%0 Journal Article %T One-dimensional Ising spin-glass with power-law interaction : real-space renormalization at zero temperature %+ Institut de Physique Théorique - UMR CNRS 3681 (IPHT) %A Monthus, Cécile %Z v2=revised version (17 pages) with new section VII concerning the Dyson hierarchical Spin-Glass model %< avec comité de lecture %Z t14/110 %@ 1742-5468 %J Journal of Statistical Mechanics: Theory and Experiment %I IOP Publishing %V 2014 %P P06015 %8 2014 %D 2014 %Z 1403.1098 %R 10.1088/1742-5468/2014/14/P06015 %Z Physics [physics]/Condensed Matter [cond-mat]/Other [cond-mat.other]Journal articles %X For the one-dimensional long-ranged Ising spin-glass with random couplings decaying with the distance $r$ as $J(r) \sim r^{-\sigma}$ and distributed with the Lévy symmetric stable distribution of index $1 <\mu \leq 2$ (including the usual Gaussian case $\mu=2$), we consider the region $\sigma>1/\mu$ where the energy is extensive. We study two real space renormalization procedures at zero temperature, namely a simple box decimation that leads to explicit calculations, and a strong disorder decimation that can be studied numerically on large sizes. The droplet exponent governing the scaling of the renormalized couplings $J_L \propto L^{\theta_{\mu}(\sigma)}$ is found to be $\theta_{\mu}(\sigma)=\frac{2}{\mu}-\sigma$ whenever the long-ranged couplings are relevant $\theta_{\mu}(\sigma)=\frac{2}{\mu}-\sigma \geq -1$. For the statistics of the ground state energy $E_L^{GS}$ over disordered samples, we obtain that the droplet exponent $\theta_{\mu}(\sigma) $ governs the leading correction to extensivity of the averaged value $\overline{E_L^{GS}} \simeq L e_0 +L^{\theta_{\mu}(\sigma)} e_1$. The characteristic scale of the fluctuations around this average is of order $L^{\frac{1}{\mu}}$, and the rescaled variable $u=(E_L^{GS}-\overline{E_L^{GS}})/L^{\frac{1}{\mu}}$ is Gaussian distributed for $\mu=2$, or displays the negative power-law tail in $1/(-u)^{1+\mu}$ for $u \to -\infty$ in the Lévy case $1<\mu<2$. %G English %2 https://cea.hal.science/cea-01053485/document %2 https://cea.hal.science/cea-01053485/file/month2.pdf %L cea-01053485 %U https://cea.hal.science/cea-01053485 %~ CEA %~ CNRS %~ DSM-IPHT %~ CEA-DRF %~ GS-MATHEMATIQUES %~ GS-PHYSIQUE