Absence of spontaneous magnetic order at non-zero temperature in one- and two-dimensional Heisenberg and XY systems with long-range interactions

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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-alpha with a sufficiently large exponent alpha. For oscillatory interactions, ferromagnetic long-range order at T>0 is ruled out if alpha >= 1 (D=1) or alpha > 5/2 (D=2). For systems with monotonically decreasing interactions ferro- or antiferromagnetic long-range order at T>0 is ruled out if alpha >= 2D.