Variational Ground-State of the One-Dimensional Heisenberg Model

We wish to study the ground state of a spin system described by the Heisenberg Hamiltonian% \[ H=-\frac{1}{2}J\sum_{l}\left\{ 2\left( \sigma _{l}^{+}\sigma _{l+1}^{+}+\sigma _{l}^{-}\sigma _{l+1}^{-}\right) +\sigma _{l}^{z}\sigma _{l+1}^{z}\right\} , \]% where $\sigma ^{\pm }=\sigma _{x}\pm i\sigma _{y}$ and $J$ is the interaction strength. We choose as our trial ket $\left\vert \psi _{0}\left( \lambda \right) \right\rangle =e^{\lambda \hat{S}}\left\vert \phi _{0}\right\rangle $ where $\left\vert \phi _{0}\right\rangle $ is chosen to be the ferromagnetic state with all spins aligned downward, and $\hat{S}$ is the operator $\hat{S}=\sum_{l}\sigma _{l}^{+}\sigma _{l+1}^{+}$ with $% \lambda $ a variational parameter. We then construct our variational basis by systematically taking derivatives of $\left\vert \psi _{0}\left( \lambda \right) \right\rangle $ with respect to $\lambda $: $\left\vert \psi _{N}\left( \lambda \right) \right\rangle =\partial _{\lambda }^{N}\left\vert \psi _{0}\left( \lambda \right) \right\rangle $. The lowest eigenvalue of the Hamiltonian matrix $E_{0}\left( \lambda \right) $ is then minimized with respect to $\lambda $. Comparisons are then made with other approximation schemes.