
### Theories of Quantum MatterQuantum Mechanics of Many Body Systems

Welcome to the Part III Physics course Theories of Quantum Matter, running in Michaelmas 2018. The goals and structure of this course are described here.

There will be four supervisions, each with their own set of problems.

## Important Announcement 1

In this course $\hbar = 1$. Momentum has the units of inverse length; energy has units of inverse time; angular momentum is dimensionless, etc..

## Important Announcement 2

In the text you will see questions contained in boxes like this one. You will be answering these questions in the lecture!

## Outline of Lectures

### Many Body Wavefunctions

Product States. Fermi gas. Density, density matrix, and pair distribution.

### Quantum Hall Effect

Landau levels. Laughlin wavefunction. Fractional charge, Fractional statistics.

### The Elastic Chain

Quantizing a chain. Ground state displacement fluctuations.

### Spin Models

Heisenberg model; Heisenberg chain; Magnons; Antiferromagnets; Symmetry breaking; Spin wave theory.

### A is for Annihilation

Product states and occupation numbers. Creation and annihilation operators. The case of fermions. Representation of operators.

### B is for Bunching

Density correlations. Hanbury Brown and Twiss effect. Hartree--Fock theory.

### Lattice Models and Strong Correlations

Tight binding models. Hubbard models and the Mott transition. Superexchange.

### Bose Gas

Gross--Pitaevskii approximation. Superfluidity. Bogoliubov theory.

### Fermi Gas

Interactions described by perturbation theory. Quasiparticles. Landau Fermi liquid.

### Superconductivity

Cooper's problem. BCS theory. The BCS-BEC crossover.

### Response and Correlation

Response functions. Structure factor. Dielectric function. Sum rules.

### Jellium

Perturbation series for partition function and free energy. Screening.

### Jordan-Wigner and Bosonization

Spin-1/2 XY model as a system of free fermions. Bosonization.

### The Kondo Effect

The Kondo model. Divergences at second order. Scaling theory.

### The Lieb-Liniger Model

Bethe's wave function. The Bethe equations. Excited states.