Spinssystem

Spinssystem (German for spin system) is a theoretical framework used to describe a collection of localized magnetic moments, or spins, that interact with each other and with external fields. In quantum spin systems the degrees of freedom at lattice sites are quantum spins, and the many-body state belongs to a tensor product of local Hilbert spaces. The Hamiltonian encodes the exchange interactions and external fields. Common models include the Ising model with nearest-neighbor coupling, the Heisenberg model with isotropic exchange, and the XXZ model with anisotropy; on some lattices, Kitaev or frustrated variants are studied as well. In classical spin systems, spins are treated as fixed-length vectors with continuous orientations and the energy is expressed in terms of pairwise and field terms.

Analysis of spin systems uses statistical mechanics and quantum many-body methods. The partition function Z = Tr exp(-βH) plays a central role, and properties are obtained by computing expectation values, correlation functions, and response functions. Techniques range from mean-field theory and Bethe ansatz in exactly solvable cases to numerical approaches such as Monte Carlo simulations for classical systems and density matrix renormalization group (DMRG) or tensor-network methods for quantum lattices.

Phenomena of interest include magnetic long-range order, thermal and quantum phase transitions, spin liquids, and spin glasses due to frustration. Experimental realizations include insulating magnets, transition-metal oxides, and engineered quantum simulators. Spin systems provide insight into magnetism, critical phenomena, and, in the quantum regime, potential applications in quantum information processing.