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Mainly on the python part. I had a quick browse through to check if the scripts were still working.
72 lines
2.6 KiB
ReStructuredText
72 lines
2.6 KiB
ReStructuredText
.. _ipt:
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Iterated perturbation theory: a simple DMFT solver
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==================================================
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Introduction
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------------
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The iterated perturbation theory (IPT) was one of the first methods used to solve the
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DMFT equations [#ipt1]_. In spite of its simplistic nature, IPT gives a qualitatively
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correct description of a Mott metal-insulator transition in the Hubbard model on
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infinite-dimensional lattices (on the quantitative level it tends to underestimate
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correlations though). In IPT one iteratively solves the DMFT equations using the
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second-order perturbation theory in Hubbard interaction :math:`U` to approximate
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the impurity self-energy. For the particle-hole symmetric case it reads
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.. math::
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\Sigma(i\omega_n) \approx \frac{U}{2} +
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U^2 \int_0^\beta d\tau e^{i\omega_n\tau} G_0(\tau)^3
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A Hartree-Fock contribution :math:`U/2` in the self-energy cancels with a term
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from :math:`G_0(i\omega_n)^{-1}` when the functions are substituted into the
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Dyson's equation. For this reason this contribution is usually omitted from
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both functions.
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The success of IPT is caused by the fact that it becomes exact not only in the
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weak coupling limit (by construction), but also reproduces an atomic-limit
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expression for :math:`\Sigma(i\omega_n)` as :math:`U` grows large [#ipt2]_.
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IPT solver
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----------
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We start by writing an IPT solver that implements the weak-coupling
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perturbation theory for a symmetric single-band Anderson model.
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All Green's functions in the calculations have just one index because
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*up* and *down* components are the same.
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.. literalinclude:: ipt_solver.py
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Visualization of a Mott transition
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----------------------------------
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We can now use this solver to run DMFT calculations and scan a range of
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values of :math:`U`. At every iteration the resulting data is plotted
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and saved into PNG files using the :ref:`TRIQS matplotlib interface<plotting>`.
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Not that :math:`G(i\omega_n)` is analytically continued to the real axis using
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:ref:`Padé approximant<GfReFreq>`.
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At the end of the script an external utility `convert` is invoked to join the
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DOS plots into a single animated GIF file which illustrates how a metallic
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solution evolves towards an insulator.
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.. literalinclude:: ipt_dmft.py
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.. only:: html
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The result of this script is the following animated gif:
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.. image:: mott.gif
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:width: 700
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:align: center
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Journal references
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------------------
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.. [#ipt1] A. Georges and G. Kotliar,
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Phys. Rev. B 45, 6479–6483 (1992).
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.. [#ipt2] X. Y. Zhang, M. J. Rozenberg, and G. Kotliar,
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Phys. Rev. Lett. 70, 1666–1669 (1993)
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