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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 [Phys. Rev. D 19, 3173 (1979)] is known to reproduce exactly the location of the zero-temperature critical point and the correlation length exponent $ν=1$. Recently, Miyazaki and Nishimori [Phys. Rev. E 87, 032154 (2013)] 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 $ν\simeq 0.625$ in $d=2$ (to be compared with the 3D classical Ising model exponent $ν\simeq 0.63$) and $ν\simeq 0.5018$ (to be compared with the 4D classical Ising model mean-field exponent $ν=1/2$). 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 $ψ\simeq 0.65$, the typical correlation exponent $ν_{typ} \simeq 0.44$ and the finite-size correlation exponent $ν_{FS} \simeq 1.25$. We discuss the similarities and differences with the Strong Disorder Renormalization results.