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 $\overline{J^2(r)}\propto r^{-2\sigma }$ 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 $J_L^{typ}(h)\propto \mathrm{{\rm Y}}(h)L^{\theta }$ with the classical droplet exponent $\theta =1-\sigma$, where the stiffness modulus vanishes at criticality $\mathrm{{\rm Y}}(h)\propto (h_c-h{)}^{\mu }$, whereas the typical renormalized transverse field decays exponentially $ h^{typ}_L(h) \propto e^{- \frac{L}{\xi}}$wherethecorrelationlengthdivergesatthetransition$\xi \propto (h_c-h)^{-\nu}$\mathrm{.}Atthecriticalpoint$h=h_c$, the typical renormalized coupling $J_L^{typ}(h_c)$ and the typical renormalized transverse field $h_L^{typ}(h_c)$ 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 $\sigma$ in the region $1\mathrm{/}2<\sigma <1$.