Thermal and Disorder Effects in the One-Dimensional Long-Range Ising Model with Random Fields

We investigate the one-dimensional long-range Ising model with 1/r^2 interactions in the presence of weak Gaussian random fields (1DLRIM-RF), a system expected by the Imry–Ma argument to be disordered at zero temperature. Using large-scale Monte Carlo simulations and exact energy-spectrum enumeration, we confirm the T = 0 state is indeed disordered. Moreover, we uncover a low-temperature re-entrant ordering mechanism arising from a quasi-degenerate band of first-excited states that is energetically isolated from single spin-flip excitations by a large gap. This "order-by-disorder" mechanism stabilizes ferromagnetic order at low but finite temperature before the system undergoes a Kosterlitz–Thouless transition at higher temperature. Our results provide numerical evidence that quenched disorder and thermal fluctuations act antagonistically rather than additively in this model, in direct qualitative agreement with the low-temperature phase structure predicted by Cardy–Ostlund for the random-field XY model.