学习规划

• 2022/10/9 这周计划学习一下 2016年的综述，以及组里的两篇投影表示的重要文章，周末把最近学的内容整理一下：TI/TSC 中的对称性，基于对称性的分类：K理论/Clifford代数简单的结论，Band crossing/inversion，拓扑不变量的定义和计算，几个简单的模型（SSH，Graphene，Kitaev chain)(主要是熟练拓扑数的计算和各种变换，对称性的应用），空间群，投影表示的原理。这部分整理一个周末肯定不够，下周继续，等这部分物理初步整理完了，可以去看Berry phase的文章，以及更详细的分类工作；继续学习Altland上拓扑场论相关的部分，之后去看Nakahara；可以开始学习Moore的书了；

一些基本概念

• Based on the idea of tensor product states, the ground state of a many body system H can be classified into SRE and LRE states, the latter is known as having intrinsic topological order. Further take symmetry into consideration, the SRE is then promoted to SPT states, as long as keeping the energy gap and preserving symmtry, the ground state can be topological: can’t be deformed into trivial states.

• The field theoretical definition of topological phases of matter is that the long-range effective theory of the ground state is a topological field theory.There are several types of topological field theories: Chern-SImons term ,WZW term , theta term, DW TQFT, TV TQFT,… The bulk-boundary correspondence implies that at the boundary there can be CFT presenting.

• The bosonic SPT states are classified by group cohomology, while it is not clear if fermionic SPT states can be classified by super group cohomology.

• For noninteracting fermionic SPT states, there’s a complete classification, known as 10-fold way.This classification result can be obtained using several ways: K theory in topological band theory, extension problem of real/complex Clifford algebra(Using Dirac Hamiltonian),Anderson localization at the boundary using NLSMs and homotopy(of classifying space), gauge anomalous response(topological field theories). Last method also applies in interacting cases.

• The 10-fold classification have several features:
  -Bott periodicity
  -The table is constructed with antiunitary symmetries(unitary ones will make the classification finer, but “degenerate” in non-interacting phases): T,C,S.
  -The 10-fold classification have geometrical meanings:Cartan symmetric spaces, and have something to do with disorder(A-Z random matrix theory)

• In each case, the phases are classified by topological invariants,these topological invariants coincide with coefficients of the topological terms and is quantized due to symmtry. They can also been obtained using topological band theory: the Berry phase,…,these topological invariants can be considered as indicating the homotopy group of the k-space band Hamiltonian.
  -There’s certain dimension reduction, descent

*Specific physical implications of topology:
-Anomalous boundary/defect states,protected by symmtry,topological(stable) against disorder(extended),these states are gapless.
-The topological invariants can identified with polarizations physically and related to boundary chiral/helical states( Weyl/Dirac/Majorana)
-Exotic boundary states when the energy gap is open.

• These feature also applies to certain gapless phases:semimetal, non-ordinary SC,… Where the topology is not captured by filled bands but by the Fermi surface/line/point.

*…

作为基础的数学和物理补充

• Nakahara: Topology,geometry and physics

• Moore: Quantum Symmetries and Compatible Hamiltonians

• Moore: Quantum Symmetries and K-Theory(a shorter version)

• Moore: Clifford algebra

• Moore: Abstract group theory(focus on group cohomology and group extension)

• Karoubi: K-theory: an introduction

• …

• Tong: Instanton (topological defect,especially index theorem)

• Altland: chapter 9

• Topology,geometry and quantum interference in condensed matter physics.

文献：Non-interacting fermionic SPT(TI,TSC,Nodal systems)：

综述及教材

• Qi_topological insulator and superconductor

• Topological insulator_Hasan

• Classification of topological quantum matter with symmetries

• Topological insulators Dirac equation in condensed matter (Shen, Shun-Qing)

• A Short Course on Topological Insulators Band Structure and Edge States in One and Two Dimensions

• Topological Insulators and Topological Superconductors by B. Andrei Bernevig Taylor L. Hughes

• Topological quantum by H.Simons

• Witten_Three Lectures On Topological Phases Of Matter

• Another note by Witten

• …

其他的笔记

• Q.R Wang Lecture notes
• Topo-Invariants by Y.X Zhao
• …

讲分类的文章和用K理论，NLSM,Clifford代数的文章

• Classification of topological insulators and superconductors in three spatial dimensions
• Classification of topological insulators and superconductors
• Topological insulators and superconductors tenfold way and dimensional hierarchy
• Kitaev_Periodic table for topological insulators and superconductors
• Topological phases classification of topological insulators and superconductors of non-interacting fermions, and beyond_Ludwig_2015
• Stability of fermi surfaces and K theory
• Topological classification and stability of fermi surfaces
• Disordered Weyl semimetals and their topological family
• Topoligical phases_isomorphism,homotopy and K-theory
• Topological classification with additional symmetries from Clifford algebras
• Twisted equivariant matter
• Clifford modules and symmetries of topological insulators
• …

Wilson loop方法

• Benalcaza_Bernevig_Hughes_2017
• Equivalent expression of Z2 topological invariant for band insulators using the non-Abelian Berry connection

其他相关的物理基础

Topological nodal line semimetals

经典原始文献

• Kane_Z2 Topological Order and the Quantum Spin Hall Effect
• Kane_Time reversal polarization and Z2 adiabatic spin pump
• Kane_Topological insulators in 3D
• Kane_Topological insulators with inversion symmetry
• Kitaev_Unpaired Majorana fermions in quantum wires
• Qi_Topological field theory of timereversal invariance topological insulators
• A&Z_Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures
• TKNN
• Laughlin
• Haldane model

组内文章

文献：General SPT:

主要是Kitaev,Wen,Witten,…的文章。