## PHY 444: Many Body Physics

3 credits | Prerequisites: PHY 203, 304

### Course rationale

Quantum Mechanics is the most accurate description of physics at a small scale. To understand solid-state physics, it is there imperative to apply the proper quantum formalism. This course will describe the application of Quantum Mechanics on many bodies. Some novel aspects of condensed matter physics will also be covered in this course.

### Course content

Introduction to Quantum Fields: collective quantum fields, harmonic oscillator-zero dimensional field theory, collective modes-phonon, the thermodynamic limit, the continuum limit; Bosons and Fermions: commutation and anti-commutation algebras, field operators, field operators as creation and annihilation operators, interaction, equivalence between many-body Schrodinger equation, identical particles in thermal equilibrium, Jordan-Wigner transformation, the Hubbard model, non-interacting particles in thermal equilibrium; Green’s Functions: driven harmonic oscillators, interaction, Wick’s theorem, generating functional, Green’s functions for bosons and fermions, Gell-Man-Low theorem, generating functional for free fermions, spectral representation; Landau Fermi-liquid Theory: quasiparticle concept, neutral Fermi liquid, Landau parameters, the equilibrium distribution of

quasiparticles, a renormalization of paramagnetism, mass renormalization, quasiparticle scattering amplitudes, Landau-Silin theory, inelastic quasiparticle scattering, the microscopic basis of Fermi-liquid theory; Zerotemperature Feynman Diagrams: the path integral-general formalism, construction of path integral, construction of many-body path integral, field integral for the quantum partition function, Feynman diagrams, symmetry factors, Feynman rules in momentum space, Hartree-Fock energy, exchange-correlation, an electron in scattering potential, response functions; Finite-temperature many-body physics: imaginary time, periodicity and antiperiodicity, Matsubara representation, the contour integral method, generating function, and Wick’s theorem, Feynman diagram, linked-cluster theorem, Hartree-Fock at finite temperature, an electron in disordered potential, electron-phonon interaction, mass renormalization, Migdal’s theorem; Linear Response Theory: fluctuation-dissipation theorem for a classical harmonic oscillator, quantum mechanical response function, spectral decomposition-correlation function, retarded response function, quantum fluctuation-dissipation theorem, calculation of response function, spectroscopy, Kramers-Kronig relation, the Kubo formula, Drude conductivity, electron diffusion, Anderson localization.

### Course objectives

- Familiarize with Quantum Field Theory in Condensed Matter Physics.
- Familiarize yourself with the path integral method.
- Use the Feynman diagram to calculate different quantities.
- Understand the basics of linear response theory.
- Apply linear transport theory to compute transport parameters.

### References

- Introduction to Many-Body Physics (1st edition) by Piers Coleman
- Condensed Matter Field Theory (2nd edition) by Alexander Atland, and Ben D. Simons
- Many-Particle Physics (3rd edition) by Gerald D. Mahan
- Modern Condensed Matter Physics (1st edition) by Steven M. Grivin, and Kun Yang