.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/tight_binding/plot_square_lattice_bands.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_tight_binding_plot_square_lattice_bands.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_tight_binding_plot_square_lattice_bands.py:


The Square Lattice: Bloch's Theorem in its Simplest Form
============================================================

The smallest complete illustration of Bloch's theorem: one orbital per
unit cell on a square lattice, nearest-neighbor hopping only. With a
single orbital the Bloch Hamiltonian :math:`H(\mathbf{k})` is a
:math:`1\times1` matrix, so "diagonalizing" it is nothing but reading it
off, and the whole band structure is one closed-form expression,

.. math::

   E(\mathbf{k}) = -2t\left(\cos k_x a + \cos k_y a\right)\, ,

periodic on the Brillouin zone torus exactly as Bloch's theorem
requires. Everything the rest of this gallery does to richer models --
paths through high-symmetry points, Brillouin-zone meshes, densities of
states -- is visible here against an answer that can be checked by hand.

.. GENERATED FROM PYTHON SOURCE LINES 20-36

.. code-block:: Python

    import numpy as np
    import matplotlib.pyplot as plt

    from tbkit.lattice import Lattice
    from tbkit.system import System
    from tbkit.plot import Plot
    from tbkit.kspace import KSpace, reciprocal_vectors
    from tbkit.dos import density_of_states


    a = 1.
    t = 1.    # H = -t sum_<ij> c_i^dag c_j, so the band minimum sits at Gamma
    unit_cell = [{'tag': 'a', 'r0': (0., 0.)}]
    prim_vec = [(a, 0.), (0., a)]









.. GENERATED FROM PYTHON SOURCE LINES 37-42

The lattice
--------------
One orbital per unit cell, repeated by :math:`\mathbf{a}_1=(a,0)` and
:math:`\mathbf{a}_2=(0,a)`. Each site has four nearest neighbors, all
at the same distance, all with the same amplitude.

.. GENERATED FROM PYTHON SOURCE LINES 42-49

.. code-block:: Python


    lat = Lattice(unit_cell=unit_cell, prim_vec=prim_vec)
    lat.get_lattice(n1=6, n2=6)
    vis = System(lat)
    vis.set_hopping([{'n': 1, 't': t}])
    fig_lat = Plot(vis).lattice(plt_hop=True, ms=11, figsize=(4.5, 4.5))




.. image-sg:: /api/gallery/tight_binding/images/sphx_glr_plot_square_lattice_bands_001.png
   :alt: plot square lattice bands
   :srcset: /api/gallery/tight_binding/images/sphx_glr_plot_square_lattice_bands_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 50-53

The reciprocal lattice is square too, with
:math:`\mathbf{b}_i = (2\pi/a)\hat{\mathbf{e}}_i`, so the Brillouin
zone is the square :math:`[-\pi/a, \pi/a]^2`.

.. GENERATED FROM PYTHON SOURCE LINES 53-59

.. code-block:: Python


    b1, b2 = reciprocal_vectors(prim_vec)
    print('b1 = ({:.4f}, {:.4f}),  b2 = ({:.4f}, {:.4f})'.format(*b1, *b2))
    assert np.allclose(b1, (2*np.pi/a, 0.))
    assert np.allclose(b2, (0., 2*np.pi/a))





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    b1 = (6.2832, -0.0000),  b2 = (-0.0000, 6.2832)




.. GENERATED FROM PYTHON SOURCE LINES 60-68

The Bloch Hamiltonian
---------------------------
:meth:`~tbkit.kspace.KSpace.set_hopping` takes one representative of
each hopping -- here the two bonds to the right and upward neighbor,
labelled by the lattice vector :math:`\mathbf{R}` that separates the
cells -- and adds each Hermitian conjugate itself. Bloch-summing the
four resulting bonds gives
:math:`H(\mathbf{k}) = -2t(\cos k_x a + \cos k_y a)` directly.

.. GENERATED FROM PYTHON SOURCE LINES 68-84

.. code-block:: Python


    sq = KSpace(lat)
    sq.set_hopping([{'i': 0, 'j': 0, 'R': (1, 0), 't': -t},
                          {'i': 0, 'j': 0, 'R': (0, 1), 't': -t}])


    def band(kx, ky):
        '''The closed form, to check the Bloch sum against.'''
        return -2. * t * (np.cos(kx * a) + np.cos(ky * a))


    rng = np.random.default_rng(0)
    ks_test = rng.uniform(-np.pi/a, np.pi/a, size=(200, 2))
    assert np.allclose(sq.get_bands(ks_test)[:, 0], band(ks_test[:, 0], ks_test[:, 1]))
    print('Bloch sum reproduces -2t(cos kx a + cos ky a) at 200 random k-points.')





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Bloch sum reproduces -2t(cos kx a + cos ky a) at 200 random k-points.




.. GENERATED FROM PYTHON SOURCE LINES 85-94

Bands along Gamma - X - M - Gamma
------------------------------------------
The standard path through the high-symmetry points of the square
Brillouin zone. The band runs from :math:`-4t` at :math:`\Gamma`
(:math:`\mathbf{k}=0`, every bond bonding) up to :math:`+4t` at
:math:`M=(\pi,\pi)/a` (every bond antibonding), crossing :math:`E=0` at
the zone edge :math:`X=(\pi,0)/a`: a total bandwidth of :math:`8t`,
which for a lattice of coordination number :math:`z` is just
:math:`2zt`.

.. GENERATED FROM PYTHON SOURCE LINES 94-106

.. code-block:: Python


    G, X, M = (0., 0.), (np.pi/a, 0.), (np.pi/a, np.pi/a)
    ks_dist, en = sq.k_path([G, X, M, G], nk=120)
    fig_bands = sq.plot_bands(node_labels=[r'$\Gamma$', 'X', 'M', r'$\Gamma$'],
                                          figsize=(6.5, 4.5))

    for label, k, expected in [('Gamma', G, -4.*t), ('X', X, 0.), ('M', M, 4.*t)]:
        value = float(sq.get_bands([k])[0, 0])
        print('E({:<5}) = {:+.4f}  (expect {:+.4f})'.format(label, value, expected))
        assert np.isclose(value, expected)
    assert np.isclose(en.max() - en.min(), 8.*t)




.. image-sg:: /api/gallery/tight_binding/images/sphx_glr_plot_square_lattice_bands_002.png
   :alt: plot square lattice bands
   :srcset: /api/gallery/tight_binding/images/sphx_glr_plot_square_lattice_bands_002.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    E(Gamma) = -4.0000  (expect -4.0000)
    E(X    ) = +0.0000  (expect +0.0000)
    E(M    ) = +4.0000  (expect +4.0000)




.. GENERATED FROM PYTHON SOURCE LINES 107-116

The band over the whole Brillouin zone
------------------------------------------------
A single smooth sheet, periodic in both directions. Its minimum is the
zone centre and its maxima are the four zone corners; the
:math:`E=0` contour is the square joining the four X points. That
contour is the half-filled Fermi surface, and it is *perfectly nested*:
the vector :math:`\mathbf{Q}=(\pi,\pi)/a` maps each of its edges onto
the opposite one, which is why the half-filled square-lattice Hubbard
model orders antiferromagnetically at arbitrarily weak coupling.

.. GENERATED FROM PYTHON SOURCE LINES 116-136

.. code-block:: Python


    nk = 201
    kx = np.linspace(-np.pi/a, np.pi/a, nk)
    kxx, kyy = np.meshgrid(kx, kx, indexing='ij')
    e_grid_2d = sq.get_bands(np.column_stack([kxx.ravel(), kyy.ravel()]))[:, 0]
    e_grid_2d = e_grid_2d.reshape(nk, nk)

    fig_bz, ax_bz = plt.subplots(figsize=(5.6, 4.6))
    mesh = ax_bz.pcolormesh(kxx, kyy, e_grid_2d, cmap='viridis', shading='auto')
    ax_bz.contour(kxx, kyy, e_grid_2d, levels=[0.], colors='w', linewidths=2.5)
    ax_bz.set_aspect('equal')
    ax_bz.set_xlabel('$k_x a$')
    ax_bz.set_ylabel('$k_y a$')
    ax_bz.set_title(r'$E(\mathbf{k})$; white: the nested $E=0$ Fermi surface')
    fig_bz.colorbar(mesh, ax=ax_bz, label='$E/t$')
    fig_bz.tight_layout()

    assert np.isclose(e_grid_2d.min(), -4.*t)     # Gamma
    assert np.isclose(e_grid_2d.max(), 4.*t)      # the four zone corners




.. image-sg:: /api/gallery/tight_binding/images/sphx_glr_plot_square_lattice_bands_003.png
   :alt: $E(\mathbf{k})$; white: the nested $E=0$ Fermi surface
   :srcset: /api/gallery/tight_binding/images/sphx_glr_plot_square_lattice_bands_003.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 137-145

Density of states: the van Hove singularity
------------------------------------------------------
The saddle point of :math:`E(\mathbf{k})` at X -- a maximum along one
zone direction and a minimum along the other -- makes the density of
states diverge logarithmically at :math:`E=0`, the van Hove
singularity of the 2D square lattice. Broadening cuts the divergence
off at a finite peak, but the pile-up of states in the middle of the
band, and the sharp edges at :math:`\pm4t`, are unmistakable.

.. GENERATED FROM PYTHON SOURCE LINES 145-168

.. code-block:: Python


    en_mesh = sq.mesh_bands(nk=400)
    energies, dos = density_of_states(en_mesh, broadening=0.05,
                                                 e_grid=np.linspace(-5., 5., 601))

    fig_dos, ax_dos = plt.subplots(figsize=(6.5, 4.))
    ax_dos.plot(energies, dos, lw=2.)
    ax_dos.axvline(0., color='k', ls='--', lw=1.)
    ax_dos.set_xlabel('$E/t$')
    ax_dos.set_ylabel(r'$\rho(E)$')
    ax_dos.set_title('van Hove singularity at the band centre')
    fig_dos.tight_layout()

    peak = energies[np.argmax(dos)]
    print('DOS peaks at E = {:+.3f} (expect 0, the van Hove energy).'.format(peak))
    assert abs(peak) < 0.1
    # and it really is a peak, not a plateau: at this broadening the centre
    # carries about three times the weight of a point midway out towards the
    # band edge (the unbroadened divergence is logarithmic, so the height of
    # the peak itself is set by `broadening`, not by the physics).
    ratio = dos[np.argmin(np.abs(energies))] / dos[np.argmin(np.abs(energies - 2.))]
    print('rho(0)/rho(2t) = {:.2f}'.format(ratio))
    assert ratio > 2.5



.. image-sg:: /api/gallery/tight_binding/images/sphx_glr_plot_square_lattice_bands_004.png
   :alt: van Hove singularity at the band centre
   :srcset: /api/gallery/tight_binding/images/sphx_glr_plot_square_lattice_bands_004.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    DOS peaks at E = +0.000 (expect 0, the van Hove energy).
    rho(0)/rho(2t) = 2.97





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 3.293 seconds)


.. _sphx_glr_download_api_gallery_tight_binding_plot_square_lattice_bands.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_square_lattice_bands.ipynb <plot_square_lattice_bands.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_square_lattice_bands.py <plot_square_lattice_bands.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_square_lattice_bands.zip <plot_square_lattice_bands.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

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