Chaos properties of the one-dimensional long-range Ising spin-glass

For the long-range one-dimensional Ising spin-glass with random couplings decaying as $J(r) \propto r^{-\sigma}$, the scaling of the effective coupling defined as the difference between the free-energies corresponding to Periodic and Antiperiodic boundary conditions $J^R(N) \equiv F^{(P)}(N)-F^{(AP)}(N) \sim N^{\theta(\sigma)}$ defines the droplet exponent $\theta(\sigma)$. Here we study numerically the instability of the renormalization flow of the effective coupling $J^R(N)$ with respect to magnetic, disorder and temperature perturbations respectively, in order to extract the corresponding chaos exponents $\zeta_H(\sigma)$, $\zeta_J(\sigma)$ and $\zeta_T(\sigma)$ as a function of $\sigma$. Our results for $\zeta_T(\sigma)$ are interpreted in terms of the entropy exponent $\theta_S(\sigma) \simeq 1/3$ which governs the scaling of the entropy difference $S^{(P)}(N)-S^{(AP)}(N) \sim N^{\theta_S(\sigma)}$. We also study the instability of the ground state configuration with respect to perturbations, as measured by the spin overlap between the unperturbed and the perturbed ground states, in order to extract the corresponding chaos exponents $\zeta^{overlap}_H(\sigma)$ and $\zeta^{overlap}_J(\sigma)$.