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Dyson Hierarchical Long-Ranged Quantum Spin-Glass via real-space renormalization

We consider the Dyson hierarchical version of the quantum Spin-Glass with random Gaussian couplings characterized by the power-law decaying variance $\overline{J^2(r)} \propto r^{-2σ}$ and a uniform transverse field $h$. The ground state is studied via real-space renormalization to characterize the spinglass-paramagnetic zero temperature quantum phase transition as a function of the control parameter $h$. In the spinglass phase $h<h_c$, the typical renormalized coupling grows with the length scale $L$ as the power-law $J_L^{typ}(h) \propto Υ(h) L^θ$ with the classical droplet exponent $θ=1-σ$, where the stiffness modulus vanishes at criticality $Υ(h) \propto (h_c-h)^μ $, whereas the typical renormalized transverse field decays exponentially $ h^{typ}_L(h) \propto e^{- \frac{L}ξ}$ where the correlation length diverges at the transition $ξ\propto (h_c-h)^{-ν}$. At the critical point $h=h_c$, the typical renormalized coupling $J_L^{typ}(h_c) $ and the typical renormalized transverse field $ h^{typ}_L(h_c)$ display the same power-law behavior $L^{-z}$ with a finite dynamical exponent $z$. The RG rules are applied numerically to chains containing $L=2^{12}=4096 $ spins in order to measure these critical exponents for various values of $σ$ in the region $1/2<σ<1$.