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) similar to r(-sigma) and distributed with the Levy symmetric stable distribution of index 1 < mu <= 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) proportional to L-theta mu(sigma) is found to be theta(mu)(sigma) = 2/mu - sigma whenever the long-ranged couplings are relevant theta(mu)(sigma) >= -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 <(E-L(GS))over bar>similar or equal to Le(0) + L(theta mu(sigma))e(1). The characteristic scale of the fluctuations around this average is of order L-1/mu, and the rescaled variable u=(E-L(GS) - <(E-L(GS))over bar>/L-1/mu is Gaussian distributed for mu = 2, or displays the negative power-law tail in 1/(-u)(1+mu) for u -> -infinity in the Levy case 1 < mu < 2. Finally we apply the zero-temperature renormalization procedure to the related Dyson hierarchical spin-glass model where the same droplet exponent appears.