.. _sphx_glr_api_gallery_condensed_correlated:

Correlated Electrons
--------------------

Interacting-electron models: the BCS/Bogoliubov-de Gennes quasiparticle gap,
and Mott suppression of itinerant motion in the 1D Hubbard model.


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


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

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    <div class="sphx-glr-thumbcontainer" tooltip="Bardeen, Cooper, and Schrieffer explained superconductivity as a condensate of Cooper pairs bound by an effective attraction, however weak. Pairing opens a gap \Delta in the single-particle excitation spectrum: unlike the normal-state band |\xi(k)|, which touches zero at the Fermi points, the BCS quasiparticle band E(k) = \sqrt{\xi(k)^2 + |\Delta(k)|^2} never does. bdg_bcs_hamiltonian builds the Bogoliubov-de Gennes Hamiltonian in Nambu (particle-hole) space, and bdg_spectrum diagonalizes it across the Brillouin zone.">

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  .. image:: /api/gallery/condensed/correlated/images/thumb/sphx_glr_plot_bcs_gap_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/correlated/plot_bcs_gap`

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      <div class="sphx-glr-thumbnail-title">BCS Theory: The Superconducting Quasiparticle Gap</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Bednorz and Muller&#x27;s 1986 discovery of superconductivity above 30 K in La-Ba-Cu-O launched the search for the cuprate superconductors, whose undoped parent compounds are not metals at all but antiferromagnetic Mott insulators -- exactly the strong-coupling (U/t \gg 1), half-filled regime of the Hubbard model already implemented in hubbard_1d_exact_diagonalization. No dedicated cuprate (multi-band Emery, or t-J) model is implemented here, but the same single-band Hubbard Hamiltonian used for the 1963 Mott-transition example, pushed to half filling and strong coupling, already reproduces the qualitative signature: short-range Neel (antiferromagnetic) spin order, the arena any theory of cuprate pairing has to live in. hubbard_spin_correlations extracts \langle S_i^z S_j^z\rangle directly from the many-body ground state, using that S^z is diagonal in the occupation-number basis the exact-diagonalization solver already works in.">

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  .. image:: /api/gallery/condensed/correlated/images/thumb/sphx_glr_plot_cuprate_afm_correlations_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/correlated/plot_cuprate_afm_correlations`

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      <div class="sphx-glr-thumbnail-title">Cuprate Physics: Antiferromagnetism from the Strong-Coupling Hubbard Model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The Hubbard model is the minimal model of interacting electrons on a lattice: band electrons that hop with amplitude t, penalized by an onsite Coulomb repulsion U whenever two opposite-spin electrons share a site. As U/t \to \infty at half filling, double occupancy is suppressed entirely and the itinerant metal is driven into a Mott insulator of localized moments. hubbard_1d_exact_diagonalization builds and diagonalizes the full Fock-space Hamiltonian in a fixed (n_\uparrow, n_\downarrow) sector for small clusters, exposing this crossover directly in the ground-state energy.">

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  .. image:: /api/gallery/condensed/correlated/images/thumb/sphx_glr_plot_hubbard_mott_transition_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/correlated/plot_hubbard_mott_transition`

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      <div class="sphx-glr-thumbnail-title">The Hubbard Model: Mott Suppression of Itinerant Motion</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/condensed/correlated/plot_bcs_gap
   /api/gallery/condensed/correlated/plot_cuprate_afm_correlations
   /api/gallery/condensed/correlated/plot_hubbard_mott_transition