The critical behavior of one-dimensional ferromagnetic chains with isotropic long-range interactions between $n$-component spins is investigated near the critical range of $r^{-2\mathrm{}}$. Different formulations of the renormalization group are used for $n>\mathrm{}1\mathrm{}$ and $n=1\mathrm{}$. It is shown that the behavior is quite different in the two cases, with very close analogies to the short-range problem near two dimensions. Critical exponents are calculated to first order.

• Received 10 May 1976

DOI:https://doi.org/10.1103/PhysRevLett.37.1577