Syllabus

Module 1 (7 hours and 20 minutes)
1st quantization, Hilbert space, fermions, bosons, 2nd quantization, Fock basis, one and two body operators, density operator formalism.


Module 2 (7 hours and 20 minutes)
Models for free fermions, tight binding models, canonical transformation of operators, Fourier transformation for diagonalization, SSH model.


Module 3 (5 hours and 30 minutes)
SSH model: topological phase transitions, winding number, trivial and topological phases, symmetries: TRS, chiral, particle-hole symmetry.


Module 4 (5 hours and 30 minutes)
Spin model: Transverse Ising model, Jordan-Wigner transformation, parity operator (even-odd subspaces), PBC and ABC for fermionic Hamiltonian, winding number.


Module 5 (7 hours and 20 minutes)
Schrodinger, Heisenberg, and interaction representation, Green's function at zero temperature, retarded and advanced Green's function, single particle for fermions, Green's function for time-dependent Hamiltonian, and Green's function for quantum gas.

Introduce the mathematical formalism of quantum mechanics using Hilbert space and quantization techniques.

Develop an understanding of second quantization and the Fock space formalism for fermions and bosons.

Explore free fermion models, including tight-binding models and canonical transformations.

Investigate topological properties of condensed matter systems through models like the SSH model and spin models and provide an introduction to Green's function approach.

At the end of the course, students will be able to:
CO1. Understand the basic mathematical framework of quantum mechanics, including Hilbert space and quantization techniques.
CO2. Apply second quantization and the Fock space formalism to describe fermionic and bosonic systems.
CO3. Analyze free fermion (quadratic) models and perform their diagonalization in real and k-space.
CO4. Explain topological phase transitions using winding numbers in the SSH and spin models.

J. J. Sakurai, Advanced Quantum Mechanics, Pearson Education India , 1st Edition (2002)

J. K. Asbóth, L. Oroszlány, A. Pályi, A Short Course on Topological Insulators, Springer , (2016)

John W. Negele & Henri Orland, Quantum Many-Particle Systems, Addison-Wesley , (1988)

P. Coleman, Introduction to Many-Body Physics, Cambridge University Press , (2015)

3. The quantum Ising chain for beginners, G. B. Mbeng, A. Russomanno, G. E. Santoro, SciPost Phys. Lect. Notes 82 (2024).