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

Abstract

.For the one-dimensional long-ranged Ising spin-glass with random couplings decaying with the distance r as J(r) ∼ r−σ and distributed with the Lévy symmetric stable distribution of index 1 < μ ≤ 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 JL∝ Lθμ(σ) is found to be whenever the long-ranged couplings are relevant θμ(σ) ≥ −1. For the statistics of the ground state energy over disordered samples, we obtain that the droplet exponent θμ(σ) governs the leading correction to extensivity of the averaged value . The characteristic scale of the fluctuations around this average is of order , and the rescaled variable is Gaussian distributed for μ = 2, or displays the negative power-law tail in 1/(−u)1+μ for u → −∞ in the Lévy case 1 < μ < 2. Finally we apply the zero-temperature renormalization procedure to the related Dyson hierarchical spin-glass model where the same droplet exponent appears.