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Description
Outline:
This module provides a basic grounding the in the analytical techniques of many body quantum mechanics as applied to condensed matter systems.
Aims:
The aim of this module is to give a theoretical introduction to a broad range of quantum mechanical phenomena occurring in condensed matter physics, the construction of model Hamiltonians to capture them and the mathematical tools required to analyze them. The emphasis will be upon canonical ideas 鈥� where possible these will be connected to questions of current research interest.
Intended Learning Outcomes:
At the end of this module you will master second quantisation of many-body quantum systems and be able to diagonalise quadratic (non-interacting) Hamiltonians using unitary transformations and Bogoliubov transformations. You will understand the concept of mean-field theory to self-consistently reduce interacting many-body Hamiltonians to quadratic Hamiltonians. These general concepts will be applied to lattice vibrations (phonons), spin waves in magnets (magnons) and interacting systems of bosons and fermions. This will enable you to formally describe superconductivity in terms of BCS theory and the Bose-Einstein condensation of weakly interacting bosonic atoms.
Teaching and Learning Methodology:
This module is delivered via weekly lectures supplemented by a series of workshops and additional discussion.
In addition to timetabled lecture hours, it is expected that students engage in self-study in order to master the material. This can take the form, for example, of practicing example questions and further reading in textbooks and online.
Indicative Topics:
'+ Overview of second quantization: introduction to the mathematical methods of second quantization. Building upon the use of raising and lowering operators for the harmonic oscillator.
Part I: Bosons: Phonons and Quantum Magnets
+ Phonons: coupling harmonic oscillators, from two to a chain. Using the Fourier transform to diagonalise the Hamiltonian.
+ Spinwaves in quantum magnets: The Heisenberg model of insulating magnets. Using the Holstein-Primakoff transform to write this Hamiltonian in terms of bosonic operators. Spinwaves in the ferromanget and anti-ferromagnet. Zero-point energy in the anti-ferromagnet. The Mermin-Wagner theorem.
+ Haldane鈥檚 Conjecture: the topological distinction between integer and half-integer anti-ferromagnet 鈥� one of the first applications of topology in quantum theory.
+ The weakly-interacting Bose gas: Calculating the superfluid fraction and thermodynamic and superfluid properties.
Part II: Fermions
+ Landau鈥檚 theory of Fermi liquids: Extending the idea of the Fermi gas to allow for interactions. The Landau-Fermi liquid phenomenology and simple calculations of its properties.
+ BCS theory of superconductivity: Interaction between electrons and phonons 鈥� attractive interaction at Fermi surface. The Cooper instability towards electron pairing. BCS theory of superconductivity.
+ The Hubbard model and Mott transition: Introduction to the Hubbard model as a simple model to account for the properties of electrons in solids. Review of systems to which it applies. Simple discussion of the Mott transition from insulating to metallic behaviour
+ Impurities in metals; the Kondo model
+ Brief intro to topology in condensed matter physics: quantum Hall effect, Haldane model and topological insulators
Module deliveries for 2024/25 academic year
Last updated
This module description was last updated on 8th April 2024.
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