Spin Models

Spin models replace continuous position degrees of freedom with finite-dimensional local Hilbert spaces. A single spin-$1/2$ site has $\mathbb C^2$; a chain of $N$ such sites has $(\mathbb C^2)^{\otimes N}$. This makes the tensor-product structure explicit and gives compact testbeds for symmetry, entanglement, phase transitions, and exact diagonalization.

Cards

Model	Main lesson	Hamiltonian reference
Two-Level System	Generic two-dimensional quantum dynamics	Two-Level System Hamiltonian
Ising Chain	Discrete symmetry, classical versus quantum spin chains	Ising Chain Hamiltonian
Heisenberg Chain	Rotationally invariant exchange and quantum magnetism	Heisenberg Chain Hamiltonian
XY Model	Anisotropic planar exchange and fermionization in one dimension	this model card

Use Common Spin Hamiltonians to translate spin-versus-Pauli normalization, bond counts, exchange tensors, fields, and anisotropies across these model families.

Shared Conventions

Spin-chain formulas are incomplete until the following choices are stated:

• spin value at each site;
• boundary conditions;
• sign convention for couplings;
• normalization of spin operators versus Pauli matrices;
• whether fields are longitudinal or transverse relative to the interaction axis.

For spin-$1/2$ sites,

$$\mathbf S_j = \frac{\hbar}{2}\boldsymbol\sigma_j.$$

Some many-body literature sets $\hbar=1$ and writes Hamiltonians directly in terms of Pauli matrices. Compare conventions before comparing spectra.

Common Mistakes

• Treating a spin chain as a collection of independent spins when coupling terms entangle sites.
• Comparing $J$ values without checking the sign convention in the Hamiltonian.
• Forgetting boundary-condition and parity-sector effects in small chains.
• Assuming a classical spin picture applies unchanged to noncommuting quantum spin operators.

References

• S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press, 2011.
• A. Auerbach, Interacting Electrons and Quantum Magnetism, Springer, 1994.
• T. Giamarchi, Quantum Physics in One Dimension, Oxford University Press, 2004.