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Title: Theoretical studies of systems with long-ranged interactions
Author: Johnson, S. R.
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 1996
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It is demonstrated that a perturbation theory for the Coulomb Gap can be developed, which is identical to the TAP approach to spin glasses. Thus chapter 2 presents the standard theoretical method used to study the Coulomb Gap, together with the analogous result in spin-glass theory. Chapter 3 begins the exposition of new work contained in the thesis. The example of the classical dipole fluid is used to demonstrate how classical spin systems with long-ranged interactions can be expanded as a series of Feynman diagrams. A similar method, taking account of the disorder in the Coulomb Gap problem, is used to derive the strong-disorder Coulomb Gap density of states as a leading set of Feynman diagrams. In the next chapter, it is shown that the assumption of single-parameter scaling enables the use of the Renormalization Group to derive such non-perturbative effects as the crossover to weak disorder, and edge effects within the Coulomb Gap fluid. In chapter 5, it is shown that the analogy with spin glasses enables the use of the replica trick to obtain an effective field theory for the Coulomb Gap, which resembles the Sine-Gordon model in statistical physics. This resemblance allows a systematic derivation of the RG scaling function to any accuracy and a study of the renormalizability of the effective field theory, leading in principle to a test of the single-parameter scaling hypothesis. The second part of the thesis considers the optical phonons in a disordered ionic crystal, and in chapter 6 it is shown that the use of projection operators together with the standard techniques of perturbation theory enables a derivation of the phonon density of states. The result is that the longitudinal and transverse optic phonon modes are broadened by disorder into semicircular bands, and is identical to the result for the disorder-broadened Landau levels in a metal. Inspired by this similarity, methods of perturbation theory and supersymmetry have been applied to deriving the transport properties of the phonons. As described in section 7.2, the lack of a small parameter makes this impossible for the disordered ionic crystal, but the simpler problem of a uniaxial ionic crystal would be suitable for the use of these techniques.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available