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## Harish-Chandra Research Institute

### Masters Program in Physics

#### Curriculum and detailed course content

###### Degree granted by Homi Bhabha National Institute

## Introduction

Harish-Chandra Research Institute (HRI) is an institute of international repute for research in Theoretical Physics and Mathematics. It is an aided institution of the Department of Atomic Energy, Government of India. For the last couple of decades HRI has had highly successful graduate programs in Physics and Mathematics. In the span of less than 20 years, the institute has produced over 30 alumni who have gone on to become faculty members at leading research institutions and universities in the country.

After its strong success in imparting training at the Ph. D. level, HRI has decided to expand into a Masters program in Physics. The aim of this program is to provide high level training at the Masters level to prepare students to pursue research careers in all branches of Physics. With a highly modern course structure including a broad range of electives and faculty engaged in cutting-edge research, our program promises a unique experience to students.

##### History of HRI

The institute was founded as the Mehta Research Institute in 1965, with an endowment from the B.S. Mehta Trust, Calcutta. The institute was initially managed by Prof. B. N. Prasad and following his death in January 1966 by Prof. S.R. Sinha (both from the Allahabad University). The first official Director of the Institute was Prof. P. L. Bhatnagar. He was followed by Prof. S.R. Sinha again.

Prof. S. S. Shrikhande joined the Institute as its Director in January 1983. The institute was facing financial difficulties, and Prof Shrikhande sought DAE support for the institute. Following the recommendations of the DAE review committee, the Government of Uttar Pradesh committed to provide a campus for HRI, while the DAE committed to provide full funding for all operational expenses.

In January 1990, the Institute was granted about 66 acres (270,000 sq. m.) of land in Jhunsi town of Allahabad district (Uttar Pradesh) and Prof. H.S. Mani took over as Director. The institute moved to its present campus in 1996. Since then, the institute has grown in facilities, scope of research as well as number of faculty and students.

Till October 10, 2000, the Institute was known as Mehta Research Institute of Mathematics and Mathematical Physics (MRI) after which it was renamed as Harish-Chandra Research Institute (HRI) after the internationally acclaimed mathematician, late Prof. Harish-Chandra. Prof. Ravi S. Kulkarni succeeded Prof. HS Mani as the Director in August 2001 and was followed by Prof. Amitava Raychaudhuri in July 2005. Prof. J. K. Bhattacharjee followed in April 2012 and at present Prof. Pinaki Majumdar is the director of HRI.

##### Present Status

Today HRI is a dream Institute for those who want to pursue research in Theoretical Physics and Mathematics. We provide an academically conducive environment that exposes students to value based quality education and all-round personality development. The institute has fostered an impressive pool of Ph.D. students who are doing very well, both in India and abroad.

We presently offer Ph.D. and Integrated Ph.D. Programs in the field of Theoretical Physics and Mathematics. Apart from mentoring by reputed faculty, the students have access to state-of-the-art infrastructure, such as a library with subscription to leading international journals, campus-wide computer network and a Cluster computing facility.

The number of permanent academic members at HRI is 35, of which 22 of them are in the Theoretical Physics department and rest are in the Mathematics department.

##### New Masters Program

HRI will be starting a new two year Masters program in Physics from August 2016. This program is a standalone two year M.Sc. program with students having no obligation of continuing at HRI for pursuing their Ph. D. The program is designed in such a way that the student not only will get a rigorous training in theoretical physics but also will get exposure to a good amount of experiments.

Although there is no obligation to do further study in HRI, based on a predetermined performance criterion, which will be informed to the students, HRI may offer Ph. D. admissions to some students at the end of first year of M.Sc. The M.Sc. curriculum and course content is explained in detail in the next few chapters.

##### Eligibility and Admission Criteria

HRI selects students from the nationwide screening test, JEST. This test acts as a first filter for admission to the Masters program. Students scoring above the JEST marks cutoff set by HRI will then appear for a written test and interview at HRI. Based on the performance in the written test and interview a list of selected students will be put up.

In addition to the above mentioned selection process students with Bachelors degree in Physics or engineering from any Indian University/Institute should have minimum 60% aggregate marks at Bachelors level. Admission to the Masters program will be automatically canceled if a student selected in the written test and interview fails to get 60% marks in Bachelors examination.

##### Evaluation Process

The theory courses will have classroom teaching and evaluation of students will be based on quizzes, home assignments, mid and end-semester examinations. As of now the grading procedure is absolute and each course will carry 100 marks. The passing criterion is 50% in each subject. First class will correspond to securing an aggregate of 60% or more marks and first class with distinction will correspond to an aggregate of 75% or more marks.

**HRI Ph. D. admission criterion will be informed to students at the time of joining the Masters program.**

##### Physics at HRI

HRI is one of the leading research institutes in India. Currently we pursue research in five areas, namely Astrophysics, Condensed Matter Physics, Particle Physics, Quantum Information and Computation, and String Theory.

The institute has a highly competitive Ph. D. program in Physics and a steady influx of post-doctoral fellows. In addition, a large number of academic activities like conferences workshops and schools are regularly organised on campus - making HRI a leading national centre for scientific interactions. Theoretical Physics at HRI also strongly benefits from interaction with the Mathematicians at HRI.

##### Contact Us

Postal Address |
Telephone |
---|---|

Harish-Chandra Research Institute | +91 (532) 2569509 |

Chhatnag Road, Jhunsi | 2569318, 2569578, |

Allahabad 211019 INDIA | |

E-mail: physjest(at) hri (dot) res (dot) in |
Fax: +91 (532) 2569576, 2567748 |

##### Faculty Members

**The list of Physics faculty members and their area of expertise:**

Name |
Field of Interest |
---|---|

Anirban Basu | String Theory |

Sudip Chakraborty | Condensed Matter Physics |

Tapas Kumar Das | Astrophysics |

Asesh Krishna Datta | Particle Physics |

Aditi Sen De | Quantum Information and Computation |

Tathagata Ghosh | Particle Physics |

Dileep Jatkar | String Theory |

Anshuman Maharana | String Theory |

Pinaki Majumdar | Condensed Matter Physics |

Tribhuvan P. Pareek | Condensed Matter Physics |

Arun Pati | Quantum Information and Computation |

Santosh Kumar Rai | Particle Physics |

Ashoke Sen | String Theory |

Prasenjit Sen | Condensed Matter Physics |

Ujjwal Sen | Quantum Information and Computation |

## Course Structure

The Masters program consists of four semesters of pedagogical lectures

Elective I : Choose one from — Advanced Statistical Mechanics, Fluid Mechanics, General Relativity, Non-linear Dynamics, Quantum Information and Computation I.

Elective II and III : Choose two from — Advanced Topics in General Relativity, Astrophysics, Computational Many Body Theory I, Computational Materials Science, Condensed Matter Physics II, Correlated Electron Systems, Cosmology, Disorder in Condensed Matter, Introduction to Electronic Structure, Matter Out of Equilibrium, Mesoscopic Physics, Quantum Field Theory II, Quantum Information and Computation II, Quantum Many Body Theory, Quantum Optics, Soft Matter, Spectroscopic Methods, String Theory I, Topological Quantum Matter, Ultra Cold Atoms.

## Semester I

##### Classical Mechanics

A rapid review/summary of Newtonian mechanics

Calculus of variations: Concept of variation - Euler equation - Applications - Variation subject to constraints and Lagrange multipliers.

The Lagrangian formulation: Generalised coordinates and velocities - The principle of least action and the Lagrange equations of motion - Extension to constrained systems.

Conservation laws: Symmetries and Noether’s theorem.

Integration of the Lagrange equations of motion: Motion in one dimension - The two body problem, reduced mass and the equivalent one-dimensional problem - Motion in a central field - Kepler’s problem - Scattering.

Small oscillations: Free, damped and forced oscillations in one dimension - Resonance - Damped and forced oscillations - Parametric resonance.

Rigid body motion: Angular velocity - The inertia tensor and angular momentum of a rigid body - The equations of motion- Eulerian angles - Motion of tops - Motion in a rotating frame - Coriolis force.

The Hamiltonian formulation: Hamiltonian and Hamilton equations - Poisson brackets - Dynamics in the phase space - Hamilton-Jacobi equation - Separation of variables and solutions - Action-angle variables - Adiabatic invariants.

Elements of non-linear dynamics: Differential equations as dynamical systems - Lyapunov exponents.

##### Quantum Mechanics I

Basic notions: states, operators, time evolution

One-dimensional problems: harmonic oscillator, periodic potential, Kronig-Penny model;

Three-dimensional problems: central force potential, the hydrogen atom

Charged particle in an electromagnetic field: gauge invariance, Landau levels

Symmetries and conservation laws in QM: Degeneracies, Discrete symmetries

Angular momentum in quantum mechanics: raising and lowering operators, angular momentum addition, Clebsch-Gordon coefficients; Tensor operators and Wigner-Eckart theorem

Time-independent perturbation theory: non-degenerate and degenerate cases, Stark and Zeeman effects

Semiclassical (WKB) approximation and variational methods

##### Classical Electrodynamics

Special theory of relativity and electrodynamics: Lorentz transformation of electromagnetic fields, Lorentz covariant formulation of electrodynamics; gauge invariance, Maxwell equations from action principle.

The electrostatic limit of Maxwell equations, multipole expansion, uniqueness, boundary value problems, solution of Poisson equation.

The magnetostatic limit of Maxwell equations, applications.

Electrodynamics: motion of charges in external fields; electromagnetic waves in vacua and propagation through continuous media; energy-momentum of electromagnetic field and Poynting theorem.

Advanced and retarded Green functions; Lienard-Wiechert potentials; dipole radiation and Larmor’s formula; spectral resolution and angular distribution of radiation from a relativistic point charge; synchrotron radiation; Rayleigh and Thomson scattering; collision problems; Bremsstrahlung and Cerenkov radiation.

Scattering of electromagnetic waves: Rayleigh and Thomson scattering, radiation damping.

##### Mathematical Methods I

Vector Analysis: operations with vectors, scalar and vector fields, gradient, curl and divergence. Line, surface, and volume integrals, Curvilinear coordinate systems, Elements of tensors.

Vector Spaces, linear transformations, scalar product and dual space, bases, linear operators, eigenvalues and eigenfunctions, unitary and hermitian operators

Complex Analysis: functions of a complex variable, analytic functions, integral calculus, contour integrals, Taylor and Laurent series, singularities, residues, principal values, Riemann surfaces, conformal mapping, analytic continuation

Ordinary differential equations: linear ODEs, Green functions, second order differential equations: classification of singularities and local solutions, special functions

Elements of statistics: probability, random walk. Probability distributions

## Semester II

##### Quantum Mechanics II

Scattering theory and applications

Schrodinger and Heisenberg pictures; postulates of quantisation

Time dependent perturbation theory, Interaction picture, Fermi golden rule

Path integrals: propagators, amplitudes as path integrals, Semiclassical methods revisited

Quantum mechanics of many particles, identical particles and symmetries of the wave-function, scattering of identical particles

Relativistic quantum mechanics, Klein-Gordon and Dirac equations and their solutions, gyromagnetic ratio of the electron, relativistic corrections to the Schrodinger equation

Entangled states and Bell inequalities

##### Research Methodology and Numerical Methods

Research Methodology including quantitative methods, communication skills, seminar presentation and review of research papers

Introduction to programming languages: F77, F90 or C

Errors in numerical calculations.

Numerical linear algebra, eigenvalue and eigenvectors.

Interpolation techniques.

Generation and use of random numbers.

Sorting and searching.

Differentiation and Integration (including Monte Carlo techniques)

Root finding algorithms

Optimisation, extrema of many variable functions.

ODEs and PDEs: including FFT and finite difference methods, integral equations.

##### Statistical Mechanics

Basics: phase space, distributions, notion of equilibrium, ensembles, Boltzmann distribution, partition function, calculating observables.

Non interacting classical systems: few level systems, ideal gases, oscillators.

Non interacting quantum systems: method of second quantisation, electrons in metals, relativistic electron systems, electrons in a strong magnetic field, lattice vibrations and phonon physics, photons, blackbody radiation, Bose condensation.

Interacting classical systems: non-ideal gases, van der Waals gas, cluster expansion, classical spin models - Ising and Heisenberg, outline of exact solutions.

Phase transitions: symmetry breaking and long range order, mean field approach, Landau theory, 2nd and 1st order transitions, Landau-Ginzburg functional, illustrative examples, estimate of fluctuations.

##### Electronics

Circuit theory: lumped circuit approximation, circuit elements, Kirchoff’s current and voltage laws, resistive networks, node and loop analysis, Thevenin and Norton’s theorem, time domain response of RL, RC and RLC circuits, frequency domain response, impedance, filters and transfer function.

Analog electronics: discrete devices, characteristics and operation - diode, Zener diode, LED, photodiode. Simple diode circuits. Bipolar junction transistor (BJT): biasing, h parameters, small and large signal response, amplifiers. Field effect transistors.

Operational amplifiers: device properties, integrator, differentiator, RC active filter, negative and positive feedback, oscillators.

Digital electronics: logic gates, truth table, multiplexer, combinatorial circuits, flip-flop, counters, programmable logic devices, microprocessors.

##### Laboratory II

Coupled Oscillator Circuits:

Thermal Equation of State and Critical point:

Lock-in Amplifier and Signal Processing:

OpAmps I: Amplifiers & Negative Feedback:

OpAmps II: Limitations & Applications:

Diodes: Clamps, Rectifiers, Power supplies

Transistors I: Switch, Common Emitter Amplifier, Push-pull Follower

Transistors II: Characteristics, Comparators, MoSFET, CMoS Inverter

Logic Gates: NAND gate, OR, AND, NOT; Adder, Oscillator

Flip-flops: as Memory element, Shift Register, Counters

Microcontroller I: Programming to MCU, using the port for input

Microcontroller II: Some Applications, Seven Segment Display

## Semester III

##### Condensed Matter Physics I

The building blocks - atoms to solids: atomic physics, Coulomb effects, crystal fields in solids, local moments and band electrons, lattice vibrations, electron-lattice coupling, electron-electron interactions.

Structure: characterising structures - crystalline/amorphous/liquids, classification of periodic structures, reciprocal space, x-ray and neutron diffraction.

Electronic structure: free electrons - spectrum, density of states, thermodynamics, band electrons - nearly free electron and tight binding limits, consequences for thermodynamics and transport.

Physics of metals: specific heat, susceptibility, impurity scattering, basic transport theory. Response to magnetic fields: Landau quantization, quantum Hall effect.

Phonons: Debye and Einstein model, spectrum of a real lattice, thermodynamics of phonons, anharmonic effects, Debye-Waller factor.

Magnetism: spin paramagnetism, itinerant-vs-localised electrons, Stoner and Heisenberg models, mean-field theory, spin waves.

Superconductivity: phenomenology, pairing interaction, BCS theory, Ginzburg-Landau theory and type II superconductors.

##### Quantum Mechanics III

Atomic physics: One electron atoms - spin-orbit interaction, fine structure, Lamb shift, Zeeman effect, Stark effect.

Two electron atoms: spin wave functions, approximate handling of electron-electron repulsion. Coupling of angular momenta, multiplet structure, gyromagnetic effects. Hyperfine and nuclear quadrupole interactions.

Many electron atoms: central field approximation, Thomas-Fermi and Hartree-Fock methods.

Molecular physics: Born-Oppenheimer approximation, molecular structure, rotation and vibration of diatomic molecules, hydrogen molecular ion, vibrational-rotational coupling, effect of vibration and rotation on molecular spectra. Electronic structure- molecular orbital and valence bond theories.

Atoms and light: transition rates, dipole approximation, Einstein coefficients, radiative damping, optical absorption, ac Stark effect.

Cold atoms: Doppler cooling, magneto-optical trap, ion traps, dipole force, evaporative cooling, optical lattice.

Collective effects: Feshbach tuning of interactions, Bose condensation of alkali atoms, BCS-BEC crossover, the unitary Fermi gas. Imaging cold atoms.

Computing with atoms: qubits and their properties, entanglement, quantum logic gates, decoherence and error correction.

##### Quantum Field Theory I

Non-relativistic quantum field theory: quantum mechanics of many particle systems; second quantisation; Schrodinger equation as a classical field equation and its quantisation; inclusion of inter-particle interactions in the first and second quantised formalism

Irreducible representations of the Lorentz group, connection to quantum fields

Symmetries and conservation laws: examples in non-relativistic and relativistic field theories; translation, rotation, Lorentz boost/Galilean transformation and internal symmetry transformations; associated conserved charges

Free Klein-Gordon equation: classical action and its quantisation; spectrum; Feynman rules for computing n-point Green functions of elementary and composite operators.

Interacting Klein-Gordon field: Feynman rules for computing Green functions; physical mass of the particle from the analysis of two point Green functions; S-matrix and its computation from n-point Green functions; relating S-matrix to cross-section.

Quantisation of free Dirac fields: spectrum; Feynman rules

Quantisation of free electromagnetic field: role of gauge invariance; gauge fixing; physical state condition; spectrum; Feynman rules

Quantum electrodynamics: coupling Dirac field to electromagnetic field; gauge invariance; quantisation; Feynman rules for computing Green functions; Spectrum and S-matrix from the Green functions.

##### Mathematical Methods II

Integral transforms, Fourier transforms, inversion and convolution, Laplace transforms

Advanced topics in ODE, Partial differential equations: classification of second order PDEs, Laplace and Poisson equations, applications to electrostatics, Heat equation, Wave equation

Group theory, definitions and examples of groups. Homomorphism, isomorphism and automorphism, Permutation groups

Group representation: reducibility, equivalence, Schur’s lemma. Lie groups and Lie algebras, SU(2) and SU(3). Representations of simple Lie algebras, SO(n), Lorentz group. Symmetries in physical systems, Young Tableau.

##### Elective I

**Choose any one of the following topics, Advanced Statistical Mechanics, Fluid Dynamics, General Theory of Relativity, Techniques in Nonlinear Dynamics, Quantum Information and Computation I.**

##### Advanced Statistical Mechanics

Critical phenomena : Liquid-gas transition and Van der Waals equation of state, Classical spin systems, Transfer matrix for one dimensional systems, Order parameters, Mean field approach, Landau theory, Universality, Critical exponents, Scaling hypothesis, Estimating fluctuations

Renormalisation : Hubbard-Stratanovich transformation and the Ginzburg-Landau-Wilson functional,Self-consistent approximation, Basic ideas of renormalisation group, Real space RG in one and two dimensions, Spherical limit, Wilsonian RG and ε-expansion, Field theoretic RG, Two dimensions and BKT transition

Equilibrium dynamics : Conserved and broken symmetry variables, Hydrodynamic approach, Dynamical critical phenomena

(Extra module : one of the following two) :

Non-equilibrium phenomena : Fluctuation-dissipation, Linear response, Kubo formula, Langevin and Fokker-Planck descriptions

Stochastic thermodynamics : Non-equilibrium work theorems (Jarzynski, Crooks, …), Non-equilibrium steady-states, Stochastic heat engines, Examples from colloidal systems and molecular motors

##### Fluid Mechanics

Ideal Fluids: Euler equation, hydrostatics, Bernoulli equation, conservation laws, incompressible fluids, waves, irrotational flows, inviscid fluids and vorticity

Viscous Fluids: Viscosity, Navier-Stokes equation, Reynolds number, laminar flow, exact solution to the eq. of motion.

Turbulence: Stability of flows, instabilities, quasi-periodic flows, Strange attractors, turbulent flows, jets, free shear layers, wakes, boundary layers

Thermal Conduction in fluids: eq. of heat transfer, conduction in incompressible fluid, law of heat transfer, convection, convective instability in static fluid

Compressible flows

Relativistic Fluid dynamics: eq. of motion, energy-momentum tensor, eq. for flow with viscosity and thermal conduction.

##### General Theory of Relativity

Review of Lorentz transformations and special theory of relativity.

Tensors and their transformation laws; Christoffel symbol and Riemann tensor; geodesics; parallel transport along open lines and closed curves; general properties of the Riemann tensor.

Equivalence principle and its applications: gravity as a curvature of space-time; geodesics as trajectories under the influence of gravitational field; generalisation to massless particles; gravitational red-shift; motion of a charged particle in curved space-time in the presence of an electric field; Maxwells equation in curved space-time.

Einsteins equation, Lagrangian formulation, Einstein-Hilbert action.

Schwarzschild solution: construction of the metric and its symmetries; motion of a particle in the Schwarzschild metric; Schwarzschild black hole; white holes and Kruskal extension of the Schwarzschild solution: construction of the metric and its symmetries; Motion of a particle in the Schwarzschild metric; precession of the perihelion; bending of light; horizon, its properties and significance.

Precession of the perihelion; bending of light; radar echo delay.

Initial value problem; extrinsic curvature; Gauss-Codacci equations;

Linearised theory, gravitational waves, field far from a source, energy in gravitational waves, quadrupole formula

Elementary cosmology: principles of homogeneity and isotropy; Friedman-Robertson-Walker metric; open, closed and flat universes; Friedman equation and stress tensor conservation, equation of state, big bang hypothesis and its successes.

##### Nonlinear Dynamics

Long time behaviour of the solutions of a system of ordinary nonlinear differential equations, fixed points and their classification according to stability.

Periodic orbit for conservative systems, periodic orbits for dissipative systems ( limit cycles ) and their stability, Bifurcations and centre manifolds.

Different kinds of perturbation theory for calculating periodic orbits, Renormalisation group aided perturbation theory, Poincare Bendixon theorem, chaos and strange attractors.

Maps, fixed points, cycles and stability, bifurcations , period doubling, intermittency and quasi periodicity, universal behavior at the onset of chaos , renormalization group and scaling behaviour.

Partial differential equations , patterns, Galerkin truncations and reduction to dynamical systems.

##### Quantum Information and Computation I

Quantum formalism: states, evolution, measurements.

Multipartite quantum systems: description and manipulation of bipartite systems and beyond.

Entanglement: quantification and detection in bipartite and multipartite systems.

Quantum communication: no-cloning theorem, quantum teleportation, quantum dense coding, multipartite communication protocols.

Quantum cryptography: essential classical cryptography, BB84, B92, Ekert, and secret sharing protocols.

Quantum computation: quantum algorithms, universal gates.

Interface of quantum information with other sciences.

Experimental realisations.

## Semester IV

##### Particle Physics

Experimental methods: fixed target and collider experiments, particle detectors.

Role of symmetries: charge conjugation, parity, time reversal, isospin and SU(2), quark model and SU(3).

Introduction to relativistic kinematics: Mandelstam variables, phase space, calculation of cross-sections and decay widths.

Basics of quantum electrodynamics: electron-positron annihilation, electron-muon scattering, Bhabha scattering, Compton scattering.

Deep inelastic scattering: Bjorken scaling, parton model, scaling violation, introduction to quantum chromodynamics and tree level processes.

Introduction to weak interactions: parity violation, V-A theory, pion and muon decay, neutrino scattering.

Standard Model: Glashow-Salam-Weinberg model, neutral current, physics of W, Z and Higgs, CKM mixing and CP violation.

Neutrino physics, neutrino oscillation

##### Elective II and III

Choose any two of the following topics,

Advanced Topics in General Relativity, Astrophysics, Computational Many Body Theory I, Computational Maretial Science, Condensed Matter Physics II, Cosmology, Disorder in Condensed Matter, Introduction to Electronic Structure, Matter out of Equilibrium, Mesoscopic Physics, Quantum Field Theory II, Quantum Optics, Soft Matter, String Theory I, Topological Quantum Matter, Ultra cold Atoms.

##### Advanced Topics in General Relativity

**Prerequisites: General Theory of Relativity**

Penrose diagrams

Hypersurface Geometry

Initial Value Problem in General Relativity

Aspects of Black Hole Physics: black hole thermodynamics and models of collapse

Brief survey of singularity theorems

Gravitational Waves

##### Astrophysics

Introduction to celestial objects, coordinates and the concept of time. Radiation transfer. Equations of radiation transfer, Black-body/thermal radiation, Opacity and optical depth, solutions of the radiation transfer equations in limiting cases, Rosseland mean opacity.

Thermal Bremsstrahlung emission, synchrotron emission. Self absorption and the emergent spectrum. Thomson scattering. Compton and Inverse-Compton scattering. Scattering in a region with magnetic field, Faraday rotation Introduction to fluid dynamics. Convection instability and transfer of energy from cores of stars. Supersonic motion, shocks.

Introduction to Magneto-hydro dynamics, flux freezing, Generation and amplification of magnetic fields in astrophysical situations.

Stellar structure. Mass-radius relation for main sequence stars, Minimum and maximum mass for nucleosynthesis, Hertzsprung-Russell diagram, Evolution of a star on the HR diagram. Novae and Supernovae, End points of stellar evolution. Inter-stellar medium. Phases of interstellar medium. Thermal, photoionisation, chemical and pressure equilibrium, Star formation, feedback and the evolution of ISM.

Orbits around massive bodies, Tidal disruption, restricted 3 body problem, Roche limit. Orbits in external potentials, potential-density pairs. An overview of models for galaxies. Accretion of matter on to a point mass, spherical accretion, Eddington limit.

Introduction to Cosmology, Friedmann models, equations. Hubble’s law. A brief overview of the thermal history of the universe.

##### Computational Many Body Theory I

**Prerequisites: QM I & II, Statistical Mechanics, Condensed Matter Physics I, Numerical Methods**

Free electrons in periodic structures: single band tight binding models in one, two and three dimensions, square, triangular and hexagonal lattices. Fermi surface and density of states. Multiband models. Spin- omrbit coupling. Response functions of the free system, incipient instabilities.

Disordered electrons: models with potential and hopping disorder, inverse participation ratio, maps of eigenfunctions, nobility edge, finite size effects, resistivity and optical conductivity using the Kubo formula. Disorder averaging.

Effect of an orbital magnetic field, Landau levels, role of disorder.

Mean field theory: implementing iterative consistency in a particle number conserving model. Competing phases.

Boguliubov-de Gennes schemes: spectrum and observables for a given pairing field, implementation of consistency, iterative scheme in the presence of disorder. Computing local observables.

Classical Monte Carlo for spin models: the Ising, XY and Heisenberg models on the square and triangular lattice, structure factor and energy, finite size effects.

##### Computational Materials Science

**Prerequisites: QM I & II, Statistical Mechanics, Condensed Matter Physics I, Numerical Methods**

Introduction: Basic ideas of modeling and simulation. Length, time, and energy scales in materials.

Computational techniques: Monte Carlo Methods: Metropolis sampling and Monte Carlo integration, Ensemble averages.

Molecular dynamics: MD in different ensembles, idea of thermostat, Nose- Hoover and Nose-Hoover chain thermostats.

Optimization techniques: Gradient-based methods, conjugate gradient method.

Atomistic model/simulation of molecules and materials :

Interatomic potentials: Motivation, Lennard-Jones, Morse, Tersoff etc. potentials. Embedded atom potentials. First principles approach: Basic ideas of Hartree-Fock and density functional theory.

Application of the above computational techniques in atomistic systems— using interatomic potential and first principles.

Materials: Applications of the above techniques and ideas to real materials. Structure optimization of molecules and solids. Electronic and magnetic properties of crystalline solids. Defect properties. Properties of solid surfaces, and two-dimensional materials. Electronic and magnetic properties of molecules and clusters.

Possible advanced topics: Evolutionary (genetic) algorithm and Monte Carlo based techniques for optimization. Application to structure optimization. Reactive force fields. Functionalizing materials for target applications such as catalysis, sensing. Adsorption of molecules and clusters on surfaces, their applications.

Length and time scales which can be addressed by the methods discussed. Elementary ideas about methods to treat longer length and time scales: Kinetic Monte Carlo, Cellular automata, Phase field models. Multi-scale modeling.

##### Condensed Matter Physics II

The course will consist of any two of A-D:

**Part A: Mesoscopics and spintronics:**

Foundation: low dimensional systems: quantum Wells, wires and quantum dots, 1D and 2D heterostructures, coupled wells and superlattices.

Charge Transport: transmission and its relation to conductance, Landauer theory, transmission function, S matrix and Green functions. Non-equilibrium Green functions and Landauer-Buttiker theory. Noise in Charge transport, scattering theory of shot noise.

Spintronics: introduction to spintronics.(Datta-Das spin transistor) equilibrium and non-equilibrium spin currents, spin Hall effect, coupled charge and spin transport, TMR, spin shot noise, entanglement generation and its detection.

**Part B: Electronic structure:**

Physics in low dimensions: surface states, reconstructions, adsorption, atomic wires and clusters.

Electron-electron interactions: Hartree-Fock approximation, electron gas, density functional theory.

Anharmonic effects in crystals: thermal expansion, lattice thermal conductivity, umklapp processes.

Phonons in metals: Kohn anomaly, dielectric constant, temperature dependence of electrical resistivity.

Dielectric properties of insulators. Plasmons, magnons etc.

**Part C: Mesoscopics and interacting systems:**

Quantum Hall effect

Quantum dots and quantum wires, Kondo effect

Fermi liquid theory and non Fermi liquids

Bosonization and Luttinger liquids.

Quantum spin systems

**Part D: Correlated electrons:**

Mott physics: electron localisation, magnetic order, doped phase, physics in the cuprates.

Kondo systems: physics of the single impurity, dense systems Kondo and Anderson lattice, heavy fermions, quantum criticality.

Metallic magnets: ferromagnetism in strongly repulsive systems, the transition metals, spin-fermion systems, the double exchange model, the classical Kondo lattice.

Electron-phonon coupling: the classical theory, polaron formation, many electron systems, polaron ordering, physics in the manganites.

Superconductivity: the BCS-BEC crossover, superconductivity in repulsive systems, competition with magnetism, effect of disorder.

##### Correlated Electron Systems

**Prerequisites: Quantum Mechanics I & II, Condensed Matter Physics I**

Mott physics: electron localisation, magnetic order, doped phase, physics in the cuprates.

Kondo systems: physics of the single impurity Anderson model, dense systems, Kondo and Anderson lattice, heavy fermions, quantum criticality.

Metallic magnets: ferromagnetism in strongly repulsive systems, the transition metals, spin-fermion systems, the double exchange model, the classical Kondo lattice.

Electron-phonon coupling: the classical theory, polaron formation, quantum theory of the ground state, many electron systems, polaron ordering, physics in the manganites.

Superconductivity: BCS and Migdal-Eliashberg theory, the BCS-BEC crossover, superconductivity in repulsive systems, competition with magnetism, effect of disorder.

##### Cosmology

Friedman-Robertson-Walker metric, Friedman equation and stress tensor conservation, equation of state: matter, radiation, cosmological constant, experimental evidence for dark matter and dark energy.

Age of the universe, cosmological horizon, expansion rate.

Thermal history of the universe, formation of hydrogen and origin of CMBR, decoupling of neutrinos, nucleosynthesis, recombination.

The horizon problem, possible resolution via inflation, slow roll condition and slow roll parameters, reheating, inflationary origin of density perturbation.

Early history, electroweak baryogenesis via leptogenesis, dark matter.

Theory of cosmological perturbations: gauge invariant scalar and tensor perturbations, spectral index, ratio of tensor to scalar fluctuation and Lyth bound, transition from quantum to classical perturbation: horizon exit and reentry, from density fluctuation to CMB fluctuations via Boltzmann transport equation, origin of the acoustic peak, origin of CMB polarisation, E and B modes.

##### Disorder in Condensed Matter

**Prerequisites: Quantum Mechanics I & II, Condensed Matter Physics I**

Origin of disorder in condensed matter: point defects. alloys, grain boundaries and dislocations. Disorder in dielectic media. Distributions of disorder. Correlated and uncorrelated disorder.

Classical waves in a disordered medium: photons and phonons in disordered media, localisation effects.

Perturbation theory and disorder average: low order scattering and results for the single particle Green’s function and the conductivity.

Quantum interference and localisation: coherent backscattering and its effects in different dimensions. The mobility edge. Anderson localisation effects in three dimensions. Scaling theory of the metal-insulator transition. Experimental survey.

Phase breaking effects: effect of inelastic scattering, spin flips and spin-orbit coupling. The effect on conductivity and magnetoresistance.

Hopping conduction: localised states and phonon assisted hopping, variable range hopping, coulomb gap, experiments on insulators.

Electron-electron interaction in disordered systems: the Altshuler-Aronov theory. Combined effects of interaction and disorder on density of states and transport properties.

Special topics: percolation theory, self consistent theory of localisation, typical medium theory, spin glasses, Anderson-Mott problem.

##### Introduction to Electronic Structure

Review of QM: variational method, identical particles, many fermion wave functions.

First-principles Hamiltonian and Born-Oppenheimer approximation.

Treating electron-electron interactions: Hartree-Fock approximation, exchange energy, correlation energy.

Density functional theory: Thomas-Fermi method, Hohenberg-Kohn theorems, Levy constrained search formulation, Kohn-Sham formulation, exchange-correlation energy, LDA and GGA functionals, spin density functional theory.

Solution of the Kohn-Sham equations, basis sets - LCAO: STO-NG, 4-31G, 6-31G etc, quality of basis sets, polarisation functions, spin-restricted calculations, Roothan equations.

Spin unrestricted calculations. Plane wave basis set.

Pseudopotentials and PAW in conjunction with plane waves.

Structure optimisation, Hellman-Feynman theorem.

Simple practical applications: band structure of standard solids, metals and semiconductors, optimisation of lattice constants, cohesive energies and other simple properties.

Possible advanced topics: hybrid functionals, van der Waals interactions, density functional perturbation theory, phonon band structure, electron-phonon coupling. CI, CCSD methods, QMC.

##### Matter Out of Equilibrium

**Prerequisites: Quantum Mechanics I & II, Statistical Mechanics, Condensed Matter Physics I, Numerical Methods**

A. Classical problems:

Recapitulation of equilibrium: Boltzmann distribution, ensemble average, solution of a few model problems.

Langevin equation: the physical argument, derivation from a system plus bath Hamiltonian, dynamical solution for free and harmonically bound particles, time dependent averages, distribution functions. decay of metastable states - Kramers escape.

Fokker-Planck equations: derivation from the Langevin equation, solution for free and harmonically bound particle, the Smoluchowski equation,

Kinetic equations: the BBGKY hierarchy, the Boltzmann equation for dilute gases, transport coefficients, approach to equilibrium.

B. Quantum problems:

Recapitulation of equilibrium Green’s functions and diagrammatic theory. Real time dynamics at equilibrium.

Schwinger-Keldysh formalism: the Keldysh contour, contour ordered Green’s functions, Wick’s theorem, Feynman rules, diagrammatics, particles in a time dependent field.

Interacting systems: electron-phonon and electron-electron interaction, low order perturbation theory, Dyson equation, skeleton diagrams, Hartree and Hartree-Fock approximation.

Examples: nonlinear electrical conduction, response to strong harmonic perturbation.

##### Mesoscopic Physics

**Prerequisites: Quantum Mechanics I & II, Condensed Matter Physics I**

Basics - time, length, energy scales. Ballistic transport, Landauer- Buttiker formalism, conductance quantisation

Diffusive transport, weak localisation, phase coherence, Aharanov- Bohm effect, general interference effects

Quantum dots, charging effects, Coulomb blockade

Landau levels and integer quantum Hall effect, edge states

Non-equilibrium Green’s functions and Landauer-Buttiker theory

Quantum wires, bosonisation, 1D Luttinger liquid physics including edge physics

Spintronics, Datta-Das spin transistor, spin currents and its detection

Noise, Nyquist-Johnson noise and shot noise

Mesoscopic superconductivity, Josephson effect

##### Quantum Field Theory II

Path integrals for Scalar and Fermionic fields: Generating Functional, Feynman rules, Loop Diagrams.

Renormalisation of scalar and Yukawa theories: power counting, Regularisation, Renormalisable and Non-renormalisable Theories, Green Functions at 1 loop of Some Prototypical Theories, Basics of Renormalisation Group (running coupling), 1PI Effective Actions.

Spontaneous Symmetry Breaking and Goldstone’s Theorem.

Path Integrals for the Maxwell field, Gauge Fixing.

Renormalisation of QED: 1 loop diagrams, Landau Pole.

Non-abelian Gauge Theories: Classical theory of Non-Abelian Gauge Theories, Quantization of Non-Abelian Gauge Theories by Path Integral Methods, Non-Abelian Gauge Theories at One Loop and Asymptotic Freedom, Spontaneous Symmetry Breaking in Non-Abelian Gauge Theories.

##### Quantum Information and Computation II

General evolution and Decoherence theory.

Master equations (Markovian and Non-Markovian, Various measure of nonmarkovianity).

Advanced entanglement theory (GM, GGM, newly proposed measures etc).

Quantum Correlation Beyond Entanglement (Quantum Discord, Geometric discord, Work-Deficit etc).

Resource theory in QI (Entanglement, Quantum Coherence, Reference Frame, Asymmetry etc).

Quantum Thermodynamics.

Advanced topics in quantum channels.

Quantum information and condensed matter systems.

##### Quantum Many Body Theory

Basics: second quantisation, the many body Hilbert space, few particle problems. Green functions: formal definition, Lehmann representation, calculation for quadratic problems, expression of observables in terms of Green functions. Finite temperature: the imaginary time formulation, analytic continuation.

Perturbation theory: the interaction representation, Wicks theorem, low order expansion and diagrammatic representation, Dyson equation and self-energy, vertex functions and Bethe-Salpeter equation, explicit calculations in the Anderson impurity model.

Resummations: random phase approximation in the electron gas, ladder summation in dilute hardcore systems, Hartree-Fock and higher order conserving approximations.

Long range order: self-consistent calculations for broken symmetry phases, static mean field and dynamical calculations, Nambu formulation and Eliashberg theory. Goldstone modes in the ordered phase - metallic antiferromagnets and superconductivity,

Functional integral methods: representing the partition function, bosons and fermions, quadratic integrals, Hubbard-Stratonovich decomposition of interactions, saddle point, gaussian fluctuations, beyond the gaussian theory, Ginzburg-Landau expansions.

##### Quantum Optics

Introduction: Quantization of the electromagnetic field, Fock states, coherent states, squeezed states, basic atom-photon interaction, density-matrix formalism.

Theory of coherence; Semiclassical theory of atom-photon interaction.

Quantum theory of atom-photon interaction.

Quantum theory of dissipation.

Quantum information in continuous variable systems; Quantum state engineering.

Quantum operations based on beam splitters, mirrors, squeezing and homodyne and heterodyne measurements and nonlinear operations such as parametric down converters.

Photon addition and subtraction operations; Elements of cavity QED.

##### Soft Matter

Forces, energies and timescales in soft matter, van der Waals force, hydrophobic and hydrophilic interactions. Basic phenomenology of liquid crystals, polymers, membranes, colloidal systems. Phase behaviour, diffusion and flow, viscoelasticity.

Order parameter, phase transitions: mean-field theory and phase diagrams, elasticity, stability, metastability, interfaces.

Colloidal systems: Poisson-Boltzmann theory, DLVO theory, sheared colloids, stability of colloidal systems, measurement of interaction.

Polymers: model systems, chain statistics, polymers in solutions and in melts, flexibility and semi-flexibility, distribution functions, self-avoidance, rubber elasticity, viscoelasticity, reptation ideas.

Membranes: fluid vs. solid membranes, energy and elasticity, surface tension, curvature, de Gennes-Taupin length, brief introduction to shape transitions.

Experimental tools and numerical approaches: Stokes limit, Rouse and Zimm Model for polymers, membranes, relaxation, computational studies, multiscale modelling.

##### Spectroscopic Methods

Probes for matter on different energy and spatial scales.

Interaction of electromagnetic radiation with matter, correlation functions in classical and quantum matter, point group symmetries and selection rules.

Electron spectroscopy in atoms and molecules: single and many electron atoms, simple molecules, vibronic transitions.

Vibration and rotational spectroscopy: infrared, Raman and microwave methods. Computing the spectrum of simple atomic and molecular systems.

Probing spin states: electron spin resonance and nuclear magnetic resonance. Mossbauer spectroscopy. Spectra of magnetic ions. Solid state effects on the spectrum.

Probe of collective effects: X-ray and neutron scattering from condensed matter. Static structure and dynamical correlations. Effect of phonons on lattice dynamical structure factor. Dynamical magnetic structure factor from ferro and antiferromagnetic spin waves. Diffuse magnetic scattering. Dynamics of classical liquids.

Extended electronic states: angle resolved photoemission spectroscopy, computing the spectrum for weakly correlated electron systems.

Ultrafast dynamics: control and probe of chemical reactions via femtosecond spectroscopy.

##### String Theory I

**Prerequisites: Quantum Field Theory I, General Theory of Relativity**

Bosonic Strings

Light Cone Quantization of Bosonic Strings

Introduction to 2D Conformal Field Theories

Vertex Operators

BRST Quantisation of Bosonic Strings

Tree Level and One Loop Amplitudes in Bosonic Strings

Compactifications, Kaluza-Klein, and Winding Modes

T Duality

D Branes

##### Topological Quantum Matter

**Prerequisites: Quantum Mechanics I & II, Condensed Matter Physics I**

Berry curvature and Berry phase, two level systems

Landau levels and integer quantum Hall effect Graphene and other Dirac materials

Unitary and anti-unitary symmetries, discrete symmetries, parity, inversion, time-reversal invariance and Kramer’s theorem

Basic ideas of topological invariants, winding numbers, Chern numbers, Z2 quantum numbers

Topological band theory and topological insulators, bulk states and surface states, toy models to realistic models

Boguliobov-De Gennes formalism and topological superconductors, Kitaev model and Majorana modes

Weyl semimetals, surface states and Fermi arcs

##### Ultra Cold Atoms

Spatial, time, and energy scales in cold atom physics.

Experimental background: trapping and cooling, Feshbach resonance, optical lattices, cold atom spectroscopies.

Basic theory: many particle physics, mean field theory, phase transitions, perturbation theory.

Continuum bosons: bosons in free space, weak interactions, Bogoliubov theory, BEC in trapped systems, Gross-Pitaevski equation.

Continuum fermions: fermions in free space, trapped fermions, Fermi liquid theory, weak attraction - BCS instability, strong attraction - BEC of pairs, the unitary Fermi gas, Stoner instability.

Optical lattices: Hubbard model - Bose/Fermi cases, superfluid-Mott transition for repulsive bosons, BCS-BEC crossover for attractive fermions, Mott transition in repulsive fermions.

Spin systems: quantum, S = 1/2, magnetism on unfrustrated and frustrated lattices. Entanglement in many body systems: pure states, mixed states, area laws, tensor network states.

Special topics: population imbalance, Anderson localisation, gauge fields, quench dynamics.

##### Project

In this semester, every student is supposed to do a project on a theoretical physics topic under the supervision of HRI faculty. Main fields in theoretical physics represented at HRI at the moment are, Astrophysics, Condensed Matter Physics, Particle Physics Phenomenology, Quantum Information and Computation, and String Theory.

##### Laboratory III

Ferro to Para Electric Phase Transition(or its Magnetic analogue)

Raman Spectroscopy

ESR

Earth’s field NMR gradient

Bragg Diffraction by Microwaves

Hall Effect

G-M counter, Counting Statistics, Gamma ray absorption cross section

Gamma ray Spectroscopy

STM with Graphene, HOPG, Gold, Semiconductors and CDW

Measurement of Speed of Light