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.

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?

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Discussion entitled 'Chern insulators' is available in the online version of the course.

Titus Neupert, Luiz Santos, Claudio Chamon, Christopher Mudry

We present a simple prescription to flatten isolated Bloch bands with
non-zero Chern number. We first show that approximate flattening of bands with
non-zero Chern number is possible by tuning ratios of nearest-neighbor and
next-nearest neighbor hoppings in the Haldane model and, similarly, in the
chiral-pi-flux square lattice model. Then we show that perfect flattening can
be attained with further range hoppings that decrease exponentially with
distance. Finally, we add interactions to the model and present exact
diagonalization results for a small system at 1/3 filling that support (i) the
existence of a spectral gap, (ii) that the ground state is a topological state,
and (iii) that the Hall conductance is quantized.

Hint: The hunt for flat bands

Quantum anomalous Hall effect in magnetic topological insulators

The search for topologically non-trivial states of matter has become an
important goal for condensed matter physics. Here, we give a theoretical
introduction to the quantum anomalous Hall (QAH) effect based on magnetic
topological insulators in two-dimension (2D) and three-dimension (3D). In 2D
topological insulators, magnetic order breaks the symmetry between the
counter-propagating helical edge states, and as a result, the quantum spin Hall
effect can evolve into the QAH effect. In 3D, magnetic order opens up a gap for
the topological surface states, and chiral edge state has been predicted to
exist on the magnetic domain walls. We present the phase diagram in thin films
of a magnetic topological insulator and review the basic mechanism of
ferromagnetic order in magnetically doped topological insulators. We also
review the recent experimental observation of the QAH effect. We discuss more
recent theoretical work on the coexistence of the helical and chiral edge
states, multi-channel chiral edge states, the theory of the plateau transition,
and the thickness dependence in the QAH effect.

Hint: Making a Chern insulator more like quantum Hall effect

Motivated by new capabilities to realise artificial gauge fields in ultracold
atomic systems, and by their potential to access correlated topological phases
in lattice systems, we present a new strategy for designing topologically
non-trivial band structures. Our approach is simple and direct: it amounts to
considering tight-binding models directly in reciprocal space. These models
naturally cause atoms to experience highly uniform magnetic flux density and
lead to topological bands with very narrow dispersion, without fine-tuning of
parameters. Further, our construction immediately yields instances of optical
Chern lattices, as well as band structures of higher Chern number, |C|>1.

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Discussion entitled 'Chern insulators' is available in the online version of the course.