.. _sphx_glr_api_gallery_condensed_topology:

Topology
--------

Topological band theory: SSH edge states and the Zak phase, exactly
quantized Chern numbers (TKNN), the Haldane model's zero-net-flux Chern
insulator and its chiral edge states, unpaired Majorana zero modes in the
Kitaev chain, graphene's massless Dirac cone, and the Kane-Mele/BHZ
:math:`\mathbb{Z}_2` topological insulators.


.. raw:: html

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


.. raw:: html

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

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Fu, Kane, and Mele&#x27;s 2007 extension of the 2D Z2 classification to three dimensions predicted a bulk insulator whose surface hosts a single, gapless Dirac cone, protected by time-reversal symmetry alone -- confirmed directly by ARPES on Bi2Se3 and Bi2Te3 in 2008-2009. topological_insulator_3d_slab_hamiltonian opens the minimal 4-band lattice model along one direction, exposing exactly this surface state: pinned to zero energy at the surface Brillouin zone center, localized on a single outer layer, and dispersing linearly away from it.">

.. only:: html

  .. image:: /api/gallery/condensed/topology/images/thumb/sphx_glr_plot_3d_ti_surface_states_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/topology/plot_3d_ti_surface_states`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The 3D Topological Insulator: A Single Surface Dirac Cone</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Thouless, Kohmoto, Nightingale, and den Nijs showed that the quantized Hall conductance is a topological invariant: the integral of the Berry curvature of the occupied Bloch bands over the Brillouin zone, now called the (first) Chern number C. Because C can only change when a bulk gap closes, it is exact and disorder-independent -- the moment band theory became topological band theory. compute_berry_curvature and compute_chern_number implement the Fukui-Hatsugai-Suzuki lattice discretization of this integral, returning exactly quantized integers for any gapped Bloch Hamiltonian.">

.. only:: html

  .. image:: /api/gallery/condensed/topology/images/thumb/sphx_glr_plot_chern_number_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/topology/plot_chern_number`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The TKNN Invariant: Exactly Quantized Chern Numbers</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Geim and Novoselov&#x27;s isolation of a single atomic layer of graphite produced the first genuinely 2D crystal -- and its low-energy quasiparticles obey a massless relativistic (Dirac) equation rather than the ordinary Schrodinger equation, turning a tabletop material into a laboratory for relativistic quantum phenomena. graphene_hamiltonian reproduces this linear dispersion near the honeycomb lattice&#x27;s Brillouin zone corners.">

.. only:: html

  .. image:: /api/gallery/condensed/topology/images/thumb/sphx_glr_plot_graphene_dirac_cone_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/topology/plot_graphene_dirac_cone`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Graphene: A Massless Dirac Cone on a Honeycomb Lattice</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="plot_graphene_dirac_cone looks at graphene&#x27;s bulk Bloch spectrum -- a linear Dirac cone with no gap. Cutting a finite flake out of the honeycomb lattice along its two primitive-lattice directions produces the classic &quot;zigzag&quot; edge termination, and with it a macroscopically degenerate band of zero-energy states with no bulk counterpart, localized entirely on the flake&#x27;s boundary and almost entirely on a single sublattice. Unlike the SSH or Haldane edge states elsewhere in this gallery, these zero modes are not protected by a bulk topological invariant -- they are a purely geometric consequence of the zigzag edge cut, and vanish under an armchair termination instead.">

.. only:: html

  .. image:: /api/gallery/condensed/topology/images/thumb/sphx_glr_plot_graphene_zigzag_flake_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/topology/plot_graphene_zigzag_flake`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Graphene: Zero-Energy Edge States on a Zigzag-Terminated Flake</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="plot_haldane_phase_transition exposes the Haldane model&#x27;s chiral edge states on a ribbon -- periodic in-plane, open along one crystallographic direction. That is the easiest cut to compute, but it leaves open whether the edge state depends on cutting along a lattice direction the way graphene&#x27;s zigzag zero modes do (plot_graphene_zigzag_flake). It does not: a nonzero Chern number guarantees a chiral state on any boundary of the sample, however it is shaped. This example carves a circular disk out of the infinite lattice -- a boundary with no crystallographic meaning at all -- using build_finite_cluster&#x27;s keep predicate, and shows the in-gap states still hug that boundary in the topological phase, and fail to in the trivial one.">

.. only:: html

  .. image:: /api/gallery/condensed/topology/images/thumb/sphx_glr_plot_haldane_edge_state_map_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/topology/plot_haldane_edge_state_map`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Haldane Model: a Chiral Edge State on an Arbitrary Boundary</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Duncan Haldane showed that the integer quantum Hall effect does not require a net magnetic field at all. Threading complex second-neighbor hopping t_2 e^{i\phi} through a honeycomb lattice, in a pattern with zero net flux per unit cell but nonzero local (loop) curvature, produces a Chern insulator with C=\pm1 -- the first quantum anomalous Hall model, and the direct template for the \mathbb{Z}_2 topological insulators discovered two decades later. haldane_model gives the Bloch Hamiltonian used to track the bulk Chern number across the phase transition, and haldane_lattice_hamiltonian builds the same model in real space to expose the chiral edge states that bulk-boundary correspondence predicts.">

.. only:: html

  .. image:: /api/gallery/condensed/topology/images/thumb/sphx_glr_plot_haldane_phase_transition_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/topology/plot_haldane_phase_transition`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Haldane Model: A Chern Insulator with Zero Net Flux</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Von Klitzing, Dorda, and Pepper&#x27;s 1980 discovery -- a two-dimensional electron gas in a strong magnetic field develops a Hall conductance quantized to extraordinary precision, \sigma_{xy} = \nu\,e^2/h with \nu an exact integer -- is reproduced here on its lattice (Bloch) incarnation, the Harper-Hofstadter model (~physicskit.condensed.models.harper_hofstadter_hamiltonian): a square lattice threaded by a uniform flux p/q per plaquette. At flux 1/3 the spectrum splits into three magnetic sub-bands; feeding each one, in turn, into compute_chern_number (the same TKNN machinery used for the Haldane model) gives the exact integer Chern number of that band, and summing the Chern numbers of every filled band below a gap gives that gap&#x27;s quantized Hall conductance directly -- the numerical content of the TKNN formula that explains why von Klitzing&#x27;s measurement came out an integer at all.">

.. only:: html

  .. image:: /api/gallery/condensed/topology/images/thumb/sphx_glr_plot_iqhe_hall_conductance_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/topology/plot_iqhe_hall_conductance`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Integer Quantum Hall Effect: Quantized Hall Conductance from Chern Numbers</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="plot_z2_topological_insulator computes the bulk \mathbb{Z}_2 invariant of the Kane-Mele model directly from its Bloch Hamiltonian. This example instead looks at what that invariant predicts on a finite sample. Whenever the Rashba term vanishes, s_z is conserved and kane_mele_hamiltonian is exactly block-diagonal in spin -- two independent Haldane models with opposite next-nearest-neighbor phase, \phi and -\phi. Building each spin&#x27;s finite flake separately with build_finite_cluster (the same disk-shaped cut used in plot_haldane_edge_state_map) shows both spins independently host a boundary-hugging state at the same real-space locations -- a Kramers pair -- while a momentum-resolved ribbon dispersion shows the two spins cross the gap with opposite group velocity, the helical counter-propagation that makes the quantum spin Hall edge robust against time-reversal-preserving disorder.">

.. only:: html

  .. image:: /api/gallery/condensed/topology/images/thumb/sphx_glr_plot_kane_mele_edge_flake_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/topology/plot_kane_mele_edge_flake`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Kane-Mele: Helical Edge States from Two Time-Reversed Haldane Copies</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Alexei Kitaev showed that a 1D spinless p-wave superconductor hosts unpaired Majorana zero modes -- particles that are their own antiparticles -- localized at the two ends of an open chain, whenever the chemical potential satisfies |\mu| &lt; 2t. A pair of spatially separated Majoranas encodes one nonlocal fermionic degree of freedom immune to local perturbations, making the Kitaev chain the founding proposal for topologically protected quantum computation. kitaev_chain_hamiltonian gives the bulk (periodic) BdG Bloch Hamiltonian used to confirm the bulk gap stays open, and kitaev_chain_bdg_real_space builds the open chain that exposes the Majorana end modes directly in the spectrum -- plotted below as a real-space structure so the two end modes are visible directly on the chain rather than as an abstract site index.">

.. only:: html

  .. image:: /api/gallery/condensed/topology/images/thumb/sphx_glr_plot_kitaev_majorana_modes_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/topology/plot_kitaev_majorana_modes`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Kitaev Chain: Unpaired Majorana Zero Modes</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Konig, Molenkamp, and coworkers measured a quantized two-terminal conductance of 2e^2/h in HgTe/CdTe quantum wells in 2007 -- the first experimentally realized topological insulator, and the direct confirmation of the band-inverted BHZ model. Opening the BHZ ribbon along one direction with bhz_ribbon_hamiltonian exposes exactly the pair of helical, counter-propagating edge states carrying that dissipationless conductance: gapless at k_x=0, spin-locked, and immune to backscattering by nonmagnetic disorder.">

.. only:: html

  .. image:: /api/gallery/condensed/topology/images/thumb/sphx_glr_plot_qsh_edge_states_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/topology/plot_qsh_edge_states`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Quantum Spin Hall Effect: Konig et al.'s Helical Edge States</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Su, Schrieffer, and Heeger showed that a dimerized chain -- alternating intracell hopping v and intercell hopping w -- is topologically nontrivial whenever v &lt; w: the bulk Zak phase is quantized to \pi, and cutting the chain open exposes a pair of protected, exponentially localized zero-energy states, one at each end. This is the earliest example of the bulk-boundary correspondence. ssh_hamiltonian gives the closed-form Bloch Hamiltonian, zak_phase computes the bulk invariant from it, and ssh_lattice_hamiltonian together with build_finite_cluster exposes the boundary modes on a finite, open chain -- and lets us plot the chain&#x27;s actual real-space structure rather than an abstract site-index axis.">

.. only:: html

  .. image:: /api/gallery/condensed/topology/images/thumb/sphx_glr_plot_ssh_edge_states_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/topology/plot_ssh_edge_states`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The SSH Model: Zak Phase and Protected Edge States</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Schnyder, Ryu, Furusaki, and Ludwig, and independently Kitaev, showed that every noninteracting Bloch or Bogoliubov-de Gennes Hamiltonian falls into one of just ten symmetry classes, fixed by whether time-reversal and particle-hole symmetry are present, the sign each squares to, and whether their product enforces a chiral symmetry -- and that the allowed topological invariant (\mathbb{Z}, \mathbb{Z}_2, or none) in a given spatial dimension depends only on which of the ten classes a Hamiltonian belongs to. Three of this package&#x27;s own topological models, already built for their own separate breakthroughs, are three different rows of that table:">

.. only:: html

  .. image:: /api/gallery/condensed/topology/images/thumb/sphx_glr_plot_tenfold_way_classification_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/topology/plot_tenfold_way_classification`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Tenfold Way: Three Topological Invariants, One Classification Scheme</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Wan, Turner, Vishwanath, and Savrasov predicted that breaking inversion or time-reversal symmetry in a 3D Dirac material splits each doubly-degenerate Dirac point into a pair of nondegenerate Weyl nodes -- momentum-space sources and sinks of Berry curvature carrying opposite chirality, the direct 3D generalization of the 2D TKNN invariant. A surface hosts open, non-closed &quot;Fermi arcs&quot; of surface states connecting the surface projections of opposite-chirality bulk nodes, later confirmed by ARPES on TaAs. weyl_semimetal_hamiltonian implements the minimal two-band cubic-lattice model with this physics,">

.. only:: html

  .. image:: /api/gallery/condensed/topology/images/thumb/sphx_glr_plot_weyl_semimetal_fermi_arcs_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/topology/plot_weyl_semimetal_fermi_arcs`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Weyl Semimetals: Momentum-Space Monopoles and Fermi Arcs</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Kane and Mele showed that adding intrinsic spin-orbit coupling to graphene -- two time-reversed copies of Haldane&#x27;s model, one per spin -- produces a time-reversal-symmetric insulator with a new, \mathbb{Z}_2-valued topological invariant protecting helical, counter-propagating edge states: the quantum spin Hall effect. Bernevig, Hughes, and Zhang then predicted the same physics in HgTe/CdTe quantum wells (the BHZ model), later observed experimentally -- the first realized topological insulator. kane_mele_hamiltonian and bhz_hamiltonian provide both microscopic realizations, and z2_invariant extracts the \mathbb{Z}_2 invariant from each via the spin-Chern-number reduction valid whenever s_z is conserved.">

.. only:: html

  .. image:: /api/gallery/condensed/topology/images/thumb/sphx_glr_plot_z2_topological_insulator_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/topology/plot_z2_topological_insulator`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Kane-Mele and BHZ: Z2 Topological Insulators</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/condensed/topology/plot_3d_ti_surface_states
   /api/gallery/condensed/topology/plot_chern_number
   /api/gallery/condensed/topology/plot_graphene_dirac_cone
   /api/gallery/condensed/topology/plot_graphene_zigzag_flake
   /api/gallery/condensed/topology/plot_haldane_edge_state_map
   /api/gallery/condensed/topology/plot_haldane_phase_transition
   /api/gallery/condensed/topology/plot_iqhe_hall_conductance
   /api/gallery/condensed/topology/plot_kane_mele_edge_flake
   /api/gallery/condensed/topology/plot_kitaev_majorana_modes
   /api/gallery/condensed/topology/plot_qsh_edge_states
   /api/gallery/condensed/topology/plot_ssh_edge_states
   /api/gallery/condensed/topology/plot_tenfold_way_classification
   /api/gallery/condensed/topology/plot_weyl_semimetal_fermi_arcs
   /api/gallery/condensed/topology/plot_z2_topological_insulator