Details of PH4206 (Spring 2020)

Level: 8 Type: Theory Credits: 4.0

Course CodeCourse NameInstructor(s)
PH4206 Quantum Many-body Theory Amit Ghosal

Syllabus

  • Review of First and Second Quantization: Quantum Mechanics with many particles; First quantization, many-particle systems; Operators in first quantization; Second quantization, basic concepts; The occupation number representation; The boson creation and annihilation operators; The fermion creation and annihilation operators; The general form for second quantization operators; Basis change in second quantization; Examples -- Operators for kinetic energy, spin, density, and current; The Coulomb interaction in second quantization.

  • Mean Field theory (the easiest method for quantum many-body systems): The art of mean field theory; Concept of broken symmetry; HartreeFock approximation in second quantization notation; Examples -- The Heisenberg model of ionic ferromagnets; The Stoner model of metallic ferromagnets; Breaking of global gauge symmetry and Bogoliubov mean field theory of BCS superconductors.

  • Green's Functions:

    1. Preamble: Time evolution pictures -- Schrodinger, Heisenberg and Dirac (interaction) Pictures; Time dependent creation and annihilation operators; ``Classical'' Greens functions; Greens function for the one-particle Schrodinger equation.

    2. Single-particle Greens functions of many-body systems: meaning and significance; Greens function of free electrons; The Lehmann representation; The spectral function; Broadening of the spectral function due to interactions; Measuring the single-particle spectral function -- spectroscopy; Two-particle correlation functions of many-body systems.

    3. Equation of motion theory of Single-particle Greens functions: Example -- Andersons model for magnetic impurities; Mean-field approximation for the single impurity Anderson model; The equation of motion of Green's functions for the Anderson model and comparison of solution with MFT results; Correlations in the Anderson model: Kondo effect; The equations of motion for the two-particle correlation function; The Random Phase Approximation (RPA).

    4. Imaginary time Greens functions: Matsubara Greens functions: definition and Fourier transform; Connection between Matsubara and retarded and advanced Green's functions; Examples -- Single-particle Matsubara Greens function; Evaluation of Matsubara sums -- Summations over functions with simple poles; Summations over functions with known branch cuts; Equation of motion for Matsubara Greens functions; Wicks theorem and example of polarizability of free electrons.


  • Feynman Diagrams (if time permits): Non-interacting particles in external potentials; Random impurities in disordered metals; Feynman diagrams for the impurity scattering; Impurity self-average.

Prerequisite
Intermediate Quantum Mechanics, Basic Statistical Mechanics

References

  1. Henrik Bruus and Karsten Flensberg, Many-body quantum theory in condensed matter physics (up to chapter 11), Oxford Univ. Press, Oxford (2007).
  2. Gerald D. Mahan, Many Particle Physics, Plenum Press, New York (1993).
  3. Piers Coleman, Introduction to Many-Body Physics, Lecture Notes, Rutgers Univ.

Course Credit Options

Sl. No.ProgrammeSemester NoCourse Choice
1 IP 2 Not Allowed
2 IP 4 Elective
3 IP 6 Not Allowed
4 MR 2 Not Allowed
5 MR 4 Not Allowed
6 MS 8 Elective
7 RS 1 Elective
8 RS 2 Elective