Block renormalization for quantum Ising models in dimension d=2: applications to the pure and random ferromagnet, and to the spin-glass

For the quantum Ising chain, the self-dual block renormalization procedure of Fernandez-Pacheco (1979 Phys. Rev. D 19 3173) is known to reproduce exactly the location of the zero-temperature critical point and the correlation length exponent nu = 1. Recently, Miyazaki and Nishimori (2013 Phys. Rev. E 87 032154) have proposed to study the disordered quantum Ising model in dimensions d > 1 by applying the Fernandez-Pacheco procedure successively in each direction. To avoid the inequivalence of directions of their approach, we propose here an alternative procedure where the d directions are treated on the same footing. For the pure model, this leads to the correlation length exponents nu similar or equal to 0.625 in d = 2 (to be compared with the 3D classical Ising model exponent nu similar or equal to 0.63) and nu similar or equal to 0.5018 (to be compared with the 4D classical Ising model mean-field exponent nu = 1/2). For the disordered model in dimension d = 2, either ferromagnetic or spin-glass, the numerical application of the renormalization rules to samples of linear size L = 4096 yields that the transition is governed by an Infinite Disorder Fixed Point, with the activated exponent psi similar or equal to 0.65, the typical correlation exponent nu(typ) similar or equal to 0.44 and the finite-size correlation exponent nu(FS) similar or equal to 1.25. We discuss the similarities and differences with the Strong Disorder Renormalization results.