Existence of Long-Range Magnetic Order in Heisenberg Spin Nanoribbon with Edge Modification

A long-range magnetic order appears on side decorated Heisenberg spin nanoribbon at nonzero temperature, although no spontaneous magnetization exists in a one- or two-dimensional isotropic Heisenberg model at any nonzero temperature according to Mermin-Wagner theorem. By use of the spin Green’s function method, we calculated the magnetizations of Heisenberg nanoribbon decorated by side spins with single-ion anisotropy and found that the system exhibits a nonzero transition temperature, whether the decorated edge spins of the system link together or separate from each other. When the width of the nanoribbon achieves infinity limit, the transition temperatures of the system tend to a same finite constant eventually whether one edge or double edges are decorated by side spins in the nanoribbon. The results reveal that the magnetism of a low-dimensional spin system is different from that of a three-dimensional spin system. When the single-ion anisotropy of edge spins in Heisenberg spin nanoribbon can be modulated by an electric field experimentally, various useful long-range magnetic orders of the system can be obtained. The works can provide a detailed theoretical basis for designing and fabricating the next-generation low-dimensional magnetic random-access memory.