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. The distribution of the (sample dependent) pseudo-critical points is found to scale with $1/\ln L$, $L$ being the linear size of the sample. Similarly, the critical magnetization scales with $(\ln L)^χ/L^d$ 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.

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