Topics for self-study

Simulations: what about other symmetries

So you’ve made it through the content of the first week. Congratulations!

Now let’s get our hands dirty.

First task: combination of particle-hole and time-reversal symmetries

Look at the notebook about topology of zero-dimensional systems, and see how we generate Hamiltonians with a spinful time-reversal symmetry

\[ H = \sigma_y H^* \sigma_y. \]

Now try to add this time reversal symmetry to a Hamiltonian which also has particle-hole symmetry. It is easiest to do in the basis where particle-hole symmetry has the form \(H = -H^*\). What do you think will happen? What will the extra symmetry do to the topological invariant? Test your guess by plotting the spectrum and calculating the Pfaffian invariant.

Second task: Su-Schrieffer-Heeger (SSH) model

Similar to the Kitaev chain, the SSH model is simple a one-dimensional model where you can see all the essential aspects of topological systems. Unlike the Kitaev chain it does correspond to a physical system: electrons in a polyacetylene chain.

Here’s such a chain:

Due to the dimerization of the chain the unit cell has two atoms and the hoppings have alternating strengths \(t_1\) and \(t_2\), so that the Hamiltonian is

\[ H = \sum_{n=1}^N t_1 \left|2n-1\right\rangle\left\langle 2n\right|+t_2 \left|2n\right\rangle \left\langle 2n+1\right| + \textrm{h.c} \]

We can choose to start a unit cell from an even-numbered site, so \(t_1\) becomes intra-cell hopping and \(t_2\) inter-cell hopping.

Now get the notebook with the Kitaev chain and transform a Kitaev chain into an SSH chain.

Now repeat the calculations we’ve done with Majoranas using SSH chain. Keep \(t_1 = 1\) and vary \(t_2\). You should see something very similar to what you saw with the Kitaev chain.

As you can guess, this is because the chain is topological. Think for a moment: what kind of symmetry protects the states at the edges of the chain. (Hint: you did encounter this symmetry in our course.)

The particle-hole symmetry, is a consequence of a mathematical transformation, and cannot be broken. The symmetry protecting the SSH chain, however, can be broken. Test your guess about the protecting symmetry by adding to your chain a term which breaks this symmetry and checking what it does to the spectrum of a finite chain and to its dispersion (especially as chain goes through a phase transition).

Now share your results:

Review assignment

For the first week we have these papers:

Topological properties of superconducting junctions (arXiv:1103.0780)

D. I. Pikulin, Yuli V. Nazarov

Motivated by recent developments in the field of one-dimensional topological superconductors, we investigate the topological properties of s-matrix of generic superconducting junctions where dimension should not play any role. We argue that for a finite junction the s-matrix is always topologically trivial. We resolve an apparent contradiction with the previous results by taking into account the low-energy resonant poles of s-matrix. Thus no common topological transition occur in a finite junction. We reveal a transition of a different kind that concerns the configuration of the resonant poles.

Hint: Topological classification is not always applied to Hamiltonians. Figure out what is the topological quantity in open systems. See this idea also applied in arXiv:1405.6896.

Wigner-Poisson statistics of topological transitions in a Josephson junction (arXiv:1305.2924)

C. W. J. Beenakker, J. M. Edge, J. P. Dahlhaus, D. I. Pikulin, Shuo Mi, M. Wimmer

The phase-dependent bound states (Andreev levels) of a Josephson junction can cross at the Fermi level, if the superconducting ground state switches between even and odd fermion parity. The level crossing is topologically protected, in the absence of time-reversal and spin-rotation symmetry, irrespective of whether the superconductor itself is topologically trivial or not. We develop a statistical theory of these topological transitions in an N-mode quantum-dot Josephson junction, by associating the Andreev level crossings with the real eigenvalues of a random non-Hermitian matrix. The number of topological transitions in a 2pi phase interval scales as sqrt(N) and their spacing distribution is a hybrid of the Wigner and Poisson distributions of random-matrix theory.

Hint: This is a study of statistical properties of topological transitions.

How to realize a robust practical Majorana chain in a quantum dot-superconductor linear array (arXiv:1111.6600)

Jay D. Sau, S. Das Sarma

Semiconducting nanowires in proximity to superconductors are promising experimental systems for Majorana fermions, which may ultimately be used as building blocks for topological quantum computers. A serious challenge in the experimental realization of the Majorana fermions is the supression of topological superconductivity by disorder. We show that Majorana fermions protected by a robust topological gap can occur at the ends of a chain of quantum dots connected by s-wave superconductors. In the appropriate parameter regime, we establish that the quantum dot/superconductor system is equivalent to a 1D Kitaev chain, which can be tuned to be in a robust topological phase with Majorana end modes even in the case where the quantum dots and superconductors are both strongly disordered. Such a spin-orbit coupled quantum dot - s-wave superconductor array provides an ideal experimental platform for the observation of non-Abelian Majorana modes.

Hint: A toy model may still be useful in practice.

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!