This page collects lecture notes from Prof. Fengcheng Wu (Wuhan University), Prof. K.T. Law, and additional contributors on topics including topological moiré bands, fractional Chern insulators, quantum Monte Carlo sign problems, higher‑form symmetries, quantum anomalies, conformal field theory, BCS superconductivity, quantum geometry, Anderson localization, and Friedel oscillations. All PDFs are stored in the PPTX/ folder.
I. Topological Moiré Bands & Fractional Chern Insulators
📘 Lecture 2.2 – Topological Moiré Bands
Theory of topological bands in moiré superlattices, including continuum models, layer‑pseudospin skyrmions, quantum geometry, and Wilson loop methods.
📄 Download PDF📘 Lecture 3.1 – Integer Chern Insulators & Competing States in tMoTe₂
Quantum anomalous Hall insulators based on the Kane‑Mele‑Hubbard model, Hartree‑Fock approximation, and phase diagrams with competing orders in tMoTe₂.
📄 Download PDF📘 Lecture 3.2 – Collective Excitations
Collective excitations in integer QAHIs: excitonic optical response (Bethe‑Salpeter equation) and topological magnons, plus spin models and domain walls.
📄 Download PDF📘 Lecture 4 – Abelian & Non‑Abelian Fractional Chern Insulators in tMoTe₂
Lattice analogs of fractional quantum Hall effects: skyrmion picture, generalized Landau levels, ideal vs. nonideal quantum geometry, and Abelian (Jain sequences) as well as non‑Abelian (\(\nu = 5/2\)) fractional states.
📄 Download PDFII. Supplementary Lecture Notes
II-A. Quantum Monte Carlo & The Sign Problem
📘 The Sign Problem in Quantum Monte Carlo
Author: Xiao Yan Xu
Pedagogical exposition covering negative/complex weights in SSE, world‑line, and DQMC; average sign and reweighting cost; structural cures (Marshall rotations, determinant pairing, fermion bags, merons, Majorana positivity); and fundamental limits (NP‑hardness, stoquastic‑basis search, topological obstructions).
II-B. Higher‑Form Symmetry & Quantum Anomalies
📘 Compact Boson, Duality, Gauging, and Anomalies
Source: Summer school handout
Self‑contained tour of 1+1D compact boson: momentum/winding \(\mathrm{U}(1)\) symmetries, mixed anomaly, T‑duality, discrete gauging, emergent dual symmetries, BF pairing, Kramers‑Wannier duality, and generalization to higher‑form symmetries in Maxwell theory (electric/magnetic 1‑form symmetries, Coulomb/Higgs/confined phases).
📘 量子反常简介（'t Hooft Anomaly）
Author: 姚元 (Yuan Yao)
Elementary introduction using a (0+1)‑dimensional Dirac fermion on a thermal circle. Shows the conflict between local \(\mathrm{U}(1)\) gauge invariance and global charge conjugation, leading to a mixed anomaly; demonstrates anomaly inflow and proves that anomalous symmetries imply ingappability.
II-C. Conformal Field Theory
📘 Conformal Field Theory – From Basics to Minimal Models
Author: 朱伟 (Wei Zhu)
Self‑contained notes covering: conformal transformations in \(d\) dimensions, primary fields and correlation functions, holomorphic nature in 2D, Witt and Virasoro algebras, central charge, radial quantization, state‑operator correspondence, OPE, highest‑weight representations, Kac determinant, unitary minimal models, and the critical Ising model as a worked example (fusion rules, critical exponents).
III. K.T. Law Lecture Series – Superconductivity, Quantum Geometry, and Disorder
III-A. Cooper Problem & BCS Theory of Superconductivity
📊 Beamer – The Cooper Problem
Type: Presentation slides
Introduction to the Cooper problem: physical setup, trial wavefunction, two‑body Schrödinger equation, model interaction, self‑consistency, and bound‑state energy \( \Delta = 2\hbar\omega_D e^{-2/N(0)V} \). Emphasizes the essential singularity and instability of the normal state.
📊 Beamer – From the Cooper Pair Problem to the BCS Ground State
Type: Presentation slides
Bridge from single‑pair Cooper problem to full BCS many‑body state: reduced BCS Hamiltonian, variational wavefunction \( |\Psi_{\mathrm{BCS}}\rangle = \prod_k (u_k + v_k c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger)|0\rangle \), energy minimization, gap parameter, coherence factors, pair amplitude \( g_k \), gap equation, and condensation energy.
📘 Lecture Note – The Cooper Problem (Concise)
Type: Lecture notes
Concise written derivation of the Cooper problem: motivation, trial wavefunction, amplitude equation, model interaction, bound‑state energy, and physical significance.
📘 Lecture Note – Cooper Problem & BCS Theory (Full)
Type: Comprehensive lecture notes
Complete written version covering both Cooper problem and full BCS variational treatment: two‑body Schrödinger equation, model interaction, bound‑state energy, reduced BCS Hamiltonian, BCS trial wavefunction, energy minimization, coherence factors, pair amplitude, gap equation, and condensation energy.
III-B. Quantum Geometry & Flat‑Band Superconductivity
📊 Beamer – The Size of a Cooper Pair in a Flat Band
Type: Presentation slides
Resolves the paradox of flat‑band superconductivity: introduces the quantum metric \( g_{\mu\nu}(\mathbf{k}) \), derives \( \xi_{\mathrm{pair}}^2 = 2\langle \operatorname{tr} g \rangle_{\mathrm{BZ}} \), and proves the topological lower bound \( \xi_{\mathrm{pair}}^2 \gtrsim |C| a^2/\pi \) for Chern bands.
📘 Lecture Note – The Size of a Cooper Pair in a Flat Band
Type: Comprehensive lecture notes
Self‑contained pedagogical introduction: conventional BCS coherence length, flat‑band Cooper problem with band projection, binding energy \( E_b = -U/N_{\mathrm{orb}} \), quantum geometric tensor \( Q_{\mu\nu} = g_{\mu\nu} - \frac{i}{2}\Omega_{\mu\nu} \), pair size \( \xi_{\mathrm{pair}}^2 = 2\langle \operatorname{tr} g \rangle_{\mathrm{BZ}} \), topological lower bound, and equivalence to Wannier spread.
III-C. Fermi‑Velocity‑Controlled Length Scales in Solids
📊 Beamer – Fermi-Velocity-Controlled Length Scales in Solids
Type: Presentation slides
Unified framework: \( L_T = \hbar v_F/2\pi k_B T \), \( \xi_0 = \hbar v_F/\pi\Delta \), \( \xi_K = \hbar v_F/k_B T_K \), \( \ell = v_F\tau \), diffusive generalization \( L_E \to \sqrt{\hbar D/E} \), and Thouless energy \( E_{\mathrm{Th}} = \hbar D/L^2 \).
📘 Lecture Note – Fermi-Velocity-Controlled Length Scales in Solids
Type: Comprehensive lecture notes
Full derivations: linearization at Fermi surface, three readings of \( L_E = \hbar v_F/E \), thermal length via Matsubara poles, SNS Josephson decay, BCS coherence from pair wavefunction, Kondo cloud from Yosida variational state, mean free path and 1D localization, diffusive generalization, and Thouless energy.
III-D. Friedel Oscillations in One Dimension
📊 Beamer – Friedel Oscillations in One Dimension
Type: Presentation slides
Green's function derivation: Lehmann representation, free Green's function \( G_0^R(x,x';E) = -\dfrac{i}{\hbar v_E} e^{ik_E|x-x'|} \), Dyson equation, separable \( T \)-matrix \( t(E) = u/(1 + iu/\hbar v_E) \), zero‑temperature result \( \delta n(x) \sim -\dfrac{u\nu_F}{2}\dfrac{\cos(2k_F x)}{|x|} \), and finite‑temperature thermal length \( \xi_T = \hbar v_F/2\pi k_B T \).
📘 Lecture Note – Friedel Oscillations in One Dimension
Type: Detailed lecture notes
Six‑section written note: physical setup, Lehmann representation and spectral function, free Green's function, Dyson equation and separable \( T \)-matrix, charge density from Green's function, zero‑temperature result, and finite‑temperature thermal envelope \( (x/\xi_T)/\sinh(x/\xi_T) \).
III-E. Anderson Localization in One Dimension
📊 Beamer – Anderson Localization in One Dimension
Type: Presentation slides
1D Anderson model: tight‑binding Hamiltonian, three‑term recurrence, transfer matrix \( T_n \in \mathrm{SL}(2,\mathbb{R}) \), Lyapunov exponent \( \gamma(E) = \lim_{N\to\infty} \frac{1}{N}\ln\|\mathcal{M}_N\| \), localization length \( \xi(E) = 1/\gamma(E) \), clean‑chain limit, weak‑disorder Thouless formula \( \xi(E) = 2(4t^2 - E^2)/\sigma^2 \), Prüfer variables, and numerical QR re‑orthogonalization.
📘 Lecture Note – Anderson Localization in One Dimension
Type: Detailed lecture notes
Complete written derivation: tight‑binding Hamiltonian, transfer matrix, Furstenberg and Oseledec theorems, clean‑chain limit, weak‑disorder expansion in Prüfer variables, Thouless formula, numerical implementation, and connection to scaling theory of localization.
III-F. Solid‑State Length Scale Overview
📊 Beamer – Solid-State Length Scale Overview
Type: Presentation slides
High‑level summary: hierarchy of length scales from \( \lambda_F \) to \( \xi_K \), diffusive generalization, Thouless energy as the inverse dictionary, and limits of the framework (flat bands, non‑Fermi liquids, Dirac materials).
