Variational Study of a Finite Heisenberg Chain

Here we wish to study the ground-state of the $1D$ Heisenberg chain% \[ H=-\frac{1}{2}J\sum_{l=1}^{N}\left[ 2\left( \sigma_{l}^{+}\sigma_{l+1}% ^{+}+\sigma_{l}^{-}\sigma_{l+1}^{-}\right) +\sigma_{l}^{z}\sigma_{l+1}% ^{z}\right] , \] where the $\sigma$'s are the usual Pauli spin matrices and $J$ is the strength of the spin-spin interaction. The purpose of our revisiting such a well known system is to use it as a benchmark for our variational ansatz in which a trial vector is chosen $\left\vert \psi_{0}\left( \alpha\right) \right\rangle =\exp\left( \alpha\sum_{l=1}^{N}\sigma_{l}^{+}\sigma_{l+1}^{z}\right) \left\vert 0\right\rangle _{N}$, where $\alpha$ is the variational parameter and $\left\vert 0\right\rangle _{N}$ is an appropriately chosen initial array of spins. We then construct a basis according to the prescription $\left\vert \psi_{m}\right\rangle =\partial_{\alpha}^{m}\left\vert \psi_{0}\left( \alpha\right) \right\rangle $ creating an energy matrix with elements $h_{ij}=h_{ij}\left( \alpha,J\right) $ whose eigenvalues are then evaluated.