Mean field theory revisited

We will build upon the mean-field theory for the Ising model by introducing a more efficient notation

🗒️ Note: for now the external magnetic field will be set to zero ie $h=0$

Effective energy - general definition

We define the effective free energy $\tilde F(T,P,m)$

🗒️ Note: it is different to the free energy in thermodynamics and statistical mechanics

• Starting from the definition of the partition function we write

  $$ \begin{aligned} Z(T)&=\sum_{\{s_i\}} \exp \left (- \frac{E[\{s_i\}]}{k_B T}\right ) \\ &=\sum_{\tilde m} \sum_{\{s_i \}_{ \tilde m}} \exp \left ( -\frac{E[\{s_i \}]}{k_B T} \right )=\sum _{\tilde m} \exp \left ( \frac{\tilde F(T,\tilde m)}{k_B T} \right )

  \end{aligned} $$

  where $\{ s_i \}_{ \tilde m }$ is the set of all spin configurations such that the average magnetisation per lattice is $-1 \le \tilde m \le 1$ ie

  $$ \frac 1 N \sum _i s_i = \tilde m $$

  🗒️ Notes: $\tilde m$ is not $m=M/N$ instead it is a variable that can assume any value that is tied to a subset of spin configurations. $\tilde m$ is quantised in units of $2/N$ and in the limit of $N\to \infin$ we can treat it continuously as

  $$ \sum_{\tilde m} \to \frac{N}{2} \int _{-1}^1 \,\text d\tilde m $$

The relation between the effective and true free energy

• We start by defining the free energy per lattice site

  $$ \tilde f (T, \tilde m ) = \frac{\tilde F (T, \tilde m )}{N} $$

• The partition function now becomes

  $$ Z(T)=\sum_{\tilde m} \exp \left ( -\frac{N \tilde f(T, \tilde m)}{k_B T} \right )=\frac N2 \int ^1 _{-1} \,\text d \tilde m \exp \left (-\frac{N \tilde f (T,\tilde m )}{k_B T} \right ) $$

  For large $N$ the integral above is dominate by the value of $\tilde m$ that minimises the exponent.

  Therefore calling $m$ the value of $\tilde m$ that minimises the free energy

  $$ \left . \frac{\partial \tilde f(T,\tilde m )}{\partial \tilde m} \right | _{\tilde m=m}=0 $$

  we have that

  $$ Z(T)=\sum_{\tilde m} \exp \left (-\frac{N\tilde f (T,\tilde m)}{k_B T} \right )\simeq \exp \left (-\frac{N \tilde f (T,m)}{k_B T} \right ) $$

• Putting this together we have

  $$ F(T)=N \tilde f(T,m) $$

  💎 Conclusions:

  • The true free energy is the minimization of the $N\tilde f(T,\tilde m)$ with respect to $\tilde m$
  • Thus there is more information in $\tilde F(T,\tilde m )$ then $F(T)$ and we can see this because the first depends on two parameters $T$ and $\tilde m$ while the second one only depends on $T$

The effective free energy for the Ising model

If we write out $\tilde f(T,\tilde m)$ we get

$$ {\tilde f(T,\tilde m)=-\frac 12Jz \tilde m^2 -h \tilde m -k_B T \left [ \ln(2) - \frac{1+\tilde m}{2} \ln (1+ \tilde m) -\frac{1-\tilde m}{2} \ln(1-\tilde m) \right ]} $$

• If we minimise this expression we get

  $$ \left . \frac{\partial \tilde f(T,\tilde m)}{\partial \tilde m} \right |_{\tilde m = m}=0 \quad \Rightarrow Jzm +h=\frac{k_BT}{2}\ln \left ( \frac{1+m}{1-m} \right ) $$

  which rearranging gives

  $$ m=\tanh \left ( \frac{Jzm+h}{k_B T}\right ) $$

  we recognise this equation

  💎 Conclusion: neglecting the fluctuations of the spin with respect to their average value, provides the same result as assuming that they provide an effective magnetic field

• Finally we can find the explicit value for

  $$ \frac{F(T)}{N}=\tilde f(T,m)=\frac 12 Jzm^2-k_B T \ln \left [ 2\cosh \left ( \frac{h_\text{eff}}{k_B T} \right ) \right ] $$

Landau theory for the Ising model

🧽 Assumptions of Landau’s theory

1. Smooth polynomial expansion: The effective free energy $\tilde f(T,\tilde m)$ can be expressed as a polynomial in the order parameter $\tilde m$ where unphysical or symmetry-violating terms are excluded
2. Analytical Coefficients: the coefficients of the polynomial expansion are assumed to be smooth function of the temperature $T$

The transition around the critical point

We can rewrite the the the expression $\tilde f(T,\tilde m )$ for $T\simeq T_C$ where $m\simeq 0$