
ξl∝∣μ−μc∣−νl with
νl=0.26±0.05. Under the assumption of hyperscaling, the compressibility data leads to a correlation length exponent
νκ=0.21±0.04. Our results show that the exponents
νκ and
νl agree within statistical uncertainty. This confirms the assumption of hyperscaling with correlation length exponent
ν=1/4 and dynamical exponent
z=4. In contrast the metal-insulator transition in the generic band insulators in all dimensions as well as in the one-dimensional Hubbard model satisfy the hyperscaling assumption with exponents
ν=1/2 and
z=2.
The correlation functions related to topological phase transitions in inversion-symmetric lattice models described by $2\ifmmode\times\else\texttimes\fi{}2$ Dirac Hamiltonians are discussed. In one dimension, the correlation function measures the charge-polarization correlation between Wannier states at different positions, while in two dimensions it measures the itinerant-circulation correlation between Wannier states. The correlation function is nonzero in both the topologically trivial and nontrivial states, and allows us to extract a correlation length that diverges at topological phase transitions. The correlation length and the curvature function that defines the topological invariants are shown to have universal critical exponents, allowing the notion of universality classes to be introduced. Particularly in two dimensions, the universality class is determined by the orbital symmetry of the Dirac model. The scaling laws that constrain the critical exponents are revealed, and are predicted to be satisfied even in interacting systems, as demonstrated in an interacting topological Kondo insulator.