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110 lines
3.4 KiB
ReStructuredText
110 lines
3.4 KiB
ReStructuredText
.. index:: Green's functions
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A little tutorial
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#########################
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In this first part, we present a couple of examples in the form of a tutorial.
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The full reference is presented in the next sections.
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A simple example
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----------------
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As a first example, we construct and plot the following Matsubara Green's function:
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.. math::
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G_{11} (i \omega_n) = \frac{1}{i \omega_n + 0.5}
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This is done with the code :
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.. plot:: reference/python/green/example.py
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:include-source:
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:scale: 70
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In this very simple example, the Green's function is just a 1x1 block. Let's go through
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the different steps of the example:
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.. literalinclude:: example.py
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:lines: 1-2
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This imports all the necessary classes to manipulate Green's functions. In this example it allows
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to use ``GfImFreq``, ``BlockGf`` :
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.. literalinclude:: example.py
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:lines: 4-5
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This creates a **block** Green's function which has just one index
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(1). ``Beta`` is the inverse temperature, ``NFreqMatsubara`` the number of
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Matsubara frequencies.
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.. literalinclude:: example.py
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:lines: 6-6
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This initializes the block with :math:`1/(i \omega_n + 0.5)`.
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Two points are worth noting here :
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* The right hand side (RHS) of this statement is a *lazy* expression : its evaluation is delayed until
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it is needed to fill the Green function.
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* The funny *<<=* operator means "set from". It fills the Green function with the evaluation of the expression
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at the right.
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.. literalinclude:: example.py
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:lines: 9-10
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These lines plot the block Green's function (both the real and imaginary parts) using
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the matplotlib plotter. More details can be found in the section
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:ref:`plotting`.
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A slightly more complicated example
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--------------------------------------------------
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Let's turn to another example. This time we consider a problem of a d-impurity
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level embedded in a flat conduction bath :math:`\Delta` of s-electrons. We want to
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construct the corresponding 2x2 Green's function:
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.. math::
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\hat{G}^\mathrm{s+d} (i \omega_n) = \begin{pmatrix} i\omega_n - \epsilon_d & V \\ V & \Delta^{-1} \end{pmatrix}^{-1}
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This is done with the code :
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.. plot:: reference/python/green/impinbath.py
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:include-source:
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:scale: 70
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Again, the Green's function is just one block, but this time it is a 2x2 block with
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off-diagonal elements. Another difference is that we use real-frequency Green's functions in
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this example:
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.. literalinclude:: impinbath.py
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:lines: 6-7
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In this constructor for the block Green's function, we specify that there are two indices s and d.
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Because it is a real-frequency Green's function we need to define the mesh on which it is
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computed. This is done with the ``MeshArray`` option.
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.. literalinclude:: impinbath.py
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:lines: 8-11
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These lines initialize specific entries of the block Green's function. Note how the elements are
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accessed with ``[index1,index2]``. These lines also show how to initialize a Green's function
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to a constant or to the inverse of a Wilson bath (constant spectral function on an interval [-D,D], with
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D=1.0 in our example).
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.. literalinclude:: impinbath.py
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:lines: 12-12
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``invert()`` inverts the entire block Green's function (as a matrix).
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.. literalinclude:: impinbath.py
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:lines: 14-17
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Here, we isolate some elements of the blocks Green's function. It is also the
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first example of an operation on the block Green's function, which is
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multiplied by a scalar. The last lines of the script just plot these new
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objects.
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