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     (I) SUMMARY OF MAIN TOPICS TO BE COVERED IN DAGOTTO'S LECTURES:

     (1) Aschroft+Mermin, chapter 17. Here we start with the analysis of screening in metals.
     We will define epsilon(q) and chi(q), using the ``linear approximation'' or ``linear response''
     assumption. Before this subject, details about the homework projects will be discussed in the first class.
   
    (2) To evaluate chi(q), we will use the Thomas-Fermi  approx. where the potential is
     assumed to change slowly with position. A formula will be derived in q-space and transformed
     to real space, showing the presence of an exponential suppression of the original 1/r potential
     of a charged impurity. The screening length associated with the cloud of electrons around
     the impurity will be estimated for standard metals and it will be found to be ~0.5 - 1 Angstrom.
    The Lindhard theory of screening is discussed next. The perturbative calculation will be sketched.
    The limit q-->0 will be shown to give Thomas Fermi. The real space total potential has oscillations
    (Friedel oscillations) related with singular behavior at q=2kF in q-space. The formula for
     epsilon(q,omega) will also be briefly discussed.

     (3) Special cases of Lindhard function. Kohn anomalies in phonons. Explanation  of singular
     behavior at 2kF based on Fermi surface arguments.  RKKY interaction between spins. Plasma
     oscillations in electrons. Exponential or oscillatory decay of electric fields inside  the sample.
     Plasmon and its energy vs. k relation. Explanation in terms of long-range forces.
     Plasmon frequency for positive ions without electrons. Inclusion of electrons and relation
     between phononic and Fermi velocities.

      (4) Dielectric function of a metal. Time-delayed interactions between electrons, mediated
      by positive ions. Retardation.  Overscreening,  intuitive meaning. Attraction between carriers,
      once retardation caused by heavy ions is considered. Switch from external charges to a pair of
      electrons within the sample (q->k-k', omega -> Delta E/hbar).  Representation in terms of Feynman
      diagramms, conservation of momentum and energy at every vertex. Discussion of Cooper's paper
      (Phys. Rev. 104, 1189 (1956)), following Kittel's book. We will end with the writing of the
      Schroedinger equation in terms of the energy e and the matrix element <e|V|e'> of the potential.

      (5) Solution of the 2 electron Cooper problem in the presence of many other inert electrons
      near the Fermi level. Relevance of other electrons to existence of bound state for weak attraction V.
      Formula for Delta as a function of omega_Debye, density-of-states at EFermi, and attraction V. Size of pair.
      Relevant Hamiltonian in the space of pairs. BCS wave function. Values of uk and vk, and gap Delta.

       (6) We will start with high temperature superconductors following Rev. Mod. Phys. 66, 763 (1994).
       Introduction on page 765: typical chemical compositions, Tc's, structures, ionic charge of
       the many elements. Coppers are in a state of spin 1/2.  We will discuss single- and bi-layer
       materials. Chemical doping such as La replaced by Sr. Phase diagram of LSCO.     Optimal,
       underdoped and overdoped doping. Parent insulating antiferromagnetic compound (x=0). 
       Doping is through holes, but it can also be electrons (NdCeCuO). Linear resistivity, as opposed
       to quadratic as in standard metals.  

      (8) We will study the instructor's notes (hardcopy to be provided) on the mean-field solution of the
      Hubbard model. Since the math is fairly clear in the notes, only the important concepts will be
      addressed. We will discuss up to page 10 approx. The presence of a gap induced by the interaction
      U is important (Hubbard gap , aka Mott gap; Hubbard-Mott insulator). Similarities with formulas for
      BCS superconductors are clear. The mean-field Hamiltonian appears to not conserve momentum
      since electrons with momentum p can be transformed into p+Q. Explanation will be provided based
      on the spin-waves of momentum Q which can be easily created. We will continue with the study
      of the one-band Hubbard model, analyzing the ``quasiparticles'' gamma which are a mixture of
      electrons with momentum p and spin sigma, and electrons with  momentum p+Q and spin sigma. 
      We will discuss the similarities with the BCS formalism.

       (9)  We will study spin-ladders as examples of materials that have a ``spin gap''. Basic ideas and
      suggestion of superconductivity upon doping. Experimental realizations.  Reference:
      Science 271, 618-623 (1996) .

       (10)  We will learn what ``manganites'' are, namely the materials with the colossal magnetoresistance (CMR).
      Hamiltonians for manganites will be presented and they will involve many active orbitals, and strong
      Hund couplings. The experimental phase diagram will be discussed, as well as the magnetoresistance
      effects. Monte Carlo results will suggest the main competing phases and also the relevance of phase
      separation ideas. Orbital order will be explained. Experimental evidence of inhomogeneities will be
     discussed. Numerical results suggesting that the inhomogeneities are crucial to understand the CMR
     will be given.

      We will follow E. Dagotto et al., Phys. Reports 1, 344 (2001). A copy can be found here.

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