Class Title: Quantum many-body condensed matter physics

Session: Spring 2017 (01/23/2017 – 05/02/2017)
Days & Times: MoWeFr 11:15AM – 12:05PM
Location: Lederle Graduate Research Center Tower room 1033

March 20 (Monday) — LGRT 419B. Note that the location change is for this day only!

Instructor: Tigran Sedrakyan
E-mail: tsedrakyan@physics.umass.edu
Office: HAS 105
Phone:  413-545-2409
Office hours: Fr. 5-6. We can meet also after the lectures or other times by appointment.

TOPICS:
1. the classical field theory
2. field quantization (first quantization),
3. the second quantization and its applications. Jordan-Wigner fermionization.
4. Green’s function method and the concept of quasiparticles, aspects of Landau Fermi-liquid theory
5. Perturbation theory and Feynman diagrams
6. Functional field integral
7. Bose-Einstein condensation, Superconductivity
8. Aspects of phase transitions and broken symmetries
9. The renormalization group analysis
10. Topological states of matter

Course Description: How much information about the properties of quantum matter can be extracted a priori by studying constituent particles and the fundamental laws that govern their interactions?  As much information as the knowledge of the basic chemical constituents tells us about the behavior of a living organism!  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.  QFT is not a prerequisite.  We will cover aspects of QFT in condensed matter physics and introduce topological states of matter.

Prerequisites: Physics 816.

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 (30%) and a 30 min presentation (30%). The project can be on any research topic related to many-body physics, with consent of instructor.
—  Homework & Exam  –40%

For Disability Accommodation and Academic Honesty policy statements see:
Academic Honesty Policy Statement
Disability Statement

Reading — Homework assignments and solutions — Notes

Week 1
Classical Hamiltonian formalism; equations of motion; Poisson brackets; canonical variables; global U(1) symmetry and the corresponding constant of motion;  complex scalar fields as canonical variables; canonical transformation; Bogoliubov transformation; Bogoliubov-de Gennes (BdG) equations. Reading: Ref. 1 (see references in the Reading section above) Chapter 1: Neutral matter field, paragraph 1: Classical Hamiltonian formalism.    Week 1 Lecture notes. 

Week 2
Non-relativistic properties for a complex valued scalar field; Klein-Gordon equation and its non-relativistic limit: Lorentz invariance; continuity equation; velocity of flow; Gross-Pitaevskii equation (nonlinear Schroedinger equation — NLSE); healing length; local density approximation; vortices: topological defects. Reading: Ref. 1 Chapter 1 paragraph 2: Basic dynamic and static properties.  Week 2 Lecture notes.

Week 3
Vortex energy; instability of a vortex with topological charge |M|>1; spontaneous breaking of U(1); normal modes/excitation spectrum; sound waves; nonlocal interaction; Reading: Ref. 1 Chapter 1; quantum physics: scales. Week 3 Lecture notes.

Week 4
Quantization of fields (first and second): quantization of a classical string, canonical commutation relations, particle exchange statistics: bosons and fermions, quantum fields and many-body wave function, uncertainty principle from quantization and prevention of the collapse of the hydrogen atom. Collective quantum fields: second quantization of the harmonic oscillator, phonons -1D free QFT, occupation numbers and Bose-Einstein distribution. Reading: Ref 2 Chapter 1 paragraphs 1.1 through 1.7, Chapter 2 paragraph 2.1. Ref 3 Chapter 2 paragraphs 2.1 through 2.6.  Week 4 Lecture notes.

Week 5
The thermodynamic limit in QFT, Example: phonons in 3D – harmonic crystal; The continuum limit in QFT, Example: quantized 1D string – cutoff and renormalization; Contrasting first and second quantizations; Examples of second quantization, Jordan-Wigner transformation; Fermion representation of 1D spin-1/2 Heisenberg chain. Reading: Ref 3 Paragraphs 2.5, 2.6; Chapter 3; paragraphs 3.1 through 3.7; Ref 5 Chapter 5 paragraph 5.2 . Week 5 Lecture notes.

Week 6 
Anisotropic Heisenberg (XXZ) model, fermion representation, Goldstone modes: magnons, 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 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. See also Week 6 Lecture notes.

Homework assignment (Due Mar 20)

Week 7 
Feynman’s path integral for bosons: coherent states for bosons, matrix elements and the completeness relation, path integral for the partition function, multi particle systems, time ordered expectation values: the two-point bosonic Green’s function. Applications of the Feynman path integral:  Gaussian path integrals, free energy of a free bosonic gas, Green’s function for free bosons. Reading: Ref 3 Chapter 12 Paragraphs 12.1 through 12.3. Ref 2 Chapter 3 Paragraphs 3.1 through 3.3.  See also Week 7 Lecture notes.

Week 8 
Spring break and APS March meeting.

Week 9 
Feynman’s path integral for fermions: coherent states for fermions, Grassman mathematics,  matrix elements and the completeness relation, path integral for the partition function: fermions, free energy of a single fermion, multi particle systems, Berry phase, Gaussian path integrals for fermions, generating functional and source terms, free energy of a free fermion gas, Green’s function for free fermions. Reading: Ref 3 Chapter 12 paragraphs 12.4, 12.5. Ref 2 Chapter 4, Paragraphs 4.1, 4.2.  See also Week 9 Lecture notes.

Week 10 
The Cooper instability, the BCS Hamiltonian, mean-field description of the condensate, Nambu spinors, the BCS ground state, quasiparticle excitations in the BCS theory. Reading: Tinkham, Introduction to superconductivity, chapter 3, Ref. 3, chapter 14, paragraphs 14.1 through 14.5; Ref. 2: 6.4. See also Week 10 Lecture Notes.

Week 11 
Path integral formulation of BCS theory of superconductivity, saddle point of the path integral: mean-field theory, superconducting gap and Tc, the Nambu-Gorkov’s Green’s function, tunneling density of states, coherence factors. Reading: Tinkham, Introduction to superconductivity, chapter 3, paragraphs 3.5.1-3.7.1, 3.8. Ref 3: paragraphs 14.6, 14.7, 14.8. See also Week 11 Lecture Notes and Notes on Spontaneous symmetry breaking and Goldstone bosons. Motivation and brief introduction to Chern-Simons theory.

Week 12 
Reminder: Maxwell’s electromagnetic theory. Introduction to Chern-Simons theory, Chern-Simons Lagrangian, gauge invariance and classical Euler-Lagrange equations. Bianchi identity and current conservation. Chern-Simons coupled to mater fields – anyons. CS gauge field and the corresponding singular gauge transformation.  Aharonov-Bohm phase and anyonic statistics. Maxwell-Chern-Simons: topologically massive gauge theory. Higgs mechanism in a Maxwell-Chern-Simons theory. Fermions in D=2+1 dimensions. Discrete symmetries: C,P,T. Ref. 10: Chapters 1 and 2. Perturbatively induced Chern-Simons terms: Fermion loop.  Chapter 5 paragraph 5.1.

Week 13 
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. Graphene and Haldane’s model in an external gauge field: the role of the Chern number (for the first part see the paper by G. W. Semenoff).  Canonical quantization of Abelian Chern-Simons theories: canonical structure of CS theories, coordinate and momentum fields, canonical commutation relations,  CS quantum mechanics, canonical quantization, quantization on the torus. Reading: Ref 10 Paragraph 3, chapters 3.1 through 3.4. Topology and Chern-Simons theory: K-matrix, integer K=m for quantum Hall states,   mutual CS with symmetric K-matrix for the Z2 spin-liquid. CS theory on a torus: operator relation for Wilson loops and degeneracy of the ground state on a torus. Edge states of the integer quantum Hall effect, edge excitations of chiral CS theory and of Z2 spin liquids. Reading: see notes by S. Sachdev.

Week 14 
Fractional quantum Hall states and chiral spin-liquids: quantum Hall states of bosons and fermions in strong magnetic field, flux attachment, gauge invariance and emergency of the CS term. Mean-field theory and fluctuations: gapped collective density fluctuations and fractionalization of vortices.  Reading: see notes by S. Sachdev. Chiral spin-liquid state in continuous moat-band systems: corresponding CS description and the energy of the state at low densities.  Lattice CS theory and spontaneous formation of a chiral-spin liquid state on a lattice. Reading: see  Phys. Rev. Lett. 115, 195301 (2015); Rapid Communication in Phys. Rev. B 89, 201112R (2014); Phys. Rev. Lett. 114, 037203 (2015). 

Week 15
Chern-Simons superconductivity.

Exam

Research Presentations

Time: Wednesday, May 3rd, 11:15am – 2pm
Location: Lederle Graduate Research Center Tower room 1033

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11:30-12:00 Salem Al Mosleh

Title: Gauge Fields in Graphene (and an analogy with stochastic semi-classical gravity).

Abstract:
It is well known that planer graphene quasi-particles behave as massless two dimensional Dirac fermions, Which gives the material many of its interesting properties. The presence of crystal defects and thermal fluctuations result in a curved sheet with non-zero gaussian curvature. The electronic structure can be studied in this case as a quantum field theory on a curved background. In this talk, I will review the theory of electronic structure of curved graphene and suggest an analogy with stochastic semi-classical gravity for understanding the back-reaction of electronic excitation on the curvature.

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12:00-12:30 Yue Qiu

Title: `Toric code model and anyons’

Abstract:
Kitaev’s toric code model is a 2D spin 1/2 exactly solvable model. It is the simplest toy model for topological order with degenerate ground states.  The energy excitations are anyonic, which can be used to perform fault tolerant quantum computation. Therefore, it was initially proposed as a system that can be considered as a quantum computer. We will present a detailed analysis of toric code model and discuss some promising future directions.

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12:45-1:15 Zhiyuan Yao

Title: Particle-hole symmetry and interaction effects in the Kane-Mele-Hubbard model

Abstract:
At half filling, the Kane-Mele-Hubbard model with purely imaginary next-nearest-neighbor hopping is shown to feature a particle-hole symmetry. This symmetry results in the absence of charge and spin currents along open edges, and the absence of sign problem in determinant quantum Monte Carlo simulations. Therefore, the interaction effects on a band topological insulator can be studied numerically in a controlled manner. Through studies of the edge as well as the bulk properties of the model, the ground-state phase diagram is determined.

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1:15-1:45 Kun Chen

Title: The Problem of Trapped Quanta in a Quantum-Critical Environment: Charge Fractionalization and Beyond

Abstract:
Quantum systems that are right at the critical point between two ground states, such as a superfluid and a Mott insulator, exhibit a variety of exotic phenomena. We recently added a new discovery to this area by studying an impurity problem in two-dimensional lattice bosons. We considered a static impurity—a repulsive center—and defined its charge as the integrated density distortion it creates in the environment. Will this charge be always an integer—no matter what is the value of the potential strength V, like in the Mott insulator, or a certain continuous function of V, like in the superfluid? Generically, the trapped charge is an integer. However, extremely nontrivial things happen when V is close to the transition point, V_c, between two adjacent integers. The integer charge is fractionalized into a half-integer core and diverging halo carrying the complementary charge of ±1/2. This effect can be studied experimentally with ultra-cold atoms in optical lattices by means of atomic microscopy.

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