Dynamical spin structure factor
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Let's solve the following Hubbard model on the square lattice.

.. math::

 H = -t \sum_{\langle i,j\rangle , \sigma}(c_{i\sigma}^{\dagger}c_{j\sigma}+{\rm H.c.})+U \sum_{i} n_{i\uparrow}n_{i\downarrow}


The input files (``samples/tutorial_4.1/stan1.in`` and ``samples/tutorial_4.1/stan2.in``) for 8-site Hubbard model are as follows ::

 stan1.in

 a0W = 2
 a0L = 2
 a1W = -2
 a1L = 2
 model = "hubbard"
 method = "CG"
 lattice = "square"
 t = 1.0
 t' = 0.5
 U = 4.0
 2Sz = 0
 nelec = 8
 EigenvecIO = "out"

:: 

 stan2.in

 a0W = 2
 a0L = 2
 a1W = -2
 a1L = 2
 model = "hubbard"
 method = "CG"
 lattice = "square"
 t = 1.0
 t' = 0.5
 U = 4.0
 2Sz = 0
 nelec = 8
 LanczosEPS = 8
 CalcSpec = "Normal"
 SpectrumType = "SzSz"
 SpectrumQW = 0.5
 SpectrumQL = 0.5
 OmegaMin = -10.0
 OmegaMax = 20.0
 OmegaIM = 0.2
 OmegaOrg = 10.0
 
You can execute HPhi as follows ::

 HPhi -s stan1.in
 HPhi -s stan2.in

After finishing calculations, the spectrum :math:`G_{S_z S_z}({\bf Q} \equiv (\pi, \pi), \omega) = \langle S_z(-{\bf Q}) \left[H-\omega-\omega_0 + i\eta\right]^{-1}S_z({\bf Q})\rangle` is outputted in `output/zvo_DynamicalGreen.dat`. Here, :math:`S_z({\bf Q})= \sum_{i}e^{i {\bf Q} \cdot {\bf r}_i} S_z^i` and the frequency :math:`\omega` moves from :math:`-10` to :math:`10`, :math:`\omega_0 = 10`, and :math:`\eta` is set as :math:`0.2`. You can check the result by executing the following command on gnuplot::
  
 gnuplot
 gnuplot> set xlabel "Energy"
 gnuplot> set ylabel "G_{SzSz}(E)"
 gnuplot> set xzeroaxis
 gnuplot> plot "output/zvo_DynamicalGreen.dat" u 1:3 w l tit "Real", \
 > "output/zvo_DynamicalGreen.dat" u 1:4 w l tit "Imaginary"

You can see the following output image.

.. image:: ../../../figs/spectrum.*
   :height: 500px
   :width: 700px
   :align: center