# 402-0888-18L Fractionalization of Particles in Physics

Semester | Spring Semester 2018 |

Lecturers | C. Chamon |

Periodicity | non-recurring course |

Language of instruction | English |

Abstract | The course will cover fractionalization phenomena in one and two spatial dimensions. It will survey the theoretical methods used to understand fractionalization, including bosonization, Chern-Simons theory, quantum anomalies, and use of topological invariants. These methods will be applied in several examples. |

Objective | In condensed matter physics, the electron need not be “fundamental” in the sense that it may have little relation to the low-lying charge excitations due to strong interaction effects. In the fractional quantum Hall effect, for example, a very strong magnetic field enhances dramatically the importance of Coulomb interactions among electrons over their kinetic energy; so much so that elementary charge excitations carry a fractional charge of the electron. The aim of this course is to explain by way of examples, often motivated but not limited to condensed matter physics, how interactions, either among particles or between particles and their background, can modify the quantum numbers of the “elementary” building blocks of matter, in short the fractionalization of particles in physics. The course will cover fractionalization phenomena in one and two spatial dimensions. It will survey the theoretical methods used to understand fractionalization, including bosonization, Chern-Simons theory, quantum anomalies, and use of topological invariants. These methods will be applied in the study of fractionalization of charge in material systems in 1D and 2D, and in the study of topological insulators and superconductors. |

Content | 1. One-dimensional (1D) systems • Fermiology on the lattice and in the continuum • Symmetries • Sublattice grading and spectral folding • A model for polyacetylene • The Peierls instability for polyacetylene • The Su-Schrieffer-Heeger (SSH) model 2. Zero-modes and fractionalization in 1D • Point defects in the dimerization • Zero modes bound to topological defects • Zero modes in the lattice and in the continuum • First encounter of charge fractionalization 3. Evaluation of the induced charge using various methods • Supersymmetry and the Witten index • The gradient expansion • The adiabatic expansion • Fractionalization from Abelian bosonization • Rational vs. irrational charges in 1D 4. Fractionalization in one-dimensional superconductors • Bogoliubov-de-Gennes Hamiltonians • The Kitaev chain • Majorana zero modes 5. 2D systems – Dirac fermions • Graphene and the Dirac fermions in 2D • Classification of masses for 2D Dirac fermions • Vortices in mass order parameters • Zero-modes tied to vortices • Confinenement and deconfinement – axial gauge fields • Rational vs. irrational charges in 2D – confinement vs. deconfinement 6. 2D systems – fractional quantum Hall systems • Quantized Hall effect • Laughlin gauge argument • Flux insertion and fractional charge quantization • Chern-Simons theory and electromagnetic response • Wire construction of 2D fractional quantum Hall states |

Lecture notes | Required Texts: • C. Chamon and C. Mudry, manuscript on Fractionalization of Particles in Physics. These notes will be made available to students in the course. |

Literature | Recommended Texts: There are many helpful references available to complement the notes, including: • E. Fradkin, Field Theories of Condensed Matter Physics, 2nd edition (Cambridge Univ. Press) • A. Tsvelik, Quantum Field Theory in Condensed Matter Physics, 2nd edition (Oxford univ. Press) • C. Mudry, Lecture notes on field theory in condensed matter physics (World Scientific Publishing) • B. A. Bernevig with T. L. Hughes, Topological Insulators and Topological Superconductors (Princeton Univ. Press) |

Prerequisites / Notice | Contact: Link |