Summary and bibliography of the lectures

• Monday, February 18, 2019

  Introduction

  • Program and objectives of the course.

  • History, development and applications of Condensed Matter Physics.

  • Materials and devices.

  • Importance in the development of Physics.

• Problems: Série 4a) 1 and 2

• Extra references

  • Video on Superconductivity

    • Quantum Levitation - Tel Aviv University Demonstration

      • https://www.youtube.com/watch?v=sFOrdHiinsc

  • Formulations of Quantum Mechanics

    • Nine formulations of quantum mechanics, D. F. Styer et al., American Journal of Physics 70, 288 (2002).

  • Second Quantization

    • Lectures on Quantum Mechanics, G. Baym

      • chapter 18, Identical particles.

      • chapter 19, Second Quantization.

    • Quantum Theory of Many-Particle Systems, A. L. Fetter and J. D. Walecka

      • chapter 1, Second Quantization.

• Thursday, February 21, 2019

• Review of Quantum Mechanics and Quantum Statistical Mechanics

  • Entropy in Physics and Information Theory.

  • Shanonn entropy.

  • Measurements and averages.

  • Pure and and statistical states. Density matrix.

  • Von Newmann entropy.

  • Micro canonical, canonical and grand canonical ensembles. Boltzmann factor.

  • Fermi-Dirac and Bose-Einstein distributions.

• Problems: Série 1c) 1 and 2

• References

  • Section: Information theory and thermodynamics, of:

    • Quantum information and phase transitions: Fidelity and state distinguishability, V. R. Vieira, Journal of Physics: Conference Series 213, 012005 (2010).

• Extra references

  • On the Bertrand Paradox

    • The well-posed problem, E. T. Jaynes, Foundations of Physics 3, 477-492 (1973).

• Monday, February 25, 2109

• Review of Quantum Mechanics and Quantum Statistical Mechanics

  • Canonical ensembles, T→0 limit. Projector.

  • Euler and Lagrange descriptions in Classical Mechanics. Analogy with the Schrödinger and Heisenberg representations.

    • Rate of change of volume and divergence of the velocity.

  • Interaction Picture: equations of motion of states, operators, density matrix and averages.

    • Method of variation of constants of a differential eaquation.

  • Free particles: quadratic Hamiltonians for Bosons and Fermions, linear Hamiltonians for spins.

  • Equilibrium average of time dependent operators.

• Problems: Série1) 2, 4, 5; and Série 1) 10, Série 1b) 3

• References

• Expansion, or Divergence of a Field, Foundations of Potential Theory, Kellogg, Oliver Dimon, Dover (1953), pgs. 34-37.

• Thursday, February 28, 2109

• Review of Quantum Mechanics and Quantum Statistical Mechanics

  • Evolution Operator.

    • Time ordering. Time ordered exponential.

    • Parametric differentiation and derivative of the evolution operator.

  • Equilibrium correlation functions.

    • Time invariant systems.

    • Thermal equilibrium and the fluctuation-dissipation theorem.

• Problems:

  • Série 1) 3, 20.

  • Série1) 5 and Série 2) 3. Verify that the correlation function satisfy the fluctuation theorem.

  • Série M) 2 (special functions).

• References

  • Principles of Condensed Matter Physics, P. M. Chaikin & T. C. Lubensky

    • Sections: 7.1, 7.6

• Monday, March 4, 2019

  • Carnival: here was no class.

• Thursday, March 7, 2019

  • Discussion and solutions to exercises

    • Parameter differentiation

  • Linear Response Theory

    • Kubo formula

  • Reference to Topological Phase Transitions

    • Geometrical invariants:

      • mappings: winding and linking numbers, Berry phase, Wess-Zumino (SU(2): area of sphere)

• Problems:

  • Série 1) 9 and Série 1a) 8.

  • Série 1c) 3.

• References

  • Principles of Condensed Matter Physics, P. M. Chaikin & T. C. Lubensky

    • Sections: 7.1, 7.6

  • Parameter differentiation

    • R. M. Wilcox, Exponential Operators and Parameter Differentiation in Quantum Physics, J. Math. Phys. 8, 962 (1967).

• Monday, March 11, 2019

  • Discussion and solutions to exercises

    • Differential equations and boundary conditions. Green’s functions.

  • Causality. Kramers- Kronig relations.

  • Fluctuation-dissipation theorem.

    • Simple example: Einstein relation: diffusion and damping.

  • Symmetries and properties of correlation functions.

• Problems:

  • Série 1) 17, 18

• References

  • Principles of Condensed Matter Physics, P. M. Chaikin & T. C. Lubensky

    • Sections: 7.1, 7.2, 7.4, 7.6, 7.7 (1, 2, 3)

  • Classical Electrodynamics, J. D. Jackson

    • Section: 7.10

• Thursday, March 14, 2019

  • Discussion and solutions to exercises

  • Description of the structure and chapters of the rest of the course

  • Materials e properties

    • Insulating, semiconductors, conductors, superconductors, magnetic, etc.

    • Properties: structural, thermodynamic, transport (charge, spin, heat, etc.), susceptibilities, etc.

  • Strongly correlated systems

    • Collective phenomena and phase transitions

      • energies and susceptibilities.

      • ordered phases, order parameter.

      • ground state and excitations.

• Monday, 18, March 2019

  • Bohr-van Leeuwen theorem

  • Gauge invariance

    • Mechanics, Electromagnetism, Quantum Mechanics and Quantum Field Theory

      • Phase invariance

      • Minimal coupling

  • Localized magnetism

    • Localized moments

      • Effective interaction: Coulomb interaction and Pauli exclusion principle

        • Heisenberg model.

        • Exchange operator and coupling of two spins 1/2.

      • Other models: XYZ, Ising, Ising in transverse field, etc.

      • Ferromagnetism, antiferromagnetism, frustration.

    • Classical spin

      • Phase space: sphere

      • Langevin function

      • Partition function, magnetization, susceptibility

    • Spin 1/2

      • Partition function, magnetization, susceptibility

      • Entropy

        • entropy of a two state system

• Problems

  • Série 1) 6, 12, Série 4a) 4

• Thursday, March 21, 2019

  • Discussion of the Colloquium at the DF in the previous day

  • Spin in a magnetic field

    • Spherical basis

    • Heisenberg spin operators

    • Time dependent correlation functions and thermodynamic averages

    • Brillouin function

      • Partition function, magnetization, susceptibility

    • Particular cases: classical limit and spin ½.

• Problems

  • Série 1) 7 , 13; série 1a) 4, 5; série 2) 1, 2, 3

• Monday, March 25, 2019

  • Spin in a magnetic field (continuation)

    • Partition function, magnetization and susceptibility (Brillouin function)

      • Small field behavior: Curie law and rotational invariance.

      • Large field behavior: saturation and deviation from it.

      • Spin ½ and classical field limit.

      • Longitudinal and transversal susceptibilities.

      • S^2= S(S+1) relation.

    • Time dependent correlation functions

      • fluctuation-dissipation theorem.

      • classical limit.

• Problems

  • Série 1) 7, 15; Série 1a) 4; série 2) 1, 2, 3

• Thursday, March 28, 2019

• Discussion and solutions to exercises

• Localized moment models

  • Type of couplings: Heisenberg, XYZ, XY, Ising, etc.

  • Dimension of space and of the order parameter.

• Ferromagnetism and antiferromagnetism, Néel state

• Ferromagnetic Heisenberg chain

  • Ground state

  • Translational invariance

    • Spin waves, dispersion relation.

• Mean Field Approximation

  • Decoupling approximation.

  • Effective field.

  • Equation of state and constitutive equation.

• Problems

  • Série 1b) 6, Série 2b) 2

• Monday, Abril 1, 2019

• Discussion and solutions to exercises

• Heisenberg Model

  • Mean Field and Random Phase Approximations

  • Mutual consistency

  • Effective field and effective interaction

• Phase Transitions

  • Phenomenology of phase transitions

  • Symmetry. Explicit and spontaneous (or hidden) violation of symmetry.

  • Critical temperature. Order parameter. Correlation function.

  • Singularities and thermodynamic limit.

  • Critical exponents.

  • Dimension of space and of the order parameter. Universality.

  • Reference to quantum and topological phase transitions.

• Problems

  • Série 1) 14, Série 2) 4, Série 2a) 3, Série 2b) 1

• References

• BE and FD functions, Quantum Mechanics, E. Merzbacher, John Wiley & Sons ,Chap 22, Sect. 5, Quantum Statistics and Thermodynamics, pg. 566

• Critical Phenomena and Landau Free Energy, Condensed Matter Physics, M. P. Marder, John Wiley & Sons, Chapter 24, Sections 24.6 and 24.6.1, pgs. 691-698

• Thursday, April 4, 2019

  • Phase Transitions

    • Mean field theory. Importance of thermal fluctuations.

    • Landau theory.

    • Dimension of space and of order parameter. Universality.

    • Critical point. Calculation of the critical exponents.

    • Reference to the scaling relations between the critical exponents.

• Problems

  • Série 2) 4, Série 2a) 2, 3, 4, Série 2b) 3

• References

  • Principles of Condensed Matter Physics, P. M. Chaikin & T. C. Lubensky

    • Sections: 4.2, 4.3

• Critical Phenomena and Landau Free Energy, Condensed Matter Physics, M. P. Marder, John Wiley & Sons, Chapter 24, Sections 24.6 and 24.6.1, pgs. 691-702.

• Monday, Abril 8, 2019

• Phase Transitions

• Universality and scaling

• Transversal correlations and Nambu-Golstone boson

  • Examples: acoustic phonon, spin waves.

• Ginzburg criterion

  • Upper critical dimension.

• Mermin-Wagner theorem

  • Lower critical dimension.

• Problems

  • Série 2a) 1

• References

  • Principles of Condensed Matter Physics, P. M. Chaikin & T. C. Lubensky

    • Sections: 4.2, 4.3.

    • Section 3.5, 3.6, Appendix 3A.

• Extra Reference

  • Lectures on Phase Transitions and the Renormalization Group, N. Goldenfeld, Perseus Books, Introduction, pgs. 5-21.

• Wednesday, Abril 10, 2019 (extra class)

• Discussion and solutions to exercises.

• Thursday, April 11, 2019

• Discussion and solutions to exercises.

• Precession of an isolated spin in a magnetic field.

• Reference to the Landau–Lifshitz–Gilbert equation and to the Slonczewski spin-transfer torque of the dynamics of a macrospin.

• Heisenberg model. Spin precession equation.

• Ferromagnetic model. Linearization of the equations of motion.

  • Dispersion relation.

  • Magnons and the Nambu-Goldstone boson.

• Problems

  • Série 1a) 4, Série 2) 5.

• References

  • Introduction to Solid State Physics, C. Kittel, John Wiley & Sons, Chapter 12. Ferromagnetism and Antiferromagnetism. Magnons.

• Monday, Abril 22, 2019

  • Discussion and solution of the test.

  • Antiferromagnetic Heisenberg model.

    • Linearization of the equations of motion.

    • Dispersion relation.

    • Magnons and the Nambu-Goldstone boson.

  • Holstein-Primakoff transformation.

    • Ferromagnetic Heisenberg model.

    • Effective quadratic Hamiltonian.

    • Dispersion relation.

• Problems

  • Série 2) 5, 6, 7, 8, 9, 10, 11, Série 2a) 1, Série 4a) 9.

• References

  • Quantum Theory of Solids, C. Kittel, John Wiley & Sons, Chapter 4.

  • Principles of the Theory of Solids, J. M. Ziman, Cambridge University Press, Sections 10.11 and 10.12.

• Monday, Abril 29, 2019

  • Holstein-Primakoff transformation.

    • Antiferromagnetic Heisenberg model.

  • Bogoliubov-Valatin transformation.

    • Preservation of the (anti)commutation relations.

    • Effective quadratic Hamiltonian.

    • Dispersion relation.

• Problems

  • Série 2) 5, 6, 7, 8, 9, 10, 11, Série 2a) 1, Série 4a) 9.

• References

  • Quantum Theory of Solids, C. Kittel, John Wiley & Sons, Chapter 4.

  • Principles of the Theory of Solids, J. M. Ziman, Cambridge University Press, Sections 10.11 and 10.12.

• Thursday, May 2, 2019

  • Rayleigh-Ritz variational method. Hellmann-Feynman theorem.

  • Pauli exclusion principle. Slater determinants.

  • The self-consistent Hartree and Hartree-Fock approximations. Direct and exchange terms.

  • The electron gas.

  • Born-Oppenheimer approximation.

  • The jellium model. Electrical neutrality. Yukawa regularization of the Coulomb potential.

  • Second quantized Hamiltonian. Background cancellation.

  • Number operator. Number and kinetic energy.

  • Fermi sea.

• Problems

  • Série 1b) 1, 5, Série 3a) 4.

• References

  • Quantum Mechanics Vol.II, Messiah A., North-Holland, Chapter XVIII, II. The Hartree and Fock-Dirac Atoms.

  • Quantum Theory of Many-Particle Systems, A. L. Fetter and J. D. Walecka, McGraw-Hill, Chapter 1, Second Quantization.

  • Solid State Physics, N. W. Ashcroft and N. D. Mermin, Harcourt Brace College Publishers, Chapter 17, Beyond the Independent Electron Approximation.

• Monday, May 6, 2019

  • The Fermi sea as a vacuum.

  • Electrons and holes.

  • Physical interpretation. Energy and momentum.

  • Wigner-Seitz radius. High density expansion.

  • First order correction to the energy.

  • Reference to the Wigner solid.

• Problems

  • Série 1b) 5, Série 3a) 1, 2, 5.

• References

  • Quantum Theory of Many-Particle Systems, A. L. Fetter and J. D. Walecka, McGraw-Hill, Chapter 1 Second Quantization.

  • Solid State Physics, N. W. Ashcroft and N. D. Mermin, Harcourt Brace College Publishers.

  • Quantum Theory of Solids, C. Kittel, John Wiley & Sons, Chapter 4.

• Thursday, May 9, 2019

  • Correlation energy. Historical Importance.

  • Self-consistent Hartree-Fock approximation. Particle energies.

  • Screening. Dielectric constant.

  • Thomas-Fermi approximation.

• Problems

  • Série 3a) 1, 2, 3, 5.

• References

  • Quantum Theory of Many-Particle Systems, A. L. Fetter and J. D. Walecka, McGraw-Hill, Chapter 1 Second Quantization.

  • Solid State Physics, N. W. Ashcroft and N. D. Mermin, Harcourt Brace College Publishers.

  • Quantum Theory of Solids, C. Kittel, John Wiley & Sons, Chapter 4.

• Monday, May 13, 2019

  • Electron gas.

    • Susceptibility and density-density correlation function.

    • Lindhard function.

    • Static limit and linear Thomas-Fermi approximation. Comparison with the Debye-Hückel theory.

    • Long wavelenght limit and drude theory. Plasma frequency.

    • Reference to the Friedel oscillations, RKKY interaction and Kohn anomaly.

  • Itinerant magnetism.

    • Hubbard model. Stoner theory.

    • Effective Hamiltonian.

    • Equations of state and constitutive equations.

    • Stoner criterion.

• Problems

  • Série 3) 2; 3a) 6, 7; 3b) 1, 2, 3, 5.

• References

  • Solid State Physics, N. W. Ashcroft and N. D. Mermin, Harcourt Brace College Publishers.

  • Quantum Theory of Solids, C. Kittel, John Wiley & Sons, Chapter 4.

  • Green’s Functions for Solid State Physicists, Doniach and Sondheimer. Benjamin, Chapter 7.

• Thursday, May 16, 2019

• Itinerant magnetism.

  • Lindhard function.

  • Transversal susceptibilities and rotational invariance.

  • Spin waves and the Nambu-Goldstone boson.

• Superconductivity.

  • Historical development. Importance of superconductivity in the development of Physics. Reference to the Higgs mechanism. Applications.

  • Phenomenology: resistivity, Meissner-Ochsenfeld effect, type I and II superconductors, Abrikosov vortices, persistent currents and flux magnetic quantization, specific heat, absorption of radiation and energy gap, isotopic effect, Josephson effect.

  • London-London theory.

    • London equations.

    • London penetration depth.

• Problems

  • Série 3a) 7; 4a) 1, 2, 4.

• References

  • Green’s Functions for Solid State Physicists, Doniach and Sondheimer. Benjamin, Chapter 7.

  • Superconductivity, Wikipedia, https://en.wikipedia.org/wiki/Superconductivity

  • Solid State Physics, G. Grosso and G. P. Parravicini, Academic Press, Chapter XVIII.

  • Quantum Theory of Many-Particle Systems, A. L. Fetter and J. D. Walecka, Chapter 13.

  • Solid State Physics, N. W. Ashcroft and N. D. Mermin, Harcourt Brace College Publishers.

  • Quantum Theory of Solids, C. Kittel, John Wiley & Sons, Chapter 4.

• Monday, May 20, 2019

• Superconductivity.

  • Ginzburg-Landau theory.

    • Non-linear Schrödinger equation.

    • Maxwell equation and the expression for the current.

    • Meissner effect, London gauge and gauge invariance.

• Problems

  • Série 4a) 2, 3, 5.

• References

  • Solid State Physics, G. Grosso and G. P. Parravicini, Academic Press.

    • Chapter XVIII. Superconductivity.

  • Quantum Theory of Many-Particle Systems, A. L. Fetter and J. D. Walecka.

    • Chapter 13. Superconductivity.

  • Feynman Lectures on Physics, vol III, R. P. Feynman, R. B. Leighton, M. Sands, Addison Wesley 1970.

    • Chapter 21. The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity.

  • I. J. R. Aitchison and A. J. G. Hey, Gauge Theories in Particle Physics, 1989.

    • Chapter 13. Hidden gauge invariance: the U(1) case.

• Thursday, May 23, 2019

• Superconductivity.

  • Ginzburg-Landau theory.

    • Bulk order parameter. Critical field.

    • Ginzburg coherence length.

    • Interfaces and Abrikosov vortices. Type I and type II superconductors.

    • Magnetic flux quantization.

  • BCS theory. Overview.

    • Attractive effective electron-electron interaction.

    • Cooper pairs. Superconducting instability.

    • Reduced Hamiltonian. BCS variational wave function for T=0º.

    • Mean field theory for T ≠ 0º and BCS decoupling.

    • Bogoliubov-Valatin transformation.

    • Equations of motion method.

    • Gap equation for T=0º and T ≠ 0º. Gap at T=0º and T_C.

• Problems

  • Série 4) 2, 3, 4; 4a) 6, 7, 8.

• References

  • Solid State Physics, G. Grosso and G. P. Parravicini, Academic Press,

    • Chapter XVIII. Superconductivity.

  • Quantum Theory of Many-Particle Systems, A. L. Fetter and J. D. Walecka,

    • Chapter 13. Superconductivity.

  • W. A. Harrison, Solid State Theory, 1970.

    • B. Superconductivity.

• Monday, May 27, 2019

  • BCS theory. Calculational detais.

    • Attractive effective electron-electron interaction.

    • Cooper pairs. Superconducting instability.

    • Reduced Hamiltonian. BCS variational wave function for T=0º.

• Problems

  • Série 4) 2, 3, 4; 4a) 6, 7, 8.

• References

  • G. Baym, Lectures on Quantum Mechanics.

    • Chapter 8, Cooper Pairs.

  • W. A. Harrison, Solid State Theory, 1970.

    • B. Superconductivity.

  • Solid State Physics, G. Grosso and G. P. Parravicini, Academic Press,

    Chapter XVIII. Superconductivity.

• Thursday, May 30, 2019

  • BCS theory. Calculational detais.

    • Mean field theory for T ≠ 0º and BCS decoupling.

    • Bogoliubov-Valatin transformation.

    • Equations of motion method.

    • Gap equation for T=0º and T ≠ 0º. Gap at T=0º and T_C.

• Problems

  • Série 4) 2, 3, 4; 4a) 6, 7, 8; 4b) 1.

• References

  • W. A. Harrison, Solid State Theory, 1970.

    • B. Superconductivity.

  • Solid State Physics, G. Grosso and G. P. Parravicini, Academic Press,

    • Chapter XVIII. Superconductivity.

  • Theory of Superconductivity, J. R. Schrieffer, Perseus Books,

    • 2-5 Quasi-Particle Excitations.