w4_assignments.ipynb

Kernel: Python 3

import sys
sys.path.append('../code')
from init_mooc_nb import *
init_notebook()

Populated the namespace with:
np, matplotlib, kwant, holoviews, init_notebook, interact, display_html, plt, pf, SimpleNamespace, pprint_matrix, scientific_number, pretty_fmt_complex
from code/edx_components:
MoocVideo, PreprintReference, MoocDiscussion, MoocCheckboxesAssessment, MoocMultipleChoiceAssessment, MoocPeerAssessment, MoocSelfAssessment
from code/functions:
spectrum, hamiltonian_array, h_k, pauli

#Table of Contents

• Simulations: more Chern insulators

• Review assignment

Simulations: more Chern insulators

As usual, start by grabbing the notebooks of this week (w4_haldane). They are once again over here.

Yet another Chern insulator

One more tight binding of a Chern insulator that you can encounter in the wild is a regular square lattice with half a flux quantum of magnetic field per unit cell. If you made the Hofstadter butterfly assignment from the previous week, it's just in the middle of the butterfly. Half a flux quantum per unit cell means that the hoppings in one direction are purely imaginary, and different rows have alternating signs

$$t_y = t,\quad t_x = (-1)^y it.$$

This model has a dispersion very similar to graphene: it has two Dirac cones without a gap. Like graphene it also has two sites per unit cell, and sublattice symmetry.

Simulate this model. Think which parameters you need to add to it to make it a Chern insulator. Check that the edge states appear, and calculate the Berry curvature.

Back to the winding

Integration of Berry curvature is just another way to calculate the same quantity: the topological invariant. Verify that the winding of reflection phase gives the same results. To do that, make the pumping geometry out of a Chern insulator rolled into a cylinder, thread flux through it, and check that the topological invariant obtained through Berry curvature integration is the same as that obtained from winding.

We know that Berry curvature is concentrated close to the Dirac points. Do you notice anything similar for the pumped charge?

MoocSelfAssessment(due=3*7 + 2, review_due=4*7 + 2)

Now share your results:

MoocDiscussion('Labs', 'Chern insulators')

Review assignment

display_html(PreprintReference('1012.4723', description="The hunt for flat bands"))
display_html(PreprintReference('1409.6715', description="Making a Chern insulator more like quantum Hall effect"))
display_html(PreprintReference('1208.4579', description="A Chern insulator without lattice"))

Bonus: Find your own paper to review!

Do you know of another paper that fits into the topics of this week, and you think is good? Then you can get bonus points by reviewing that paper instead!

MoocPeerAssessment(due=3*7 + 2, review_due=4*7 + 2)

Do you have questions about what you read? Would you like to suggest other papers? Tell us:

MoocDiscussion("Reviews", "Chern insulators")