Absence of Spontaneous Magnetic Order at Nonzero Temperature in One- and Two-Dimensional Heisenberg andXYSystems with Long-Range Interactions

The Mermin-Wagner theorem is strengthened so as to rule out magnetic long-range order at $T>0$ in one- or two-dimensional Heisenberg and XY systems with long-range interactions decreasing as R^-α with a sufficiently large exponent α. For oscillatory interactions, ferromagnetic long-range order at $T>0$ is ruled out if α≥1(D0ex0ex=0ex0ex1) or α>5/2(D0ex0ex=0ex0ex2). For systems with monotonically decreasing interactions, ferro- or antiferromagnetic long-range order at $T>0$ is ruled out if α≥2D.