J1-J2 Heisenberg model
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Here, we solve the :math:`J_{1}-J_{2}` Heisenberg model on the square lattice, 
which is a canonical example of the frustrated magnets.
Its Hamiltonian is defined as

.. math::

  {\mathcal H}=J_{1}\sum_{\langle i,j\rangle }{\bf S}_{i}\cdot{\bf S}_{j}+J_{2}\sum_{\langle\langle i,j\rangle\rangle }{\bf S}_{i}\cdot{\bf S}_{j},

where :math:`J_{1}` (:math:`J_{2}`) represents the nearest (next-nearest) neighbor interactions.

An input file (``samples/tutorial_1.5/stan1.in``) for treating :math:`J_{1}-J_{2}` Heisenberg model is given as ::

 model = "Spin" 
 method = "CG" 
 lattice = "square" 
 L = 4
 W = 4
 J = 1
 J' = 1
 2S = 1
 2Sz = 0
 exct = 4

Here, J (J') represents :math:`J_{1}` (:math:`J_{2}`).

Calculations of spin structure factors for ground state
""""""""""""""""""""""""""""""""""""""""""""""""""""""""""
First, we calculate the spin structure factors, which are defined as

.. math::

  S({\bf q})=\frac{1}{N_{\rm s}}\sum_{i,j} {\bf S}_{i}\cdot{\bf S}_{j}

To calculate :math:`S({\bf q})`, it is necessary to prepare
a proper input file for two-body Green functions.
By using a python script **MakeGreen.py** (``samples/tutorial_1.5/MakeGreen.py``),
you can generate **greentwo.def** for calculating :math:`S({\bf q})`.
To use  **MakeGreen.py** , an input file for specifying lattice geometry (**input.txt**, ``samples/tutorial_1.5/input.txt``) is necessary,
whose form is given as follows ::

 Lx 4
 Ly 4
 Lz 1
 orb_num 1

Here, Lx (orb_num) represents the length of the x direction (number of orbitals).

By using a python script **CalcSq.py** (``samples/tutorial_1.5/CalcSq.py``),
you can calculate :math:`S({\bf q})` from **output/zvo_cisajscktalt_eigen0.dat**.

Procedure for calculating and visualizing :math:`S({\bf q})` is given as follows ::
 1. HPhi -sdry stan.in
 2. python3 MakeGreen.py (input.txt is necessary)
 3. HPhi -e namelist.def 
 4. python3 CalcSq.py (input.txt is necessary)
 5. You can obtain **Sq_eigen0.dat** !!
 6. splot "Sq_eigen0.dat" u 1:2:3 (using gnuplot)

Following the procedure, please see how :math:`S({\bf q})` changes
by changing J'.
As an example, we show :math:`S({\bf q})` for J1/J2=0 and 1 below.

.. image:: ../../../figs/J1J2_Sq.*
   :align: center

Calculations of spin structure factors for excited states
"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""
By changing exct in stan.in, you can obtain several excited states.
For those excited states, by changing **max_num=1** in CalcSq.py as,
for example, **max_num=4**,
you can obtain  :math:`S({\bf q})` for the excited states.
As a practice for editing file, please try to edit CalcSq.py manually 
(open CalcSq.py, finding variable max_num =1, and changing max_num = 4).

Please see how :math:`S({\bf q})` in the excited states 
changes by changing J'. For example, please check 
what is the nature of the
first excited state J'=0,0.5, and 1.