1D Ising model: no phase transition at positive temperature

Transfer-matrix solution of the 1D Ising chain showing analyticity of the free energy for T>0, exponential decay of correlations, and absence of spontaneous magnetization.

1D Ising model: no phase transition at positive temperature

Prerequisites and notation

Lattice spin system and Hamiltonian: lattice Hamiltonian
Model definition: Ising model , ferromagnetic Ising interaction
Canonical formalism: canonical ensemble , canonical partition function
Correlations: two-point correlation function , correlation length
Order parameter language: order parameter , spontaneous magnetization

We consider spins $\sigma_i\in\{-1,+1\}$ on a 1D ring of $N$ sites (periodic boundary conditions).

Example: 1D ferromagnetic Ising chain

Hamiltonian

H_N(\sigma) \;=\; -J\sum_{i=1}^N \sigma_i\sigma_{i+1} \;-\; h\sum_{i=1}^N \sigma_i, \qquad \sigma_{N+1}=\sigma_1,

with $J>0$ and external field $h\in\mathbb{R}$ .

Transfer matrix and partition function

The canonical partition function is

Z_N(\beta,h)=\sum_{\sigma\in\{\pm1\}^N} e^{-\beta H_N(\sigma)}.

Introduce the $2\times 2$ transfer matrix

T_{\sigma,\sigma'}=\exp\!\left(\beta J\,\sigma\sigma' + \frac{\beta h}{2}(\sigma+\sigma')\right), \qquad \sigma,\sigma'\in\{\pm1\}.

Then

Z_N(\beta,h)=\mathrm{Tr}\, T^N = \lambda_+^N + \lambda_-^N,

where the eigenvalues are

\lambda_{\pm} = e^{\beta J}\cosh(\beta h) \pm \sqrt{e^{2\beta J}\sinh^2(\beta h)+e^{-2\beta J}}.

Thermodynamic limit and analyticity

The (canonical) free energy per site is

f(\beta,h)= -\frac{1}{\beta}\lim_{N\to\infty}\frac{1}{N}\log Z_N(\beta,h) = -\frac{1}{\beta}\log \lambda_+.

For every $\beta<\infty$ (i.e. $T>0$ ), $\lambda_+>\lambda_->0$ and $\lambda_+$ is a real-analytic function of $(\beta,h)$ , hence $f(\beta,h)$ is analytic for $T>0$ . In particular, there is no finite-temperature phase transition (no nonanalyticity of thermodynamic potentials), in contrast with the 2D Ising model .

No spontaneous magnetization for $T>0$

Define magnetization per site

m(\beta,h)= -\frac{\partial f}{\partial h}(\beta,h).

Because $f(\beta,h)$ is analytic in $h$ around $h=0$ for every $T>0$ , the one-sided limits agree:

m(\beta,0^+)=m(\beta,0^-)=0 \quad (T>0),

so there is no spontaneous magnetization at positive temperature.

Exponential decay of correlations and correlation length

At $h=0$ , the two-point function satisfies (in the infinite-volume limit)

\langle \sigma_0\sigma_r\rangle = (\tanh(\beta J))^{r}.

Thus correlations decay exponentially with

\xi(\beta)=\frac{1}{-\log(\tanh(\beta J))},

which is finite for every $T>0$ and diverges only as $T\downarrow 0$ .

Interpretation via Gibbs measures

In 1D with finite-range interaction, the infinite-volume Gibbs state is unique for all $T>0$ , matching the absence of multiple equilibrium states seen in phase transitions as non-uniqueness of Gibbs measures .