A first external code
=====================

.. highlight:: c

As a first exercise you can try to write a Monte Carlo code for an Ising chain
in a magnetic field. Your goal is to write this code as an external project and
to use the Monte Carlo class provided by TRIQS.

Take some time to read the :ref:`Monte Carlo <montecarlo>` chapter, but don't
read the complete example at the end of the chapter because it is precisely
what you need to do here. You can check your implementation later.

.. _isingex:

Ising chain in magnetic field
-----------------------------

Here's the Hamiltonian for the problem of Ising spins in a magnetic field

.. math::

  \mathcal{H} = -J \sum_{i=1}^N \sigma_i \sigma_{i+1} - h \sum_{i=1}^N \sigma_i.

The goal is to find the magnetization per spin :math:`m` of the system for 
:math:`J = -1.0`, a magnetic field :math:`h = 0.5` as a function of
the inverse temperature :math:`\beta`. You can see how the results
change with the length of the chain :math:`N`.

Implementation hints
--------------------

Here are a couple of implementation hints that you might want to follow.

* In most Monte Carlo programs there is a *configuration* which is modified
  along the simulation. Take enough time to think how this configuration
  can be efficiently described and implement it in a separate file, say
  :file:`configuration.hpp`. In this example, the configuration is a
  collection of spins that can e.g. be described by a vector of integers.
  +1 would be a spin up and -1 a spin down. If you're worried with memory
  space, you could use a vector of booleans (true for up spins, false for
  down spins).

* More to come...


Solution
--------

In the limit :math:`N \rightarrow \infty`, the solution for the magnetization
is

.. math::

  m = \frac{\sinh(\beta h) + \frac{\sinh(\beta h)\cosh(\beta h)}{\sqrt{\sinh^2(\beta h) + e^{-4\beta J}}}}
           {\cosh(\beta h) + \sqrt{\sinh^2(\beta h) + e^{-4\beta J}}}.

Here's a plot of :math:`m` versus :math:`\beta` for different values of :math:`N`:

.. image:: m_vs_beta.png
   :width: 700
   :align: center