Absence of Spontaneous Magnetic Order at Nonzero Temperature in One- and Two-Dimensional Heisenberg and $\mathit{XY}$ Systems with Long-Range Interactions

The Mermin-Wagner theorem is strengthened so as to rule out magnetic long-range order at $T>\mathrm{}0\mathrm{}$ in one- or two-dimensional Heisenberg and XY systems with long-range interactions decreasing as $R^{-\alpha }$ with a sufficiently large exponent $\alpha$. For oscillatory interactions, ferromagnetic long-range order at $T>\mathrm{}0\mathrm{}$ is ruled out if $\alpha \ge 1(D=1\mathrm{})$ or $\alpha >5/2\mathrm{}(D=2\mathrm{})$. For systems with monotonically decreasing interactions, ferro- or antiferromagnetic long-range order at $T>\mathrm{}0\mathrm{}$ is ruled out if $\alpha \ge 2D$.