Heisenberg model (quantum)

The Heisenberg model is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spins 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. Except the coupling between magnetic dipole moments, there is also a multipolar version of Heisenberg model called the multipolar exchange interaction.

Overview

For quantum mechanical reasons (see exchange interaction or the subchapter "quantum-mechanical origin of magnetism" in the article on magnetism), 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

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

where $J$ is the coupling constant for a 1-dimensional model consisting of N dipoles, represented by classical vectors (or "spins") σ_j, subject to the periodic boundary condition $\sigma_{N+1} = \sigma_1$. 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 3-dimensions, the Hamiltonian is given by

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

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

$\sigma^x = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}$

$\sigma^y = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}$

$\sigma^z = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}$

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.

A simplified version of Heisenberg model is the one-dimensional Ising model, where the transverse magnetic field is in the x-direction, and the interaction is only in the z-direction:

$\hat H = -J_z \sum_{j =1}^{N} \sigma_j^z \sigma_{j+1}^z - gJ_z \sum_{j =1}^{N} \sigma_j^x$

At small g and large g, the ground state degeneracy is different, which implies that there must be a quantum phase transition in between. It can be solved exactly for the critical point using the duality analysis. The duality transition of the Pauli matrices is $\sigma_i^z = \prod_{j \leq i}S^x_j$ and $\sigma_i^x = S^z_i S^z_{i+1}$, where $S^x$ and $S^z$ are also Pauli matrices which obey the Pauli matrix algebra. Under periodic boundary conditions, the transformed Hamiltonian can be shown is of a very similar form:

$\hat H = -gJ_z \sum_{j =1}^{N} S_j^z S_{j+1}^z - gJ_z \sum_{j =1}^{N} S_j^x$

but for the $g$ attached to the spin interaction term. Assuming that there's only one critical point, we can conclude that the phase transition happens at $g=1$.

The most widely known type of Heisenberg model is the Heisenberg XXZ model, which occurs in the case $J = J_x = J_y \neq J_z = \Delta$. The spin 1/2 Heisenberg model in one dimension may be solved exactly using the Bethe ansatz, while other approaches do so without Bethe ansatz.

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 ferromagnetic. At negative $J$ the ground state is antiferromagnetic in two and three dimensions, it is from this ground state that the Hubbard model is given. 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 exhibits quasi-long range order.

Applications

• Another important object is entanglement entropy. One way to describe it is to subdivide the unique ground state into a block (several sequential spins) and the environment (the rest of the ground state). The entropy of the block can be considered as entanglement entropy. At zero temperature in the critical region (thermodynamic limit) it scales logarithmically with the size of the block. As the temperature increases the logarithmic dependence changes into a linear function. For large temperatures linear dependence follows from the second law of thermodynamics.

See also

• Classical Heisenberg model
• Dmrg of Heisenberg model
• Quantum rotor model
• t-J model
• J1 J2 model
• Majumdar–Ghosh model
• AKLT model
• Multipolar exchange interaction

References

• R.J. Baxter, Exactly solved models in statistical mechanics, London, Academic Press, 1982
• H. Bethe, Zur Theorie der Metalle, Zeitschrift für Physik A, 1931 doi:10.1007/BF01341708