One-dimensional Ising spin-glass with power-law interaction : real-space renormalization at zero temperature

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 L'evy symmetric stable distribution of index $1<\mu \le 2$ (including the usual Gaussian case $\mu =2$), we consider the region $\sigma >1\mathrm{/}\mu$ 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 ${\theta }_{\mu }(\sigma )=\frac{2}{\mu }-\sigma \ge -1$. 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 $u=(E_L^{GS}-\overline{E_L^{GS}})\mathrm{/}{L}^{\frac{1}{\mu }}$ is Gaussian distributed for $\mu =2$, or displays the negative power-law tail in $1\mathrm{/}(-u{)}^{1+\mu }$ for $u\to -\mathrm{\infty }$ in the L'evy case $1<\mu <2$.