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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=σyH∗σy.H = \sigma_y H^* \sigma_y.
(1)

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∗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 simply 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 t1t_1 and t2t_2, so that the Hamiltonian is

H=∑n=1Nt1∣2n−1⟩⟨2n∣+t2∣2n⟩⟨2n+1∣+h.cH = \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}
(2)

We can choose to start a unit cell from an even-numbered site, so t1t_1 becomes intra-cell hopping and t2t_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 the SSH chain. Keep t1=1t_1 = 1 and vary t2t_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 the chain goes through a phase transition).

Now share your results:

Review assignment

For the first week we have these papers:

Pikulin & Nazarov (2011)

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 Mi et al. (2014).

Beenakker et al. (2013)

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

Sau & Sarma (2011)

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!

References
  1. Pikulin, D. I., & Nazarov, Y. V. (2011). Topological properties of superconducting junctions. 10.48550/ARXIV.1103.0780
  2. Mi, S., Pikulin, D. I., Marciani, M., & Beenakker, C. W. J. (2014). X-shaped and Y-shaped Andreev resonance profiles in a superconducting quantum dot. 10.48550/ARXIV.1405.6896
  3. Beenakker, C. W. J., Edge, J. M., Dahlhaus, J. P., Pikulin, D. I., Mi, S., & Wimmer, M. (2013). Wigner-Poisson statistics of topological transitions in a Josephson junction. 10.48550/ARXIV.1305.2924
  4. Sau, J. D., & Sarma, S. D. (2011). How to realize a robust practical Majorana chain in a quantum dot-superconductor linear array. 10.48550/ARXIV.1111.6600
Topology in toy models
Bulk-edge correspondence in the Kitaev chain
Majoranas I
From Kitaev chain to a nanowire