Statistical mechanics of the Ising model

2.1 The canonical ensemble

A system in contact with a heat bath at temperature $T$ is found in configuration $\sigma$ with the Boltzmann probability

The partition function $Z$ contains all the thermodynamics, through the free energy $F = -k_B T \ln Z$. The average of any observable $A$ is $\left\langle A \right\rangle = \sum_\sigma A(\sigma) P(\sigma)$.

2.2 Thermodynamic quantities and fluctuations

The quantities we will compute are the energy and the magnetization,

and their responses: the specific heat $C = \partial E / \partial T$ and the magnetic susceptibility $\chi = \partial M / \partial h$. They are all derivatives of $\ln Z$:

where $\mathcal{M} = \sum_i s_i$. These fluctuation–dissipation relations are very useful in simulations: they give the responses $C$ and $\chi$ from the fluctuations of $H$ and $\mathcal{M}$ at fixed temperature and field, without having to take numerical derivatives. Since $E$, $M$, $C$ and $\chi$ are proportional to $N$, we give their values per spin: $e = E/N$, $m = M/N$, $c = C/N$ and $\chi/N$, which we still call $\chi$.

The difficulty is the number of terms in $Z$. A $10 \times 10$ lattice already has $2^{100} \approx 1.3 \times 10^{30}$ configurations: summing over all of them, even at $10^9$ configurations per second, would take $4 \times 10^{13}$ years. Exact results require mathematical tricks (Sections 3 and 5); otherwise we must sample the configurations (Section 6).

2.3 Symmetry and spontaneous symmetry breaking

At $h = 0$, the energy (1) does not change when all the spins are flipped, $s_i \to -s_i$. For every configuration with magnetization $\mathcal{M}$ there is one with $-\mathcal{M}$ and the same probability, so

A finite system therefore never has a spontaneous magnetization in the strict sense. A phase transition can only occur in the thermodynamic limit $N \to \infty$. There, the spontaneous magnetization is defined by taking the limit of a small field in the right order,

If $m_0 \neq 0$, the system has chosen one of the two directions although the energy has no preferred one: the symmetry is spontaneously broken. The magnetization is the order parameter of the transition. Mathematically, a phase transition is a point where the free energy per spin, $f = F/N$ in the limit $N \to \infty$, is not an analytic function of $T$ and $h$. Since each finite $Z$ is a finite sum of exponentials, the non-analyticity can only appear in the limit.

In a large but finite system below the transition, the magnetization does not vanish at a given moment: it fluctuates around $+m_0$ or $-m_0$, and very rarely jumps from one to the other. This is why simulations measure $\left\langle |m| \right\rangle$ rather than $\left\langle m \right\rangle$. For the same reason, the 2D simulations at $h = 0$ measure the fluctuations of $|m|$,

instead of Eq. (4), which in a finite system at $h = 0$ reduces to $N\left\langle m^2 \right\rangle/k_BT$ and grows like $N$ below $T_c$ because $m$ flips between $\pm m_0$. Below $T_c$, $\chi'$ measures the response within one ordered state; above $T_c$, where $m$ has Gaussian fluctuations, $\chi' = (1 - 2/\pi)\chi$. Both diverge at $T_c$ with the same exponent. All the 2D results at $h = 0$ below use $\chi'$; the 1D results, and all results in a field, use Eq. (4).

2.4 Warm-up: independent spins

Without interactions ($J = 0$), the spins are independent:

This is a paramagnet: the field aligns the spins against thermal agitation, and the susceptibility follows Curie's law $\chi \propto 1/T$. There is no spontaneous magnetization. Interactions are needed for order.