cea-01053485
https://cea.hal.science/cea-01053485
https://cea.hal.science/cea-01053485/document
https://cea.hal.science/cea-01053485/file/month2.pdf
arxiv:1403.1098
doi:10.1088/1742-5468/2014/14/P06015
[CEA] CEA - Commissariat à l'énergie atomique
[CNRS] CNRS - Centre national de la recherche scientifique
[DSM-IPHT] IPHT
[CEA-DRF] Direction de Recherche Fondamentale
[GS-MATHEMATIQUES] Graduate School Mathématiques
[GS-PHYSIQUE] Graduate School Physique
One-dimensional Ising spin-glass with power-law interaction : real-space renormalization at zero temperature
Monthus, Cécile
[PHYS.COND.CM-GEN] Physics [physics]/Condensed Matter [cond-mat]/Other [cond-mat.other]
ART
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 $1<\mu<2$.
2014
2022-10-03
en
Journal of Statistical Mechanics: Theory and Experiment
IOP Publishing