Ising Chain

One-Sentence Description

The Ising chain is a lattice spin model with a preferred $z$ axis, nearest-neighbor alignment energy, and, in its transverse-field form, noncommuting quantum dynamics.

Physical Setup

Place a spin-$1/2$ degree of freedom on each site of a one-dimensional lattice. The Ising interaction favors neighboring spins being parallel or antiparallel along one chosen axis. A transverse field introduces spin flips and makes the model quantum.

The model is a minimal laboratory for discrete $\mathbb Z_2$ symmetry, domain walls, order parameters, finite-size spectra, and quantum phase transitions.

Hilbert Space

For $N$ spin-$1/2$ sites,

$$\mathcal H = \bigotimes_{j=1}^{N}\mathbb C^2.$$

The computational basis diagonalizes all $\sigma_j^z$ operators.

Hamiltonian

A common transverse-field Ising chain convention is

$$H = -J\sum_{j} \sigma_j^z\sigma_{j+1}^z - h\sum_j\sigma_j^x.$$

The classical Ising chain keeps only the commuting alignment term,

$$H_{\mathrm{classical}} = -J\sum_j s_j s_{j+1}, \qquad s_j=\pm1.$$

Boundary conditions decide whether the sum includes a term coupling site $N$ back to site $1$.

Parameters

Symbol	Meaning
$J$	nearest-neighbor Ising coupling
$h$	transverse-field strength
$N$	number of sites
$\sigma_j^\alpha$	Pauli matrix acting on site $j$

Solvability

The one-dimensional transverse-field Ising chain is exactly solvable by the Jordan–Wigner transformation followed by a fermionic Bogoliubov transformation. The classical one-dimensional Ising chain is solved by transfer matrices. Higher-dimensional or perturbed versions are not generally exactly solvable.

In the infinite one-dimensional transverse-field model, the standard convention above has a quantum critical point at

$$\lvert h\rvert=\lvert J\rvert,$$

up to convention-dependent factors.

Hamiltonian conventions, limiting states, exact quasiparticles, finite-size sectors, and model-specific entanglement are developed in Transverse-Field Ising Model. Quantum Phase Transitions owns the general scaling interpretation.

Key Observables

• Magnetization along the Ising axis.
• Transverse magnetization.
• Spin-spin correlation functions.
• Domain-wall excitations.
• Energy gap and finite-size scaling.

What It Teaches

The Ising chain teaches how a simple tensor-product Hamiltonian can encode symmetry breaking, competing noncommuting terms, quantum criticality, and exactly solvable many-body structure.

It is also a benchmark model for exact diagonalization because the Hilbert space grows as $2^N$ while symmetries can reduce the calculation.

Canonical Links

• Transverse-Field Ising Model dossier
• Transverse-Field Ising Model
• Ising Chain Hamiltonian
• Quantum Phase Transitions
• Spin Operator
• Tensor Product Ordering
• Interactions and Coupling Terms
• XY Model

Variants

• classical Ising chain;
• transverse-field Ising chain;
• longitudinal-field Ising chain;
• two-dimensional classical Ising model;
• disordered Ising chains;
• long-range Ising models.

Common Mistakes

• Confusing the classical Ising model with the transverse-field quantum Ising model.
• Forgetting that $\sigma^x$ and $\sigma^z$ terms do not commute.
• Quoting the critical point without stating the Hamiltonian convention.
• Inferring thermodynamic phases from a very small chain without finite-size analysis.

Quick Checks

1. Why is the transverse-field Ising chain quantum while the classical Ising chain is diagonal in a spin basis?

Solution

The Ising interaction is diagonal in the $\sigma^z$ basis, but the transverse field uses $\sigma^x$. Since $\sigma^x$ does not commute with $\sigma^z$, the transverse-field term mixes the classical spin configurations.

2. What information is missing from $H=-J\sum_j\sigma_j^z\sigma_{j+1}^z$ before a finite-chain spectrum is well defined?

Solution

One must specify the number of sites, boundary conditions, and sign convention for $J$. Open and periodic chains differ in the number of bonds and in their symmetry sectors.

References

• S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press, 2011.
• P. Pfeuty, “The one-dimensional Ising model with a transverse field”, Annals of Physics 57, 79-90, 1970.
• E. Lieb, T. Schultz, and D. Mattis, “Two soluble models of an antiferromagnetic chain”, Annals of Physics 16, 407-466, 1961.