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 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.

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") σj, 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, spin matrices 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. 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,1 while other approaches do so without Bethe ansatz.2

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.3 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 (in 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 temperature linear dependence follows from the second law of thermodynamics.

See also

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

Notes








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