Heisenberg model

Heisenberg model. In condensed matter physics, the Heisenberg model is a fundamental lattice model used to describe the magnetic properties of systems where the dominant interaction is the quantum mechanical exchange interaction between localized magnetic moments. It was introduced in 1928 by Werner Heisenberg to explain the phenomenon of ferromagnetism in materials like iron and nickel. The model's Hamiltonian captures the essence of spin-spin interactions and serves as a cornerstone for understanding quantum magnetism, phase transitions, and many-body physics.

Definition and Hamiltonian

The model is defined on a lattice, such as the square lattice or honeycomb lattice, with a spin operator, typically representing a spin-1/2 degree of freedom, residing on each site. The Hamiltonian for the isotropic Heisenberg model is given by \( H = -J \sum_{\langle i,j \rangle} \mathbf{S}_i \cdot \mathbf{S}_j \), where \( \mathbf{S}_i \) is the spin operator at site \( i \), the sum runs over nearest-neighbor pairs \( \langle i,j \rangle \), and \( J \) is the exchange coupling constant. The sign of \( J \) dictates the magnetic order: \( J > 0 \) favors ferromagnetic alignment, as studied in the context of the Heisenberg ferromagnet, while \( J < 0 \) favors antiferromagnetic alignment, leading to models like the Heisenberg antiferromagnet on various lattices. Anisotropic versions, such as the XXZ model, introduce directional preferences in the interactions, bridging to simpler models like the Ising model.

Physical motivation and applications

The primary physical motivation stems from the exchange interaction, a quantum mechanical effect arising from the interplay of the Pauli exclusion principle and Coulomb interaction between electrons. This interaction is responsible for the magnetic ordering in insulating magnetic materials, known as Mott insulators. The model has been extensively applied to understand the properties of real materials, including the parent compounds of high-temperature cuprate superconductors like La2CuO4, which are described by a two-dimensional Heisenberg antiferromagnet. It also provides insights into the magnetic susceptibility and spin wave excitations observed in neutron scattering experiments on materials such as K2NiF4.

Exact solutions and integrability

Despite its simplicity, the model is notoriously difficult to solve exactly in more than one dimension due to strong quantum fluctuations. A landmark achievement was Hans Bethe's 1931 exact solution for the one-dimensional antiferromagnetic chain using the Bethe ansatz, which revealed the absence of long-range order and the presence of a spinon continuum. The model is integrable in one dimension, and its properties have been deeply explored in connection with Yang-Baxter equation and quantum inverse scattering method. Other exactly solvable cases include the Majumdar–Ghosh model and certain limits of the Haldane model, which predict unique ground states like the valence bond solid.

Approximation methods and results

For higher dimensions and frustrated geometries, various approximation methods are employed. Spin wave theory, also known as linear spin wave theory, treats deviations from ordered states as bosonic excitations and successfully predicts the dispersion of magnons. For strongly fluctuating systems, quantum Monte Carlo simulations, particularly using the stochastic series expansion algorithm, provide numerical insights into thermodynamic properties and critical exponents. The density matrix renormalization group is powerful for quasi-one-dimensional systems. Key results include the establishment of the Néel state as the ground state for the square lattice antiferromagnet and the identification of quantum spin liquid phases in highly frustrated lattices like the kagome lattice.

Extensions and related models

Numerous extensions modify the original Hamiltonian to capture more complex physics. The t-J model incorporates charge degrees of freedom and is central to the theory of high-temperature superconductivity. The Hubbard model, at strong coupling, maps onto the Heisenberg model. Including longer-range interactions or Dzyaloshinskii–Moriya interaction leads to models for spin glasses or chiral magnets. For systems with orbital degrees of freedom, the Kugel–Khomskii model is used. The study of quantum criticality near pressure-induced phase transitions often involves anisotropic Heisenberg models. These related frameworks continue to be tested against experiments on materials like Herbertsmithite and through quantum simulators using ultracold atoms in optical lattices.

Category:Lattice models Category:Condensed matter physics Category:Quantum magnetism