.. highlight:: none

Finite Temperature Calculations for the Transverse Field Ising Model
----------------------------------------------------------------------

In this section, we present a calculation example of the ferromagnetic Ising model on a square lattice subjected to a transverse magnetic field, denoted by ``hx``, at finite temperatures.
The Hamiltonian is

.. math::

   \begin{aligned}
   H = J^z \sum_{\langle i,j \rangle} {S}_i^{z} {S}_j^{z} - h^x \sum_i S_i^x.
   \end{aligned}

Please note that the model is defined using spin operators of size 1/2, not Pauli operators.
The input and script files used in this tutorial are located in the ``sample/03_finite_temperature`` directory. Below is a sample input file (``simple_ft_strong.toml``):

.. literalinclude:: ../../../../sample/03_finite_temperature/simple_ft_strong.toml

To perform finite temperature calculations, set the ``mode`` to ``finite``.
Here, the transverse magnetic field is set to ``hx = 2.0`` with ``tau = 0.01`` (the inverse temperature step size is 2 times ``tau``).
Once preparing an input file of the simple mode, execute ``tenes_simple``, ``tenes_std``, and ``tenes`` in the same way as for the ground state calculation.

The results of the finite temperature calculations are output to the ``output_ft_strong`` directory.
Basically, the output is the same as the ground state calculation, but the inverse temperature is added to the first column.
For example, ``FT_density.dat`` is as follows::

    # The meaning of each column is the following:
    # $1: inverse temperature
    # $2: observable ID
    # $3: real
    # $4: imag
    # The meaning of observable IDs are the following:
    # 0: Energy
    # 1: Sz
    # 2: Sx
    # 3: Sy
    # 4: bond_hamiltonian
    # 5: SzSz
    # 6: SxSx
    # 7: SySy

    0.00000000000000000e+00 0  0.00000000000000000e+00  0.00000000000000000e+00
    0.00000000000000000e+00 1  0.00000000000000000e+00  0.00000000000000000e+00
    0.00000000000000000e+00 2  0.00000000000000000e+00  0.00000000000000000e+00
    0.00000000000000000e+00 3  0.00000000000000000e+00  0.00000000000000000e+00
    0.00000000000000000e+00 4  0.00000000000000000e+00  0.00000000000000000e+00
    0.00000000000000000e+00 5  0.00000000000000000e+00  0.00000000000000000e+00
    0.00000000000000000e+00 6  0.00000000000000000e+00  0.00000000000000000e+00
    0.00000000000000000e+00 7  0.00000000000000000e+00  0.00000000000000000e+00

       ... continued ...

The second column indicates the type of physical quantity, and for example, 0 represents energy.
Thus, you can extract the temperature dependence by extracting only the energy with ``awk``::

    awk '$2 == 0 {print $1, $3, $4}' output_ft_strong/FT_density.dat > energy_strong.dat

To observe the behavior at different transverse magnetic fields, we've provided additional sample input files: ``simple_ft_middle.toml`` (``hx = 0.8``), ``simple_te_weak.toml`` (``hx = 0.5``), and ``simple_ft_zero.toml`` (``hx = 0.0``).
Moreover, a script named ``run.sh`` has been set up to execute all these calculations simultaneously.
Ensure you've added tools like ``tenes`` to your PATH, then initiate the calculations with:

::

    sh run.sh

The computation should complete in about a minute.
Since specific heat is difficult to calculate directly, it is calculated from the energy by numerical differentiation.
``calcspec.py`` is a script to calculate specific heat from the energy by using the spline interpolation::

    python3 calcspec.py

To visualize the results, scripts have been prepared to plot energy, heat capacity, and magnetization (:math:`S_x`, :math:`S_z`): ``plot_e.plt``, ``plot_c.plt``, ``plot_mx.plt``, and ``plot_mz.plt``.
Running the following:

::

    gnuplot -persist plot_e.plt
    gnuplot -persist plot_c.plt
    gnuplot -persist plot_mx.plt
    gnuplot -persist plot_mz.plt

will display plots for energy, heat capacity, and magnetizations (:math:`m_x` and :math:`m_z`). The resulting plots are illustrated in :numref:`fig_tutorial3_finitetemperature`. For comparison, results obtained using Quantum Monte Carlo calculations are also shown (using ``DSQSS/dla``).

.. figure:: ../../img/tutorial_03_finitetemperature.*
    :name: fig_tutorial3_finitetemperature
    :width: 600px

    Graphs for the finite temperature calculations of the Ising model: (a) energy, (b) heat capacity, (c) transverse magnetization, and (d) longitudinal magnetization. The vertical axis represents the physical quantity, and the horizontal axis denotes temperature.
    Curves are results by TeNeS and squares are results by DSQSS/dla (longitudinal magnetization is zero from symmetry).