Topics for self-study

Simulations: more Chern insulators

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?

Now share your results:

Review assignment

Fractional quantum Hall states at zero magnetic field (arXiv:1012.4723)

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 (arXiv:1409.6715)

Jing Wang, Biao Lian, Shou-Cheng Zhang

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.

Designing Topological Bands in Reciprocal Space (arXiv:1208.4579)

N. R. Cooper, R. Moessner

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.

Hint: 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!