PHY 745 Group Theory 11-11:50 AM MWF Olin 102 Plan for Lecture 30:

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Presentation transcript:

PHY 745 Group Theory 11-11:50 AM MWF Olin 102 Plan for Lecture 30: Topological analysis of band structures and their relationships to group theory Hasan and Kane, RMP 82, 3045-3067 (2010) What is the notion of topological order Examples 4/5/2017 PHY 745 Spring 2017 -- Lecture 30

Topic for presentation 4/5/2017 PHY 745 Spring 2017 -- Lecture 30

4/5/2017 PHY 745 Spring 2017 -- Lecture 30

pgs. 3045-3067 4/5/2017 PHY 745 Spring 2017 -- Lecture 30

Topology is a mathematical term concerned with the properties of space that are preserved under continuous deformations. 4/5/2017 PHY 745 Spring 2017 -- Lecture 30

Typical band structure of an insulator EF 4/5/2017 PHY 745 Spring 2017 -- Lecture 30

Typical band structure of a metal 4/5/2017 PHY 745 Spring 2017 -- Lecture 30

Much more exciting band structure -- graphene -- From: Castro Neto et al. Rev. Mod. Phys. 81 109-162 (2009) 4/5/2017 PHY 745 Spring 2017 -- Lecture 30

Digression – what happens when electrons confined to two dimensions are placed in a magnetic field? 4/5/2017 PHY 745 Spring 2017 -- Lecture 30

Spectrum B>0 B=0 E n=3 n=2 n=1 n=0 4/5/2017 PHY 745 Spring 2017 -- Lecture 30

Topological similarity of insulator and 2-dimensional Landau-level (quantum Hall) system 4/5/2017 PHY 745 Spring 2017 -- Lecture 30

These two donuts illustrate two different topologies Weird sense of humor These two donuts illustrate two different topologies 4/5/2017 PHY 745 Spring 2017 -- Lecture 30

From Nature 464, 194-198 (2010) 4/5/2017 PHY 745 Spring 2017 -- Lecture 30

Examples of these surface states 4/5/2017 PHY 745 Spring 2017 -- Lecture 30

Productive strategy for studying these effects Model Hamiltonian  First principles calculations 4/5/2017 PHY 745 Spring 2017 -- Lecture 30

4/5/2017 PHY 745 Spring 2017 -- Lecture 30

Many assumptions  quadratic model in region of interest Model Hamiltonian Many assumptions  quadratic model in region of interest 4/5/2017 PHY 745 Spring 2017 -- Lecture 30

Band structure of LaBiTe3 4/5/2017 PHY 745 Spring 2017 -- Lecture 30

4/5/2017 PHY 745 Spring 2017 -- Lecture 30

Additional review articles -- pgs. 1057-1110 4/5/2017 PHY 745 Spring 2017 -- Lecture 30