QFT-temp-04

QFT-temp-04 - Physics 243 Lecture Notes Chetan Nayak 2...

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Unformatted text preview: Physics 243 - Lecture Notes Chetan Nayak January 27, 2004 2 Contents 1 Impurities in Solids 1 1.1 Impurity States . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Density of states*** . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.1 Anderson Model . . . . . . . . . . . . . . . . . . . . . 3 1.3.2 Locator expansion*** . . . . . . . . . . . . . . . . . . 5 1.3.3 Anderson Insulators vs. Mott Insulators . . . . . . . . 5 1.4 Physics of the Insulating State . . . . . . . . . . . . . . . . . 7 1.4.1 Variable Range Hopping . . . . . . . . . . . . . . . . . 7 1.4.2 AC Conductivity . . . . . . . . . . . . . . . . . . . . . 9 1.4.3 Effect of Coulomb Interactions . . . . . . . . . . . . . 10 1.4.4 Magnetic Properties . . . . . . . . . . . . . . . . . . . 12 1.5 Physics of the Metallic State . . . . . . . . . . . . . . . . . . 16 1.5.1 Disorder-Averaged Perturbation Theory . . . . . . . . 16 1.5.2 Lifetime, Mean-Free-Path . . . . . . . . . . . . . . . . 18 1.5.3 Conductivity . . . . . . . . . . . . . . . . . . . . . . . 20 1.5.4 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5.5 Weak Localization . . . . . . . . . . . . . . . . . . . . 28 1.5.6 Quasiclassical Approach to Weak Localization*** . . . 33 1.5.7 Weak Magnetic Fields and Spin-orbit Interactions: the Unitary and Symplectic Ensembles . . . . . . . . . . . 33 1.6 The Metal-Insulator Transition . . . . . . . . . . . . . . . . . 33 1.6.1 Percolation . . . . . . . . . . . . . . . . . . . . . . . . 33 1.6.2 Mobility Edge, Minimum Metallic Conductivity . . . . 35 3 4 CONTENTS 1.6.3 Scaling Theory . . . . . . . . . . . . . . . . . . . . . . 37 CHAPTER 1 Impurities in Solids 1.1 Impurity States In the previous parts of this book, we have discussed the low-energy excita- tions which result from broken symmetry, criticality, or fractionalization. In this final part of the book, we discuss the low-energy excitations which result from the presence of ‘dirt’ or ‘disorder’ in a solid. By ‘dirt’ or ‘disorder’, we mean impurities which are frozen into the solid in some random way. Consider phosphorous impurities in silicon. Presumably, the true ground state of such a mixture is one in which the phosphorus atoms form a super- lattice within the silicon lattice. However, this equilibrium is never reached when the alloy is made: it is cooled down before equilibrium is reached, and the phosphorus impurities get stuck (at least on time scales which are relevant for experiments) at random positions. These random static spatial fluctuations have interesting effects on the electronic states of the system: they can engender low-lying excitations and they can dramatically change the nature of such excitations. To see the significance of this, recall that, in the absence of a broken continuous symmetry, a system will generically form a gap between the ground state and all excited states. By tuning to a critical state – such as a Fermi liquid – we can arrange for a system to...
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QFT-temp-04 - Physics 243 Lecture Notes Chetan Nayak 2...

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