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Random Transverse Field Ising model on the Cayley Tree : analysis via Boundary Strong Disorder Renormalization

Strong Disorder Renormalization for the Random Transverse Field Ising model leads to a complicated topology of surviving clusters as soon as $d>1$. Even if one starts from a Cayley tree, the network of surviving renormalized clusters will contain loops, so that no analytical solution can been obtained. Here we introduce a modified procedure called 'Boundary Strong Disorder Renormalization' that preserves the tree structure, so that one can write simple recursions with respect to the number of generations. We first show that this modified procedure allows to recover exactly most of the critical exponents for the one-dimensional chain. After this important check, we study the RG equations for the quantum Ising model on a Cayley tree with a uniform ferromagnetic coupling $J$ and random transverse fields with support $[h_{min},h_{max}]$. We find the following picture (i) for $J>h_{max}$, only bonds are decimated, so that the whole tree is a quantum ferromagnetic cluster (ii) for $J<h_{min}$, only sites are decimated, so that no quantum ferromagnetic cluster is formed, and the ferromagnetic coupling to the boundary coincides with the partition function of a Directed Polymer model in a random medium (iii) for $h_{min}<J<h_{max}$, both sites and bonds can be decimated : the quantum ferromagnetic clusters can either remain finite (the physics is then similar to (ii), with a quantitative mapping to a modified Directed Polymer model) or an infinite quantum ferromagnetic cluster appears. We find that the quantum transition can be of two types : (a) either the quantum transition takes place in the region where quantum ferromagnetic clusters remain finite, and the singularity of the ferromagnetic coupling to the boundary involves the typical correlation length exponent $ν_{typ}=1$ (b) or the quantum transition takes place at the point where an extensive quantum ferromagnetic cluster appears.