.. _sphx_glr_api_gallery_floquet:

Floquet
-------

Periodically driven lattices: quasienergies, the Floquet Hamiltonian,
dynamic localization and coherent destruction of tunnelling, the
photovoltaic Hall effect of graphene in circularly polarized light, a
Chern insulator engineered with that light, and anomalous Floquet
phases, whose edge states the winding number counts while every Chern
number vanishes.


.. 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="In a static insulator the chiral edge states in a gap are counted by the Chern numbers of the bands below it. Kitagawa, Berg, Rudner and Demler (2010) found that a periodically driven lattice escapes this rule: since quasienergies are only defined modulo \omega = 2\pi/T, edge states can wind around the whole Floquet zone, and a driven system can have chiral edge states although every Floquet band has C = 0. Rudner, Lindner, Berg and Levin (2013) built the minimal example -- a five-step drive of the bipartite square lattice in which the hopping J is on along one bond direction \mathbf{b}_s per step (right, up, left, down, then a pause), all under a sublattice potential \delta_{AB} = \epsilon_A - \epsilon_B -- and the invariant that counts these states: the winding number">

.. only:: html

  .. image:: /api/gallery/floquet/images/thumb/sphx_glr_plot_anomalous_floquet_phases_thumb.png
    :alt:

  :doc:`/api/gallery/floquet/plot_anomalous_floquet_phases`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Anomalous Floquet Topological Phases: Edge States With Zero Chern Numbers</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Dunlap and Kenkre (1986) found that an AC field F\cos\omega t on a chain does not simply shake its electrons: averaged over a period, it renormalizes the hopping,">

.. only:: html

  .. image:: /api/gallery/floquet/images/thumb/sphx_glr_plot_dynamic_localization_thumb.png
    :alt:

  :doc:`/api/gallery/floquet/plot_dynamic_localization`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Dynamic Localization and Coherent Destruction of Tunnelling</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="T. Oka and H. Aoki (2009) predicted that graphene illuminated by circularly polarized light becomes a Hall insulator; Kitagawa, Oka, Brataas, Fu and Demler (2011) showed it is a Floquet realization of Haldane&#x27;s model. A time-periodic drive H(t+T) = H(t) acts, stroboscopically, as the static Floquet Hamiltonian H_F, e^{-iH_FT} = \mathcal{T}e^{-i\int_0^TH(t)dt}. Light enters the hoppings through the Peierls phase of its vector potential, t_{ij} \to t_{ij}e^{i\mathbf{A}(t)\cdot\mathbf{d}_{ij}} with \mathbf{A}(t) = A_0(\cos\omega t, \sin\omega t). At high frequency,">

.. only:: html

  .. image:: /api/gallery/floquet/images/thumb/sphx_glr_plot_floquet_chern_insulator_thumb.png
    :alt:

  :doc:`/api/gallery/floquet/plot_floquet_chern_insulator`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Floquet Engineering: A Chern Insulator From Circularly Polarized Light</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Oka and Aoki (2009) asked what a DC field does to graphene that is illuminated by circularly polarized light, and found a Hall current with no magnetic field: the photovoltaic Hall effect. An electron near a Dirac point is driven around a circle in momentum space once per period, and the geometric (Aharonov-Anandan) phase it picks up acts as a Berry curvature of the Floquet states |u_\alpha(\mathbf{k})\rangle. The Hall conductivity follows from the Floquet extension of the Kubo formula,">

.. only:: html

  .. image:: /api/gallery/floquet/images/thumb/sphx_glr_plot_photovoltaic_hall_effect_thumb.png
    :alt:

  :doc:`/api/gallery/floquet/plot_photovoltaic_hall_effect`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Photovoltaic Hall Effect in Graphene</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/floquet/plot_anomalous_floquet_phases
   /api/gallery/floquet/plot_dynamic_localization
   /api/gallery/floquet/plot_floquet_chern_insulator
   /api/gallery/floquet/plot_photovoltaic_hall_effect