.. _sphx_glr_api_gallery_correlations:

Correlations
------------

Interacting electrons, in the Hartree-Fock mean-field approximation of
the Hubbard model: the magnetism of graphene's zigzag edges and of
triangular nanoislands, Lieb's theorem and the ferrimagnet of the Lieb
lattice, and flat-band ferromagnetism (Nagaoka, Mielke, Tasaki). The
non-collinear mean field finds the 120-degree order of the frustrated
triangular lattice.


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    <div class="sphx-glr-thumbcontainer" tooltip="Itinerant ferromagnetism is hard to get from the Hubbard model: the repulsion U favours aligned spins, which cannot share a site, but aligning them costs kinetic energy. Three exact results say when the spins align fully anyway.">

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  .. image:: /api/gallery/correlations/images/thumb/sphx_glr_plot_flat_band_ferromagnetism_thumb.png
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  :doc:`/api/gallery/correlations/plot_flat_band_ferromagnetism`

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      <div class="sphx-glr-thumbnail-title">Flat-Band Ferromagnetism: Nagaoka, Mielke and Tasaki</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="J. Hubbard (1963) -- together with Gutzwiller and Kanamori, the same year -- reduced the problem of interacting electrons in narrow bands to its bare bones: hopping between neighbouring sites, and a repulsion U between two electrons on the same site,">

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  .. image:: /api/gallery/correlations/images/thumb/sphx_glr_plot_hubbard_edge_magnetism_thumb.png
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  :doc:`/api/gallery/correlations/plot_hubbard_edge_magnetism`

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      <div class="sphx-glr-thumbnail-title">The Hubbard Model: Magnetism of Graphene's Zigzag Edges</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="E. Lieb proved in 1989 one of the few exact results on the Hubbard model: on a bipartite lattice at half filling, for any repulsion U &gt; 0, the ground state has total spin">

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  .. image:: /api/gallery/correlations/images/thumb/sphx_glr_plot_lieb_theorem_thumb.png
    :alt:

  :doc:`/api/gallery/correlations/plot_lieb_theorem`

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      <div class="sphx-glr-thumbnail-title">Lieb's Theorem: The Ferrimagnet of the Lieb Lattice</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="On a triangle, three antiparallel spins cannot all be satisfied: the antiferromagnet is frustrated. G. Wannier (1950) found that the Ising triangular antiferromagnet never orders; P. W. Anderson (1973) proposed that the quantum Heisenberg one is a spin liquid; the answer, settled numerically in the late 1980s (Huse and Elser 1988), is a compromise: the spins order non-collinearly, at 120 degrees from each other on the three sublattices, with zero total moment.">

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  .. image:: /api/gallery/correlations/images/thumb/sphx_glr_plot_noncollinear_120_degree_order_thumb.png
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  :doc:`/api/gallery/correlations/plot_noncollinear_120_degree_order`

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      <div class="sphx-glr-thumbnail-title">Frustrated Magnetism: the 120-Degree Order of the Triangular Hubbard Model</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="J. Fernández-Rossier and J. J. Palacios (2007) asked which graphene flakes are magnets, and found the answer in their shape. A triangle with zigzag edges has all three edges on the same sublattice, A, and more A than B sites. A bipartite Hamiltonian then has (at least) N_Z = |A| - |B| states at exactly zero energy, all on sublattice A and concentrated on the edges. At half filling they hold N_Z electrons: like the open shell of an atom, they are filled with parallel spins (Hund&#x27;s rule), and the flake carries a spin S = N_Z/2 -- for any repulsion U, however small, while a hexagonal flake needs U above a threshold. This is Lieb&#x27;s theorem made concrete: the moment grows with the size of the triangle. Triangular nanographenes (the &quot;triangulenes&quot;) were synthesized and seen to be magnetic in 2017-2021.">

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  .. image:: /api/gallery/correlations/images/thumb/sphx_glr_plot_triangular_graphene_nanoislands_thumb.png
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  :doc:`/api/gallery/correlations/plot_triangular_graphene_nanoislands`

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      <div class="sphx-glr-thumbnail-title">Magnetism of Triangular Graphene Nanoislands</div>
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.. toctree::
   :hidden:

   /api/gallery/correlations/plot_flat_band_ferromagnetism
   /api/gallery/correlations/plot_hubbard_edge_magnetism
   /api/gallery/correlations/plot_lieb_theorem
   /api/gallery/correlations/plot_noncollinear_120_degree_order
   /api/gallery/correlations/plot_triangular_graphene_nanoislands