:tocdepth: 2

Tight-Binding Lieb
==========================

.. toctree::
    :hidden:

    codeLiebTB
    examplesLieb

Motivations
-------------------

**liebTB** is a module providing a Python implementation of a face centered square lattice within the tight-binding framework. It can be used to build up and solve tight-binding models with complex valued onsite energies and/or hoppings. As such, this package can be used to analyze the concepts of Parity-Time symmetry, Parity-Time symmetry breaking, zero-modes, and topologically protection.


The lattice is defined by:

    * three sublattices named A, B and C. The unit cell is given by:

    .. image:: ../TBfig/lieb/unit_cell.png
        :height: 300px
        :width:  300px
        :align: center


    * four hoppings:

        * In the :math:`x` direction: the intradimer coupling :math:`t_{ab}` which links the A sites to the B sites, and the interdimer hopping :math:`t_{ba}` which links the B sites to the A sites.

        * In the :math:`y` direction: the intradimer coupling :math:`t_{ac}` which links the A sites to the B sites, and the interdimer hopping :math:`t_{ca}` which links the C sites to the A sites.


**liebTB** can

    * obtain the spectrum (eigenenergies of the tight-binding Hamiltonian) and the probability densities of the states of the system (absolute value squared eigenvectors of the Hamiltonian).
    * obtain the polarization of the A sublattice (the sum of the probability densities of the A sites)
    * select states by introducing a condition on the A sublattice polarization (revealing zero modes and/or localized states).
    * test the robustness to disorder by implementing hopping disorder.
    * get the time evolution of the field (using the Crank-Nicolson method).
    * get the time evolution of the field with adiabatic pumping (smooth variation of the hoppings).


Installation
---------------

**liebTB** is available at https://github.com/cpoli/TB

To use **liebTB**,  you need to install the programming language python and three additional packages:

    * python 3.x
    * numpy
    * scipy
    * matplotlib

See https://cpoli.github.io/python-doc.html for details, and the TB module https://github.com/cpoli/TB:

    * latticeTB
    * eigTB
    * plotTB
    * propTB

Feedback
-----------------

Please send comments or suggestions for improvement to cpoli83 at hotmail dot fr