CMSP course for DTP students
(Monsoon '08)
Instructors: Kedar Damle & Vikram Tripathi
From Aug 11 to Sept 19 (KD)Timings: Monday-Thursday 1.00pm-2.30pm
(Extra class if needed: Tues. 1.00pm-2.30pm)
From Sept 22 to Oct 31 (VT)
Timings: To be decided.
This course is self-contained: i.e for students arriving at TIFR with a BSc, the TIFR `Core' courses in Statistical Mechanics and Quantum Mechanics (I) suffice as prerequisite; MSc students will have learnt equivalent preparatory material during their Masters elsewhere. First year students interested in attending this course for credit are strongly advised to first check with the Subject Board in Physics about the latest rules that apply to them, since these may be in a state of flux.
Course Outline
(Initials in bracket refer to lecturer. 00 refers to the fact that we most likely not have time to do this in the Monsoon semester, and may teach a follow up course sometime later to introduce the material being left out)I: Basic conceptual/technical stuff. (KD)
Linear response theory Why do we care about correlation functions? (Using statistical mechanics to understand linear response measurements, spectral representations, correlation functions, fluctuation-dissipation theorem.)
Conceptual Preview
Emergent properties of many-body systems (rigidity etc), characterizing phases (broken symmetry, order parameters, confinement/deconfinement of vortices, transport properties etc), universality---usefulness and limitations, semiclassical dynamics of order parameters (or: why is symmetry broken only in infinitely large systems?).
Review: Quantum---Classical Correspondence
Quantum statistical mechanics of a single particle---path-integral formulation and connection with classical statistical mechanics, reverse mapping: D=1 classical Ising modelt---equivalence to a single quantum spin in a transverse field, connection between physical quantities in both pictures. (review material)
Ordered and paramagnetic phases in simple models: Discrete vs
continuous symmetry cases.
Ising Case
Equivalence of 1d quantum Ising model with D=2 classical Ising model (Transfer matrix), Stability of ordered state, stability of disordered phase, qualitative properties of both phases, `soft-spin' Ginzburg-Landau effective theory, dynamical properties of quantum model.
O(3) Case
Equivalence of 1d quantum rotor model with D=2 classical Heisenberg model, Instability of ordered state of rotors for 1d quantum rotor model, stability of ordered state of 2d quantum rotors & qualitative properties of ordered state: spin-waves and Goldstone's theorem, stability of disordered phase in all dimensions (strong-coupling expansion), qualitative properties, sketch of dynamical properties.
II: Examples from condensed matter physics. (KD)
Preliminaries: The language used for spins Coherent states, path integral, Berry phase for spin, geometric interpretation.
Quantum antiferromagnets
From strongly-correlated electrons to effective spin model: A sketch
One-band Hubbard-model description for some magnetic insulators, Mott-Insulating state at zero doping, spin Hamiltonian for the Mott Insulator.
Low-energy long-distance theory for antiferromagnets with (at least) short-range Neel order.
Coherent-state path integral, large-S approximation, continuum limit: 1d case and 2d cases, geometric interpretation of `topological' term in 1d half-integer case.
Applying rotor-model results to quantum antiferromagnets
Mapping between operators, general criterion for information that can be reliably extracted, some examples.
Many body language for bosons
Creation and annihilation operators and many-body language
Interacting Bosons in a lattice potential (KD)
Physical realizations
Josephson Junction Arrays, atom-trap BECs in an optical potential, effective model (Boson-Hubbard model)
Superfluid phase of the Boson-Hubbard model
Representation in terms of planar rotors, `universal' hydrodynamic properties: stiffness and sound mode, vortex excitations and connection to earlier classical discussion, qualitative discussion of properties of superfluid phase, superfluid phase with long-range interactions between particles: The Anderson-Higgs phenomenon (sound ---> plasmon)
Insulating phase of the Boson-Hubbard model
Strong-coupling expansion, spectrum: gapped particle-hole excitations etc.
III: Introduction to renormalization group ideas (KD)
`Poor man's scaling' approach to ordered states with continuous symmetry-breakin
g: Example
of O(N) rotors
Conceptual introduction
Continuum theory for ordered phase, the RG idea, a road-map of steps: eliminate short-distance modes, change cutoff, rescale, compare coupling constants.
Actual calculations to lowest non-trivial order
Results and their interpretation
Conclusions for stability of phase and various properties at T=0 and low-T.
T=0 Superfluid-Insulator transition at integer density in two spatial dimensions:
Epsilon expansion.
IV: Fermi Liquid Phenomenology (VT)
The interacting Fermi liquid phenomenology
Landau quasiparticles
Notion of quasiparticle, low energy effective theory for quasiparticles, ADD/CHANGE HERE
Low energy modes of quasiparticle liquid
ADD/CHANGE HERE
instabilities of quasiparticle liquid.
ADD/CHANGE HERE
V: Fermi Liquid: Calculational basis (VT)
Many-body language for fermions Anticommuting creation and annihilation operators, grassmann variable path integrals ADD/CHANGE HERE
Perturbation theory for many-fermion system Formulation of perturbation theory, Green's functions, diagrams, self-energy, dyson equation. ADD/CHANGE HERE
Perturbation theory for many-fermion system Sketch of Nozieres `derivation' of fermi-liquid theory. ADD/CHANGE HERE
VI Local moment physics (VT)
Local moments Origin of local moments
Simple theory, examples. ADD/CHANGE HERE
Interaction of local moments
RKKY coupling in simple metals, examples. ADD/CHANGE HERE
Quenching of local moments
Interaction of local moments with conduction band electrons, screening of moment (qualitative preview) ADD/CHANGE HERE
Kondo Effect and Kondo Problem
Examples of Kondo effect, statement of Kondo Problem, sketch of solution. ADD/CHANGE HERE
Competion: RKKY vs Kondo
ADD/CHANGE HERE
VII: Other topics (00)
One, two, three...infinity: The large-N approximation to quantum rotors
Basic idea behind the approximation, technical details of calculation, results at large-N: static and dynamic properties of each phase, properties at and near critical point (as an introduction to the phenomenology of (quantum) critical points)
The Kosterlitz-Thouless transition
Putative ordered phase and vortices
Naive continuum theory for ordered phase of 2d xy model---Gaussian model and properties, vortices, energetics of a single vortex, entropy-energy balance and stability of confined phase, interaction between two vortices, qualitative picture of transition: deconfinement of vortex pairs.
Technical tools
Villian approximation, Poisson-resummation and dual representations, height model, Sine-Gordon model
Some calculations
Vortex corrections to correlation functions, RG approach, partial derivation of flow equations, Kosterlitz-Thouless phase transition and properties.
BCS theory of superconductivity
One dimensional systems: Luttinger liquids and bosonization
Non-fermi liquids: Review of some well-known experimental
examples (no theory)
Quantum transport theory.
References
(List ordered alphabetically by first author):Abrikosov Gorkov Dzyloshinksii `Methods of QFT in Statistical Physics'
Anderson's `Basic Notions of Condensed Matter Physics'
Auerbach's `Interacting Electrons and Quantum Magnetism'
Chaikin & Lubensky's `Principles of Condensed Matter Physics'
Goldenfeld `Lectures on Phase transitions and RG'
Hewson
Kogut's Reviews of Modern Physics article `Lattice-gauge Theories and Quantum Spin Systems'
Landau and Lifshitz's `Statistical Physics: Part I'
Pines and Nozieres `Theory of Quantum liquids I, II' Negele and Orland's `Quantum Many-particle Systems'
Nozieres `Interacting Fermi systems'
Sachdev's `Quantum Phase Transitions'