Heisenberg Chain Hamiltonian

This card provides a quick Hamiltonian lookup. The Heisenberg Chain dossier is the convention-complete spin-$1/2$ chain record; Heisenberg Model owns the broader model treatment.

Hamiltonian

The nearest-neighbor Heisenberg chain is

$$H = J\sum_i \mathbf s_i\cdot\mathbf s_{i+1}.$$

The spins are dimensionless and $J$ has units of energy. For spin $1/2$,

$$\mathbf s_i = \frac{1}{2}\boldsymbol\sigma_i.$$

With physical angular momenta $\mathbf S_i=\hbar\mathbf s_i$, the exchange term is $(J/\hbar^2)\mathbf S_i\cdot\mathbf S_{i+1}$.

An anisotropic version is

$$H = \sum_i \left( J_xs_i^xs_{i+1}^x + J_ys_i^ys_{i+1}^y + J_zs_i^zs_{i+1}^z \right).$$

The uniform spin-$1/2$ specialization $J_x=J_y=J$ and $J_z=J\Delta$ is developed in XXZ Spin Chain.

Assumptions

• A spin degree of freedom lives on each lattice site.
• Boundary conditions and spin value are specified.
• $J>0$ is commonly antiferromagnetic in the convention shown.
• Additional fields, anisotropies, and longer-range couplings are omitted unless stated.

Common Mistakes

• Comparing sign conventions for $J$ without checking the Hamiltonian definition.
• Confusing Heisenberg and Ising interactions.
• Forgetting that spin operators on different sites commute.
• Treating finite-size spectra as thermodynamic phases without scaling checks.

Canonical Links

• Heisenberg Chain dossier
• Heisenberg Model
• XXZ Spin Chain
• Spin Operator
• Angular Momentum Algebra
• Math Needed for Quantum Matter
• Tensor Product Ordering

References

• H. Bethe, “Zur Theorie der Metalle. I.”, Zeitschrift für Physik 71, 205–226 (1931).
• A. Auerbach, Interacting Electrons and Quantum Magnetism, Springer (1994).