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65 lines
2.0 KiB
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65 lines
2.0 KiB
ReStructuredText
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A first external code
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=====================
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.. highlight:: c
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As a first exercise you can try to write a Monte Carlo code for an Ising chain
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in a magnetic field. Your goal is to write this code as an external project and
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to use the Monte Carlo class provided by TRIQS.
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Take some time to read the :ref:`Monte Carlo <montecarlo>` chapter, but don't
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read the complete example at the end of the chapter because it is precisely
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what you need to do here. You can check your implementation later.
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.. _isingex:
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Ising chain in magnetic field
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-----------------------------
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Here's the Hamiltonian for the problem of Ising spins in a magnetic field
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.. math::
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\mathcal{H} = -J \sum_{i=1}^N \sigma_i \sigma_{i+1} - h \sum_{i=1}^N \sigma_i.
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The goal is to find the magnetization per spin :math:`m` of the system for
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:math:`J = -1.0`, a magnetic field :math:`h = 0.5` as a function of
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the inverse temperature :math:`\beta`. You can see how the results
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change with the length of the chain :math:`N`.
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Implementation hints
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--------------------
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Here are a couple of implementation hints that you might want to follow.
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* In most Monte Carlo programs there is a *configuration* which is modified
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along the simulation. Take enough time to think how this configuration
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can be efficiently described and implement it in a separate file, say
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:file:`configuration.hpp`. In this example, the configuration is a
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collection of spins that can e.g. be described by a vector of integers.
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+1 would be a spin up and -1 a spin down. If you're worried with memory
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space, you could use a vector of booleans (true for up spins, false for
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down spins).
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* More to come...
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Solution
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--------
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In the limit :math:`N \rightarrow \infty`, the solution for the magnetization
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is
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.. math::
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m = \frac{\sinh(\beta h) + \frac{\sinh(\beta h)\cosh(\beta h)}{\sqrt{\sinh^2(\beta h) + e^{-4\beta J}}}}
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{\cosh(\beta h) + \sqrt{\sinh^2(\beta h) + e^{-4\beta J}}}.
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Here's a plot of :math:`m` versus :math:`\beta` for different values of :math:`N`:
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.. image:: m_vs_beta.png
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:width: 700
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:align: center
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