| Heisenberg model | |
|---|---|
| Name | Heisenberg model |
| Field | Quantum mechanics; Condensed matter physics |
| Introduced | 1928 |
| Introduced by | Werner Heisenberg |
| Key concepts | spin, Exchange interaction, Hamiltonian |
| Applications | Quantum magnetism, Spin chain, Quantum computing |
Heisenberg model
The Heisenberg model is a paradigmatic quantum lattice model describing interacting spins via an exchange interaction. It captures essential physics of magnetism in solids, provides an exactly solvable example in one dimension, and serves as a starting point for understanding quantum phase transitions and many-body entanglement in condensed matter.
The Heisenberg model was originally proposed by Werner Heisenberg to explain ferromagnetism through an exchange mechanism arising from the Pauli exclusion principle and Coulomb interaction. On a lattice (for example the square lattice or linear chain), localized magnetic moments on sites are represented by quantum spin operators that interact with nearest neighbors. The model is physically relevant to magnetic insulators such as transition-metal oxides and to engineered systems like cold atoms in optical lattices and quantum dot arrays. It provides a microscopic bridge from the Hubbard model in the large-U limit to low-energy spin wave descriptions like linear spin-wave theory and the nonlinear sigma model.
The prototypical isotropic Heisenberg Hamiltonian on a lattice Λ is H = J ∑_{⟨i,j⟩} S_i · S_j, where S_i = (S_i^x,S_i^y,S_i^z) are spin-S operators at lattice site i and the sum runs over pairs of interacting sites ⟨i,j⟩. The coupling constant J>0 (antiferromagnetic) or J<0 (ferromagnetic) determines the favored alignment. Variants include the ferromagnetic/antiferromagnetic sign, longer-range couplings (e.g. next-nearest neighbor J_2), and anisotropic exchange. The Hamiltonian conserves total spin in the isotropic case and commutes with global SU(2) generators, connecting the model to representations of Lie algebras and to the algebraic structure used in exact solution techniques.
The one-dimensional spin-1/2 Heisenberg chain (the XXX model) was solved exactly by Hans Bethe in 1931 using the Bethe ansatz, yielding quantized rapidities that determine eigenstates and energies. The Bethe ansatz construction led to the development of the quantum inverse scattering method and connections to Yang–Baxter equation and integrable system theory. Exact thermodynamics were later obtained via methods such as the thermodynamic Bethe ansatz and quantum transfer matrix. Integrability breaks down for generic perturbations (e.g. higher dimensions or strong anisotropy), but the Bethe framework remains foundational for understanding 1D quantum many-body physics.
For the spin-1/2 chain, the ground state depends on J: the ferromagnetic chain has a fully polarized ground state while the antiferromagnetic chain exhibits a gapless spin-fluid (Tomonaga–Luttinger liquid) with fractionalized spin-1/2 excitations called spinons. In higher dimensions, the square-lattice spin-1/2 antiferromagnet develops Néel order at zero temperature in many cases, described by broken SU(2) symmetry and Goldstone modes (magnons). Integer-spin chains (e.g. spin-1) can display the Haldane gap predicted by F. D. M. Haldane, a topological distinction tied to the AKLT model and symmetry-protected phases. The phase diagram is enriched by frustration (e.g. J1–J2 model), which can stabilize quantum spin liquids, valence-bond solids, or noncollinear orders.
Anisotropic generalizations include the XXZ model (with U(1) symmetry) and the XYZ model (no continuous symmetry), controlled by different exchange strengths in spin components. The XXZ chain interpolates between Ising-like gapped phases and XY-like critical phases and is integrable via Bethe ansatz. Higher-spin (S>1/2) Heisenberg models alter quantum fluctuations and stability of ordered states; for example, spin-1 chains exhibit the Haldane phase while large-S approaches connect to classical classical behavior via the Holstein–Primakoff transformation and 1/S expansions.
The Heisenberg model is central to theories of antiferromagnetism and ferromagnetism in materials such as cuprates (parent compounds of high-temperature superconductivity) and Mott insulators. It arises as the low-energy limit of the Hubbard model at strong coupling and underpins descriptions of low-dimensional magnets studied in experiments on materials like KCuF3 and organic magnets. The model also informs designs in quantum simulation platforms (ultracold atoms, trapped ions) and proposals for quantum information processing using spin chains for state transfer. Experimental probes include neutron scattering, nuclear magnetic resonance, and inelastic electron tunneling spectroscopy.
Because exact solutions are limited, many numerical techniques have been developed: exact diagonalization for small clusters, density matrix renormalization group (DMRG) for 1D and quasi-1D systems, quantum Monte Carlo (QMC) for unfrustrated models, and tensor network methods like matrix product states and projected entangled pair states (PEPS) for higher dimensions. These methods have characterized order parameters, excitation spectra, entanglement measures, and finite-temperature behavior. Benchmark results such as the ground-state energy and spin correlations for the square-lattice spin-1/2 Heisenberg model are now standard tests for algorithms and inform comparisons with experimental data from materials and inelastic neutron scattering studies.
Category:Quantum spin models Category:Condensed matter physics