Chapter 36 Further Reading: Topological Phases of Matter

Tier 1: Essential References

These are the primary references that cover the material of this chapter at a level closely matching our treatment. You should consult at least one of these.

Bernevig, B. A. & Hughes, T. L. — Topological Insulators and Topological Superconductors (2013)

The most pedagogical graduate-level textbook on topological phases. Chapters 1–8 cover the quantum Hall effect, Berry phase, Chern numbers, and the TKNN formula. Chapters 9–15 cover topological insulators and the $\mathbb{Z}_2$ invariant. Clear derivations, many worked examples, and excellent physical intuition. - Best for: Students who want a complete, self-contained treatment at the level of this chapter.

Hasan, M. Z. & Kane, C. L. — "Colloquium: Topological insulators," Reviews of Modern Physics 82, 3045 (2010)

The definitive review article on topological insulators. Covers the theoretical foundations (Berry phase, TKNN, $\mathbb{Z}_2$ invariant) and the experimental discoveries in HgTe and Bi$_2$Se$_3$. Well-written and accessible to advanced undergraduates with a solid quantum mechanics background. - Best for: A comprehensive overview of the field as of 2010, which remains highly relevant.

Qi, X.-L. & Zhang, S.-C. — "Topological insulators and superconductors," Reviews of Modern Physics 83, 1057 (2011)

A companion review to Hasan & Kane, with more emphasis on topological superconductors, Majorana fermions, and topological quantum computation. More mathematically sophisticated. - Best for: Students interested in the connection between topological phases and quantum computing.

Tong, D. — Lectures on the Quantum Hall Effect (2016)

David Tong's Cambridge lecture notes, freely available at damtp.cam.ac.uk/user/tong/qhe.html. Covers the integer and fractional quantum Hall effects with characteristic clarity and wit. Includes the TKNN formula, edge states, and Laughlin's argument. - Best for: Students who want a thorough treatment of the quantum Hall effect specifically, from a field-theoretic perspective.

Tier 2: Supplementary and Enrichment

These sources provide deeper historical context, alternative perspectives, or advanced treatments of specific topics.

Textbooks and Monographs

Vanderbilt, D. — Berry Phases in Electronic Structure Theory (2018) A comprehensive treatment of Berry phases in condensed matter physics, including topological invariants, polarization, orbital magnetization, and Wannier functions. More mathematical than our treatment but excellent for students who want the rigorous foundations.

Asboth, J. K., Oroszlany, L., & Palyi, A. — A Short Course on Topological Insulators (2016) A concise, lecture-note style introduction freely available on arXiv (arXiv:1509.02295). Covers the SSH model, Chern insulators, and $\mathbb{Z}_2$ topological insulators with numerical exercises. Excellent for self-study. - Best for: Students who want a shorter, more computational introduction than Bernevig & Hughes.

Wen, X.-G. — Quantum Field Theory of Many-Body Systems, 2nd ed. (2004) An advanced text that treats topological order from the perspective of quantum field theory. Chapters on the fractional quantum Hall effect, Chern-Simons theory, and topological quantum computation. Graduate-level.

Nayak, C. et al. — "Non-Abelian anyons and topological quantum computation," Reviews of Modern Physics 80, 1083 (2008) The definitive review of topological quantum computing. Covers anyonic systems, braiding, the connection to topological quantum field theory, and the experimental platforms. Essential reading for anyone interested in topological quantum computing. - Best for: Students who want the full story on non-abelian anyons and their use in quantum computation.

Historical and Contextual

Von Klitzing, K. — "The quantized Hall effect," Reviews of Modern Physics 58, 519 (1986) Von Klitzing's Nobel Lecture, describing the discovery and its implications for metrology. A beautiful example of how unexpected experimental precision demanded a theoretical explanation.

Haldane, F. D. M. — "Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the 'parity anomaly'," Physical Review Letters 61, 2015 (1988) Haldane's original paper proposing what we now call the Haldane model. Remarkably concise and insightful. The paper that launched the field of Chern insulators.

Kane, C. L. & Mele, E. J. — "$\mathbb{Z}_2$ topological order and the quantum spin Hall effect," Physical Review Letters 95, 146802 (2005) The paper that predicted time-reversal invariant topological insulators. Introduced the $\mathbb{Z}_2$ invariant and the Kane-Mele model.

König, M. et al. — "Quantum spin Hall insulator state in HgTe quantum wells," Science 318, 766 (2007) The experimental discovery of the 2D topological insulator. Clear presentation of the transport measurements that confirmed the theoretical prediction.

Online Resources

MIT OpenCourseWare — 8.513 Many-Body Physics (Fall 2016) Prof. Senthil Todadri's graduate course. Several lectures cover topological phases, the quantum Hall effect, and topological insulators. Video lectures and notes available.

Topocondmat.org — Online Course on Topology in Condensed Matter An interactive online course with numerical exercises. Covers the major topics in topological phases with Python-based computational tools. - Best for: Students who learn best through hands-on computation.

Nobel Prize lectures (2016) The Nobel lectures of Thouless, Haldane, and Kosterlitz are available at nobelprize.org. All three are excellent. Haldane's lecture, in particular, provides a personal perspective on the development of the Haldane model.

Original Papers (for the historically inclined)

  • Klitzing, K. v., Dorda, G., & Pepper, M. (1980). "New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance." Physical Review Letters, 45, 494.
  • Laughlin, R. B. (1981). "Quantized Hall conductivity in two dimensions." Physical Review B, 23, 5632.
  • Thouless, D. J., Kohmoto, M., Nightingale, M. P., & den Nijs, M. (1982). "Quantized Hall conductance in a two-dimensional periodic potential." Physical Review Letters, 49, 405.
  • Haldane, F. D. M. (1988). "Model for a quantum Hall effect without Landau levels." Physical Review Letters, 61, 2015.
  • Kane, C. L. & Mele, E. J. (2005). "$\mathbb{Z}_2$ topological order and the quantum spin Hall effect." Physical Review Letters, 95, 146802.
  • Bernevig, B. A., Hughes, T. L., & Zhang, S.-C. (2006). "Quantum spin Hall effect and topological phase transition in HgTe quantum wells." Science, 314, 1757.
  • König, M. et al. (2007). "Quantum spin Hall insulator state in HgTe quantum wells." Science, 318, 766.
  • Kitaev, A. (2003). "Fault-tolerant quantum computation by anyons." Annals of Physics, 303, 2.
  • Mourik, V. et al. (2012). "Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices." Science, 336, 1003.

Reading Strategy

For Chapter 36, we recommend:

  1. Everyone: Read Hasan & Kane (2010), Sections I–III, for the big picture of topological insulators and their theoretical foundations. It is accessible and well-illustrated.

  2. If you want the quantum Hall effect in depth: Read Tong's lecture notes, Chapters 1–4. Clear, rigorous, and often amusing.

  3. If you want computational practice: Work through the Asboth, Oroszlany, & Palyi short course (free on arXiv). It includes Python exercises closely aligned with our code modules.

  4. If you are interested in topological quantum computing: Read the Nayak et al. (2008) review. It is long but comprehensive and remains the standard reference.

  5. If you want the full mathematical framework: Bernevig & Hughes (2013) is the most complete textbook. Start with Chapters 3 (Berry phase), 8 (Chern insulator), and 10 ($\mathbb{Z}_2$ insulator).