The effective boson–boson interaction U eff versus the critical boson density < n c >. U eff at < n c > → 0 ( H = H s , cross) was estimated from the linear T- dependence of < n >. For H < H s , U eff was estimated from < n c > at T c ( H ) in a (triangles) and at H c ( T ) in Supplementary Fig. 1c (squares) by using U eff = gμ B ( H s − H )/2< n c >. The broken line represents the < n c > dependence of U eff with 2D logarithmic renormalization and a cut-off by J ⊥ , phenomenologically expressed by U eff = − U 0 (1/ln< n c > + 1/ln( J ⊥ / J )). A bare boson–boson interaction U 0 = 6 K = 1.2 J and J ⊥ / J = 2 × 10 −5 are used.
