# Bulletin of the American Physical Society

# APS March Meeting 2011

## Volume 56, Number 1

## Monday–Friday, March 21–25, 2011; Dallas, Texas

### Abstract: K1.00047 : Variational Study of a Finite Heisenberg Chain

#### Author:

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.

To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2011.MAR.K1.47

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