Course info

Venue: Emmy Noether Seminar Room

Class Timings: 11:20 AM to 12:50 PM, Tuesdays and Thursdays

First Meeting: 11 August 2022

Course description: The objective of this course is to enable you to develop ideas and techniques required to understand many-body systems and critical phenomena starting from basic ideas of thermodynamics.

Tentative Topics for the course

• Topic 1 : Basics of Statistical approaches to many-body systems

1. Recap of Fundamentals of thermodynamics 

2. Basics of statistical mechanics for classical systems : Microstate and macrostate, phase space, Liouville’s equation, equal a priori probability, ensembles as tools in statistical mechanics, Entropy and irreversibility.

3. The partition function and thermodynamic free energies, correlation and response functions.

4. Experiments on many-body systems : Thermodynamic, transport and spectroscopic measurements.

5. Classical Many-body systems and application of statistical ideas (recap of relevant parts of the probability theory) : Classical ideal gas, Harmonic Oscillator,two-level systems etc.

6. Role of symmetries : Vibrational modes of molecules, Dilute polyatomic gases etc.

7. Brief introduction to second quantization (Optional)

8. Formulation of quantum statistical mechanics - Quantum microstates and macrostates, density matrix and imaginary time path integrals.

9. Classical and quantum Identical particles, Gibb’s paradox

• Topic 2 : Statistical Mechanics of Non-interacting and weakly interacting Quantum systems

1. Fermi gas, Pauli Paramagnetism, Landau Diamagnetism

2. Bose gas, Vibrations of solid (phonons), Black body radiation, Bose Einstein Condensation.

• Topic 3 : Statistical Mechanics of interacting many-body systems : Mean field theory and fluctuations

1. Microscopic models of interacting many-body systems : spin models of magnetism, exact and approximate solutions.

2. Idea of spontaneous symmetry breaking and phase transitions, emergent phenomena.

3. Curie-Weiss mean field theory of phase transitions in magnets.

4. Fluctuations.

• Topic 4 : Critical phenomena and Landau free energies

1. Role of long wave-length fluctuations and thermodynamic/continuum limit : Coarse grained Landau free energies.

2. Mean field theory of Landau free energies and spontaneous symmetry breaking.

3. Critical exponents and universality.

4. Experimentally relevant correlation functions from Landau free energies.

• Topic 5 : Renormalisation group approaches to critical phenomena

1. Block-spin and Real space RG as a procedure to understand critical phenomena

2. Effective field theories

3. Momentum-shell RG : mode integration, renormalisation, beta-function and Wilson-Fisher fixed points and connection to critical phenomena.

References: 

1. Lecture notes for the course.
   
2. M. Kardar, Statistical Physics of Particles
   
3. R. K. Pathria, Statistical mechanics
   
4. K. Huang, Statistical mechanics
   
5. J. M. Sethna, Statistical Mechanics: Entropy, Order Parameters and Complexity
   
6. M. Kardar, Statistical Physics of fields
   
7. N. Goldenfeld, Lectures of Phase Transitions and renormalisation group
   
8. S. K. Ma, Modern Theory of Critical Phenomena
   
9. Some other books and papers, references of which will be provided in the class.

Course evaluation: Assignments (Typically once every two weeks): 60%, Midterm Exam (in class): 20% and Endterm Exam (take home/in class): 20%

Prerequisites:

1. Familiarity with basic non-relativistic classical and quantum mechanics will be assumed.
2. Familiarity with basic thermodynamics will be assumed.
   
3. Some idea of Probability theory will be helpful.