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Lecture Topics and Background References

Prof. Anton Akhmerov (1-5 October)
Title: Numerics of topological devices

Learning goals:

  • Carry out numerical simulations of topological band structures and mesoscopic devices
  • Use numerical simulations to answer research questions about these systems

Plan of the lectures:

  1. Introduction to mesoscopic physics: continuum and tight-binding band structures, discretization error, spurious modes, scattering problem, conductance
  2. Majorana devices: conductance, Josephson junctions, breaking the protection
  3. Josephson junctions: long and short junctions, junction transparency, Majoranas in a Josephson junction
  4. Chiral and helical modes: quantum Hall effect, quantum spin Hall effect, alternative pathways to Majoranas.
  5. Topology and disorder: what is disorder strength, band structure renormalization, density of states, kernel polynomial method


Most of the physics I am going to discuss is covered in the references provided by other lecturers, however I recommend several additional sources focusing on numerical simulations.

  • Because all of the programming is going to be done using the Python programming language, I recommend to quickly follow an introduction course to Python and scientific Python packages numpy, scipy, matplotlib:
  • For simulations I am going to use the Kwant quantum transport library, it would be useful for you to read the paper describing it(New J. Phys. 16, 063065) as well as its tutorial:
  • An inspiration for a part of my lectures is the online course on topology in condensed matter:

Prof. Carlo Beenakker (9-11 October)
Title: Random-matrix theory of Majorana fermions and topological superconductors

Plan of the lectures:

1. Ten-fold-way classification of topological states of matter
2. Signatures of Majorana zero-modes
3. Quantum transport in Majorana edge-modes


  • Tutorial: A road to reality with topological superconductors  arXiv:1606.09439
  • General overview: Search for Majorana fermions in superconductors arXiv:1112.1950
  • Specialized review: Random-matrix theory of Majorana fermions and topological superconductors arXiv:1407.2131 (no need to study this in detail)


Prof. Hélène Bouchiat (8-12 October)
Title: Probing experimentally the Andreev spectrum in topological insulator based Josephson junctions

Plan of the lectures:

1. Basics of proximity induced superconductivity in SNS nanostructures
2. Spin dependent and interaction effects in SNS junctions
3. Josephson current in topological SNS junctions
4. AC phase biased SNS junctions: kinetic inductance and phase dependent dissipation
5. AC phase biased SNS junctions: probing the Andreev spectrum in topological junctions

1. Basics of proximity induced superconductivity in nanostructures

Bogoliubov de Gennes equations, phase dependent Andreev spectrum, phase dependent Josephson current, short and long junctions, role of disorder and barriers at the NS interface


  • nde Gennes P.G. (1999) “Superconductivity of Metals and Alloys” Westview Press.
  • Nazarov, Yuli V., and Yaroslav M. Blanter. 2009. “Quantum Transport: Introduction to Nanoscience”. 1 edition. 
    Cambridge, UK ; New York: Cambridge University Press.
  • Tero Heikkila “The Physics of Nanoelectronics  2013 “ Oxford University Press.
  • H.van Houten and C.W.J. Beenakker. Andreev reflection and the josephson effect in a quantum point contact: An analogy with phase-conjugating resonators. Physica B 175(1–3):187 – 197, 1991.
  • Supriyo Datta and Philip F.Bagwell « Can the Bogoliubov–de Gennes equation be interpreted as a ‘one-particle’ wave equation? » Superlattices and Microstructures 25, 1233 (1999).
  • “Flux sensitivity of a piecewise normal and superconducting metal loop”, M. Büttiker and T. M. Klapwijk, Phys. Rev. B 33, 5114(R) (1986).
  • M. L. Della Rocca, M. Chauvin, B. Huard, H Pothier, D. Esteve, and C. Urbina. “Measurement of the current-phase relation of superconducting atomic contacts”. Phys.Rev. Lett. 99, 127005 (2007).
  • Tero T. Heikkilä, Jani Särkkä, and Frank K. Wilhelm « Supercurrent-carrying density of states in diffusive mesoscopic Josephson weak links » T Phys. Rev. B 66, 184513 (2002).
  • H. le Sueur, P. Joyez, H. Pothier, C. Urbina, and D. Esteve. Phase Controlled Superconducting Proximity Effect Probed by Tunneling Spectroscopy. Physical Review Letters, 100(19) (2008).
  • A.Golubov, I.Illichev, M.Kuprianov “The current-phase relation in Josephson junctions” REVIEWS OF MODERN PHYSICS,  76,  (2004).

2. Spin dependent and interaction effects in  Josephson SNS junctions

Spin Orbit and Zeeman effect in SNS junctions, 0/pi transitions, phi junctions, Josephson effect in quantum dots, Coulomb Blockade and Kondo effect . Application to carbonnanotubes 


  • F. S. Bergeret, A. F. Volkov, K. B. Efetov Rev. Mod. Phys. 77, 1321 (2005).
  • S. De Franceschi, L. P. Kouwenhoven, C Schonenberger, and W. Wernsdorfer. “Hybrid superconductorâquantum dot devices”. Nat. Nanotechnol. 5, 10 (2010), pp. 703–711.
  • J.-D. Pillet, C. H. L. Quay, P. Morfin, C. Bena, A. L. Yeyati, and P. Joyez. “Andreev bound states in supercurrent-carrying carbon nanotubes revealed”. Nat. Phys. 6, 12  965 (2010).
  • E. J. H. Lee, X. Jiang, M. Houzet, R. Aguado, C. M. Lieber, and S. De Franceschi. “SpinresolvedAndreev levels and parity crossings in hybrid superconductor-semiconductor nanostructures.” Nat. Nanotechnol. 9, 1 79 (2014).
  • E. Vecino, A. Martín-Rodero, and A. Levy Yeyati. “Josephson current through a correlated quantum level: Andreev states and junction behavior”. Phys. Rev. B 68, 035105 (2003).
  • M.-S. Choi, M. Lee, K. Kang, and W. Belzig. “Kondo effect and Josephson current through a    quantum dot between two superconductors”. Phys. Rev. B 70, 020502 (2004).
  • C Karrasch, A. Oguri, and V Meden. “Josephson current through a single Anderson impurity coupled to BCS leads”. Phys. Rev. B 77, 024517 (2008).
  • R. Delagrange, D. J. Luitz, R. Weil, A. Kasumov, V. Meden, H. Bouchiat and R. Deblock. "Manipulating the magnetic state of a carbon nanotube Josephson junction using the superconducting phase". Phys. Rev. B 91, 241401 (2015).
  • A. Buzdin. Direct coupling between magnetism and superconducting current in the josephson ϕ0 junction. Phys. Rev. Lett., 101:107005, Sep 2008.
  • Tomohiro Yokoyama, Mikio Eto, and Yuli V. Nazarov. Anomalous josephson effect induced by spin-orbit interaction and zeeman effect in semiconductor nanowires. Phys. Rev. B, 89:195407, (2014).
  • Szombati DB, Nadj-Perge S, Car D, Plissard SR, Bakkers EPAM, Kouwenhoven LP. Josephson ϕ0-junction in nanowire quantum dots. Nat Phys. 2016;12(May):568-573.

3. Josephson current in topological SNS Josephson junctions

Examples of experimental realizations, topological protection and current phase relation, phi junction behavior.


  • “Unpaired Majorana fermions in quantum wires”, A.Kitaev, Phys.-Usp. 44 131(2001).
  • R. M. Lutchyn, J. D. Sau, and S. D. Sarma, Phys. Rev. Lett. 105, 077001 (2010). 
  • V. Mourik , K. Zuo, S. M.Frolov, S. R. Plissard, E. P. A. M. Bakk and L. P.Kouwenhoven, Science 336 1003 (2012).
  • SM Albrecht, AP Higginbotham, M Madsen, F Kuemmeth, TS Jespersen, ...Exponential protection of zero modes in Majorana islands Nature 531 (7593), 206
  • M. Hasan and C. Kane, “ Topological insulators”,Rev. Mod. Phys. 82, 3045 (2010).
  • Kane CL, Mele EJ. Quantum Spin Hall Effect in Graphene. 2005;226801(November):1-4. doi:10.1103/PhysRevLett.95.226801.
  • Bernevig BA, Zhang SC. Quantum Spin Hall Effect. Phys Rev Lett. 2006;96(March):106802. doi:10.1103/PhysRevLett.96.106802.
  • König M, Wiedmann S, Brüne C, Roth A, Buhmann H, Molenkamp W, Qi X-L, Zhang S-C, Quantum Spin Hall Insulator State in HgTe Quantum Wells. Science (5851);318.
  • L. Fu and C. Kane,“Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator”,  Phys. Rev. Lett. 100, 096407 (2008).
  • L. Fu and C. Kane,“Probing Neutral Majorana Fermion Edge Modes with Charge Transport”, Phys. Rev. Lett. 102, 216403 (2009).
  • Beenakker CWJ, Pikulin DI, Hyart T, Schomerus H, Dahlhaus JP. Fermion-parity anomaly of the critical supercurrent in the quantum spin-Hall effect. Phys Rev Lett. 2013;110(1):1-5.
  • Hart S, Ren H, Wagner T, Philipp Leubner, Mathias Mühlbauer, Christoph Brüne, Hartmut Buhmann, Laurens W. Molenkamp & Amir Yacoby, Induced superconductivity in the quantum spin Hall edge. Nat Phys. 2014.
  • R.Aguado Riv. Nuovo Cimento 40, 523-593 (2017). (Revue paper).
  • C. Li, A. Kasumov, A. Murani, S. Sengupta, F. Fortuna, K. Napolskii, D. Koshkodaev, G. Tsirlina, Y. Kasumov, I. Khodos, R. Deblock, M. Ferrier, S. Guéron, and H. Bouchiat, “Magnetic field resistant quantum interferences in Josephson junctions based on bismuth nanowires” Phys. Rev. B 90, 245427 (2014).
  • A. Murani, A.Kasumov, S. Sengupta, Y. Kasumov, V. Volkov, I. Khodos, F. Brisset, R. Delagrange, A. Chepelianskii, R. Deblock, H. Bouchiat and S. Guéron, “Ballistic edge states in Bismuth nanowires revealed by SQUID interferometry” Nature Communications 8, 15941 (2017).
  • F. Schindler, Z. Wang, M. Vergniory, A. Cook, A. Murani, S. Sengupta, A. Kasumov, R. Deblock, S. Jeon, I. Drozdov, H. Bouchiat, S. Guéron, A. Yazdani, B. Bernevig, T. Neupert, “Higher-Order Topology in Bismuth”Nature Physics Juil.30 (2018) arXiv:1802.02585.
  • Dolcini, F., Houzet, M. & Meyer, J. S., Topological Josephson 0 junctions. Phys. Rev. B 92, 035428 (2015).

 4.   AC phase biased SNS junctions: kinetic inductance and  phase dependent dissipation

 Linear response and fluctuation dissipation relation in SNS junctions, diagonal and non diagonal susceptibility, experiments in long diffusive junctions, spectroscopy of Andreev levels in small junctions


  • N. Trivedi and DA Browne. Mesoscopic ring in a magnetic  field: Reactive and dissipative response. Physical Review B, 38(14), 1988.
  • B. Dassonneville, M. Ferrier, S. Guéron, and H. Bouchiat, “Dissipation and Supercurrent Fluctuations in a Diffusive Normal-Metal–Superconductor Ring”,Phys. Rev. Lett. 110, 217001(2013).
  • M. Ferrier, B. Dassonneville, S. Guéron, and H. Bouchiat, “Phase-dependent Andreev spectrum in a diffusive SNS junction: Static and dynamic current response,” Phys. Rev. B . 88, no. 17, pp. 1–8, 2013.
  • B. Dassonneville, A. Murani, M. Ferrier, S. Guéron, and H. Bouchiat. Coherence-enhanced phase-dependent dissipation in long SNS junctions. Phys. Rev. B 97, 184505 (2018).
  • P. Virtanen, F. S. Bergeret, J. Cuevas, and T. Heikkila, Physical Review B 83, 144514 (2011).
  • K. S. Tikhonov and M. V. Feigel'man, Physical Review B 91, 054519 (2015).
  • “Coherent manipulation of Andreev states in superconducting atomic contacts”, C. Janvier, L. Tosi, L. Bretheau, Ç. Girit, M. Stern, P. Bertet, P. Joyez, D. Vion, D. Esteve, M. Goffman, H. Pothier, C. Urbina, Science 349 (6253), 1199 (2015).

5. AC phase biased SNS junctions: probing the Andreev spectrum in topological junctions

Revealing  the spin dependent  Andreev spectrum and  topological crossings


  • L. Fu and C. Kane, “Josephson current and noise at a superconductor/quantum-spin-Hall-insulator/superconductor junction”,Phys. Rev. B 79, 161408(R) (2009).
  • Doru Sticlet and Jerome Cayssol Phys. Rev. B 90, 201303(R) (2014).
  • J. I. Väyrynen, G. Rastelli, W. Belzig, and L. I. Glazman. Microwave signatures of Majorana states in a topological Josephson junction. Phys. Rev. B -  92, no. 13, pp. 1–5, (2015).
  • B. van Heck, J. I. Väyrynen, and L. I. Glazman Phys. Rev. B 96, 075404 (2017).
  • Wiedenmann, E. Bocquillon, R. S. Deacon, S. Hartinger,O. Herrmann, T. M. Klapwijk, L. Maier, C. Ames, C. Brune, C.Gould, A. Oiwa, K. Ishibashi, S. Tarucha,H. Buhmann, and L. W. Molenkamp, Nat Commun, 7,10303, (2016).
  • A. Murani, A. Chepelianskii, S. Guéron, and H. Bouchiat  “Andreev spectrum with high spin-orbit interactions: Revealing spin splitting and topologically protected crossings”  Phys. Rev. B 96, 165415 (2017).
  • Mircea Trif, Olesia Dmytruk, Helene Bouchiat, Ramón Aguado, Pascal Simon, Physical Review B 97,041415(R)(2017)
  • A. Murani, B. Dassonneville, A. Kasumov, M. Ferrier, R. Deblock, S. Guéron, H. Bouchiat, “Microwave signature of topological Andreev level crossings in superconducting-proximitized crystalline Bi nanowires”, arXiv:1808.09726.
  • Wiedenmann, E. Bocquillon, R. S. Deacon, S. Hartinger,O. Herrmann, T. M. Klapwijk, L. Maier, C. Ames, C. Brune, C.Gould, A. Oiwa, K. Ishibashi, S. Tarucha,H. Buhmann, and L. W. Molenkamp, Nat Commun, 7,10303, (2016).


Prof. Felix von Oppen (1-5 October)
Title: From Majorana bound states to topological quantum computation

Plan of the lectures:

1. Majorana bound states in p-wave superconductors and their nonabelian statistics
2. Physical realizations of 1d p-wave superconductors
3. Building blocks of a Majorana-based quantum computer
4. Introduction to quantum error correction
5. Fault-tolerant Majorana quantum computer

Basics: Books on superconductivity, especially BCS theory &Bogoliubov-deGennes formalism, see, e.g.,
• M. Tinkham, Introduction to Superconductivity, 2nd Edition, Ch. 3 and 10
• P.G. de Gennes, Superconductivity of Metals and Alloys, Ch. 4 and 5

Lecture 1: Majorana bound states in p-wave superconductors and their nonabelian statistics

Lecture 2: Physical realizations of 1d p-wave superconductors
In addition to literature under 1., also

  • R. M. Lutchyn, E. P. A. M. Bakkers, L. P. Kouwenhoven, P. Krogstrup, C. M. Marcus & Y. Oreg, Majorana zero modes in superconductor–semiconductor heterostructures,Nature Reviews Materials 3, 52 (2018).

Literature 3: Building blocks of a Majorana-based quantum computer

  • D. Aasen,M. Hell, R.V. Mishmash, A.Higginbotham, J. Danon, M. Leijnse, T. S. Jespersen, J. A. Folk, C. M. Marcus, K. Flensberg, and J. Alicea, Milestones Toward Majorana-Based Quantum Computing, Phys. Rev. X 6, 031016 (2016).
  • S. Plugge, A. Rasmussen, R. Egger, and K. Flensberg,Majoranabox qubits, New J. Phys. 19, 012001 (2017).
  • T. Karzig, C. Knapp, R. M. Lutchyn, P. Bonderson, M. B. Hastings, C. Nayak, J. Alicea, K. Flensberg, S. Plugge, Y. Oreg C. M. Marcus, and M. H. Freedman, Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes, Phys. Rev. B 95, 235305 (2017).
  • D. Litinski, F. von Oppen, Quantum Computing with Majorana Fermion Codes, Phys. Rev. B 97, 205404 (2018).

Literature 4: Introduction to quantum error correction

  • B. M. Terhal, Quantum error correction for quantum memories, Rev. Mod. Phys. 87, 307 (2015).

Literature 5: Fault-tolerant Majorana quantum computer

  • S.Vijay, T. H. Hsieh, and L. Fu, MajoranaFermion Surface Code for Universal Quantum Computation, Phys. Rev. X 5, 041038 (2015).
  • S. Plugge, L. A. Landau, E. Sela, A. Altland, K. Flensberg, andR. Egger, Roadmap toMajorana surface codes, Phys. Rev. B 94, 174514 (2016).
  • D. Litinski, F. von Oppen, Quantum Computing with Majorana Fermion Codes, Phys. Rev. B 97, 205404 (2018).


Prof. Carmine Ortix (8 October)
Title: Weyl and topological nodal line semimetals

Plan of the lectures:

1) Weyl and Dirac semimetals in three-dimensional solids, Review of Modern Physics 90, 015001 (2018);
2) Topological Materials: Weyl semimetals, Annu. Rev. Condens. Matter Phys. 8, 337 (2017);
3) Topological nodal line semimetals with and without spin-orbital coupling, Physical Review B 92, 081201(R) (2015).

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