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

### Abstract

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 $J_L^{typ}(h) \propto \Upsilon(h) L^{\theta}$ with the classical droplet exponent $\theta=1-\sigma$, where the stiffness modulus vanishes at criticality $\Upsilon(h) \propto (h_c-h)^{\mu}$, whereas the typical renormalized transverse field decays exponentially $h^{typ}_L(h) \propto e^{- \frac{L}{\xi}}$ where the correlation length diverges at the transition $\xi \propto (h_c-h)^{-\nu}$. 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 $\sigma$ in the region $1/2<\sigma<1$.Comment: 9 pages, 7 figure

Topics: Condensed Matter - Disordered Systems and Neural Networks
Publisher: 'IOP Publishing'
Year: 2015
DOI identifier: 10.1088/1742-5468/2015/10/P10024
OAI identifier: oai:arXiv.org:1506.06012

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