Mean-Field Theory and Spontaneous Symmetry Breaking

The Ising model resists exact solution above one dimension because each spin is coupled to its neighbors, which are coupled to theirs, and the sum over configurations does not factorize. Mean-field theory cuts the coupling by a single approximation: replace the fluctuating neighbors of a given spin by their average value, so that each spin responds not to its actual environment but to a uniform effective field set self-consistently by the average magnetization. The interacting many-spin problem collapses to one spin in an external field, solvable in closed form. The price is that fluctuations are discarded, and the theory is quantitatively right only where fluctuations are weak — in high dimension.

The Weiss mean field

Write each spin as its average plus a fluctuation, $s_{i} = m + δ s_{i}$ , where $m = ⟨ s_{i} ⟩$ is the magnetization per spin and $δ s_{i} = s_{i} - m$ . A nearest-neighbor product becomes

s_{i} s_{j} = (m + δ s_{i}) (m + δ s_{j}) = - m^{2} + m (s_{i} + s_{j}) + δ s_{i} δ s_{j} .

The mean-field approximation drops the product of two fluctuations $δ s_{i} δ s_{j}$ , the term that carries the correlations. Each bond then contributes $- m^{2} + m (s_{i} + s_{j})$ . On a lattice where every site has $z$ nearest neighbors there are $N z /2$ bonds, and the Hamiltonian reduces to a sum of single-spin terms,

H_{MF} = \frac{N z J}{2} m^{2} - (J z m + h) i \sum s_{i} .

Every spin now sees the same effective field

h_{eff} = J z m + h,

the applied field $h$ augmented by the Weiss molecular field $J z m$ from the $z$ averaged neighbors. The spins are decoupled, so the partition function factorizes into $N$ copies of a single two-state system, and the average of one spin in field $h_{eff}$ is $tanh (β h_{eff})$ . Demanding that this average equal the $m$ that produced the field closes the loop.

Spontaneous magnetization and the critical temperature

Set the applied field to zero and study $m = tanh (β J z m)$ . The line $y = m$ and the curve $y = tanh (β J z m)$ always intersect at the origin, the disordered solution $m = 0$ . Whether they intersect anywhere else is fixed by the slope of the tanh at the origin, which is $β J z$ .

If $β J z < 1$ , the tanh is shallower than the diagonal everywhere and the only solution is $m = 0$ : the system is disordered.
If $β J z > 1$ , the tanh starts above the diagonal and bends back to cross it at a nonzero $\pm m_{0}$ : two ordered solutions appear, and $m = 0$ becomes unstable.

The borderline $β J z = 1$ defines the mean-field critical temperature

k_{B} T_{c} = J z .

The graphical solution of

m = tanh (β J z m)

; below

T_{c}

the tanh (steep curve) is steeper than the diagonal at the origin and crosses it at

\pm m_{0}

, while above

T_{c}

(shallow curve) the only crossing is at the origin.

Near the transition the ordered root is small, and expanding $tanh x = x - x^{3} /3 + \dots$ gives $m = β J z m - \frac{1}{3} (β J z m)^{3}$ . With $β J z = T_{c} / T$ , the leading balance is

m^{2} \approx 3 \frac{T _{c} - T}{T _{c}} ⟹ m \propto (T_{c} - T)^{1/2} .

The spontaneous magnetization turns on continuously at $T_{c}$ with the mean-field exponent $β = 1/2$ , the value the exact two-dimensional result $1/8$ contradicts. As $T \to 0$ the tanh saturates and $m \to 1$ : every spin aligns.

The mean-field spontaneous magnetization rises continuously from zero at

T_{c}

(T_{c} - T)^{1/2}

and saturates toward full alignment as

T \to 0

; above

T_{c}

it is identically zero.

The Bragg-Williams free energy

The self-consistency equation is the stationarity condition of a free energy in the trial variable $m$ . The mean-field energy per spin is $u (m) = - \frac{1}{2} J z m^{2} - hm$ , and the entropy per spin of a state with magnetization $m$ is fixed by counting configurations with fraction $(1 + m) /2$ up:

\frac{s ( m )}{k _{B}} = - \frac{1 + m}{2} ln \frac{1 + m}{2} - \frac{1 - m}{2} ln \frac{1 - m}{2} .

The Bragg-Williams free energy per spin is $f (m) = u (m) - T s (m)$ , and $\partial f / \partial m = 0$ reproduces $m = tanh (β J z m + β h)$ . Expanding the entropy for small $m$ , $s (m) / k_{B} = ln 2 - m^{2} /2 - m^{4} /12 + \dots$ , collects the free energy into a polynomial in the order parameter,

f (m) = f_{0} - hm + \frac{1}{2} k_{B} (T - T_{c}) m^{2} + \frac{1}{12} k_{B} T m^{4} + \dots .

The coefficient of $m^{2}$ changes sign at $T_{c}$ . Above $T_{c}$ it is positive and $f (m)$ has a single minimum at $m = 0$ . Below $T_{c}$ it is negative, $m = 0$ turns into a local maximum, and the quartic term stabilizes two symmetric minima at $\pm m_{0}$ . The Landau expansion of the next lesson is this polynomial, here obtained from a microscopic model.

The Bragg-Williams free energy at zero field; above

T_{c}

a single well at

m = 0

, below

T_{c}

a double well with symmetric minima at

\pm m_{0}

separated by a maximum at the origin.

Spontaneous symmetry breaking

The zero-field Hamiltonian is invariant under flipping every spin, $s_{i} \to - s_{i}$ , which sends $m \to - m$ . Above $T_{c}$ the equilibrium state shares that symmetry: $m = 0$ is invariant. Below $T_{c}$ the two equilibrium states $+ m_{0}$ and $- m_{0}$ each violate the symmetry, and the system must occupy one of them. Which one it picks is not determined by the Hamiltonian; an infinitesimal field, or a chance fluctuation frozen in as the system cools, selects the branch. The symmetry of the laws is not shared by the state. This is spontaneous symmetry breaking, and it is the mechanism of every ordered phase: the ferromagnet picks a direction, the crystal picks a lattice origin, the superfluid picks a phase.

In the field-temperature plane the ordered region is bordered by a line of first-order transitions. For $T < T_{c}$ the segment $h = 0$ is a coexistence line: crossing it flips the magnetization discontinuously between $+ m_{0}$ and $- m_{0}$ , a first-order jump with the field as control variable. The line ends at the critical point $(T_{c}, 0)$ , beyond which no discontinuity remains and the magnetization is a smooth function of $h$ through zero.

The mean-field phase boundary in the field-temperature plane; the segment

h = 0

for

T < T_{c}

is a first-order line across which the magnetization jumps, terminating at the critical point

(T_{c}, 0)

The two minima are separated by a free-energy barrier that scales with system size, so in the thermodynamic limit the system cannot tunnel between them. This is why the symmetry is broken in practice and not merely on paper: a macroscopic magnet stays magnetized in one direction for astronomically long times. A finite system, by contrast, samples both minima and averages to $m = 0$ , another statement that a sharp transition lives only in the thermodynamic limit.

Range of validity

Mean-field theory neglects the fluctuation term $δ s_{i} δ s_{j}$ , so it is trustworthy only when fluctuations of the order parameter are small compared with its mean. Each spin feels $z$ neighbors; the more neighbors, the better an average represents them, so the approximation improves as $z$ (hence the dimension $d$ ) grows. Two limits make this precise.

Upper critical dimension. Above $d = 4$ the fluctuation corrections to mean-field theory are finite and do not alter the exponents: mean-field exponents are exact for $d > 4$ . The infinite-range model (every spin coupled to every other) is mean-field exact in any dimension.
Ginzburg criterion. The self-consistency of neglecting fluctuations requires that the mean-square fluctuation of the order parameter within a correlation volume be small next to $m^{2}$ . Evaluating this near $T_{c}$ gives a reduced-temperature window $∣ t ∣ ≳ t_{G}$ inside which fluctuations dominate and mean-field theory fails. The window closes ( $t_{G} \to 0$ ) for $d > 4$ and is finite for $d < 4$ .¹

Below four dimensions — including the physically important cases $d = 2$ and $d = 3$ — mean-field exponents are wrong near $T_{c}$ , and the exact Ising value $β = 1/8$ in two dimensions is the sharpest demonstration. Getting the exponents right requires summing the fluctuations, the achievement of the renormalization group. Mean-field theory nonetheless captures the qualitative structure correctly: a transition at a definite $T_{c}$ , a continuously vanishing order parameter, symmetry breaking, and a double-well free energy. Those features survive; only the numbers shift.

Summary

Mean-field theory replaces each spin's neighbors by their average, dropping the fluctuation product $δ s_{i} δ s_{j}$ ; the effective field $h_{eff} = J z m + h$ decouples the spins into $N$ single-spin problems.
The self-consistency equation $m = tanh (β J z m + β h)$ has only $m = 0$ above $T_{c}$ and gains ordered roots $\pm m_{0}$ below, with $k_{B} T_{c} = J z$ and $m \propto (T_{c} - T)^{1/2}$ (exponent $β = 1/2$ ).
The Bragg-Williams free energy $f (m) = f_{0} - hm + \frac{1}{2} k_{B} (T - T_{c}) m^{2} + \frac{1}{12} k_{B} T m^{4}$ is single-welled above $T_{c}$ and double-welled below, the Landau form derived microscopically.
Selecting one of the two minima breaks the up-down symmetry spontaneously; a size-scaling barrier locks the choice in the thermodynamic limit.
Neglecting fluctuations makes the theory exact for $d > 4$ (upper critical dimension) and the infinite-range model, but wrong exponents for $d < 4$ ; the Ginzburg criterion sets the temperature window where fluctuations take over.

Kardar (Fields), §3.3, and Pathria & Beale, §12.6. The Ginzburg criterion compares the fluctuation integral $\int d^{d} r ⟨ δ m (0) δ m (r)⟩$ over a correlation volume with $m^{2}$ ; the two scale with the same power of $∣ t ∣$ only at $d = 4$ . ↩

The Weiss mean field

Spontaneous magnetization and the critical temperature

The Bragg-Williams free energy

Spontaneous symmetry breaking

Range of validity

Summary

Footnotes