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77 lines
2.9 KiB
ReStructuredText
.. _ipt:
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Iterated perturbation theory: an extended DMFT example
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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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Our sample implementation of IPT includes two files: `ipt.py` and
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`mott.py`.
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IPT solver and self-consistency loop
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------------------------------------
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The file `ipt.py` implements the weak coupling perturbation theory for a
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symmetric single-band Anderson model (`Solver` class) and obeys
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:ref:`the solver concept<solver_concept>`. It also runs a DMFT loop with this
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solver and with a self-consistency condition provided from outside (function `run`).
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All Green's functions in the calculations consist of one Matsubara block
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(there is no need for spin indices, since only paramagnetic solutions are sought).
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.. literalinclude:: ipt.py
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Visualization of a Mott transition
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----------------------------------
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In `mott.py` the IPT module is used to run DMFT many times and scan a range of
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values of :math:`U`. On every run the resulting data are saved in
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an :ref:`HDF5 archive<hdf5_base>` and the density of states is plotted into
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a PNG file using the :ref:`TRIQS matplotlib interface<plotting>`
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(:math:`G(i\omega_n)` is analytically continued to the real axis by the
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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:: mott.py
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The result of this script is an 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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