1D Ising Model

$N$ spins $s_{1}, s_{2}, \dots, s_{N}$ , each $= \pm 1$ on a regular grid; only neighbors interact. 1D = chain, 2D = grid.

Energy

E (s_{1}, s_{2}, \dots) = - h \sum_{i = 1}^{N} s_{i} - J \sum_{i = 1}^{N - 1} s_{i} s_{i + 1}

$- h \sum s_{i}$ : coupling to external field. Energy decreases when spins align with $h$ .
$- J \sum s_{i} s_{i + 1}$ : neighbor coupling. $J > 0$ → ferromagnetic; $J < 0$ → anti-ferromagnetic.

Studied in the canonical ensemble. Define $\hat{h} = h / k T$ , $\hat{J} = J / k T$ . Boltzmann factor:

e^{- E / k T} = e^{\hat{h} \sum s_{i} + \hat{J} \sum s_{i} s_{i + 1}}

Partition function:

Z (\hat{h}, \hat{J}) = \sum_{s_{1} = \pm 1, s_{2} = \pm 1, \dots} e^{\hat{h} \sum s_{i} + \hat{J} \sum s_{i} s_{i + 1}}

Single spin ( $J = 0$ , $N = 1$ )

Z_{0} (\hat{h}) = \sum_{s_{1} = \pm 1} e^{- \hat{h} s_{1}} = e^{- \hat{h}} + e^{\hat{h}} = \cosh (\hat{h})

Average spin:

⟨ s_{1} ⟩ = \frac{1}{Z_{0} (\hat{h})} \sum_{s_{1} = \pm 1} s_{1} e^{- \hat{h} s_{1}} = \frac{1}{2 \cosh \hat{h}} (- e^{- \hat{h}} + e^{\hat{h}}) = \frac{\sinh \hat{h}}{\cosh \hat{h}} = \tanh (\hat{h})

Response saturates at $⟨ s ⟩ = \pm 1$ for large $| \hat{h} |$ .

Non-interacting spins ( $J = 0$ , $N$ spins)

Boltzmann factor factorizes → partition function factorizes:

Z (\hat{h}, N) = Z_{0} (\hat{h})^{N} = \cosh (\hat{h})^{N}

Magnetization density $m = \frac{1}{N} \sum_{i} s_{i}$ , and by the same argument per spin:

⟨ m ⟩ = \tanh \hat{h}

Strong coupling limit ( $J \to \infty$ )

$\hat{J} \to \infty$ locks spins into alignment; anti-aligned configurations have Boltzmann factor $\to 0$ . System behaves as a single dipole of strength $N$ :

⟨ m ⟩_{J \to \infty} = \tanh (N \hat{h})

As $N \to \infty$ : sharp step — any nonzero $h$ fully magnetizes the system. ( $J \to \infty$ equivalent to $T \to 0$ at fixed $J$ .)

$h = 0$ , general $J$ — link variable

Define $l_{i} = s_{i} s_{i + 1} = \pm 1$ (aligned vs. anti-aligned). Energy becomes:

E = - J \sum_{i = 1}^{N - 1} s_{i} s_{i + 1} = - J \sum_{i = 1}^{N - 1} l_{i}

The $N - 1$ link variables are independent — this is just a non-interacting Ising model for the links with external field $J$ . This is a duality: the interacting spin model maps exactly onto a non-interacting link model.

General result ( $N \to \infty$ , transfer matrix)

⟨ m ⟩ = \frac{\sinh \hat{h}}{\sqrt{\sinh^{2} \hat{h} + e^{- 4 \hat{J}}}}

Recovers $\tanh \hat{h}$ at $\hat{J} = 0$ and sharp step as $\hat{J} \to \infty$ . No finite- $T$ phase transition in 1D for any $J > 0$ . The 2D Ising model does have a finite critical $T_{c} > 0$ .

Energy

Single spin (J=0, N=1)

Non-interacting spins (J=0, N spins)

Strong coupling limit (J→∞)

h=0, general J — link variable

General result (N→∞, transfer matrix)

Single spin ( $J = 0$ , $N = 1$ )

Non-interacting spins ( $J = 0$ , $N$ spins)

Strong coupling limit ( $J \to \infty$ )

$h = 0$ , general $J$ — link variable

General result ( $N \to \infty$ , transfer matrix)