.. _sphx_glr_api_gallery_condensed_tight_binding:

Tight-Binding
-------------

The generic Bloch-Hamiltonian machinery: recovering the exact tight-binding
dispersion from a real-space chain, a Hofstadter-like spectrum from
Peierls-substituted Landau levels, and a two-orbital Slater-Koster LCAO band
structure.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="Felix Bloch&#x27;s theorem turns the Schrodinger equation for an electron in an infinite periodic crystal into a finite eigenvalue problem H(k) u_{nk} = E_n(k) u_{nk} at each crystal momentum k, periodic on the Brillouin zone. Hamiltonian builds H(k) by Bloch-summing real-space hoppings, and bands diagonalizes it -- reproducing the monatomic chain&#x27;s exact dispersion E(k) = -2t\cos k with no continuum limit or approximation involved.">

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  .. image:: /api/gallery/condensed/tight_binding/images/thumb/sphx_glr_plot_bloch_bands_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/tight_binding/plot_bloch_bands`

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      <div class="sphx-glr-thumbnail-title">Bloch's Theorem: From an Infinite Chain to H(k)</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Lev Landau showed that a charged particle in a uniform magnetic field has a kinetic-energy spectrum that collapses into discrete, macroscopically degenerate levels E_n = \hbar\omega_c(n + 1/2). On a lattice, the same continuum physics is reached through the Peierls substitution: each bond acquires a phase proportional to the flux threading the plaquettes it borders. apply_peierls_phase implements exactly this, turning a uniform flux into a Hofstadter-like spectrum whose low-energy states bunch into near-degenerate groups -- the lattice remnant of the continuum Landau tower.">

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  .. image:: /api/gallery/condensed/tight_binding/images/thumb/sphx_glr_plot_peierls_landau_levels_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/tight_binding/plot_peierls_landau_levels`

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      <div class="sphx-glr-thumbnail-title">Peierls Substitution: A Hofstadter-Like Spectrum from Landau's Physics</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Slater and Koster showed how to build realistic band structures from a minimal empirical basis: a linear combination of atomic orbitals (LCAO), with hopping matrix elements between orbitals treated as fitting parameters rather than computed from first principles. Lattice and Hamiltonian are a direct, general-purpose realization of this philosophy: declare orbitals, declare empirical hoppings between them, and Bloch-sum the result into a multi-band H(k).">

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  .. image:: /api/gallery/condensed/tight_binding/images/thumb/sphx_glr_plot_slater_koster_bands_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/tight_binding/plot_slater_koster_bands`

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      <div class="sphx-glr-thumbnail-title">Slater-Koster LCAO: Empirical Multi-Orbital Band Structure</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /api/gallery/condensed/tight_binding/plot_bloch_bands
   /api/gallery/condensed/tight_binding/plot_peierls_landau_levels
   /api/gallery/condensed/tight_binding/plot_slater_koster_bands