---
title: Mean-Field Theory and Spontaneous Symmetry Breaking
module: Phase Transitions
moduleNumber: 11
lessonNumber: 3
order: 1103
summary: >
  Mean-field theory replaces the neighbors of each spin by their average,
  turning the interacting Ising model into a single spin in a self-consistent
  field. The resulting equation m = tanh(beta J z m + beta h) has only the zero
  solution above a critical temperature and gains a nonzero root below it, giving
  spontaneous magnetization and a mean-field critical temperature k T_c = J z.
  The Bragg-Williams free energy turns single-welled above T_c and double-welled
  below, the picture of spontaneous symmetry breaking. The approximation is exact
  in high dimension and fails below the upper critical dimension four, quantified
  by the Ginzburg criterion.
topics: [Phase Transitions]
sources:
  - book: Kardar (Statistical Physics of Fields)
    ref: "Ch. 2 — Statistical Fields; §2.4–2.6"
  - book: Pathria & Beale
    ref: "Ch. 12 — Phase Transitions; §12.5–12.6"
  - book: Schroeder
    ref: "Ch. 8 — Systems of Interacting Particles; §8.2 The Ising Model"
draft: false
---

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, $s_i = m + \delta s_i$, where
$m = \langle s_i\rangle$ is the magnetization per spin and $\delta s_i = s_i - m$.
A nearest-neighbor product becomes

$$
s_i s_j = (m + \delta s_i)(m + \delta s_j)
= -m^2 + m(s_i + s_j) + \delta s_i\,\delta s_j .
$$

The **mean-field approximation** drops the product of two fluctuations $\delta s_i
\,\delta s_j$, 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 $Nz/2$ bonds, and the Hamiltonian reduces to a sum of single-spin terms,

$$
H_{\mathrm{MF}} = \frac{N z J}{2}\,m^2 - (J z m + h)\sum_i s_i .
$$

Every spin now sees the same **effective field**

$$
h_{\mathrm{eff}} = J z m + h ,
$$

the applied field $h$ augmented by the Weiss molecular field $Jzm$ 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
$h_{\mathrm{eff}}$ is $\tanh(\beta h_{\mathrm{eff}})$. Demanding that this average
equal the $m$ that produced the field closes the loop.

> **Definition (Mean-field self-consistency).** The magnetization solves
> $$
> m = \tanh\!\big(\beta J z m + \beta h\big),
> $$
> the condition that the average spin computed in the effective field reproduce
> the magnetization assumed in building that field.

## Spontaneous magnetization and the critical temperature

Set the applied field to zero and study $m = \tanh(\beta J z\, m)$. The line $y =
m$ and the curve $y = \tanh(\beta J z\, m)$ 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 $\beta J z$.

- If $\beta J z < 1$, the tanh is shallower than the diagonal everywhere and the
  only solution is $m = 0$: the system is disordered.
- If $\beta J z > 1$, 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 $\beta J z = 1$ defines the **mean-field critical temperature**

$$
k_B T_c = J z .
$$

$$
% caption: The graphical solution of $m=\tanh(\beta Jz\,m)$; below $T_c$ the tanh (steep curve) is steeper than the diagonal at the origin and crosses it at $\pm m_0$, while above $T_c$ (shallow curve) the only crossing is at the origin.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (-2.6,0)--(2.6,0) node[right,black]{$m$};
\draw[->,black] (0,-2.2)--(0,2.2) node[above,black]{};
\draw[black] (-2.2,-2.2)--(2.2,2.2);
\node[black,above left] at (2.0,2.0) {$y=m$};
% below Tc: steep tanh, a=1.9
\draw[acc,very thick] plot[domain=-2.2:2.2,samples=80] (\x,{1.75*(exp(2*1.9*\x/1.75)-1)/(exp(2*1.9*\x/1.75)+1)});
\node[acc,right] at (2.05,1.62) {$T<T_c$};
% above Tc: shallow tanh, a=0.6
\draw[black,very thick,dashed] plot[domain=-2.2:2.2,samples=80] (\x,{1.75*(exp(2*0.6*\x/1.75)-1)/(exp(2*0.6*\x/1.75)+1)});
\node[black,right] at (2.05,0.85) {$T>T_c$};
\filldraw[acc] (1.5,1.5) circle (2.0pt);
\filldraw[acc] (-1.5,-1.5) circle (2.0pt);
\node[acc,below right] at (1.5,1.45) {$m_0$};
\end{tikzpicture}
$$

Near the transition the ordered root is small, and expanding $\tanh x = x -
x^3/3 + \cdots$ gives $m = \beta J z\, m - \tfrac{1}{3}(\beta J z\, m)^3$. With
$\beta J z = T_c/T$, the leading balance is

$$
m^2 \approx 3\,\frac{T_c - T}{T_c}
\quad\Longrightarrow\quad
m \propto (T_c - T)^{1/2} .
$$

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

$$
% caption: The mean-field spontaneous magnetization rises continuously from zero at $T_c$ as $(T_c-T)^{1/2}$ and saturates toward full alignment as $T\to 0$; above $T_c$ it is identically zero.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(6.2,0) node[right,black]{$T$};
\draw[->,black] (0,0)--(0,3.0) node[above,black]{$m$};
\draw[black,dashed] (0,2.4)--(4.2,2.4);
\node[black,left] at (0,2.4) {$1$};
\draw[acc,very thick] (0.3,2.4) .. controls (2.2,2.35) and (3.3,2.05) .. (3.9,1.2);
\draw[acc,very thick] (3.9,1.2) .. controls (4.05,0.7) and (4.15,0.25) .. (4.2,0);
\draw[black,very thick] (4.2,0) -- (6.0,0);
\filldraw[black] (4.2,0) circle (1.7pt);
\node[black,below] at (4.2,-0.05) {$T_c$};
\node[acc,left] at (1.5,2.55) {ordered};
\node[black,above] at (5.1,0.05) {disordered};
\end{tikzpicture}
$$

## 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 $u(m) = -\tfrac{1}{2}Jz
m^2 - hm$, 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 $m = \tanh(\beta Jz m + \beta h)$. Expanding
the entropy for small $m$, $s(m)/k_B = \ln 2 - m^2/2 - m^4/12 + \cdots$, collects
the free energy into a polynomial in the order parameter,

$$
f(m) = f_0 - h m + \tfrac{1}{2}\,k_B(T - T_c)\,m^2
+ \tfrac{1}{12}\,k_B T\,m^4 + \cdots .
$$

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.

$$
% caption: The Bragg-Williams free energy at zero field; above $T_c$ a single well at $m=0$, below $T_c$ a double well with symmetric minima at $\pm m_0$ separated by a maximum at the origin.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
% left: above Tc single well
\draw[->,black] (-2.2,0)--(2.2,0) node[right,black]{$m$};
\draw[->,black] (0,-0.5)--(0,2.6) node[above,black]{$f$};
\draw[very thick] plot[domain=-1.9:1.9,samples=60] (\x,{0.5*\x*\x+0.3});
\node at (-1.15,2.3) {$T>T_c$};
% right: below Tc double well
\begin{scope}[xshift=6.0cm]
\draw[->,black] (-2.2,0)--(2.2,0) node[right,black]{$m$};
\draw[->,black] (0,-0.9)--(0,2.6) node[above,black]{$f$};
\draw[acc,very thick] plot[domain=-1.95:1.95,samples=80] (\x,{0.32*\x*\x*\x*\x-0.9*\x*\x+0.9});
\node[acc] at (-1.2,2.3) {$T<T_c$};
\filldraw[acc] (1.19,{0.32*1.19*1.19*1.19*1.19-0.9*1.19*1.19+0.9}) circle (2.0pt);
\filldraw[acc] (-1.19,{0.32*1.19*1.19*1.19*1.19-0.9*1.19*1.19+0.9}) circle (2.0pt);
\node[acc,below] at (1.19,-0.05) {$m_0$};
\end{scope}
\end{tikzpicture}
$$

## 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.

$$
% caption: The mean-field phase boundary in the field-temperature plane; the segment $h=0$ for $T<T_c$ is a first-order line across which the magnetization jumps, terminating at the critical point $(T_c,0)$.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,-2.0)--(0,2.2) node[above,black]{$h$};
\draw[->,black] (0,0)--(6.2,0) node[right,black]{$T$};
\draw[acc,very thick] (0.2,0) -- (4.0,0);
\filldraw[black] (4.0,0) circle (2.4pt);
\node[black!75] at (4.5,1.25) {critical point};
\draw[black,thin] (4.45,1.05)--(4.05,0.12);
\node[black,below] at (4.0,-0.05) {$T_c$};
\node[acc] at (2.0,1.2) {$m>0$};
\node[acc] at (2.0,-1.2) {$m<0$};
\node[black] at (5.4,0.55) {no transition};
\node[acc,below left] at (1.9,-0.05) {coexistence line};
\end{tikzpicture}
$$

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 $\delta s_i\,\delta s_j$, 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 $|t| \gtrsim t_G$ inside which fluctuations dominate
  and mean-field theory fails. The window closes ($t_G \to 0$) for $d > 4$ and is
  finite for $d < 4$.[^ginzburg]

[^ginzburg]: Kardar (Fields), §3.3, and Pathria & Beale, §12.6. The Ginzburg
criterion compares the fluctuation integral $\int \d^d r\,\langle \delta m(0)\,
\delta m(r)\rangle$ over a correlation volume with $m^2$; the two scale with the
same power of $|t|$ only at $d = 4$.

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 $\beta =
1/8$ 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 $\delta s_i\,\delta s_j$; the effective field $h_{\mathrm{eff}}
  = Jzm + h$ decouples the spins into $N$ single-spin problems.
- The self-consistency equation $m = \tanh(\beta Jzm + \beta h)$ has only $m=0$
  above $T_c$ and gains ordered roots $\pm m_0$ below, with $k_B T_c = Jz$ and
  $m \propto (T_c - T)^{1/2}$ (exponent $\beta = 1/2$).
- The Bragg-Williams free energy $f(m) = f_0 - hm + \tfrac12 k_B(T-T_c)m^2 +
  \tfrac{1}{12}k_B T m^4$ 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.