# Polarons

**A short overview**

In this advanced theory lecture we will learn about the formation of so-called quasiparticles. As a key example we will consider polaron quasiparticles, which emerge due to the interaction of single impurities with their many-body environment. Polarons are quite ubiquitous in physics and they determine the properties of ultracold atomic system and solid state materials alike. In this lecture I aim to provide the theoretical tools and knowledge to understand and practically describe state-of-the-art experiments so that students can dive right into current theoretical and experimental research in this highly active field of condensed matter physics.

Note: For this specialized lecture there will be no homework assignments.

**Contents**

**1. Introduction and motivation**

1.1 The concept of quasiparticles

1.2 The idea behind polaron quasiparticles

1.3 An example from the classical world

1.4 Our motivation: Current research on ultracold atoms and atomically thin materials

**2. Theoretical foundation: Second Quantization**

2.1 Motivation: Identical particles and many-body wavefunction

2.1.1 Example: Fermions in a box

2.2 Fock space and occupation numbers

2.3 Creation operators, their algebra and basis transformation

2.4 Representation of operators in second quantization

2.4.1 One-body operators

2.4.2 Two-body operators

2.4.3 Many-body Hamiltonian for electronic system

**3. Paradigm model: The Fröhlich polaron **

3.1 Motivation: Electrons in a crystalline lattice

3.2 Phonons

3.2.1 Uncoupled oscillators

3.2.2 Phonons in solids

a) 1D case — acoustic vs. optical phonons

b) 3D case

3.3 Electron-phonon interaction

3.4 Special case: Polar crystal

3.5 Fröhlich model

3.6 Perturbative Solution

3.6.1 Free case

3.6.2 Perturbation theory

3.6.3 Application to Fröhlich model

3.6.4 Polaron energy and effective mass

3.6.5 Polaron Quasiparticle weight

3.6.6 Polaron Lifetime

**4. Polarons beyond the Fröhlich paradigm**

4.1 Atomic Impurity Problems

4.2 Atomic polaron Hamiltonian

4.3 Atomic interactions — Concepts of scattering theory

4.3.1 Two-body scattering theory — approaches

4.3.2 Wave function approach

4.3.3 Diluteness and reduction to s-wave scattering

4.3.4 Scattering length

**5. Fermi polarons and molecules**

5.1 Variational principle

5.2 The Chevy variational ansatz

5.2.1. Calculation of variational energy

5.2.2. Variation

5.2.3. Relating contact interaction strength to scattering length

5.2.4. Solution of the variational equations

5.3. Fermi polaron properties

5.3.1. Energy

5.3.2. Effective mass

5.3.3. Quasiparticle weight

5.4 Molecular ansatz

5.5 The polaron-to-molecule transition

5.6 Implications for the many-body phase diagram

**6. The Anderson Orthogonality catastrophe**

6.1 Background: Infinite Mass Limit

6.2 Hamiltonian

6.2.1 Basis Transformation

6.2.2 Single-Particle Ansatz

6.3 Energy

6.4 Analytical Derivation of Quasiparticle Weight

6.4.1 Calculation of Slater Determinant

6.5. The Anderson Orthogonality Catastrophe