.. 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_tutorial8_finitetemperature`. For comparison, results obtained using Quantum Monte Carlo calculations are also shown (using ``ALPS/looper``).

.. figure:: ../../img/tutorial_08_finitetemperature.*
    :name: fig_tutorial8_finitetemperature
    :width: 600px

    Graphs for the finite temperature calculations of the Ising model: (a) energy, (b) heat capacity, (c) :math:`m_x`, and (d) :math:`m_z`. The vertical axis represents the physical quantity, and the horizontal axis denotes temperature.