= Solution
Near zero the <Langevin function> has expansion
$$
\mathcal L(x)=\frac x3-\frac{x^3}{45}+O(x^5).
$$
For positive $x$, $\mathcal L(x)$ rises toward one and its ratio $\mathcal L(x)/x$ decreases from $1/3$ to zero. Therefore a line $\tau x$ has only the zero intersection for $\tau\ge1/3$, and also a nonzero positive intersection for $0<\tau<1/3$. In a fixed signed component there is a corresponding negative solution; in the isotropic vector problem the nonzero solutions have all spin-space orientations.
The mean-field transition temperature is
$$
\boxed{k_BT_c^{\rm MF}=\frac{z_cJ}{3}=\frac{4J}{3}.}
$$
At the <critical temperature> the line is tangent at the origin. Just below it, $S\simeq\sqrt{(5/3)(1-T/T_c^{\rm MF})}$, the usual mean-field square-root onset. The ordered minimum is stable below this temperature while $S=0$ becomes unstable.
\b[This is a mean-field prediction, not a genuine finite-temperature ordering transition for the ideal two-dimensional short-range isotropic model.] Long-wavelength <spin> fluctuations invalidate the predicted nonzero thermodynamic <magnetization> at every $T>0$. The relevant obstruction is the <Mermin-Wagner theorem> for continuous <spin> symmetry; anisotropy, long-range interactions or a finite sample change the assumptions. Its original paper establishes the absence of finite-temperature magnetic ordering for one- and two-dimensional finite-range isotropic <Heisenberg models>.<Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models by N. D. Mermin and H. Wagner>[Physical Review Letters 17, 1133 (1966), abstract; physical limitation of finite-temperature mean-field order]{ref}
\Image[/past-exam-of-the-theoretical-physics-i-course-of-the-university-of-cambridge/2015/paper-1-langevin-mean-field.png]
{title=Original Langevin self-consistency plot and the mean-field order parameter, explicitly distinguished from true two-dimensional isotropic long-range order}
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