Logo
Institute
Research
Teaching
Members
Contact
Department of Physics

Back to University of Cologne's home page

14756.2030 | Entangled Phases of Matter (6LP)

  • Lectures: Tuesdays and Thursdays, from 16.00pm to 17.30pm, in the virtual room via Zoom
  • Tutorials: Each 2nd week on Thursdays, from 16.00pm to 17.30pm, via Zoom
  • The lectures and tutorial sessions will be given by Dr. Dmitry Bagrets (e-mail: dbagrets [at] uni-koeln.de) and Dr. Carolin Wille (e-mail: carolin.wille [at] fu-berlin.de ).
  • The first introductory lecture is on Tuesday, November 3d, 2020. Zoom link for the virtual meeting will distributed via e-mail.

Overview

In this theoretical course we will discuss unconventional phases of condensed matter where the entanglement plays a conceptional role. The remarkable feature of entangled quantum matter is that the large number of atomistic constituents forming a solid may mutually protect each other against the unwanted effects of decoherence. This opens a prospective of using such phases for (topological) quantum computation which exploits the so-called non-Abelian anyons to be introduced and discussed in details within this course.

The list of topics to be covered includes:

  • Abelian & non-Abelian Berry phases
  • Majorana fermions in p-wave superconductors
  • anyons in fractional quantum Hall effect (FQHE)
  • fusion and braiding rules for anyons
  • topological quantum computation
  • Kitaev's toric code and Kitaev's honeycomb model
  • topological entanglement entropy

Prerequisites

The familiarity with a course of advanced quantum mechanics is a basic prerequisite. On other hand, the knowledge of QFT (quantum field theory) is not at all required.

Exercise sheets

Lecture notes

  • Introductory lecture
  • Lecture 1, "Berry phase" - Script, Sections 1.5.1, 1.5.2 & 1.5.4 from [2]
  • Lecture 2, "The integer QHE" - Section 7.2.1 from [4]
  • Lecture 3 & 4, "The fractional QHE, Abelian anyons " - Script, Sections 3.1.1, 3.1.2 & 3.2 from [2]
  • Lecture 5 "BCS theory of superconductivity" - Section 6.4 from [5]
  • Lecture 6 "Spinless p-wave superconductors, Majorana fermions in 1D" - Script, see also Section II.A in [6]
  • Lecture 7 "Half-quantum vortexes in p-wave superconductors, fusion and braiding of Majorana fermions in 2D" - Script, Sections II.B and VI in [6], Ref. [7]
  • Lectures 8 & 9 "Kitaev's toric code" - Section 5.2.1 from [2]. Original paper by A. Kitaev is [8].
  • Lecture 10 "Tensor networks and entanglement entropy " - Script, see also Chapter 9 in [1] and Section 1 in [9]. Original papers are [10,11]
  • Lecture 11 "Projected Entangled Pair States (PEPS), entanglement entropy of PEPS and toric code" - Script, section 7 in Ref. [12]
  • Lecture 12 "Non-Abelian anyon models, fusion rules, F-matrix" - Script, sections 4.1.1-4.1.2 in Ref. [1], 8.1-8.3, 8.6 and 9.1-9.2 in Ref. [3].
  • Lecture 13 "Quantum dimensions, Pentagon equation for F-matrix" - Script, sections 4.1.3, 4.1.5 in Ref. [1], 4.3.1, 4.3.2 in Ref. [2].
  • Lecture 14 "Braiding of non-Abelian anyons" - Script, sections 4.1.4 in Ref. [1], 4.3.3 in Ref. [2], 10.1 in Ref. [3].
  • Lecture 15 "Braiding, hexagon equations for R-matrix" - Script, sections 4.1.5 in Ref. [1], 4.3.3 in Ref. [2], 13.2, 13.3 in Ref. [3], 2.5 in Ref. [13].
  • Lecture 16 "Braiding in Fibonacci & Ising anyon models; Introduction to quantum computing" - Script, sections 4.4, 6.3 in Ref. [1], 11.1 in Ref. [3].
  • Lecture 17 "Quantum vs Classical Computation", sections 1.4.5 and 5.1 in Ref. [14].
  • Lecture 18 "Quantum computing with Fibonacci anyons", - Script, Mathematica file, sections 11.3 and 11.4 in Ref. [3].
  • Lecture 19 "Topologically non-trivial ground states, entanglement entropy and anyons", sections 9.1 and 9.2.1 in Ref. [1].
  • Lecture 20 "Phases of matter: long-range vs short-range entanglement", section 7 in Ref. [15].

Videos of lectures are available on ILIAS

Literature

  1. Jiannis K. Pachos, "Introduction to Topological Quantum Computation"
  2. David Tong, "Lectures on the Quantum Hall Effect", https://www.damtp.cam.ac.uk/user/tong/qhe.html
  3. Steven H. Simon, "Topological Quantum: Lecture Notes and Proto-Book", https://www-thphys.physics.ox.ac.uk/people/SteveSimon/topological2020/TopoBookOct27hyperlink.pdf
  4. Xiao-Gang Wen, "Quantum Field Theory of Many-Body Systems"
  5. A. Altland and B. Simons, "Condensed Matter Field Theory", 2nd edition
  6. J. Alicea, "New directions in the pursuit of Majorana fermions in solid state systems", https://arxiv.org/abs/1202.1293
  7. D. A. Ivanov, "Non-Abelian Statistics of Half-Quantum Vortices in p-Wave Superconductors", Phys. Rev. Lett. 86, 268 (2001)
  8. A. Yu. Kitaev, "Fault-tolerant quantum computation by anyons", https://arxiv.org/abs/quant-ph/9707021
  9. J. C. Bridgeman, C. T. Chubb, "Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks", https://arxiv.org/abs/1603.03039
  10. A. Kitaev and J. Preskill, "Topological entanglement entropy", https://arxiv.org/abs/hep-th/0510092v2
  11. M. Levin and Xiao-Gang Wen, "Detecting topological order in a ground state wave function", https://arxiv.org/abs/cond-mat/0510613v2
  12. N. Schuch, I. Cirac, and D. Perez-Garcia, "PEPS as ground states: degeneracy and topology", https://arxiv.org/pdf/1001.3807.pdf
  13. S. Trebst, M. Troyer, Z. Wang, A.W.W. Ludwig, "A short introduction to Fibonacci anyon models", https://arxiv.org/pdf/0902.3275.pdf
  14. Isaac Chuang, Michael Nielsen, "Quantum Computation and Quantum Information"
  15. Bei Zeng, Xie Chen, Duan-Lu Zhou, Xiao-Gang Wen, "Quantum Information Meets Quantum Matter - From Quantum Entanglement to Topological Phase in Many-Body Systems", https://arxiv.org/pdf/1508.02595.pdf