Block renormalization for quantum Ising models in dimension d = 2: applications to the pure and random ferromagnet, and to the spin-glass. (19th January 2015) - Monthus, Cécile
Abstract: For the quantum Ising chain, the self-dual block renormalization procedure of Fernandez-Pacheco (1979 Phys. Rev. D19 3173 ) is known to reproduce exactly the location of the zero-temperature critical point and the correlation length exponent ν  = 1. Recently, Miyazaki and Nishimori (2013 Phys. Rev. E87 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 ν  ≃ 0.625 in d  = 2 (to be compared with the 3D classical Ising model exponent ν  ≃ 0.63) and ν  ≃ 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  ≃ 0.65, the typical correlation exponent ν typ  ≃ 0.44 and the finite-size correlation exponent ν FS  ≃ 1.25. We discuss the similarities and differences with the Strong Disorder Renormalization results.