Heisenberg model (quantum)

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The Heisenberg model is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spin of the magnetic systems are treated quantum mechanically. In the prototypical Ising model, defined on a d-dimensional lattice, at each lattice site, a spin $\sigma_i \in \{ \pm 1\}$ represents a microscopic magnetic dipole to which the magnetic moment is either up or down.

For quantum mechanical reasons (see exchange force), the dominant coupling between two dipoles may cause nearest-neighbors to have lowest energy when they are aligned. Under this assumption (so that magnetic interactions only occur between adjacent dipoles) the Hamiltonian can be written in the form

$H = -J \sum_{j =1}^{N} \sigma_j \sigma_{j+1} - H \sum_{j =1}^{N} \sigma_j$

for a 1-dimensional model consisting of N dipoles, subject to the periodic boundary condition $σ N + 1 = σ 1$ . The Heisenberg model is a more realistic model in that it treats the spins quantum-mechanically, by replacing the spin by a quantum operator (Pauli spin-1/2 matrices at spin 1/2), and the coupling constants $J x, J y,$ and $J z$ . As such in 1-dimension, the Hamiltonian is given by

$H = -\frac{1}{2} \sum_{j=1}^{N} (J_x \sigma_j^x \sigma_{j+1}^x + J_y \sigma_j^y \sigma_{j+1}^y + J_z \sigma_j^z \sigma_{j+1}^z - H\sigma_j^{z})$

where the H on the right-hand side indicates the external magnetic field, with periodic boundary conditions, and at spin $s = 1 / 2$ , spin matrices given by

$\sigma_1 = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}$

$\sigma_2 = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}$

$\sigma_3 = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}$

The Hamiltonian then acts upon the tensor product $(\mathbb{C}^2)^{\otimes N}$ , of dimension $2 N$ . The objective is to determine the spectrum of the Hamiltonian, from which the partition function can be calculated, from which the thermodynamics of the system can be studied. The most widely known type of Heisenberg model is probably the XXZ Heisenberg model, which occurs in the case $J = J_x = J_y \neq J_z = \Delta$ . The Bethe ansatz is a common way of solving the Heisenberg model in one dimension.

The physics of the Heisenberg model strongly depends on the sign of the coupling constant $J$ and the dimension of the space. For positive $J$ the ground state is always ferromangetic. At negative $J$ the ground state is antiferromagnetic in two and three dimensions. In one dimension the nature of correlations in the antiferromagnetic Heisenberg model depends on the spin of the magnetic dipoles. If the spin is integer then only short range order is present. A system of half-integer spins ehibits quasi-long range order.