Class Title: **Topics in Many-Body Physics**

Session: Spring 2019 (01/22/2019 - 05/01/2019)

Days & Times: TuTh 10:00-11:15AM

Location: Hasbrouck Laboratory room 242

Instructor: Tigran Sedrakyan

E-mail: tsedrakyan@physics.umass.edu

Office: HAS 403B

Phone: 413-545-2409

Office hours: Tu. 11:15AM-12:15PM, Fr. 11:00AM-12:30PM. We can also meet at other times 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.

Textbook: "Methods of Quantum Field Theory in Statistical Physics", A. A. Abrikosov, L. P. Gorkov, & I. E. Dzyaloshinski.

Reading:

1. "Superfluid states of matter" by B. Svistunov, E. Babaev, N. Prokof'ev, CRC Press (2015)

2. "Condensed Matter Field Theory" by A. Altland and B. Simons, Cambridge University Press, Second Edition (2013)

3. "Introduction to many-body physics" by P. Coleman, Cambridge University Press (2015)

4. "Field theory of non-equilibrium systems" by A. Kamenev, Cambridge University Press (2012)

5. "Field theories of condensed matter systems" by E. Fradkin, Cambridge University Press, Second Edition (2013)

6. "Quantum phase transitions" by S. Sachdev, Cambridge University Press, Second Edition (2014)

7. "Topological insulators and topological superconductors" by B. A. Bernevig with T. L. Hughes, Princeton University Press (2013)

8. "Advanced topics in quantum field theory" by M. Shifman, Cambridge University Press (2012)

9. "Introduction to superconductivity," Tinkham

10. "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 (35%)

— Final Exam (35%)

For Disability Accommodation and Academic Honesty policy statements see:

*Academic Honesty Policy Statement*

*Disability Statement*

*Reading -- Homework assignments and solutions -- Notes*

*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. 3 Paragraphs 2.5, 2.6; Chapter 3; paragraphs 3.1 through 3.7; Ref 5 Chapter 5 paragraph 5.2. See also Week 1 Lecture notes.

**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; 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 also Week 2 Lecture notes.

**Week 3**

Self-consistent block summation of Feynman diagrams: Single-particle Green's function, self-energy diagram, Dyson's equation. Poles of the Green's function and quasiparticle spectrum; quasiparticle lifetime. Two-body Green's function, reducible and irreducible interaction vertices, Bethe-Salpeter equation. Example: Polaron in the weak coupling approximation: self-energy, stability and mass renormalization. See** **Week 3 Lecture notes.

Homework assignment 1 (Due Feb 22)

Solutions

Problem 1: Chirality and the Heisenberg ladder

Problem 2: Polaron in the weak coupling limit

**Week 4**

Collective quantum fields, phonons in a crystal, the thermodynamic and the continuum limits. Examples: 1D string, phonon propagator. Bound states of two-particles: interaction potential with no retardation, Bethe-Salpeter equation and reduction to a single particle scattering picture. See Week 4 Lecture notes.

**Week 5**

Fermi gas; Kubo Formula for dynamical susceptibilities; susceptibility in terms of Green's functions; causal propagators in two and three dimensions. The problem of a heavy particle (impurity) in a Fermi gas: dynamically induced scattering potential.

See Week 5 Lecture notes.

**Week 6**

The density-density correlation function of the Fermi gas: calculation of the polarization operator in 3D. The dynamical spin-susceptibility in 3D. Density-density response in 1D; Implications this divergence for phonon propagator and its dispersion: Instability (Peierls ).

See Week 6 Lecture notes.

**Week 7**

No class on Mar 5 -- APS March meeting.

Peierls instability in 1D and Peierls transition. See Week 7 Lecture notes.

**Week 8**

Spring break.

**Week 9**

Fermi liquid (FL) theory: FL Hamiltonian and examples: nucleons in a nucleus, liquid 3He (short-range interactions), and electrons in metals (Coulomb interaction). The ground state properties of FL. Landau quasiparticle excitations in FL and long-lived quasiparticles. Landau functional. Kinetic equation and collective modes. The difference between quasiparticle and collective excitations: zero sound mode. Collective modes in metals: plasma oscillations.

**Week 10**

Random-phase approximation (RPA), the method of bose-operators of particle-hole pairs. Boson representation of the RPA Hamiltonian and its diagonalization. The spectra of normal modes: particle-hole and collective excitations. RPA as a dynamical screening of the interaction potential. Dispersion relation of plasmons. The endpoint of the spectrum (calculation of the maximal momentum, corresponding to the merging of the plasmon mode with the continuum).

**Week 11**

Finite temperature many-body physics: Imaginary time; Schroedinger, Heisenberg, and interaction representations. Imaginary time propagator; periodicity and antiperiodicity; Matsubara representation; Feynman rules at finite-T. Hartree-Fock corrections to Free energy. An electron in a disordered potential; Fluctuation-dissipation theorem. Experimentally measurable quantities (spectroscopies). Reading: Ref. 3. Chapters 8 and 9. See also Week 11 Lecture notes.

**Week 12**

The quantum Hall effect: classical effect, and the motion in a magnetic field; the Drude model. Integer and fractional quantum Hall effects, Landau levels, Landau and centrally symmetric gauges. Berry phase and Berry connection. The fractional quantum Hall effect. The Laughlin nu=1/m state and its wavefunction. Reading: see notes by D.Tong.

**Week 13**

Introduction to anyons; the braid group and Yang-Baxter relation; Fractional statistics of holes in the nu=1/m Laughlin state. Reading: see notes by D.Tong. Reminder: Maxwell's electromagnetic theory. Introduction to Chern-Simons theory, Chern-Simons Lagrangian, gauge invariance and classical Euler-Lagrange equations. Chern-Simons gauge field coupled to matter fields - anyone. Reading: Ref. 10: Chapter 1 and Chapter 2 Paragraph 2.1.

Homework assignment 12 (Due Apr 23)