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Random Transverse Field Ising Model in dimension $d=2,3$ : Infinite Disorder scaling via a non-linear transfer approach

The 'Cavity-Mean-Field' approximation developed for the Random Transverse Field Ising Model on the Cayley tree [L. Ioffe and M. Mézard, PRL 105, 037001 (2010)] has been found to reproduce the known exact result for the surface magnetization in $d=1$ [O. Dimitrova and M. Mézard, J. Stat. Mech. (2011) P01020]. In the present paper, we propose to extend these ideas in finite dimensions $d>1$ via a non-linear transfer approach for the surface magnetization. In the disordered phase, the linearization of the transfer equations correspond to the transfer matrix for a Directed Polymer in a random medium of transverse dimension $D=d-1$, in agreement with the leading order perturbative scaling analysis [C. Monthus and T. Garel, arxiv:1110.3145]. We present numerical results of the non-linear transfer approach in dimensions $d=2$ and $d=3$. In both cases, we find that the critical point is governed by Infinite Disorder scaling. In particular exactly at criticality, the one-point surface magnetization scales as $\ln m_L^{surf} \simeq - L^{ω_c} v$, where $ω_c(d)$ coincides with the droplet exponent $ω_{DP}(D=d-1)$ of the corresponding Directed Polymer model, with $ω_c(d=2)=1/3$ and $ω_c(d=3) \simeq 0.24$. The distribution $P(v)$ of the positive random variable $v$ of order O(1) presents a power-law singularity near the origin $P(v) \propto v^a$ with $a(d=2,3)>0$ so that all moments of the surface magnetization are governed by the same power-law decay $\bar{(m_L^{surf})^k} \propto L^{- x_S}$ with $x_S=ω_c (1+a)$ independently of the order $k$.