Low-energy excitation gap and phase transitions in one-dimensional long-range Ising model with random fields

We report numerical evidence for a novel order-by-disorder mechanism in the one-dimensional long-range Ising model with random fields (1DLRIM-RF). Using Monte Carlo simulations and exact enumeration of energy eigenstates, we demonstrate that while weak random fields destroy the T=0 ground state order as predicted by the Imry-Ma argument, the system exhibits re-ordering at finite temperatures. This unexpected behavior arises from nearly degenerate first excited states separated from single spin-flip excitations by a large energy gap. The energy spectrum supports a phase diagram featuring disorder at T=0, ferromagnetic order at low finite temperatures, and a Kosterlitz-Thouless transition at higher temperatures. Our findings provide numerical support for earlier predictions by Cardy and Ostlund regarding the critical behavior of random field XY models, while revealing that temperature and disorder act antagonistically rather than additively in destroying long-range order.