Simulations: Gaps and invariants

As usual, start by grabbing the notebooks of this week (w8_general). They are once again over here.

A different analytic continuation

You have learned how to map a winding number onto counting the zeros of an eigenproblem in a complex plane. This can be applied to other symmetry classes as well.

Let's try to calculate the invariant in the 1D symmetry class DIII. If you look in the table, you'll see it's the same invariant as the scattering invariant we've used for the quantum spin Hall effect,

$$ Q = \frac{\textrm{Pf } h(k=0)}{\textrm{Pf } h(k=\pi)} \sqrt{\frac{\det h(k=\pi)}{\det h(k=0)}} $$

In this paper (around Eq. 4.13), have a look at how to use analytic continuation to calculate the analytic continuation of $\sqrt{h}$, and implement the calculation of this invariant without numerical integration, like we did before.

In order to test your invariant, you'll need a topologically non-trivial system in this symmetry class. You can obtain it by combining a Majorana nanowire with its time-reversed copy.

This is a hard task; if you go for it, try it out, but don't hesitate to ask for help in the discussion below.

Finding gaps

The analytic continuation from $e^{ik}$ to a complex plane is also useful in telling if a system is gapped.

Using the mapping of a 1D Hamiltonian to the eigenvalue problem, implement a function which checks if there are propagating modes at a given energy.

Then implement an algorithm which uses this check to find the lowest and the highest energy states for a given 1D Hamiltonian $H = h + t e^{ik} + t^\dagger e^{-ik}$ (with $h$, $t$ arbitrary matrices, of course).

MoocSelfAssessment description

In the live version of the course, you would need to share your solution and grade yourself.

Now share your results:

Discussion entitled 'Topological invariants' is available in the online version of the course.

Review assignment

Topological Defects and Gapless Modes in Insulators and Superconductors

Jeffrey C. Y. Teo, C. L. Kane

We develop a unified framework to classify topological defects in insulators and superconductors described by spatially modulated Bloch and Bogoliubov de Gennes Hamiltonians. We consider Hamiltonians H(k,r) that vary slowly with adiabatic parameters r surrounding the defect and belong to any of the ten symmetry classes defined by time reversal symmetry and particle-hole symmetry. The topological classes for such defects are identified, and explicit formulas for the topological invariants are presented. We introduce a generalization of the bulk-boundary correspondence that relates the topological classes to defect Hamiltonians to the presence of protected gapless modes at the defect. Many examples of line and point defects in three dimensional systems will be discussed. These can host one dimensional chiral Dirac fermions, helical Dirac fermions, chiral Majorana fermions and helical Majorana fermions, as well as zero dimensional chiral and Majorana zero modes. This approach can also be used to classify temporal pumping cycles, such as the Thouless charge pump, as well as a fermion parity pump, which is related to the Ising non-Abelian statistics of defects that support Majorana zero modes.

Hint: The most general classification

Anomalous Topological Pumps and Fractional Josephson Effects

Fan Zhang, C. L. Kane

We discover novel topological pumps in the Josephson effects for superconductors. The phase difference, which is odd under the chiral symmetry defined by the product of time-reversal and particle-hole symmetries, acts as an anomalous adiabatic parameter. These pumping cycles are different from those in the "periodic table", and are characterized by $Z\times Z$ or $Z_2\times Z_2$ strong invariants. We determine the general classifications in class AIII, and those in class DIII with a single anomalous parameter. For the $Z_2\times Z_2$ topological pump in class DIII, one $Z_2$ invariant describes the coincidence of fermion parity and spin pumps whereas the other one reflects the non-Abelian statistics of Majorana Kramers pairs, leading to three distinct fractional Josephson effects.

Hint: Beyond classification

Topological Insulators and C^*-Algebras: Theory and Numerical Practice

M. B. Hastings, T. A. Loring

We apply ideas from $C^*$-algebra to the study of disordered topological insulators. We extract certain almost commuting matrices from the free Fermi Hamiltonian, describing band projected coordinate matrices. By considering topological obstructions to approximating these matrices by exactly commuting matrices, we are able to compute invariants quantifying different topological phases. We generalize previous two dimensional results to higher dimensions; we give a general expression for the topological invariants for arbitrary dimension and several symmetry classes, including chiral symmetry classes, and we present a detailed $K$-theory treatment of this expression for time reversal invariant three dimensional systems. We can use these results to show non-existence of localized Wannier functions for these systems. We use this approach to calculate the index for time-reversal invariant systems with spin-orbit scattering in three dimensions, on sizes up to $12^3$, averaging over a large number of samples. The results show an interesting separation between the localization transition and the point at which the average index (which can be viewed as an "order parameter" for the topological insulator) begins to fluctuate from sample too sample, implying the existence of an unsuspected quantum phase transition separating two different delocalized phases in this system. One of the particular advantages of the $C^*$-algebraic technique that we present is that it is significantly faster in practice than other methods of computing the index, allowing the study of larger systems. In this paper, we present a detailed discussion of numerical implementation of our method.

Hint: The non-commutative invariants

Scattering theory of topological insulators and superconductors

I. C. Fulga, F. Hassler, A. R. Akhmerov

The topological invariant of a topological insulator (or superconductor) is given by the number of symmetry-protected edge states present at the Fermi level. Despite this fact, established expressions for the topological invariant require knowledge of all states below the Fermi energy. Here, we propose a way to calculate the topological invariant employing solely its scattering matrix at the Fermi level without knowledge of the full spectrum. Since the approach based on scattering matrices requires much less information than the Hamiltonian-based approaches (surface versus bulk), it is numerically more efficient. In particular, is better-suited for studying disordered systems. Moreover, it directly connects the topological invariant to transport properties potentially providing a new way to probe topological phases.

Hint: All about scattering

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!

Read one of the above papers and see how it is related to the current topic.

In the live version of the course, you would need to write a summary which is then assessed by your peers.

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

Discussion entitled 'General classification' is available in the online version of the course.