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## Summary:

- How does a virtual two-step tunneling process explain antiferromagnetic ordering?

## Main Question or Discussion Point

Hi all, thanks in advance for your help!

For context, I'm generally new to condensed matter and many-body QM and am working through Altland and Simons'

I've seen a Heisenberg-like spin Hamiltonian derived by starting with a Hubbard Hamiltonian

$$H = -t\sum_{<ij>}a^{\dagger}_{i\sigma}a_{j\sigma} + U\sum_{j}n_{j+}n_{j-}$$

where the first sum is over nearest neighbors. As long as ##U/t >> 1## we can treat the tunneling terms as a perturbation. This gives us something like

$$H = -2\frac{t^{2}}{U}(1 + a^{\dagger}_{i\sigma}a^{\dagger}_{j\sigma'}a_{i\sigma'}a_{j\sigma}) = 4\frac{t^2}{U}(\vec{S}_{i}\cdot\vec{S}_{j}-\frac{1}{4})$$

From the first equality, it makes sense to me that this Hamiltonian is accounting for a process in which we essentially flip two spins at sites ##I## and ##j##. Furthermore, because our model only accounts for tunneling which a spin does not change, I think this process has to be one in which one of the spins hops to the neighboring site, and the other hops to the site that was left unoccupied (please correct me if I'm wrong!). The second equality makes it clear to me that the ordering of the ground state will be antiferromagnetic, due to the positive coupling constant.

This is all fine mathematically. I think what I am confused by is the

For context, I'm generally new to condensed matter and many-body QM and am working through Altland and Simons'

*Condensed Matter Field Theory.*I'm thinking in general about magnetic ordering.I've seen a Heisenberg-like spin Hamiltonian derived by starting with a Hubbard Hamiltonian

$$H = -t\sum_{<ij>}a^{\dagger}_{i\sigma}a_{j\sigma} + U\sum_{j}n_{j+}n_{j-}$$

where the first sum is over nearest neighbors. As long as ##U/t >> 1## we can treat the tunneling terms as a perturbation. This gives us something like

$$H = -2\frac{t^{2}}{U}(1 + a^{\dagger}_{i\sigma}a^{\dagger}_{j\sigma'}a_{i\sigma'}a_{j\sigma}) = 4\frac{t^2}{U}(\vec{S}_{i}\cdot\vec{S}_{j}-\frac{1}{4})$$

From the first equality, it makes sense to me that this Hamiltonian is accounting for a process in which we essentially flip two spins at sites ##I## and ##j##. Furthermore, because our model only accounts for tunneling which a spin does not change, I think this process has to be one in which one of the spins hops to the neighboring site, and the other hops to the site that was left unoccupied (please correct me if I'm wrong!). The second equality makes it clear to me that the ordering of the ground state will be antiferromagnetic, due to the positive coupling constant.

This is all fine mathematically. I think what I am confused by is the

*physical*connection between these two equalities. I've seen the antiferromagnetic ordering explained as originating from a virtual process in which spins reduce their energy by briefly hopping to a neighboring site. How exactly does the possibility of this process lower the energy of the ground state? Maybe asked another way, why would a ferromagnetic state, in which this virtual process is unavailable due to Pauli exclusion, necessarily raise the energy of the system?