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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 $J(r) \sim r^{-σ}$ and distributed with the Lévy symmetric stable distribution of index $1 <μ\leq 2$ (including the usual Gaussian case $μ=2$), we consider the region $σ>1/μ$ 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 $J_L \propto L^{θ_μ(σ)}$ is found to be $θ_μ(σ)=\frac{2}μ-σ$ whenever the long-ranged couplings are relevant $θ_μ(σ)=\frac{2}μ-σ\geq -1$. For the statistics of the ground state energy $E_L^{GS}$ over disordered samples, we obtain that the droplet exponent $θ_μ(σ) $ governs the leading correction to extensivity of the averaged value $\overline{E_L^{GS}} \simeq L e_0 +L^{θ_μ(σ)} e_1$. The characteristic scale of the fluctuations around this average is of order $L^{\frac{1}μ}$, and the rescaled variable $u=(E_L^{GS}-\overline{E_L^{GS}})/L^{\frac{1}μ}$ is Gaussian distributed for $μ=2$, or displays the negative power-law tail in $1/(-u)^{1+μ}$ for $u \to -\infty$ in the Lévy case $1<μ<2$.