The photospheric radius is one of the fundamental parameters governing the radiative equilibrium of a star. We report new observations of the nearest solar-type stars α Centauri A (G2V) and B (K1V) with the VLTI/PIONIER optical interferometer. The combination of four configurations of the VLTI enable us to measure simultaneously the limb darkened angular diameter θ LD and the limb darkening parameters of the two solar-type stars in the near-infrared H band (λ = 1.65 µm). We obtain photospheric angular diameters of θ LD (A) = 8.502 ± 0.038 mas (0.43%) and θ LD (B) = 5.999 ± 0.025 mas (0.42%), through the adjustment of a power law limb darkening model. We find H band power law exponents of α(A) = 0.1404 ± 0.0050 (3.6%) and α(B) = 0.1545 ± 0.0044 (2.8%), which closely bracket the observed solar value (α = 0.15027). Combined with the parallax π = 747.17 ± 0.61 mas previously determined, we derive linear radii of R A = 1.2234 ± 0.0053 R (0.43%) and R B = 0.8632 ± 0.0037 R (0.43%). The power law exponents that we derive for the two stars indicate a significantly weaker limb darkening than predicted by both 1D and 3D stellar atmosphere models. As this discrepancy is also observed on near-infrared limb darkening profile of the Sun, an improvement of the calibration of stellar atmosphere models is clearly needed. The reported PIONIER visibility measurements of α Cen A and B provide a robust basis to validate the future evolutions of these models.