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Dyson hierarchical quantum ferromagnetic Ising chain with pure or random transverse fields

The Dyson hierarchical version of the quantum Ising chain with Long-Ranged power-law ferromagnetic couplings $J(r) \propto r^{-1-σ}$ and pure or random transverse fields is studied via real-space renormalization. For the pure case, the critical exponents are explicitly obtained as a function of the parameter $σ$, and are compared with previous results of other approaches. For the random case, the RG rules are numerically applied and the critical behaviors are compared with previous Strong Disorder Renormalization results.