---
title: The Ising Model and Exact Results
module: Phase Transitions
moduleNumber: 11
lessonNumber: 2
order: 1102
summary: >
  The Ising model reduces cooperative ordering to spins on a lattice coupled to
  their neighbors, and the same Hamiltonian describes uniaxial magnets, the
  liquid-gas critical point through the lattice gas, and binary alloys. This
  lesson solves the one-dimensional chain exactly with the transfer matrix,
  shows by a domain-wall argument why one dimension has no ordered phase at any
  positive temperature, contrasts the survival of order in two dimensions, and
  quotes Onsager's exact two-dimensional results: the critical temperature, the
  logarithmically divergent heat capacity, and the magnetization exponent one
  eighth.
topics: [Phase Transitions]
sources:
  - book: Kardar (Statistical Physics of Fields)
    ref: "Ch. 2 — Statistical Fields; §2.1–2.3, and Ch. 6 §6.1–6.3"
  - book: Pathria & Beale
    ref: "Ch. 12 — Phase Transitions; §12.3, §12.5"
  - book: Baxter
    ref: "Exactly Solved Models in Statistical Mechanics — Ch. 7 (Ising 1D), Ch. 7–9 (Ising 2D)"
draft: false
---

Cooperative ordering — a magnet acquiring a spontaneous moment, a liquid
condensing from its vapor, an alloy segregating into ordered sublattices — needs
a model simple enough to solve yet rich enough to have a transition. The Ising
model is that model. It places a two-valued spin on each site of a lattice and
couples neighboring spins so that alignment lowers the energy. Whether the
resulting competition between energy and entropy produces long-range order
depends on the dimension of the lattice, and the answer can be computed exactly in
one dimension and, for the zero-field case, in two.

## The Ising Hamiltonian

To each site $i$ of a lattice assign a spin $s_i \in \{+1, -1\}$. Neighboring
spins interact, and an external field $h$ couples to each spin:

$$
H = -J \sum_{\langle i j\rangle} s_i s_j - h \sum_i s_i ,
$$

where $\langle ij\rangle$ runs over nearest-neighbor pairs and $J$ is the coupling
constant. For $J > 0$ aligned neighbors ($s_i s_j = +1$) lower the energy, favoring
**ferromagnetic** order; for $J < 0$ the model favors antialignment. The partition
function is the sum over all $2^N$ spin configurations,

$$
Z = \sum_{\{s_i\}} e^{-\beta H}, \qquad \beta = \frac{1}{k_B T} .
$$

The same Hamiltonian carries three physical readings, and every exact result
transfers among them without change.

- **Uniaxial ferromagnet** — $s_i$ is a spin that points along or against an easy
  axis, $h$ is an applied magnetic field, and the order parameter is the
  magnetization per spin $m = \langle s_i\rangle$.
- **Lattice gas** — set $n_i = (1+s_i)/2 \in \{0,1\}$ for an empty or occupied
  cell. Nearest-neighbor attraction between occupied cells maps onto the spin
  coupling, and the field maps onto the chemical potential. The liquid-gas
  critical point becomes the Ising critical point.
- **Binary alloy** — $s_i = \pm 1$ labels which of two metals sits at site $i$,
  and $J$ measures whether like or unlike neighbors are preferred, controlling
  order-disorder transitions in the alloy.

$$
% caption: A slice of the Ising lattice with a domain wall (dashed) separating a region of up spins from a region of down spins; each broken bond across the wall costs energy $2J$.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\foreach \x in {0,1,2} {
  \foreach \y in {0,1,2} {
    \draw[acc,very thick,->] (\x,\y-0.3) -- (\x,\y+0.3);
  }
}
\foreach \x in {3,4,5} {
  \foreach \y in {0,1,2} {
    \draw[black,very thick,->] (\x,\y+0.3) -- (\x,\y-0.3);
  }
}
\draw[black,thick,dashed] (2.5,-0.6) -- (2.5,2.6);
\node[acc,below] at (1,-0.7) {aligned up};
\node[black,below] at (4,-0.7) {aligned down};
\node[black,above] at (2.5,2.6) {domain wall};
\end{tikzpicture}
$$

## The one-dimensional chain by transfer matrix

On a ring of $N$ sites the energy is a sum of identical nearest-neighbor terms,
which lets the Boltzmann weight factorize into a product of two-site pieces.
Symmetrize the field between the two sites of each bond and define a $2\times 2$
**transfer matrix** whose rows and columns are indexed by the spin values,

$$
\mathsf{T}_{s\,s'} = \exp\!\Big[\beta J s s' + \tfrac{1}{2}\beta h (s + s')\Big],
\qquad
\mathsf{T} =
\begin{pmatrix}
e^{\beta(J+h)} & e^{-\beta J} \\[2pt]
e^{-\beta J} & e^{\beta(J-h)}
\end{pmatrix}.
$$

Summing the product of bond weights around the ring is a matrix trace:

$$
Z = \sum_{\{s_i\}} \prod_{i} \mathsf{T}_{s_i\,s_{i+1}}
= \operatorname{Tr}\,\mathsf{T}^N = \lambda_+^N + \lambda_-^N ,
$$

with $\lambda_\pm$ the two eigenvalues of $\mathsf{T}$. In the thermodynamic limit
the larger eigenvalue dominates, $Z \to \lambda_+^N$, and the free energy per spin
is

$$
f = -k_B T \lim_{N\to\infty}\frac{1}{N}\ln Z = -k_B T \ln \lambda_+ .
$$

At zero field the eigenvalues are $\lambda_\pm = e^{\beta J} \pm e^{-\beta J}$,
that is $\lambda_+ = 2\cosh\beta J$ and $\lambda_- = 2\sinh\beta J$, so

$$
f(T, h{=}0) = -k_B T \ln\!\big(2\cosh\beta J\big).
$$

This function is analytic for every $T > 0$: $\cosh\beta J$ is smooth and strictly
positive, so no derivative of $f$ ever diverges at finite temperature. The
one-dimensional Ising chain has **no phase transition** except at $T = 0$.

The eigenvalue ratio fixes the **correlation length**. The connected correlation
of two spins $r$ sites apart decays as $\langle s_0 s_r\rangle_c = (\lambda_-/
\lambda_+)^r \equiv e^{-r/\xi}$, so

$$
\xi^{-1} = \ln\frac{\lambda_+}{\lambda_-} = \ln\coth(\beta J).
$$

As $T \to 0$, $\beta J \to \infty$ and $\coth\beta J \to 1$, driving $\xi \to
\infty$: order becomes long-ranged only in the zero-temperature limit. At any
positive temperature $\xi$ is finite and the chain is disordered on scales beyond
it.

$$
% caption: The two transfer-matrix eigenvalues at zero field; their ratio sets the correlation length through $\xi^{-1}=\ln(\lambda_+/\lambda_-)$, and the gap closes only as $T\to 0$, where $\xi$ diverges.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(6.4,0) node[right,black]{$T$};
\draw[->,black] (0,0)--(0,2.6) node[above right,black]{eigenvalue};
\draw[acc,very thick] plot[domain=1.0:6.0,samples=60] (\x,{0.275*(exp(1.6/\x)+exp(-1.6/\x))});
\draw[black,very thick,dashed] plot[domain=1.0:6.0,samples=60] (\x,{0.275*(exp(1.6/\x)-exp(-1.6/\x))});
\draw[acc,very thick] (4.2,2.15)--(4.7,2.15);
\node[acc,right] at (4.75,2.15) {larger};
\draw[black,very thick,dashed] (4.2,1.7)--(4.7,1.7);
\node[black,right] at (4.75,1.7) {smaller};
\draw[black,<->] (4.0,{0.275*(exp(1.6/4.0)-exp(-1.6/4.0))}) -- (4.0,{0.275*(exp(1.6/4.0)+exp(-1.6/4.0))});
\node[black,right,align=left] at (4.1,0.42) {gap};
\end{tikzpicture}
$$

## Why one dimension cannot order

The absence of a transition in one dimension has a direct physical cause,
Landau's domain-wall argument. Start from the perfectly ordered ground state, all
spins up. Introduce a single **domain wall**: everything to the left of one bond
stays up, everything to the right flips down. Only the one bond across the wall is
now unsatisfied, costing energy

$$
\Delta E = 2J .
$$

The wall can be placed on any of the $N$ bonds of the chain, so inserting it
raises the entropy by

$$
\Delta S = k_B \ln N .
$$

The free-energy change of adding one wall is therefore

$$
\Delta F = \Delta E - T\,\Delta S = 2J - k_B T \ln N .
$$

For any fixed $T > 0$, the entropy term grows without bound as $N \to \infty$, so
$\Delta F \to -\infty$: creating domain walls always lowers the free energy. Walls
proliferate, the chain breaks into arbitrarily many domains, and long-range order
is destroyed. The order survives only at exactly $T = 0$, where the entropy term
vanishes.

$$
% caption: The one-dimensional domain wall costs a fixed energy $2J$ but gains entropy $k_B T\ln N$; the entropy term overtakes the energy cost for any $T>0$ as the chain grows, so walls proliferate and order cannot persist.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(6.4,0) node[right,black]{$\ln N$};
\draw[->,black] (0,0)--(0,3.6) node[above right,black]{free energy};
\draw[acc,very thick] (0.3,2.4) -- (6.0,2.4);
\node[acc,above] at (1.4,2.42) {energy cost $2J$};
\draw[black,very thick,dashed] (0.3,0.2) -- (6.0,3.3);
\node[black] at (5.15,1.1) {entropy term $k_BT\ln N$};
\coordinate (x) at (3.75,2.4);
\filldraw[black] (x) circle (1.8pt);
\draw[black,dashed] (3.75,0)--(3.75,2.4);
\node[black,below] at (3.75,0) {walls free};
\end{tikzpicture}
$$

The same accounting explains why two dimensions differ. A domain wall in two
dimensions is a closed loop of length $\ell$, with energy $2J\ell$. The number of
loops of length $\ell$ grows roughly as $\mu^\ell$ with a lattice-dependent
constant $\mu$ (near $3$ on the square lattice), giving entropy $k_B \ell \ln\mu$.
The free-energy change per unit wall length,

$$
\frac{\Delta F}{\ell} \approx 2J - k_B T \ln\mu ,
$$

stays **positive** for $T < 2J/(k_B\ln\mu)$: at low enough temperature long walls
are suppressed, and the ordered phase is stable. This is Peierls's argument, and
it establishes that a genuine finite-temperature transition exists in two
dimensions. The competition that one dimension resolves in favor of disorder,
two dimensions resolves in favor of order below a critical temperature.

## Onsager's exact two-dimensional results

Onsager solved the two-dimensional Ising model on the square lattice at zero
field, and the result is the benchmark against which every approximate theory is
measured. The critical temperature separating ordered from disordered phases is
fixed by the condition $\sinh(2\beta_c J) = 1$, giving[^onsager]

$$
\frac{k_B T_c}{J} = \frac{2}{\ln\!\big(1+\sqrt{2}\,\big)} \approx 2.269 .
$$

[^onsager]: Kardar (Fields), §6.3, and Baxter, Ch. 7. Onsager announced the free
energy in 1944; the spontaneous magnetization exponent $\beta = 1/8$ was obtained
by Yang, quoted in Pathria & Beale, §12.5.

The heat capacity does not jump at $T_c$; it **diverges logarithmically** on both
sides,

$$
C(T) \sim -A \ln\big|\,T - T_c\,\big| ,
$$

a symmetric spike with no finite peak height. This is the exponent $\alpha = 0$ in
its logarithmic realization, and it already contradicts the finite discontinuity
that the Ehrenfest scheme and mean-field theory both predict. Below $T_c$ the
spontaneous magnetization rises as a power of the reduced temperature,

$$
m(T) \propto (T_c - T)^{\beta}, \qquad \beta = \frac{1}{8},
$$

where here $\beta$ denotes the magnetization critical exponent, not the inverse
temperature. The value $1/8$ differs sharply from the mean-field $1/2$ of the next
lesson, and the discrepancy is the experimental fingerprint that mean-field theory
misses the physics of a two-dimensional critical point.

$$
% caption: Onsager's two-dimensional heat capacity diverges logarithmically at $T_c$ from both sides, a symmetric spike with no finite peak, unlike the finite jump predicted by mean-field theory.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(6.4,0) node[right,black]{$T$};
\draw[->,black] (0,0)--(0,3.9) node[above,black]{$C$};
% Tc at x=3.4; two log-like branches rising to a tall spike
\draw[very thick] (0.4,0.35) .. controls (1.8,0.7) and (2.7,1.2) .. (3.15,2.0);
\draw[very thick] (3.15,2.0) .. controls (3.28,2.6) and (3.34,3.1) .. (3.4,3.5);
\draw[very thick] (3.4,3.5) .. controls (3.46,3.1) and (3.52,2.6) .. (3.65,2.0);
\draw[very thick] (3.65,2.0) .. controls (4.1,1.1) and (5.0,0.7) .. (6.0,0.5);
\draw[acc,dashed] (3.4,0)--(3.4,3.5);
\node[acc,below] at (3.4,0) {$T_c$};
\node[acc,right,align=left] at (4.2,3.0) {$C\sim{}$log\\divergence};
\end{tikzpicture}
$$

## Universality across the three interpretations

The three physical systems packed into one Hamiltonian share not only the model
but its critical numbers. The magnet, the lattice gas, and the binary alloy all
have the same $T_c$ structure, the same logarithmic heat-capacity divergence, and
the same magnetization exponent $1/8$ in two dimensions, because the exponents
depend only on the dimensionality and the symmetry of the order parameter, not on
the microscopic details of the coupling. This is the first concrete instance of
**universality**: the liquid-gas critical point of a real fluid and the Curie
point of a real magnet fall into the same class as the two-dimensional (or, in
nature, three-dimensional) Ising model and share its exponents.

| System | Spin variable $s_i$ | Field $h$ | Order parameter |
| --- | --- | --- | --- |
| Uniaxial ferromagnet | up/down moment | magnetic field | magnetization $\langle s\rangle$ |
| Lattice gas | occupied/empty cell | chemical potential | density difference |
| Binary alloy | metal A / metal B | composition bias | sublattice order |

The exact solutions bound what any theory must reproduce: in one dimension no
transition at positive temperature, in two dimensions a transition at $k_B T_c/J
= 2.269$ with a logarithmic heat capacity and $\beta = 1/8$. The mean-field
approximation of the next lesson gets the existence of a transition right in high
enough dimension but the exponents wrong, and reconciling the two is the task of
Landau theory and the renormalization group.

## Summary

- The Ising Hamiltonian $H = -J\sum_{\langle ij\rangle} s_i s_j - h\sum_i s_i$
  models a uniaxial ferromagnet, a lattice gas (hence the liquid-gas critical
  point), and a binary alloy, with the same exact results in each reading.
- The one-dimensional chain is solved by the transfer matrix: $Z =
  \lambda_+^N + \lambda_-^N$, free energy $f = -k_B T\ln(2\cosh\beta J)$, analytic
  for all $T>0$, so there is no finite-temperature transition; the correlation
  length $\xi^{-1} = \ln\coth\beta J$ diverges only as $T\to 0$.
- Landau's domain-wall argument gives the reason: in one dimension a wall costs
  fixed energy $2J$ but gains entropy $k_B T\ln N$, so walls always proliferate;
  in two dimensions a wall is a loop whose free energy per length stays positive
  below a critical temperature, allowing order.
- Onsager's two-dimensional solution gives $k_B T_c/J = 2/\ln(1+\sqrt 2)\approx
  2.269$, a logarithmically divergent heat capacity, and magnetization exponent
  $\beta = 1/8$ — exponents shared by every system in the same universality class.