Long-range random transverse-field Ising model in three dimensions

We consider the random transverse-field Ising model in d=3 dimensions with long-range ferromagnetic interactions which decay as a power α>d with the distance. Using a variant of the strong-disorder renormalization group method we study numerically the phase-transition point from the paramagnetic side. We find that the fixed point controlling the transition is of the strong-disorder type, and based on experience with other similar systems, we expect the results to be qualitatively correct, but probably not asymptotically exact. The distribution of the (sample dependent) pseudocritical points is found to scale with 1/lnL, L being the linear size of the sample. Similarly, the critical magnetization scales with (lnL)χ/Ld and the excitation energy behaves as L-α. Using extreme-value statistics we argue that extrapolating from the ferromagnetic side the magnetization approaches a finite limiting value and thus the transition is of mixed order.