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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 |
Semester III |
---|---|---|

Elective I | Elective II | Project II |

Project I | Elective III | Project III |

Mathematical Methods II | Statistical Mechanics | |

Quantum Field Theory I | Numerical Methods |

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 — Accretion Process in Astrophysics, Advanced Topics in General Relativity, Advanced Topics in Quantum Field Theory, Astronomical Data Analysis, Astrophysics, Astrophysical Fluid Dynamics, Collider Physics, Computational Astrophysics, Computational Many Body Theory I, Computational Many Body Theory II, Computational Materials Science, Condensed Matter Physics II, Correlated Electron Systems, Cosmology, Dark Matter and Particle Astrophysics, Disorder in Condensed Matter, Flavour Physics and CP Violation, Grand Unified Theories, Introduction to Electronic Structure, Matter Out of Equilibrium, Mesoscopic Physics, Neutrino Physics, Particle Physics I, Particle Physics II, Quantum Field Theory II, Quantum Information and Computation II, Quantum Many Body Theory, Quantum Optics, Radiative Transfer Phenomena in Astrophysics, Relativistic Astrophysics, Soft Matter, Spectroscopic Methods, String Theory I, String Theory II, Supersymmetry, Topological Quantum Matter, Ultra Cold Atoms.

Elective III can also be done in semester III.

## Semester I

##### 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.

##### Project I

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.

##### 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

##### 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.

##### 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.

##### Accretion Process in Astrophysics

**Prerequisites: Astrophysical Fluid Dynamics, Radiative Transfer Phenomena in Astrophysics**

Astrophysical accretion as a source of energy.

Accretion from binary systems.

Accretion discs in astrophysics at various length scales.

Accretion onto compact objects.

Accretion power and accretion disc in active galactic nuclei.

##### 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

##### Advanced Topics in Quantum Field Theory

**Prerequisites: Quantum Field Theory I and II**

Solitons in scalar and gauge theories

Monopoles and Instantons

Large N: soluble models and applications in QCD

Anomalies: global and gauge

Introduction to Supersymmetric Field Theories (including brief discussion of phenomenological applications)

Aspects of Finite Temperature Field Theory

##### Astronomical Data Analysis

**Prerequisites: Astrophysics**

Overview of data analysis in astronomy.

Types of astronomical data- images, catalogues, spectra, polarization, time-series.

Recapitulation of basic statistics - sources of error. Probability distributions. Bayes’ theorem.

Data acquisition - sampling. Fourier methods.

Parameter estimation - model fitting.

Astronomical data archives. VO tools for analysis of archived data.

##### 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.

##### Astrophysical Fluid Dynamics

**Prerequisites: Classical Mechanics, Electrodynamics, Astrophysics**

Transition from the microscopic theory of matter to the fluid properties, equivalence of the Boltzmann equation with the Navier Stokes equation.

Blast waves and shock waves in compressible fluid under the influence of strong gravity, application to the theory of Supernova and other large scale astrophysical explosions in the universe.

Various instabilities. Theory of turbulence.

Time dependent perturbation of fluid under strong gravity and the stability analysis of various fluid structures in astrophysics.

Detailed study of Magneto-hydrodynamics.

##### Collider Physics

**Prerequisites: Quantum Field Theory I, Particle Physics I**

Introduction to colliders and its types, LEP, Tevatron and LHC

Particle Kinematics, Collider observables

Parton distribution functions, Parton model, parametrisation of quark distribution, parton model for hadron-hadron collisions, gluon distribution, fragmentation function

Jets, jet characteristics, quark jets, gluon jets, jet clustering algorithm

Review of the Standard Model, Particle Searches at colliders, weak boson production and decay, two jet production, multi-jet production

Statistics and Analysis, Monte Carlo simulations, event generators, introduction to HEP tools, brief introduction to machine learning methods

##### Computational Astrophysics

**Prerequisites: Numerical Methods, Astrophysical Fluid Dynamics, Astrophysics**

Introduction to computational astrophysics.

The N-body problem. Numerical algorithms.

Integrators for solving time-dependent nonlinear partial differential equations.

Particle and mesh related approaches. Numerical stochastic techniques.

Scopes of robust computational packages. Parallel computing.

Applications to astrophysical systems.

##### 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 Many Body Theory II

**Prerequisites: Computational Many Body Theory I**

Exact diagonalisation based Monte Carlo: path integral formulation, implementation for the Holstein and double exchange models. Thermal averaging, locating phase transitions, distribution of observables.

Molecular dynamics: Hamiltonian dynamics for single degree of freedom,linear and nonlinear models. Langevin scheme for a classical particle in a harmonic potential, match with analytic results, double well potential, Kramers escape problem. Noise driven coupled linear oscillators. Coupled nonlinear oscillators. Langevin equation for classical fields coupled to electrons.

Self consistent diagrammatic schemes: iterative perturbation theory for impurity models, fluctuation exchange schemes, Migdal-Eliashberg.

Exact diagonalisation and Lanczos: setting up the many particle states for the Hubbard and S=1/2 Heisenberg models. Computing matrix elements. ED on small finite geometries. Computing spectral functions. Lanczos implementation for the ground and excited states, spectral functions.

Variational MC for ground states: determination of variational parameters by Monte Carlo minimisation of energy. Projected wave functions for correlated superconducting states.

Quantum Monte Carlo for fermions: implementing determinantal Monte Carlo for the symmetric Anderson impurity model, Ising representation. Lattice models, the sign problem.

Special topics: Monte Carlo for bosons, density matrix renormalisation group and its 2D variants, dynamical mean field theory.

##### 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.

##### Dark Matter and Particle Astrophysics

**Prerequisites: Quantum Field Theory I, Particle Physics I**

Review of Cosmology basics

Dark matter relics and their density

WIMPS, Indirect and Direct searches for dark matter

Collider Searches for dark matter

Supernova physics

Ultra-high energy neutrinos

Portals to dark matter

##### 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.

##### Flavour Physics and CP Violation

**Prerequisites: Quantum Field Theory I, Particle Physics I**

The Discrete symmetries of the Standard model, C, P, and T symmetries, CP and CPT transformations

Flavour structure of the standard model, lepton masses and mixing, quark masses and mixing, running masses

Flavour changing Neutral currents, CKM matrix and measuring its observables

Meson mixings and mixing parameters, Kaon and B-physics, rare K and B decays

CP Violation, CPV in decays, Minimal Flavour violation

Flavour physics beyond the Standard Model.

##### Grand Unified Theories

**Prerequisites: Group Theory, QFT I and II, Particle Physics I**

Motivation and introduction, Standard Model and its limitations

Gauge symmetries, Standard Model of Particle physics, Unification of SM Forces

Grand Unified Theories, the gauge group SU(n), Georgi-Glashow model, SU(5) and SO(10)

Implications of Unifications, proton decay, fermion masses, renormalisation group equations, baryon number

GUT phenomenology, massive neutrinos

##### 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

##### Neutrino Physics

**Prerequisites: Quantum Field Theory I, Particle Physics I**

Neutrino interactions in the SM at low and high energies

Neutrino cross sections

Neutrino oscillation in vacuum and matter, MSW effect

Dirac and Majorana neutrinos

UHE neutrinos, Solar, atmorpheric, and Supernova Neutrinos

Present and future Neutrino experiments and their status.

##### Particle Physics I

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 Oscillations

##### Particle Physics II

**Prerequisites: Quantum Field Theory I, Particle Physics I**

Review of the Standard Model of Particle Physics and Gauge Theories of Particle Physics

Higgs Physics and Phenomenology

C,P,T symmetries and the CPT theorem

Anomalies

Topics in Electroweak Physics

Topics in Neutrino Physics

Selected topics in Physics beyond the Standard Model

##### 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.

##### Radiative Transfer Phenomena in Astrophysics

**Prerequisites: Electrodynamics, Astrophysics**

General overview of the field. The Einstein coefficient. Scattering and random walk. Rosseland and Eddington approximation, two stream approximation. Basic theory of radiation fields.

Bremsstrahlung.

Synchrotron radiation and radio astronomy.

Compton scattering and astrophysical spectra.

General theory of radiative transition. Line broadening and its manifestation in spectra emerging from or the vicinity from astrophysical objects.

##### Relativistic Astrophysics

**Prerequisites: General Theory of Relativity, Accretion processes in Astrophysics, Astrophysical Fluid Dynamics**

Elements of special and general relativistic fluid dynamics and thermodynamics. Astronomical Data Analysis

Extended body in general relativity. Rotation in general relativity.

Dynamics and thermodynamics of non-self gravitating matter in curved space-time.

Black hole astrophysics.

##### 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

##### String Theory II

**Prerequisites: String Theory I**

Superstrings

Quantization of Superstrings

Two Dimensional Superconformal Theories

Superstring Amplitudes

D branes and Ramond-Ramond Fields

Orbifolds and Orientifolds

String Dualities, M theory and F theory

The AdS/CFT Correspondence

AdS/CFT at finite Temperature

Calabi-Yau Compactifications

Construction of Semi-realistic Models of Particle Physics

##### Supersymmetry

**Prerequisites: Quantum Field Theory I and II**

Supersymmetry Algebra,

Chiral Superfields, Vector Superfields, and FI terms

Superspace

Quantum Corrections in Supersymmetric Theories

Theories with Extended Supersymmetry

BPS States

Introduction to Supergravity

Phenomenological Applications: the MSSM and Cosmological Implications

##### 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.