Physics 490 - Spring 2000

Suggested Topics for Term Paper

Consult with Prof. Martin before you decide on a topic, whether it is on this list or not. It is important that the topics for the different papers do not overlap too much.

Papers are due during finals week at the time of presentation to class.  (Normal final exam time to be announced.)
 

The paper should be typed in the REVTEX format in the style of a Phys Rev paper, with equations, figures (if needed), and references to the literature just as in a paper to be published. The final version should include figures and be in a single postscript file. I also want to get the latex file and (postscript figure files) sent by e-mail. That is the way you would submit it to Phys Rev. The staff will help you with REVTEX and making the postscript files. The term papers will be collected in a volume, reproduced and distributed to all students. The postscript version will also be published electronically on the class WWW pages

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Suggested Topics for Term Paper
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1. Hubbard Model. A general review is given in the review paper by Imada in the reserve list. For a term paper, you could start from references given there and discuss in more detail the sense in which the Hubbard-type models represent some important types of physical systems, and solutions of the Hubbard-type models.  References include the book by Fradkin and "The Hubbard Model".
2. Heisenberg Model. The Heisenberg model describes spins which interact with each other on different sites in a solid. The spin dependent interaction ultimately are caused by the Coulomb interactions between electrons. For a term paper, show the way that Heisenberg models can be derived from interacting electron models (e.g. Anderson or Hubbard type models). One possible topic would be "superexchange" which is the name given by Anderson to the interactions between spins centered on neighboring metal sites in insulating compounds like the transition metal oxides. Another topic would be the Ruderman-Kittel interactions between spins in metals. In either case, the basic principles and examples should be given.
3. t-J Model. This is the model for added holes in the CuO superconductor and antiferromegnetic insulator systems. It is argued to be derived by simplifying the the periodic generalized Anderson and Hubbard-type models. References include the review by Imada, et al., and the review by Dagotto, Rev. Mod Phys 66, 763 (1994).
4. Luttinger "Theorem" on the volume enclosed by the Fermi surface. Luttinger gave the argument which is very often quoted that in a many-body interacting electron system the volume of the Fermi surface is the same as in a non-interacting system. His proof was based upon equating two expressions for the total number of electrons. Luttinger was careful to state that his demonstration relies upon the assumption that perturbation theory converges starting from some non-interacting state - the conditions for validity of Fermi Liquid Theory. For a term paper restate Luttinger's proof in terms of diagrammatic summations and the state any assumptions made.  Uses linked cluster expansion decsribed in Mahan 3.6.
   J. M. Luttinger, PR 119, 1153 (1960).
   Abrikosov, et. al. textbook.
   R. M. Martin, PRL 48, 362 (1982), and references therein.

5. Friedel Sum rule in many-body systems. The Friedel Sum Rule relation phase shifts at the Fermi energy to the number of electrons around an impurity. For a term paper state the general proof based upon the S matrix ( given by Langer and Ambegaokar) that phase shifts can be defined at the Fermi energy of an interacting system and that the phase shift agrees with the Friedel sum rule which was originally derived in a strictly non-interacting electron approximation. Also give the application to the Anderson model as discussed by Langreth. (Closely related to Luttinger Theorem )
   .J. Langer and V. Ambegaokar, PR 121, 1090 (1961).
   D. C. Langreth, PR 150, 516 (1966).
   R. M. Martin, PRL 48, 362 (1982), and references therein.
6. Density Functional Theory. Proof that the exact total energy of an interacting electron system is a unique functional of the density following P. Hohenberg & W. Kohn, and the Kohn-Sham formulation in terms of single body methods. A paper would require a more in-depth study than given in class of the many-body theory that is behind the functional formexchange and correlation.  See, for example, the ook by Dreizler and Gross, and earlier references.
7. The Zaanen-Sawatzky-Allen classification scheme of the transition metal compounds into "charge transfer" insulators and "Mott-Hubbard" insulators. This is argued to be important in the properties of the added carriers in the CuO and related materials, the basis for the t-J model, etc. Discusssed in the review by Imada, et al.
8. Evidence for stripe instabilities in strongly correlated systems. Main references are to Emergy, Kivelson,and Fradkin. Prof. Martin can help with references.
9. Double Exchange. This is the mechanism proposed for the ferromagnetism in the "Collosal Magneto Resistance" (CMR) materials studied recently. It is an old model due to Zener, but has been considered in more detail recently. It is discussed in the review by Imada, et al. Here one could describe the model and give experimental evidence for the properies of the CMR materials.
10. Mott metal-insulator transition. There are many aspects of this very general problem which is reviewed by Imada, et al.
11. Materials systems in which electron correlation causes metal-insulator transitions. Several classes of materials are discussed in the review by Imada, et al.
12. Spin waves. For a term paper describe the Holstein-Primikoff transformation from a spin hamiltonian to derive spin waves (magnons). (Is this an exact or approximate transformation to Bosons?) What is the resulting specific heat at low T? Are there magnon-magnon interactions that are analogous to anharmonicity in phonon problems?
13. The observation of the Fermi surface and the superconducting gap in the high-Tc materials by photoemission. Also dispersion of bands in the related insulators. Much work has been done by many people including Z. X. Shen and coworkers. Many references are in the review by Imada, et al.   Recent articles in Science give very different interpretations.  Prof. Martin can help with references.
14. Analysis of the Kubo formulas for conductivity in interacting systems at finite T following Mahan Ch 3.  This would mean a discusiion in terms of the finite T greens functions beyond what we could do in class.
15. Review of the "Bethe Ansatz" that allows exact solutions of certain 'integrable" 1-d models. An xample is the 1-d Hubbard model by Lieb and Wu.
16. "GW" Approximation for quasiparticle energies. For a term paper describe Hedin's formulation of the "GW" approximation for quasiparticle self-energies and the Plasmon Pole Approximation for evaluation of this term in a nearly-free-electron-like system. Give the results from at least two recent applications of the GW approximation to solids. L. Hedin & S. Lundquist, Solid State Phys. vol. 23, 1 (1968). Many recent papers (often by S. G. Louie and co-workers) on solids.
17. Comparison of experiment and theory for bands in the simplest nearly-free-electron metals, Na, etc. Experiments by E. W. Plummer, Surface Sci. 152/153, 162 (1985) have lead to several different theories of the discrepancy with standard band calculations. A paper by Northrup, at. al., says a "GW" calculation of the electron self-energy resolves the discrepancy. For a term paper describe the main effects of the "GW" as given by Northrup and papers referred to by him, as well as some of the controversies among the papers referred to in the section near the end of Northrup's paper entitled "comparison with other theories". J. E. Northrup, et.al., PR 39, 8198 (1989).
18. Infinite dimensional methods. As pointed out first by Metzner and Vollhardt, mean field theory is exact infinite dimensions. It is not the usual static mean field theory but a dynamic mean field theory that includes effects of correlation. It is very powerful and suggestive for finite dimensions. An excellent review is by Georges, et al., Rev. Mod. Phys. 68, 13 (996) and there is discussion in the review by Imada, et al.
19. Bosonization. Transformation of degrees of freedom of a many-electron system into Boson variables. The plasmons of RPA are an example, but it is a more general technique. There is much work by E. Fradkin and by D. Haldane. Discussion can be found in the notes of P. Phillips on reserve.
20. A paper P. W. Anderson ( PRL 67, 660 (1991)) referred to an paper from the 1950’s (before BCS) which showed that known experimental information on thermodynamics of superconductors could be used to derive the change in kinetic energy of the nuclei at the superconducting transition. Anderson says this paper "demonstrated ... it is the phonon frequencies which are lowered, and this accounts for the condensation energy [of the superconducting state]." Give the proof on the change in the kinetic energy and comments on Anderson's statement that the lowering of the phonon energies "accounts for the condensation energy ".
21. Superconducting Instability in any metallic system. Kohn and Luttinger (PRL 15, 524 (1965)) proposed that in any condensed matter system of fermions there will be long range attractions (e.g. like those in 4He) that can lead to pairing for high angular momentum pair states and thus to superconductivity or superfluidity. A term paper should carefully describe the arguments and check for any recent work on this question.
22. Maximum Tc in a superconductor? An estimate was given by M. L. Cohen and P. W. Anderson for the maximum possible transtion temperature in a superconductor using "current theories" (i.e. BCS) based upon phonons. (The work is many years old. I have a copy of a paper from a conference. You may copy it since I do not know the proper reference!) A term paper could be to describe their arguments which lead to a maximum value of Tc and their numerical estimates, and to find at least one recent reference that addresses whether or not Cohen and Anderson really found an upper bound for superconducting Tc due to phonon mechanisms.
23. A famous paper by W. Kohn (PR A 171 (1964)) entitled "Theory of the Insulating State" showed that electronic states can rigorously be considered to be localized in an insulator whereas they cannot in a metal. This is well known in a crystal: because there is a band gap separating empty from filled bands, the latter can be described by localized Wannier states. Kohn showed that in a disordered insulator with no gap in the density of states the eigenfunctions are also strictly localized. A term paper could restate his theoretical analysis and describe the relation to metallic vs. insulating behavior in disordered materials.
24. Dielectric polarization in materials and Berry's phases. The formulation of changes of electric polarization in terms of a Berry's phases involving the variation of the phases of the Bloch functions as a function of the wave vector k has resolved an old problem. A review is in Resta, Rev. Mod. Phys., and a short review by R. M. Martin and Gerardo Ortiz "Recent Developments in the Theory of Electric Polarization in Solids", Solid State Communications 102, 121 (1997). by R. M. Martin in Solid State Communications, and the many-body formulations has been given by G.Ortiz, and R.M.Martin, "Macroscopic polarization as a geometric quantum phase: many-body formulation", Phys.Rev.B 49, 14202 (1994).
25. "Orthogonality catastrophe" and x-ray edge singularity in metals. Mahan and Nozieres & deDominicis studied the problem of the shape of the edge in x-ray absorption in metals. They stated the problem in terms of the response of the metal to a sudden potential and they showed that there is vanishing matrix element between the states created in the absorption process and the exact ground state of the final system. This is discussed in the texts by Mahan, Doniach, and Anderson. For a term paper state the problem, the theoretical arguments, and the form of the edge shape predicted. Give at least two references to either a) comparison with experiment or b) further theoretical connections to other problems.
26. The Schrieffer-Wolfe transformation relating the Anderson impurity model and the Kondo effect. This in effect isolates the low energy degrees of freedom causes by the strong Coulomb interactions between electrons on an impurity site.
27. Solitons and Fractional Charge. Linear chains can have excitations with quantum numbers fundamentally different from those of the bare particles. An example is provided by one of the simplest possible linear chains formed of carbon, which apparently exists under special circumstances based upon experimental evidence. For this paper you should describe the ideas of solitons and "charge fractionalization" and also one of the following: a) the experimental evidence for these carbon chains; or b) give other references and work on fractional charges in chains. Reference: M. J. Rice, et. al., PRL 23, 2136 (1983).
28. Charge density/spin density wave instabilities in 1d and "nested" systems. There is a great variety of work in this area, involving lattice distortions, magnetism, and other instabilities. A term paper could be to analyze the instability in lower dimensional systems and give examples where such instabilities occur.
29. Sum Rules on the dielectric function in a crystal. There are sum rules for a crystal whixch generalize the famous plasma sum rule in a homogeneous system, which relate integrals over the inverse dielectric function to Fourier components of the density. These are sometimes known as the Johnson sum rules (D. L. Johnson, PR B 9, 4475 (1974)) and are also given by M. Hybertsen and S. G. Louie, PR B 34, 5390 (1986), Eq. (29). A term paper could derive these sum rules following the references given.
30. Quantum Hall Effect. Among many possible topics for a term paper could be:
    1) A summary of the proof based upon gauge invariance of the integer QHE and the zero resistance effect. The main reference is the short paper by R. G. Laughlin.
    2) Description of the way that 2-d electron states in a magnetic field can go around impurities with no scattering, which is crucial for the zero resistance effects. One analysis has been given by R. Prange.
    3) Experimental studies of quantization, especially the effects of finite temperature.
    4) The role of edge currents in the QHE. Edge effects are essential in the definitions of the voltages in the QHE. Perceptive analyses have been given by B. I. Halperin.
    5) The fractional QHE caused by electron-electron interactions.