spinchain

A spin chain is a one-dimensional lattice model in which each site carries a quantum spin degree of freedom that interacts with its neighbors. The simplest cases involve spin-1/2 degrees of freedom and nearest-neighbor exchange interactions. Spin chains are used to study quantum magnetism, strong fluctuations, and the emergence of collective behavior in low dimensions.

The most studied models include the Heisenberg chain, the XXZ chain, and the transverse-field Ising chain. The Heisenberg (XXX) chain has the Hamiltonian H = J sum_i S_i · S_{i+1}, where S_i are spin operators. The XXZ chain adds anisotropy: H = J sum_i (S_i^x S_{i+1}^x + S_i^y S_{i+1}^y + Δ S_i^z S_{i+1}^z). The transverse-field Ising chain is H = -J sum_i S_i^z S_{i+1}^z - h sum_i S_i^x. These models exhibit a range of behaviors, from gapless to gapped phases, depending on parameters and spin value.

Theoretically, many spin chains are integrable and solvable by Bethe ansatz, especially the spin-1/2 Heisenberg chain. Their low-energy physics is described by Luttinger liquid theory, while critical points often correspond to conformal field theories with characteristic central charges. For integer spins, the Haldane conjecture predicts a finite energy gap, exemplified by the AKLT model.

Experimentally, spin chains appear in quasi-one-dimensional magnetic materials, organics, and engineered systems such as ultracold atoms in optical lattices and trapped-ion chains. These platforms enable studies of real-time dynamics, entanglement, and quantum simulations of many-body phenomena in one dimension.