Condensed Matter

Below are notes from a course (Special topics in condensed matter physics) that I taught together with Ashvin as his TA during my PhD.

• Localized majorana zero modes
• Classification of 1D time-reversal symmetric fermion chain
• Mean field is qualitatively correct with a rank-1 MPS but a wrong answer with a rank-2 MPS
• Kitaev’s honeycomb model
• Linear response

SL2 in Physics

Exact solutions are rare in many-body physics. They often rely on delicate mathematical structures. One object that keeps showing up in exact solutions is the SL2 group. Here are some notes collecting related results.

• SL2 group and unitary representations
• SL2 = Harmonic oscillator
• SL2 and AdS2
• SL2 and SYK
• Particle in a hyperbolic space with magnetic field

CFT

Conformal field theory (CFT), another much larger class of solvable theories, can be regarded as an upgraded version of the above SL2 story. Here are some notes on these topics.

• Driven oscillator and Rep of Virasoro can meet at Hill’s eqn
• Notes for a journal club
• Coulomb gas formalism
• Global quench in CFTs
• Spectral form factor of CFTs
• CFT approach to bipartite entanglement of gapped states

QFT

CFT is a small subset of a larger class of theory, quantum field theory (QFT).

• Analytical structure of two point functions
• Average null energy condition: an example
• Quantizing Chern-Simons theory
• Higher-form symmetry and some toy examples of its anomalies

Stat-Mech and Thermodynamics

• Probabilities
• Free quantum gases
• Ising model on planar graphs

• Yang’s conjecture around 2018 (When I visited Tsinghua in 2018, Yang was working on a mathematical conjecture and would like a few students to verify it for him. Here is a MMA note stating that conjecture and some numerical tests. It might be related to Lee-Yang zeros.)
• Dirichlet theorem
• ABC of Teichmuller space and Fenchel-Nielsen coordinate