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About
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*****
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The TRIQS Sum-Of-Exponentials bold HYBridization expansion impurity solver was originally developed by Z. Huang, J. Kaye, and H. U.R. Strand. We thank N. Wentzell for contributions, support and guidance in using the `TRIQS library <https://github.com/TRIQS/triqs>`_ and its related packages.
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The TRIQS arbitrary Crossing Approximation (XCA) impurity solver has been developed by F. Rilloraza, Z. Huang, J. Kaye, and H. U.R. Strand. We thank N. Wentzell for contributions, support and guidance in using the `TRIQS library <https://github.com/TRIQS/triqs>`_ and its related packages.
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For compact representation of imaginary time propagators we use the C++ implementation of the Discrete Lehmann Representation provided by the ` ``cppdlr`` library <github.com/flatironinstitute/cppdlr/>`_. The TRIQS application structure is inherited from the `app4triqs <https://github.com/TRIQS/app4triqs>`_ framework.
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For compact representation of imaginary time propagators we use the C++ implementation of the Discrete Lehmann Representation provided by the `cppdlr library <github.com/flatironinstitute/cppdlr/>`_. The TRIQS application structure is inherited from the `app4triqs framework. <https://github.com/TRIQS/app4triqs>`_
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License
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=======
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The TPRF application is published under the `GNU General Public License, version 3. <http://www.gnu.org/licenses/gpl.html>`_
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The ``triqs_xca`` application is published under the `GNU General Public License, version 3. <http://www.gnu.org/licenses/gpl.html>`_
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Note that it implies that applications using TPRF must also be GPL.
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Note that it implies that applications using ``triqs_xca`` must also be GPL.
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Disclaimer
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==========
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The program is provided as is, i.e. WITHOUT ANY WARRANTY of any kind, as stated in the license. In particular, its authors and contributors will take no responsability for any possible bugs or any improper use of these programs, including those resulting in incorrect scientific publications.
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The program is provided as is, i.e. WITHOUT ANY WARRANTY of any kind, as stated in the license. In particular, its authors and contributors will take no responsibility for any possible bugs or any improper use of these programs, including those resulting in incorrect scientific publications.
Copy file name to clipboardExpand all lines: doc/background/ppsc.rst
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.. math::
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(1 - G_0\ast\Sigma\ast) G = G_0 \\
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(1 - G_0\ast( \Sigma + \eta)\ast) G = G_0 \\
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where :math:`G_0(\tau)` is the atomic many-body propagator and :math:`\Sigma(\tau)` is the pseudo-particle self-energy truncated at a finite expansion order :math:`n` in the hybridization function :math:`\Delta(\tau)`.
Once convergence is reached physical response functions like the single particle Green's function :math:`g(\tau) = \langle\mathcal{T} c(\tau) c^\dagger(0) \rangle` can be evaluted by a separate diagrammatic series (similar to :math:`\Sigma`).
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Once convergence is reached physical response functions like the single particle Green's function :math:`g(\tau) = -\langle\mathcal{T} c(\tau) c^\dagger(0) \rangle` can be evaluated by a separate diagrammatic series (similar to :math:`\Sigma`).
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To stabilize the self consistency the pseudo particle Green's function :math:`G(\tau)` is normalized in between each new evaluaton of the self-energy :math:`\Sigma` by enforcing that :math:`\textrm{Tr}[G(\beta)] = -1` by tuning the pseudo particle chemical potential :math:`\eta` in the Dyson equation. This is essential to avoid numerical overflow.
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The ``triqs_xca`` solver implements the bold hybridization expansion using the Discrete Lehmann Representation (DLR) [#dlr]_ [#cppdlr]_ for compact representation of propagators in imaginary time :math:`\tau` and a separate hybridization function compression approach [#xca]_ (based on the famous AAA algorithm) to evaluate the diagram series for :math:`\Sigma` with lower computational complexity than standard quadrature integration. [#dlrhyb]_
The TRIQS Sum-Of-Exponentials bold HYBridization expansion impurity solver (triqs_xca) can solve generic multiband Anderson impurity problems approximately by perturbative expansion in the coupling to the environment (a.k.a. the hybridization function :math:`\Delta(\tau)`).
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The TRIQS arbitrary Crossing Approximation (**XCA**) impurity solver ``triqs_xca`` can solve generic multiband Anderson impurity problems approximately by perturbative expansion in the coupling to the environment (a.k.a. the hybridization function :math:`\Delta(\tau)`).
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This implementation is leveraging the sum-of-exponentials (SOE) trick to acheive parametrically better computational complexity compared to direct integration and uses the Discrete Lehmann Representation (DLR) for representing response functions.
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This implementation is leveraging the sum-of-exponentials (SOE) trick to achieve parametrically better computational complexity compared to direct integration and uses the Discrete Lehmann Representation (DLR) for representing response functions.
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Learn how to use triqs_xca in the :ref:`documentation`.
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.. [#cppdlr] `J. Kaye, H. U.R. Strand, N. Wentzell, J. Open Source Softw., 9(100), 6297, (2024) <https://doi.org/10.21105/joss.06297>`_
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.. [#dlr] `J. Kaye, K. Chen, O. Parcollet, Phys. Rev. B 105, 235115 (2021) <https://doi.org/10.1103/PhysRevB.105.235115>`_
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Note that this application uses the TRIQS library please also see its `citation page.<https://triqs.github.io/triqs/latest/about.html#citation>`_
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Note that this application uses the TRIQS library, please also see its `citation page <https://triqs.github.io/triqs/latest/about.html#citation>`_,
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as well as the Adapol package, see its `citation page. <https://flatironinstitute.github.io/adapol/latest/index.html#references>`_
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Prerequisites
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-------------
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#. The :ref:`TRIQS <triqslibs:welcome>` library, see :ref:`TRIQS installation instruction <triqslibs:triqs_install>`.
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#. The :ref:`TRIQS <triqslibs:welcome>` library, see :ref:`TRIQS installation instructions. <triqslibs:triqs_install>`
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We are currently depending on the ``unstable`` development branch of TRIQS that has to be compiled from source. (*Note that the packaged versions of TRIQS are not supported.*)
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In the following, we assume that TRIQS is installed in the directory ``path_to_triqs``.
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We are currently depending on the `unstable` development branch of TRIQS that has to be compiled from source. (Note that the packaged versions of TRIQS are not supported.)
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#. The `adapol <https://github.com/flatironinstitute/adapol>`_ package (Adaptive Pole Fitting for Quantum Many-Body Physics) has to be installed on the `triqs_xca` branch. This can be achieved using a local `pip` install, e.g.::
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#. The `Adapol <https://github.com/flatironinstitute/adapol>`_ package (Adaptive Pole Fitting for Quantum Many-Body Physics) has to be installed on the ``triqs_xca`` branch. This can be achieved using a local ``pip`` install, e.g.::
.. note:: The `pyed` python module has to be installed manually. Trying to install it using `pip` will not work since there is another python module with the same name (doing something completely different) registered in the PIP package index.
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.. note:: The ``pyed`` python module has to be installed manually. Trying to install it using ``pip`` will not work since there is another python module with the same name (doing something completely different) registered in the PIP package index.
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Installation steps
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------------------
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Checkout the version of the code that you want::
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$ git checkout 2.1.0
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$ git checkout block_sparse
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and follow steps 2 to 4 above to compile the code.
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