Past exam of the theoretical physics I course of the University of Cambridge / 2015 / Paper 1 / 6

Mean-field theory replaces fluctuating interactions with the average field produced by the other degrees of freedom. For an interacting spin system, write ${\mathbf{S}}_i=\mathbf{S}+\delta {\mathbf{S}}_i$ and neglect connected products $\delta {\mathbf{S}}_i\cdot \delta {\mathbf{S}}_j$. This reduces the many-body partition function to independent single-site factors coupled through a self-consistency equation for $\mathbf{S}$.

The approximation is most reliable when many effectively independent neighbors contribute, for sufficiently long-range interactions, or above the upper critical dimension away from the fluctuation-dominated critical region. It usually becomes least reliable near a continuous transition in low dimensions, where correlations extend over long distances. It can estimate ordering scales and reveal candidate broken symmetries while giving incorrect critical exponents or even a spurious transition. The two-dimensional isotropic model in the later parts is an especially important example of this limitation.

Let the coordination number be $z_c=4$, and choose the spin-space $z$ axis along $\mathbf{S}=S\hat{\mathbf{z}}$. The mean-field decoupling of a bond isThe original directed neighbor sum counts each bond twice, so its factor $1/2$ cancels that double counting. The resulting energy isThe constant prevents double counting of the mean interaction energy and is necessary if free energy is minimized with respect to $S$. With the source's unnormalized solid-angle measure, the single-site integral isThereforeAt $S=0$ take $\sinh x/x\to 1$. A normalized solid-angle measure would remove the factor $(4\pi {)}^N$ without changing the self-consistency equation. These are three-component classical spins on a two-dimensional positional lattice, not spins constrained to a plane.

The PDF's hint is a correct angular-integration identity for a term with coefficient $\beta J/2$. That coefficient is not the full single-site molecular field after summing all four neighbors and decoupling both entries of each bond. The net coefficient is $\beta z_{c}J$, as follows from the explicitly stated Hamiltonian; inserting the hint's coefficient as the total field would lose the coordination and double-counting factors.

In mean-field theory all sites have the same normalized single-site distribution $e^{x\cos \theta }/Z_1(x)$. Consequentlywhere $\mathcal{L}$ is the Langevin function. Requiring the average used to construct the field to equal the average in that field givesSetSince $S=\tau x$, the self-consistency equation is exactly $\tau x=\coth x-1/x$. It also follows by stationarity of $F_{\mathrm{MF}}/N=z_{c}J{S}^2/2-k_{B}T\log Z_1(\beta z_{c}JS)$. This independent route checks the constant term and all neighbor-counting factors in the partition function.

Near zero the Langevin function has expansionFor 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 $\tau \ge 1/3$, 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 isAt the critical temperature the line is tangent at the origin. Just below it, $S\simeq \sqrt{(5/3)(1-T/T_c^{\mathrm{MF}})}$, the usual mean-field square-root onset. The ordered minimum is stable below this temperature while $S=0$ becomes unstable.

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.^[ref]

For uniform unconstrained boundary conditions, the positive gradient coefficient favors a spatially constant minimizer. Put $u=|\mathbf{m}{|}^2\ge 0$. The remaining free-energy difference is $a(T-T_c)u+b{u}^2$. Its minimum isBelow $T_c$ the amplitude is fixed but its direction is arbitrary; the set of uniform minima is a two-sphere. The free-energy minimum is $f_0-a^2(T_c-T{)}^2/(4b)$ below $T_c$ and $f_0$ above. The order parameter vanishes continuously, so this is a continuous, second-order transition within Landau theory.

The free energy has internal $O(3)$ spin-rotation/reflection symmetry; a chosen nonzero vector leaves an $O(2)$ subgroup invariant. Restricting to proper rotations, this is $SO(3)\to SO(2)$ spontaneous symmetry breaking. There are two transverse orientation directions and a massive amplitude direction below the transition. Spatially varying directions cost gradient energy, and imposed boundaries or defects can prevent a uniform minimizer. In strictly two-dimensional equilibrium with short-range interactions, the fluctuation caveat of part (d) still applies: minimizing this polynomial functional at mean-field level does not overturn the absence of finite-temperature spontaneous magnetization.