Random transverse field Ising model in dimensionsd = 2, 3: infinite disorder scaling via a non-linear transfer approach

Abstract

The ‘cavity-mean-field’ approximation developed for the random transverse field Isingmodel on the Cayley tree (Ioffe and Mézard 2010 Phys. Rev. Lett. 105 037001) hasbeen found to reproduce the known exact result for the surface magnetization ind = 1 (Dimitrova and Mézard 2011 J. Stat. Mech. P01020). In the presentpaper, we propose to extend these ideas in finite dimensionsd > 1 via a non-linear transfer approach for the surface magnetization. In the disorderedphase, the linearization of the transfer equations corresponds to the transfermatrix for a directed polymer in a random medium of transverse dimensionD = d − 1, in agreement with the leading order perturbative scaling analysis (Monthus and Garel 2011arXiv:1110.3145). We present numerical results of the non-linear transfer approach in dimensionsd = 2 and 3. In both cases, we find that the critical point is governed by infinite disorderscaling. In particular, exactly at criticality, the one-point surface magnetizationscales as , where ωc(d) coincides withthe droplet exponent ωDP(D = d − 1) of the corresponding directed polymer model, withωc(d = 2) = 1/3 and . The distribution P(v) of thepositive random variable v of order O(1) presents a power-law singularity near the origin with a(d = 2, 3) > 0, so all moments of the surface magnetization are governed by the same power-lawdecay with xs = ωc(1 + a) independentlyof the order k.