Teaching

Quantum Phases of Matter ( Varsha semester 2025 )

An introductory course on modern quantum condensed matter theory aimed at advanced undergraduates, master and PhD students. It aims to acquaint the students with modern understanding of various phases of matter that are manifestly quantum mechanical.

The course assumes only basic knowledge of quantum mechanics and some statistical mechanics. Even the concepts of second-quantisation and band theory are taught in the preparatory module. Give this assumed background, it gives the core concepts and an overview into topics to the extant of sufficiently preparing the interested students to undertake further self-study into the nitty-gritty details and advanced topics.

 It has three modules each covering important paradigms of our current understanding:

 0. Preparation: Second quantization and band theory.

1. Symmetry protected topological phases.

2. Interacting systems and spontaneous symmetry breaking.

3. Topological order.

Following are the lecture notes. Disclaimer: The notes might have many errors given that this course was 'experimented' for the first time. Any feedback would be appreciated.

1. Lecture note 0 : Introduction

2. Lecture note 1: Second quantization

3. Lecture note 2: Lattices and tight-binding models - Lattice translational symmetry in QM, Bloch's theorem, tight-binding models, Bloch Hamiltonians.

4. Lecture note 3: Geometry and Topology of Bloch Bands - Dirac Monopole and topological quantisation, Gauge structure in projective Hilbert space, Berry structures in bands- Berry phase, Zak phase, Berry curvature and Chern number, relation to Hall conductance.

5. Lecture note 4: Symmetry Protected Topological Phases - Anti-unitary symmetries in QM, action on single-particle lattice Hamiltonians and k-space Bloch Hamiltonians, Ten-fold classification, Consequences for Berry connection and Curvature, Dirac fermions and TR-broken phases in 2D, Chern insulator.

6. Lecture note 5: One-dimensional topological phases - Su-Schrieffer-Heeger model (symmetries, winding, Zak phase, edge modes), Kitaev model for Majorana wire (BdG Hamiltonian, Phase diagram, Majorana zero modes, Braiding).

7. Lecture note 6:  Interacting electron models - Four-fermion interaction Hamiltonian- Direct, exchange and Hubbard terms, Symmetries of Hubbard model, Half-filling and super-exchange, Metal insulator transition, t-J model and high Tc superconductor phase diagram, Mean field theory for magnetic instability.

8. Lecture note 7: LSM theorem -  Importance of exact results and constraints, Gap vs Gapless, Lieb-Schultz-Mattis theorem for spin chains - Brief proof in 1D. Oshikawa's version for fermions.

9. Lecture note 8: One-dimensional spin chain -  Jordan-Wigner transformation and exact solution for transverse field XY spin chain, Phase diagram, quantum phase transitions and relation to Kitaev spin chain.

10. Lecture note 9: Topological order - Ground state degeneracy, anyons, Toric code - ground state and excitations. 

List of useful resources

List of reading projects taken up by students for end-sem presentations:

1. Quantum Hall Effect.

2. Fractional Quantum Hall phases.

3. Quantum Hall Anyons.

4. Haldane Model for Chern insulator.

5. Weyl semimetals.

6. Aharnov-Anandan phase.

7. Topological photonics.

8. Aubry-Andre-Harper Model.

9. Spin Waves.

10. Floquet phases

11. Thouless Pumping.

12. Macroscopic Quantum tunneling.

13. Luttinger's theorem.

14. Landau-Ginzburg theory.

15. Majorana Nanowire realisations.

16. Quantum Error correction in Toric code.

Electrodynamics and Special theory of Relativity, (Landau-Lifshitz course, Vol2, Classical Theory of Fields) (Vasanth Sem 2024, Varsha Sem 2025).