1. Dirac equation. Lorentz group and covariance. Interaction with an external electromagnetic field. Non-relativistic limit. Hole theory. Motivation for using field theory.

2. Free field theory. Representations of the Poincare group. One particle states for spin zero, one half and one, massive and non-massive states. Many particle systems. Second quantization. Classical field theory. Noether's theorem. Scalar field. Photon field. Proca field. Dirac field. Discrete transformations. PCT.

3. Interacting fields. Schrödringer, Heisenberg and interaction representations in quantum mechanics (QM). Relativistic scattering theory. S matrix. Green functions. Contractions and Wick's theorem. Reduction formulae. Feynman rules (FR) for j#. FR for quantum electrodynamics. Calculation of cross sections. Compton effect.

4. Functional integrals (FI). FI in non-relativistic QM. Comments on asymptotic conditions and coherent states. Scalar field with FI. Fermionic field with FI, Grassmann variables. Electromagnetic field with FI, Fadeev-Popov trick.

5. Applications. BCS theory of superconductivity with FI. Calculation of critical exponents for the Ising model using Feynman diagrams.
 
  Bibliography:

The following list is intended to give an idea of the level required in each subject.

General:

Itzykson, C., Quantum field theory, New York, McGraw Hill, 1980

Relativistic quantum mechanics:

Sakurai, J. J. ,Advanced quantum mechanics, Addison Wesley, 1967

Second quantization:

Weinberg, S., The quantum theory of fields, Cambridge University Press, 1995-1996

 Functional Integrals:

Ramond, P., Field Theory: A modern primer, 2nd Ed., Reading, AddisonWesley,1990

Applications to solid state physics:

Sakita, B., Quantum theory of many variable systems, Singapore, World Scientific,1985.

Applications to critical phenomena:

Parisi, G., Statistical field theory, Addison-Wesley, 1988 .
 
 

1.Quantization of constrained systems. Non-abelian gauge theories, functional integral approach. BRST symmetry. Spontaneously broken gauge symmetries.

2.Generating functionals. 1 particle irreducible diagrams. Slavnov-Taylor identities.

3.Renormalization. Degree of divergence. Regularization methods. Subtractions. One loop renormalization. Renormalizability and unitarity.

4.Renormalization group. Renormalization group equation. Solution. Runing coupling constants. Fixed points and renormalization group flow. Relation to critical phenomena and Wilson's approach.
 
 

Bibliography:

The following list is intended to give an idea of the level required in each subject.

Quantization of constrained systems and BRST:

N. Nakanishi y I. Ojima: Covariant operator formalism of gauge theories and quantum gravity. World Scientific, 1990.

Non-abelian gauge theories:

L.D. Faddeev y A.A. Slavnov: Gauge Fields, introduction to quantum theory. Benjamin-Cummings, 1980.

Renormalization

N.N. Bogoliubov and D.V. Shirkov, Introduction to the Theory of Quantized Fields, John Wiley & Sons, New York, 1980.

Relation to critical phenomena:

D.J.Amit, Field Theory, the Renormalization Group and Critical Phenomena.World Scientific, Singapore, 1984.
 
 

1. Basic topological notions: Topological spaces. Homeomorphisms. Homotopy.

2. Differentiable manifolds: Differential forms. Tensor fields.

3. Lie groups: Lie groups. Lie algebras. Transformation groups.

4. Fiber bundles. Principal fiber bundles. Associate bundles. Connections. Riemannian manifolds.

5. Classification and representation theory of Lie groups. Root systems. Dynkin diagrams. Representations of the permutation group. Young tableaux.
 
 

Note: An application of the corresponding mathematical ideas to a physical problem is envisaged for each of the items in this program.
 
 

Bibliography:

The overall level would be a little above the following reference:

G. Nash and S. Sen, Topology and geometry for physicist, Academic Press 1983.

Some complementary bibliography will be:

S. Kobayashi and K. Nomizu, Foundations of differential geometry, Interscience publishers 1963.

J. P. Serre, Linear representations theory of finite groups, Springer 1977.

B. Simon, Representations of finite and compact groups, American Mathematical Society 1996.
 
 

1. Energy scales. Experimental methods. Accelerators and detectors.

2. Structure of particles and their interactions. Form factors and structure functions. Bjorken's scaling.

3. Symmetries and conservation laws. Quantum numbers of particles. Absolute and partial conservation.

4. Breakdown of discrete symmetries. Two-component theory of the electron. Parity non-conservation. CP non-conservation. Strangeness oscillations.

5. Fermi theory of weak interactions. Electron-muon universality. CVC hypotesis.

6. Leptonic processes. Muon decay. Mean life.

7. Weak interactions of hadrons. Pion and kaon decays. Quark-lepton universality. PCAC.

8. Quarks and chromodynamics. Color. QCD and gluons. Confinement, asymptotic freedom. Jets.

9. Weak interactions of quarks. Charged and neutral currents. Weak isospin. The intermediate vector bosons.

10. The standard model. Gauge theories. Abelian and non-abelian case. Spontaneous symmetry breakdown. Mass of vector gauge bosons.
 
 

Bibliography:

D. Perkins: Introduction to High Energy Physics. Addison-Wesley (1972)

K. Gottfried & V. Weisskopf: Concepts of Particle Physics. Vol.1,2 Oxford University Press (1984)

O.Nachtmann: Elementary Particle Physics-Concepts and Phenomena. Springer Verlag (1990)

E. Commins: Weak Interactions. McGraw Hill (1973)

F. Close: An Introduction to Quarks and Partons. Academic Press (1979)

T. Cheng & L. Li: Gauge Theory of Elementary Particle Physics, Oxford University Press (1984)

I. Aitchison & A. Hey: Gauge Theories in Particle Physics. IOP (1989).
 
 

1.Why Susy?.

2. Dirac, Weyl and Majorana spinors.

3. Supersymmetry algebra. Extended supersymmetry. Supermultiplets.

4. Susy in different dimensions.

5. Supersymmetric quantum field theory. Chiral fields. Lagrangians for chiral fields.

Susy breaking by F-terms.

6. Vector superfields. Supersymmetric gauge theories. D-term susy breaking. Fayet-Iliopoulos term.

7. The Minimal Supersymmetric Standard Model (MSSM). Lagrangian. Coupling constant unification.

8. Soft terms. Susy and electroweak symmetry breaking. Some phenomenological issues.
 
 

Bibliography:

D. Bailin y A. Love, Supersymmetric Gauge Field Theory and String Theory, Institute of Physics publishing, 1994

J. Wess and J. Bagger, Supersymmetry and Supergravity. Princeton Series in Physics 1992.

S. Gates, M. Grisaru, M. Rocek and W. Siegel Superspace or One thousand and one lessons in Supersymmetry, Frontiers in Physics, 1983.

P. West, Introduction to Supersymmetry and supergravity, 2nd edition, World Scientific, Singapore, 1990.

G. Ross, Grand Unified Theories, Frontiers in Physics, 1984.

S. Weinberg, The Quantum theory of fields, Vol.III, Cambridge Univ. Press, 2000.
 
 

1.- Introduction

Space-time in pre-relativistic physics and in special relativity - The metric of space-time.

Space-time in general relativity. Manifolds, vectors and tensors - Derivative and parallel transport - Curvature - Geodesics.

2.- The Einstein equations

General Covariance - The Einstein equations.

Linearized gravitation: the Newtonian limit; gravitational radiation

3.- Cosmological models

Dynamics of a homogeneous and isotropic universe.

The cosmological red shift; horizons.

The evolution of the universe

4.- Relativistic stars - the Schwarzschild solution

Coordinates and metrics for a static and spherical system. Schwarzschild coordinates; interpretation.

The star matter. Structure equations.

Schwarzschild geodesics: gravitational red shift, perihelion precession, refraction of light, time delay.

5.- Gravitational collapse and black holes

Collapse for massive stars. End point for the stellar evolution. Neutron stars and black holes.

The gravitational radius. Behavior of Schwarzschild coordinates in the gravitational radius. Kruskal coordinates. Relationship between Schwarzschild coordinates and Kruskal coordinates.

General properties of black holes. Kerr charged black holes. Black hole energy emission. Black holes and thermodynamics

6.- Gravitational waves

The linearized theory. Plane wave solutions. Geodesic deviations in a linearized gravitational wave. Polarization. Quadrupolar nature of the gravitational wave.

7.- Quantum gravity and quantum effects in gravitational fields

Quantum gravity. Quantization of the linearized theory. The graviton. Propagators and vertices

Quantum fields in curved spaces. Creation of particles at the horizon of a black hole. Thermodynamics of black holes.
 
 

Bibliography:
 
 

R. M. Wald, General Relativity (The University of Chicago Press, 1984)

C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (Freeman, 1973)

S. Weinberg, Gravitation and Cosmology (John Wiley & Sons, New York, 1972).

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon, 1962)

N.D. Birrel, P.C. Davies, Quantum Fields in Curved Spaces (Cambridge University Press,1982)
 
 

1.- Cosmology: phenomenological aspects

The structure of the Universe: from stars to superclusters. The expansion of the Universe. Matter content and composition. Large scale isotropy and homogeneity. Cosmic background of microwave radiation.

2.- Standard cosmology

The cosmological principle. Evolution of the Universe. The Friedmann equation. The Hubble parameter. Horizons. Determination of the cosmological parameters. Thermal history of the Universe. Thermodynamics and entropy. Distribution functions in the Early Universe. Primordial nucleosynthesis. Decoupling of weakly interacting particles. Neutrino background. Problems of the standard cosmology.

3.- Inflation

The paradigm. Kinematics and dynamics of inflation. Inflationary models. Perturbations and fluctuations of density.

5.- Microwave background

Spectrum and anisotropies. Origin of the anisotropies. Multipolar expansion. Dependence on the cosmological parameters.

6.- Structure formation in the Universe

>The density field and its fluctuations. Linear theory of perturbations. Gravitational instability. Spectrum of the perturbations. Cold and hot dark matter. Peculiar velocities.

7.- Large scale structures

Galaxy distribution. Characterisation and observations. Clusters and superclusters.

8.- Galaxies

Types. Structure and composition. Mass determination.

9.- Gravitational lenses

Deflection of light by a gravitational field. Time delay. Strong and weak lenses. Microlensing.

10.- Dark matter

Evidence at different scales. Candidates to dark matter.

11.- Active galaxies and quasars

Characteristics. Red shift. Characteristic energies. Models. Supermassive black holes. Spectra of absorption. Intergalactic medium: Gunn-Peterson. Lyman-forest.

12.- Stars

Magnitudes and distances. Temperatures and colours. Hertzprung-Russell diagram.

13.- Basic equations of the stellar structure

Hydrostatic equilibrium. Energy conservation. Energy transfer. Gravitational effects. Generic star structures.

14.- Star evolution

The main sequence. Red giants. The horizontal sequence. White dwarfs. Binary system and type I supernovas. Intermediate mass stars. Variable stars. Massive stars and type II supernovas. Neutron stars and pulsars. Black holes.

15.- Selected topics

Phase transitions: topological defects. High energy cosmic rays. X-ray bursts.
 
 

Bibliography:
 
 

E.W. Kolb and M.S. Turner, The Early Universe, Addison Wesley .

P.J.E. Peebles, Principles of Physical Cosmology, Princeton

S. Weinberg, Gravitation and Cosmology, J. Wiley

G.G. Raffelt , Stars as Laboratories of Fundamental Physics, Chicago Univ. Press.

S.L. Shapiro and S.A. Teukolsky, Black Holes, White Dwarfs and Neutron Stars, J. Wiley.
 
 

1. Path Integrals in Quantum Mechanics. Perturbation theory and semiclassical approximation S matrix elements in the functional formalism.

2. Path integrals for many body systems. Coherent states. Path integrals in Euclidean space. Grassmann variables.

3. Path integrals in field theory. Feynman diagrams. Fermionic field theories. Symmetries and Ward-Takahashi identities. Semiclassical expansion. Gauge theories.

Finite Temperature Quantum Field Theory. Imaginary time formalism. Perturbative expansion for the free energy. Bosons and fermions. Bose-Einstein condensation. IR divergencies. Applications to the real scalar field and QED.
 
 

Bibliography:
 
 

J. Zinn-Justin, Quantum Field Theory and Critical Phenomena', Oxford Science Publications, 1994.

L. D, Faddeev y A. A. Slavnov, Gauge Fields: Introduction to Quantum Theory', Benjamin, Reading, MA 1980.

R.P. Feynman and H. Hibbs, Quantum Mechanics and Path Integrals, Mc. Graw Hill, New York, 1965.

F.W. Wiegel, Introduction to Path-Integral Methods in Physics and Polymer Sciences, World Scientific, 1986.
 
 

1. Review of Statistical Mechanics: Thermodynamic limit, phase boundaries, Ising model, Analytic properties of the free energy, phase transitions, fluid and lattice gases.

2. Transfer Matrix and one dimensional systems. Low temperature expansion.

3. Mean Field Theories: Landau Theory. Spatial variations, fluctuations and Ginzburg criterion.

4. The Static Scaling hypothesis: Scaling Laws and Widom scaling, high temperature series expansions.

5. Kadanoff Block Spins.

6. Renormalization Group: general concepts. RG flows, fixed points, linearized RG and critical exponents, Real space RG, First-order Phase transitions, cross over behaviour, corrections to scaling, finite size scaling.

7. Critical phenomena near four dimensions and the O(n) Model. Differential RG recursion relations, epsilon expansion, 1/n expansion, O(n) model below the critical temperature and Goldstone's theorem, Kosterlitz-Thouless transition.
 
 

Bibliography:
 
 

J. Zinn-Justin, Quantum Field Theory and Critical Phenomena', Oxford Science Publications, 1994.

D. Amit, Field Theory, the renormalization Group, and Critical Phenomena, World Scientific, 1989.

J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press, 1996.

N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Addison Wesley, 1992.
 
 

1. Introduction to the geometry of Hamiltonian systems: Hamiltonian systems. Symplectic Geometry. Poisson manifolds. Symplectic foliations. Symmetries: Lie group actions symplectic manifolds. Momentum map. Hamiltonian systems on Lie groups.

2. Lie-Poisson groups and Lie bialgebras: Lie-Poisson groups. Lie bialgebras. Manin triples. Coboundary Lie bialgebras. Classical Yang-Baxter equation. Factorizable Lie bialgebras. Coboundary Lie-Poisson groups. Classical integrable systems. Lax pairs. Integrable systems from r matrices.

3. Deformation of Lie algebras: Hopf algebras and the universal envelope of a Lie algebra. Duality with the algebra of functions on a Lie group. Lie-Poisson algebras. Deformation quantization. Quantization of the Borel algebra. Quantum double. Quantization of the universal envelope.
 
 

Bibliography:
 
 

Chari - Pressley, A guide to quantum groups, Cambridge Univ. Press, 1994.

Kassel, C., Quantum Groups, Springer-Verlag, 1995.

Majid, S., Foundations of quantum groups theory, Cambridge Univ. Press, 1995.

Montani, H., Lectures notes from the course given at the Department of Mathematics, Universidad Nacional del Sur, Bahia Blanca, Argentina (april, 2000).
 
 

1.Cosmic ray observations: spectrum, composition, anisotropies.

2.Origin of cosmic rays: acceleration processes, the second and first order Fermi mechanisms, one-shot acceleration in pulsars, possible astrophysical sources, top-down mechanisms for ultra-high energy CR production.

3. Cosmic ray propagation: low energy cosmic ray spallation processes, isotope abundances. The leaky box model. Magnetic diffusion and drift and relevance to explain the knee in the spectrum. Interactions with the CMBR and the GZK cutoff for protons. Possible non conventional mechanisms to evade the GZK cutoff. Infrared background and photodisintegration of nuclei. Magnetic lensing effects upon ultra-high energy CRs.

4. Extended Air Showers, Experimental techniques for CR detection. Extended air shower development. Pair production, bremsstrahlung, hadronic interactions. Longitudinal and lateral profile. The Heitler model. Primary mass and composition resolution. Experiments for detection of UHECR: Volcano Ranch, Haverah Park, Agasa, Fly's Eye, Hi Res, Auger and beyond. Montecarlo simulations and shower reconstruction techniques. Fluorescence light propagation in the atmosphere.
 
 

Bibliography:
 
 

T. Gaisser, Cosmic Rays and Particle Physics, Cambridge University Press (1990).

V. Berezinsky et al., Astrophysics of Cosmic Rays, (North Holland, Amsterdam, 1990).

P. Sokolsky, Introduction to UHECR Physics, Frontiers in Physics Series Vol.76, Addison-Wesley Publ. Co. (1989)

Selected papers and review articles.
 
 

I) 1.Second quantization: Bosons: creation and destruction operators. Fermions. Field operators. One and two particle operators in second quantization.

2.Interacting boson gas: He₄. Bogoliubov's approximation.

3. Lattice dynamics. Lyddane-Sachs-Teller relation. Phonon-phonon

interactions.

II) Electron-phonon interaction: Fröhlich's model. Polarons, effective mass. Spectral function of a localized electron interacting with a phonon. Effective attractive interaction between electrons. Phonon-frequency renormalization.

III) 1. Interacting electron gas: Hartree and Hartree-Fock approximations. Koopmans' theorem. Physical interpretation of the exchange hole and exchange energy, ferromagnetic instability of the interacting electron gas. Problems with second order perturbation theory. Correlation energy. Wigner crystal.

2. Interacting electron gas: Screening and collective excitations in the electron gas. Thomas-Fermi approximation. Dielectric response of the electron gas. Random Phase approximation (RPA). Fermi liquid theory: quasiparticles.

3. Density functional theory: Hohenberg and Kohn theorem. Local density functional approximation. Kohn and Sham equations.

IV) 1.Magnetism: diamagnetism and paramagnetism. Hund's rules. Crystal field. Exchange. Molecular field approximation. Curie-Weiss law. Hubbard model. Mott transition. Superexchange. Holstein-Primakoff approximation. Spin waves: ferro- and antiferromagnetic cases.

V)1.Transport: Boltzmann's equation. Relaxation times. Transport coefficients. Electrical conductivity. Thermal conductivity. Thermoelectric effects.

2. Optical properties: Optical conductivity and dielectric function. Reflectivity. Excitons. Interband transitions. Drude model.
 
 

Bibliography:

- P.W. Anderson, "Concepts in Solids", Reading, W.A.Benjamin, 1963.

- D. Pines, " Elementary excitations in solids",.

-D. Pines and P. Noziéres, "The theory of quantum liquids", Reading, Addison-Wesley, 1966.

-C. Kittel, "Quantum Theory of Solids", New York, Wiley, 1987.

G. Mahan, "Many-particle physics", New York, Plenum, 1981.

I. BCS Theory.

Electron-phonon interaction. Cooper pairs. The BCS ground state. Canonical transformations. Superconducting gap. Quasiparticle excitations. Thermodynamic quantities. Electron tunneling. Ultrasonic attenuation. Electromagnetic absorption. Electrodynamics. Meissner effect and penetration depth.

II. Ginzburg-Landau Theory.

The Ginzburg-Landau equations. Microscopic derivation. Domain wall energy. Flux quantization. London model. Type I superconductors. Intermediate state.

III. Vortex Physics.

Type II superconductors. Critical fields. Abrikosov vortex state. Flux pinning. Flux flow. Critical State. Vortex lattice melting. Collective pinning. Vortex glass. Bose glass.

IV. Josephson Effect.

The Josephson effect. Critical current. The RCSJ model. Shapiro steps. Interference effects. Long Josephson junctions. SQUID devices. Small junctions and quantum effecs. Macroscopic quantum tunneling. Bloch oscillations. Single electron tunneling effects.

Josephson junction arrays. Granular superconductors.

V. Special Topics.

The Bogoliubov-de Gennes equations. Microscopic structure of one vortex. Effect of magnetic impurities. Gapless superconductors. Nonequilibrium superconductivity. Unconventional superconductivity. d-wave superconductors and high Tc.
 
 

Bibliography:
 
 

- J. R. Schrieffer, "Theory of Superconductivity", Frontiers in physics, New York, 1964.

- G. Rickayzen, "Theory of Superconductivity", New York, Interscience, 1965.

- M.Tinkham, "Introduction to Superconductivity", New York, McGraw Hill, 1975.

- P. G. de Gennes, "Superconductivity of Metals and Alloys", New York, Benjamin, 1966.
 
 

I) Numerical methods and chaos.

Ordinary differential equations. Euler method, Runge-Kutta, leapfrog and implicit methods. Numerical calculation of integrals. Fast Fourier transform. Power spectrum. Application to chaos problems: logistic map, Hamiltonian chaos, Lyapunov exponent, Poincare's sections.

II) Molecular dynamics and Langevin dynamics.

Molecular dynamics.Calculation of thermodynamical magnitudes. Application to systems of interacting particles (noble gases). Verlet algorithms. Linked-list algorithms. Isokinetic and Nose-Hoover methods. Random number generation. Integration of stochastic differential equations. Helfand-Greenside algorithm. Langevin dynamics. Comparison with molecular dynamics. Hybrid methods.

III) Monte Carlo method.

Monte Carlo integration: simple sampling and importance sampling. Generation of distributions. Markov processes: definition. Limit properties, detailed balance and thermal bath. Metropolis algorithm. Glauber dynamics. Configuration correlations. Ergodicity. Ising models. Measurement of mean values and correlation functions. Critical exponents, Binder cumulants, finite size scaling.Variational Monte Carlo. Ground state of the Helium atom.

IV) Neural networks.

Formal neurons. Hebb's rule. Hopfield model (memories). Layer models. Perceptron and multilayers. Perceptron algorithm. Back-propagation. Generalization.
 
 

Bibliography:
 
 

- D.W. Heermann, "Computer Simulation Methods in Theoretical Physics", Berlin, Springer, 1990.

- W.K. Binder and D.W.Heerman, "Monte Carlo Simulation in Statistical Physics", Berlin , Springer , 1988.

- Hertz, Krogh and Palmer, "Introduction to the Theory of Neural Computation".

W.H. Press et al, "Numerical Recipes", Cambridge, CUP, 1992.

- Koonin, "Computational Physics", Menlo Park, Benjamin-Cummings, 1986.

- J.M.Thijssen, "Computational Physics".

- Allen and Tildesley, "Computer Simulation of Liquids".
 
 

I- Structure of Matter and Order Parameters.

Liquids and gases. Crystalline solids. Symmetry and crystal structure. Liquid crystals. Incommensurate structures. Quasicrystals. Magnetic order. Correlation functions and order parameters. Discrete and continuous symmetries and models.

II - Phase Transitions.

Mean field theory: Landau. Magnetic transitions and second order transitions. Solid-liquid transitions and first order transitions. Breakdown of mean field theory. Critical phenomena. Critical exponents. Scaling. Universality. Notions of renormalization group.

III. Generalized Elasticity.

Elastic energy. The XY model. Fluctuations. Long range order and quasi-long range order. Systems with O(n) symmetry. Liquid crystals: elasticity in nematics and smectics. Elasticity of solids.

IV. Topological Defects and Domain Walls.

Characterization of topological defects. Order parameter space and homotopy. Vortices. Dislocations. Disclinations. Grain boundaries. Energy of vortices and dislocations. Kosterlitz-Thouless transition. Dislocation mediated melting. Solitons and sine-Gordon equation. Frenkel-Kontorova model. Roughness transitions.

V. Dynamics: Linear Response.

Dynamic correlations and response functions. Elastic waves and phonons. Diffusion. Langevin theory. Fluctuation-dissipation theorem. Properties of response functions.

VI. Hydrodynamics and Critical Dynamics.

Conserved variables and broken symmetry variables. Hydrodynamics of: spin systems, fluids (Navier-Stokes equations), solids, liquid crystals and superfluids. Dynamic critical phenomena. Dissipative dynamics. Time dependent Ginzburg-Landau equation. Dynamic scaling. Nucleation and spinodal decomposition.
 
 

Bibliography:
 
 

- Chaikin and Lubensky, "Principles of Condensed Matter Physics", Cambridge, Cambridge University Press, 1995.

- P. W. Anderson, "Basic Notions of Condensed Matter Physics", Reading, Addison-Wesley, 1984.
 
 

1.Classical Green's functions: examples of static and time-dependent Green's functions.

2. Representations of quantum mechanics: Schrödinger, Heisenberg, interaction.

3. Second quantization for fermions: creation and destruction operators, field operators, representation of observables. Example: tight-binding Hamiltonian for electrons in a solid, introduction of the one-electron Green function: connection with energy levels and density of states, tight-binding Green functions. Second quantization for bosons: creation and destruction operators, observables. Phonons.

4. Green functions and linear response theory.

i) Response to adiabatically applied external fields: response function, relaxation function, generalized susceptibility. Properties of response functions. Kubo formula.

ii) Isothermal response, isothermal generalized susceptibility. Comparison of adiabatic and istothermal magnetic susceptibilities: Curie's law.

iii) Correlation functions and Green functions (retarded, advanced, causal).

Equations of motion for Green functions. Spectral representation.

Non-interacting systems of fermions, bosons and paramagnets.

Interactions in fermion, boson or spin systems: decoupling schemes for Eqs. of motion. Examples: random phase approximation for spin ½ ferromagnet; Hubbard model: RPA susceptibility of the interacting electron gas, Stoner instability criterion.

Calculation of the Hall coefficient for interacting electrons.

Normal mode frequencies, quasiparticles and their lifetimes.

iv) Fluctuation-dissipation theorem.

v) Neutron scattering cross section in terms of correlation functions. Scattering of electromagnetic radiation.

5. Green's functions in disordered systems.

i) Introduction.Examples: disorder in electron systems, phonons, magnetic excitations.

ii) Dyson's equation in disordered media: for the propagator and locator. Self-energy.

iii) Non-selfconsistent theories for the treatment of disorder: the virtual crystal approximation; average t-matrix approximation.

iv) The coherent potential approximation for systems with diagonal disorder. Examples. Desirable properties and limitations of the CPA. Connection with effective medium theories for infinite-dimensional systems.

Non-diagonal disorder: effective single-site theory by Blackman, Esterling & Berk.

- S. Doniach & E.H. Sondheimer, "Green's Functions for Solid State Physicists", Imperial College Press ( 1998 ).

- E.N. Economou, "Green's Functions in Quantum Physics", Springer Series in Solid State Physics, Vol.7 ( 1979 ).

- A. Gonis, "Green Functions for Ordered and Disordered Systems", North-Holland, Elsevier Science Publ. ( 1992 ).

- G.D.Mahan, "Many-Particle Physics", Plenum Press ( 1990 ).

- G. Rickayzen,"Green's Functions and Condensed Matter", Academic Press(1980).

- R.B. Stinchcombe, "Kubo and Zubarev Formulations of Response Theory", in "Correlation Functions and Quasiparticle Interactions in Condensed Matter", Plenum Publ.Corp. (1978).

- D.N. Zubarev, "Double-Time Green Functions in Statistical Physics", Sov. Phys. Usp. 3, 320 (1960).
 
 

Representations: Schrödinger, interaction, Heisenberg. Adiabatic turning on of perturbations. Gell-Mann and Low theorem.

Green's functions: definition, relations with observables, physical interpretation. Green function for free fermions. Excitation spectrum.

Wick's theorem.

Diagrammatical analysis of perturbation theory: Feynman's diagrams, Dyson equation, Goldstone theorem.

Hartree-Fock approximation.

Imperfect Fermi gas; degenerate Fermi gas.

RPA approximation, correlation energy, screening in the electron gas. Plasma oscillations in the electron gas.

Bosonic systems, Bogoliubov's approximation. Boson Green's functions, Feynman's rules. Weakly interacting boson gas.

Finite temperature Green's functions, relation with observables. Non-interacting Green's function.

Perturbation theory and Wick's theorem at finite temperature. Diagrammatical analysis, Dyson's equation at finite temperature.

Superconductivity: Gorkov's equations.

Microscopic derivation of Landau-Ginzburg's equations.
 

Bibliography:

- Fetter and Walecka, Quantum Theory of Many-Particle Systems, New York, McGraw Hill, 1971.

- Abrikosov, Gorkov, and Dzyaloshinski, Methods of Quantum Field Theory, Englewood Cliffs, NJ, Prentice Hall, 1963.

- Mahan, Many-Particle Physics, New York, Plenum, 1981.
 
 

1. Spin chains:

Thermodynamical properties.

Exact solutions.

Integer and half-integer spin chains.

Dimerized systems (spin-Peierls).

2. Fermions in one dimension:

Bosonization.

Luttinger liquids.

Correlation functions. Charge and spin separation.

Thermodynamical properties

Particle-hole transformation (Jordan Wigner).

3. Organic conductors in one dimension:

Polyacetilene.

Conductivity, superconductivity, antiferromagnetism and charge density waves

Solitons.

Transport properties.

4. Ladder systems:

Theoretical and experimental aspects

Equivalence with spin chains

Charge excitations

5. Mesoscopic and nanoscopic systems:

Quantum dots, quantum corrals, nanotubes and nanowires: Theoretical and experimental aspects.

Transport, localization and mesoscopic interference effects.
 
 

Bibliography:
 
 

- Mattis, Daniel C. Ed., "The many body problem: an encyclopedia of exactly solved

models in one dimension", Singapore, World Scientific, 1993.

- Fradkin, E, "Field theories of condensed matter systems", Redwood City, CA, Addison Wesley, 1991.

- Baskaran, G.; Ruckenstein, A.E.; Tosatti, E.; Lu., eds. , "Strongly correlated electron systems II", Adriatico Research Conference and Miniworkshop on Strongly Correlated Electron Systems II, Trieste, 18 June 27 July 1990.

- Devreese, J. T.; Evrard, R. eds., "Highly conducting one dimensional solids", New York, Plenum, 1979.

- Fulde, Peter, "Electron correlations in molecules and solids", Berlin, Springer, 1995.

- Baeriswyl, D.; Campbell, D.K.; Carmelo, J.M.P.; Guinea, F.; Louis, E. eds., "The Hubbard model: its

physics and mathematical physics", NATO Advanced Research Workshop on the Physics and Mathematical Physics of the Hubbard Model, San Sebastian, Spain, 3-8 October, 1993, New York, Plenum, 1995.

- Jérome, D.; Caron, L.G. eds., "Low dimensional conductors and superconductors", NATO Advanced Study Institute on Low Dimensional Conductors and Superconductors, Magog, Canada, 24 Agosto - 6 Setiembre, 1986, Nueva York, Plenum, 1987
 
 

Diamagnetism, Langevin formula, mononuclear systems, susceptibility of aromatic molecules. Quantum theory of diamagnetism.

Paramagnetism: atomic paramagnetism, Hund's rules, magnetic impurities in insulators, Van Vleck paramagnetic susceptibility, magnetism of rare-earths and transition metals. Crystal field, spin-orbit interaction. Metals, Pauli paramagnetism, impurities in metals.

Ferromagnetism: exchange, Weiss molecular field, Curie law, magnetization as a function of temperature and magnetic field. Ferromagnetic magnons. Itinerant magnetism. Antiferromagnetism and ferrimagnetism. Mean field theory. Magnons.

Magnetic moment formation, Anderson model, itinerant ferromagnetism, Stoner model, dilute magnetic impurities in a metal, RKKY, spin glass concept.

Conductivity and electrical resistivity in metals. Dependance of resistivity with temperature, metallic multilayers, difuse scattering, interlayer interactions, magnetic coupling between layers, transport in multilayers, giant magnetoresistance (GMR). Nanostructured systems, technological applications.

Nanoparticle magnetism, magnetic domains, superparamagnetism, Bloch walls, critical size, magnetic anisotropies. Magnetic relaxation, zero field cooling and field cooling magnetization (ZFC/FC).

Introduction to perovskites (LaMnO₃ ), magnetic coupling, Goodenough-Kanamori rules, colossal magnetoresistance (CMR), double exchange, effect of disorder, extended and localized states. Competition between super- and double exchange, orbital ordering, Jahn-Teller effect.

Bibliography:
 

- C.Kittel, "Introduction to Solid State Physics", New York, Wiley, 1986.

- C.Kittel, "Quantum Theory of Solids", New York, Wiley, 1987.

- N.F.Mott, "Metal-Insulator Transitions", London, Taylor & Francis, 1974.

- J.B. Goodenough, "Magnetism and the chemical bound", New York, Wiley, 1963.

- J.P.Ansermet, J.of Phys.Cond.Mat. 10, 6027 (1998). (Review GMR)

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