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## Harish-Chandra Research Institute

### Doctoral Program in Physics

The Ph.D. program consists of course work and projects for the first three semesters, followed by research work leading to a Ph.D. degree.

#### Curriculum and detailed course content

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

## Course Structure

The instructional part of the doctoral program consists of two semesters of pedagogical lectures followed by two projects in the third semester.

Semester I |
Semester II |
---|---|

Elective I | Elective II |

Advanced Statistical Mechanics | Elective III |

Numerical Methods | Project |

Quantum Field Theory I | Mathematical Methods II |

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

Elective II and III : Choose two from — Astrophysics, Condensed Matter Physics II, Cosmology, Introduction to Electronic Structure, Quantum Field Theory II, Quantum Information and Computation II, Quantum Optics, Soft Matter, Ultra Cold Atoms.

## Semester 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

##### 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.

##### 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.

##### Elective I

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

##### 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.

##### 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.

## Semester II

##### Elective II and III

Choose any two of the following topics,

Astrophysics, Condensed Matter Physics II, Cosmology, Introduction to Electronic Structure Calculations, Particle Physics, Quantum Field Theory II, Quantum Information and Computation II, Quantum Optics, Soft Matter, Ultra cold Atoms.

##### 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.

##### 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.

##### 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.

##### 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.

##### 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

##### 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 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.

##### 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.