Greater Bay Area
Summer School
A guided archive of lecture notes and presentations, following a conceptual path from the Cooper instability and BCS theory to quantum geometry, mesoscopic physics, and Majorana modes in topological superconductors.
One physical narrative, four connected chapters.
This collection is organized by physical questions, rather than by file name. We start with why electrons pair, ask how band geometry affects pairing, introduce real-space scales and scattering, and finish with topological superconductivity and its possible signatures.
Where both a written note and a presentation are available, they appear together. Read the notes for derivations and use the slides for a faster conceptual overview.
From the Cooper Problem to BCS Theory
Begin with the instability of a Fermi sea under an attractive interaction, then move from a single bound pair to the coherent, self-consistent many-body BCS ground state.
Guiding question How does a two-electron bound-state instability become a macroscopic superconducting condensate?
The Cooper Problem: Full Derivation
Cooper Problem
In 1956 Leon Cooper showed that the normal Fermi sea of a metal is unstable against the formation of a bound pair of electrons, provided there is any net attractive interaction. This note provides the full derivation of the bound-state energy.
The Cooper Problem: Instability of the Fermi Sea
Cooper Problem
Lecture slides covering the physical setup, trial wave function, Schrödinger equation, and Cooper's model interaction leading to the bound-state energy of the pair.
From Cooper Pair to BCS Ground State
BCS Theory
A document developing both the single-pair instability and the full BCS variational treatment, working in the language of second quantization.
From Cooper Pair to BCS Ground State
BCS Theory
Slides illustrating the promotion of the single-pair instability into a self-consistent many-body ground state of a coherent condensate of pairs.
Flat Bands & Quantum Geometry
The flat-band limit raises a subtle question: when ordinary band velocity is suppressed, what fixes the spatial extent of a Cooper pair? These materials focus on the role of the quantum metric.
Guiding question What geometric information survives when a band becomes nearly dispersionless?
Size of a Cooper Pair in a Flat Band
Flat-Band
In a flat band, electrons do not move, and the familiar estimate predicts a vanishing pair size. These notes resolve the paradox by showing a new length scale encoded in the quantum metric \(g_{\mu\nu}(\mathbf{k})\).
The Size of a Cooper Pair in a Flat Band
Flat-Band
Graduate lecture slides on quantum geometry and superconductivity, demonstrating how the quantum metric governs the physical size of pairs in flat bands.
Mesoscopic Physics & Localization
Develop physical intuition for electronic length scales, then examine how scattering produces Friedel oscillations and how disorder leads to localization in one dimension.
Guiding question How do energy scales, scattering, and interference reappear as measurable real-space structures?
Fermi-Velocity-Controlled Length Scales
Length Scales
Every energy scale \(E\) in a metal defines a characteristic length \(L_E \sim \hbar v_F / E\). These notes develop this master relation and tour its realizations in superconductors, Kondo screening, and localization.
Solid State Length Scales
Length Scales
Lecture slides explaining how \(\hbar v_F \approx 0.66 \text{ eV} \cdot \text{nm}\) converts every energy scale in a metal into a length, unifying Josephson junctions and Kondo physics.
Friedel Oscillations in One Dimension
Friedel Oscillations
A self-contained derivation of Friedel oscillations produced by a single delta-function impurity in a 1D gas of non-interacting electrons, using the T-matrix/Green's function approach.
Friedel Oscillations in 1D (L2P2)
Friedel Oscillations
Slides on the Lehmann representation, Dyson equation, and the thermal length cutoff for Friedel oscillations in one dimension.
Anderson Localization in One Dimension
Anderson Localization
An introduction to Anderson localization through the exactly tractable 1D tight-binding model, deriving the transfer-matrix formulation and computing the localization length in the weak-disorder limit.
Anderson Localization in 1D (L2P3)
Anderson Localization
Lecture slides covering transfer matrices, Lyapunov exponents, numerical QR re-orthogonalization, and the scaling theory of localization.
Topological Superconductivity & Majorana Modes
Close the series with the Kitaev-chain viewpoint, Majorana-induced selective equal-spin Andreev reflection, and quantum-anomalous-Hall-based superconducting platforms.
Guiding question Which transport signatures and device geometries can probe Majorana physics?
Introduction to Topological Superconductors
Topological SC
An overview of topological superconductivity, covering the 1D Kitaev model, realizing p-wave SC in condensed matter, and detection schemes like selective equal spin Andreev reflections.
Majorana Induced Equal Spin Andreev Reflections
Majorana
A detailed presentation on the selective equal spin Andreev reflections induced by Majorana fermions in topological superconductors, exploring the underlying scattering matrices.
Quantum Anomalous Hall Insulators
Majorana Platforms
Exploring Quantum Anomalous Hall Insulators (QAHI) as Majorana platforms. Covers quasi-1D QAHI/SC junctions, devices with cuts, and flat-band topological superconductors.
