Topological Qunautm Matters

Time:

Every Mon. & Wed. 13:30-15:05, from 2021-9-13 to 12-17 (weeks 1-14 of the fall semester in Tsinghua)

Venue:

It will be a combination of offline (宁斋W11, Ning Zhai W11) and online (腾讯会议tencent meeting: 995 283 0954, password: 654321)

Description:

In this course, we will use topology to understand some exotic quantum phases of matter. The topics will include topological insulators, topological orders, symmetry-protected topological phases, etc. The course will cover both condensed matter models in physics and general mathematical descriptions (such as group cohomology theory and modular tensor categories of knots).

Prerequisites:

Basic topology and quantum mechanics. We will try to make a compromise between mathematics and physics by introducing relevant concepts self-consistently, as there are audience from both sides.

My email:

wangqrATmail.tsinghua.edu.cn

Syllabus:

It may vary depending on the actual speed of the course. (4*45 min/week * 12 weeks)

(1) introduction to TQM, different classes of TQM (bosonic/fermionic, long/short range entangled, with/without symmetry),
1st example: Kitaev's toric code model (homology enters),
2nd example: Haldane's honeycomb model (homotopy enters)

Part I. Bosonic topological orders

(2) quantum double model, twisted quantum double model = Dijkgraaf-Witten gauge theory

(3) introduction to fusion categories, Levin-Wen model = Turaev-Viro model

(4) introduction to modular tensor categories, general description of anyon models by Kitaev

Part II. Topological insulators (fermionic symmetry-protected topological phases without interactions)

(5) introduction to band theory, integer quantum Hall effect, Thouless–Kohmoto–Nightingale–den Nijs number, Chern insulator

(6) examples: Kitaev's Majorana chain, Su-Schrieffer-Heeger model, 2+1D and 3+1D topological insulators, edge theories

(7) symmetries in free fermion system, 10-fold way classification

Part III. Symmetry-protected topological phases

(8) introduction to symmetry-protected topological (SPT) phases, Haldane chain, classification of 1+1D bosonic SPT

(9) Levin-Gu model, introduction to group cohomology, bosonic SPT model from group cohomology

(10) introduction to fermionic SPT phases, other related topics

Lecture notes:

Lecture 1 - introduction

Lecture 2 - (twisted) quantum double model

Lecture 3 - fusion categories and Turaev-Viro model = Levin-Wen model

Lecture 4 - modular tensor categories, Drinfeld center and anyon models

Lecture 5 - integer quantum hall effect and Chern insulator

Lecture 6 - more examples of topological insulators and superconductors

Lecture 7 - symmetries and classifications of free-fermion topological insulators and superconductors

Lecture 8 - introduction to symmetry-protected topological phases

Lecture 9 - group cohomology models

Lecture 10 - introduction to fermionic symmetry-protected topological phases

Videos:

Lecture 1.1 - introduction (2021-09-13)

Lecture 1.2 - introduction (2021-09-15)

Lecture 2.1 - Haldane honeycomb model, quantum double model (2021-09-18)

Lecture 2.2 - quantum double model (2021-09-22)

Lecture 2.3 - twisted quantum double model and Dijkgraaf-Witten gauge theory (1) (2021-10-04)

Lecture 2.4 - twisted quantum double model and Dijkgraaf-Witten gauge theory (2) (2021-10-06)

Lecture 3.1 - fusion categories (1) (2021-10-11)

Lecture 3.2 - fusion categories (2) (2021-10-13)

Lecture 3.3 - Levin-Wen model (2021-10-18)

Lecture 3.4 - Turaev-Viro model (2021-10-27)

Lecture 4.1 - modular tensor categories and Drinfeld center (2021-11-01)

Lecture 4.2 - excitations of Levin-Wen models (2021-11-03)

Lecture 5.1 - band theory, Berry phase, integer quantum hall effect (2021-11-08)

Lecture 5.2 - Laughlin argument, TKNN number and Chern insulator (2021-11-10)

Lecture 6.1 - examples of topological insulators and superconductors (1) (2021-11-15)

Lecture 6.2 - examples of topological insulators and superconductors (2) (2021-11-17)

Lecture 7.1 - symmetries of free-fermion systems (2021-11-22)

Lecture 7.2 - classifications of free-fermion topological insulators and superconductors (1) (2021-11-24)

Lecture 7.3 - classifications of free-fermion topological insulators and superconductors (2) (2021-11-29)

Lecture 8.1 - introduction to symmetry-protected topological phases (2021-12-01)

Lecture 8.2 - Affleck-Kennedy-Lieb-Tasaki model (2021-12-06)

Lecture 9.1 - Levin-Gu model (2021-12-08)

Lecture 9.2 - group cohomology models (2021-12-13)

Lecture 10.1 - introduction to fermionic symmetry-protected topological phases (2021-12-15)

In case that the tencent platform has problems, the recordings are also available at the tsinghua cloud.