Example for *moller* calculation with HPhi
------------------------------------------

What’s this sample?
~~~~~~~~~~~~~~~~~~~

This is an example of ``moller`` with
`HPhi <https://github.com/issp-center-dev/HPhi>`__, which is an
open-source software package for performing the exact diagonalization
method for quantum many-body problems. In this example, we will
calculate the system size dependence of the excitation gap
:math:`\Delta` of the :math:`S=1/2` (``2S_1`` directory) and :math:`S=1`
(``2S_2``) antiferromagnetic Heisenberg chain under the periodic
boundary condition. By using ``moller``, calculations with different
system sizes are performed in parallel. This is corresponding to
`section
1.4 <https://issp-center-dev.github.io/HPhi/manual/develop/tutorial/en/html/zero_temperature/spin_chain.html>`__
of HPhi's official tutorial.

Preparation
~~~~~~~~~~~

Make sure that ``moller`` (HTP-tools) package and ``HPhi`` are
installed. In this tutorial, the calculation will be performed using the
supercomputer system ``ohtaka`` at ISSP.

How to run
~~~~~~~~~~

1. Prepare dataset

   Run the script ``make_inputs.sh`` enclosed within this package.

   .. code:: bash

      $ bash ./make_inputs.sh

   Working directories ``L_8``, ``L_10``, …, ``L_24`` (up to ``L_18``
   for ``2S_2``)) will be generated. A list of the directories is
   written to a file ``list.dat``. Additionally, a shell script,
   ``extract_gap.sh``, to gather energy gaps from working directories is
   generated.

2. Generate job script using ``moller``

   Generate a job script from the job description file using ``moller``,
   and store the script as a file named ``job.sh``.

   .. code:: bash

      $ moller -o job.sh input.yaml

3. Run batch job

   Submit a batch job with the job list as an argument.

   .. code:: bash

      $ sbatch job.sh list.dat

4. Check status

   The status of task execution will be summarized by ``moller_status``
   program.

   .. code:: bash

      $ moller_status input.yaml list.dat

5. Gather results

   Once the calculation finishes, gather energy gaps from jobs as

   .. code:: bash

      $ bash extract_gap.sh

   This script writes pairs of the length :math:`L` and the gap
   :math:`\Delta` into a text file, ``gap.dat``.

   To visualize the results, a Gnuplot file ``gap.plt`` is available. In
   this file, the obtained gap data are fitted by the expected curves,

   .. math:: \Delta(L; S=1/2) = \Delta_\infty + A/L

   and

   .. math:: \Delta(L; S=1) = \Delta_\infty + B\exp(-CL).

   The result is plotted as follows:

   .. code:: bash

      $ gnuplot --persist gap.plt

   .. figure:: ../../../../images/tutorial_hphi_gap.*
      :alt: Finite size effect of spin gap

      Finite size effect of spin gap

   Note that the logarithmic correction causes the spin gap for
   :math:`S=1/2` to remain finite. On the other hand, for :math:`S=1`,
   the extrapolated value :math:`\Delta_\infty = 0.417(1)` is consistent
   with the previous results, e.g., :math:`\Delta_\infty = 0.41048(6)`
   by QMC (Todo and Kato, PRL **87**, 047203 (2001)).