If I take the evolution of the Hamiltonian of the Ising model in terms of the Pauli operators, i.e. $\exp(-it(\sigma^z_i\otimes\sigma^z_j)/\hbar)$ where $\sigma_i$, $\sigma_j$ are the Pauli operators for the spin of an electron and a nuclear spin respectively, how do I then apply an external magnetic field in the z direction only and incorporate this into my model? I am looking at a particular problem with a system of 2 spins that evolves according to the Ising Hamiltonian and how application of an external magnetic field will affect the system.
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https://en.wikipedia.org/wiki/Ising_model#Definition – By Symmetry Jul 05 '16 at 19:29
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In general, the evolution of a quantum system is given by $exp(\frac{-i H t}{\hbar})$. When a magnetic field is added to the Ising model, $H$ will have an additional term for the interaction of the spins with the magnetic field.
The potential energy of a magnetic moment $\vec{\mu} $ in a magnetic field $\vec{B}$ is $U = - \vec{\mu} \cdot \vec{B}$. If we have a magnetic field $B$ along the z-axis, then the additional term in the Hamiltonian will be $\mu B \sigma_i^z $. Remember, the external field term should simply be summed over all lattice points, and the spin-spin interaction should be summed over all neighbouring pairs of lattice points.
Note that the standard Ising model has one spin per lattice point, whereas you seem to be suggesting a more complicated model with nuclear spins and electron spins.
Sergei Patiakin
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