Class Title: Topics in Many-Body Physics
Session: Spring 2021
Days & Times: TuTh 2:30-3:45PM
Zoom based: https://umass-amherst.zoom.us/j/98834363013
Instructor: Tigran Sedrakyan
E-mail: tsedrakyan@umass.edu
Office: HAS 403B
Phone: 413-545-2409
Office hours: Fridays or we can meet by appointment.
TOPICS:
1. The second quantization and its applications.
2. Green’s function method and the concept of quasiparticles,
3. Perturbation theory and Feynman diagrams
4. Landau Fermi-liquid theory
5. Topological states of matter
6. Quantum Hall effect (QHE); fractional QHE and composite fermions
7. Chern-Simons Theory
Course Description: Many-body physics puts together the fundamental microscopic laws and produces new emergent principles that describe the macroscopic behavior of the system. Quantum field theory is a universal language that describes the emergent behavior, typical examples of which are “quasiparticle excitations.” This is one of the notions that make quantum condensed matter systems so fascinating. We will cover many topics including Green’s functions and Feynman diagrams, Landau’s Fermi liquid theory, the quantum Hall effect, topological states of matter, fermionization, and more. We will spend some time performing analytical calculations using these methods.
Prerequisite: knowledge of second quantization, for example from PHYSICS 615.
Textbooks:
- “Methods of Quantum Field Theory in Statistical Physics”, A. A. Abrikosov, L. P. Gorkov, & I. E. Dzyaloshinski.
- . “Superfluid states of matter” by B. Svistunov, E. Babaev, N. Prokof’ev, CRC Press (2015)
- “Condensed Matter Field Theory” by A. Altland and B. Simons, Cambridge University Press, Second Edition (2013)
- “Introduction to many-body physics” by P. Coleman, Cambridge University Press (2015)
- “Field theory of non-equilibrium systems” by A. Kamenev, Cambridge University Press (2012)
- “Field theories of condensed matter systems” by E. Fradkin, Cambridge University Press, Second Edition (2013)
- “Quantum phase transitions” by S. Sachdev, Cambridge University Press, Second Edition (2014)
- “Topological insulators and topological superconductors” by B. A. Bernevig with T. L. Hughes, Princeton University Press (2013)
- “Advanced topics in quantum field theory” by M. Shifman, Cambridge University Press (2012)
- “Introduction to superconductivity,” Tinkham
- “Diagrammatics” by M. V. Sadovskii, World Scientific Publishing (2020)
- “Many-Body Theory of Condensed Matter Systems” by M. G. Cottam, Z. Haghshenasfard, Cambridge University Press (2020)
- “Quantum Many-Particle Systems” by J. W. Negele and H. Orland, CRC Press (2018)
- “Aspects of Chern-Simons Theory” Les-Houches lectures by Gerald V. Dunne.
Grading: grades from the various components of the course will determine the final grade. These are weighted as follows:
— A short paper at the end of the semester and/or 30 min presentation (30%). The project can be on any research topic related to many-body physics, with the consent of the instructor.
— Homework (70%)
For Disability Accommodation and Academic Honesty policy statements see:
Academic Honesty Policy Statement
Disability Statement
Reading — Homework assignments and solutions — Notes
Week 1
Second quantization; Hamiltonian of a many-body system; Canonical transformations in second quantization; Bogolyubov transformation; quasiparticles; Examples of second quantization; Fermionic chain, Jordan-Wigner transformation; one-dimensional Heisenberg spin-1/2 magnet. Reading Ref. 4 Paragraphs 2.5, 2.6; Chapter 3; paragraphs 3.1 through 3.7; Ref 6 Chapter 5 paragraph 5.2. See also Week 1 Lecture notes.
Homework assignment 1 (Due February 18 by email)
Week 2
One-dimensional Heisenberg spin-1/2 magnet. Reading Ref. 3 Paragraphs 2.5, 2.6; Chapter 3; paragraphs 3.1 through 3.7; Ref 5 Chapter 5 paragraph 5.2 . Anisotropic Heisenberg (XXZ) model, fermion representation, Goldstone modes: magnons, the excitation energy of magnons, example: magnon dispersion in (anti)ferromagnets, quadratic and linear dispersion. Reading: Ref 2 Paragraph 2.2; Ref 3 Paragraphs 4.1,4.2; See also Week 2 Lecture notes.
Week 3
Topological superconductivity in 1D and Majorana fermions, Kitaev chain, localized edge Majorana fermions, and the degenerate ground state, properties of Majorana fermions. Reading: see the original paper by A. Kitaev. Interacting systems, interaction Hamiltonian, two body interaction Hamiltonian, Jellium model. Reading: Ref 2 Chapter 2.2; Ref 3 Paragraphs 3.5,3.6. Perturbation theory: quantum mechanics of a single particle; Green’s function; Construction of Feynman diagrams for interacting particles. See the second half of Lecture notes. Graphene: single particle dispersion on a honeycomb lattice (band structure) Dirac spectrum at K and K’ points of the BZ. Haldane’s model for a quantum Hall effect without Landau levels: building block of a topological insulator. See also Notes.
Homework assignment 2 (Due March 4 by email)
HW 2 Part 1
HW2 Part 2