University of Illinois at Urbana-Champaign · Department of Physics

Physics 598

Topological Phases in Quantum Condensed Matter

Academic Year 2015/2016

Spring Semester 2016

Instructor: Professor Eduardo Fradkin

Department of Physics
University of Illinois at Urbana-Champaign
Room 2119 ESB
1110 West Green Street, Urbana, IL 61801-3080
Phone: 217-333-4409
Fax: 217-333-9819
E-mail efradkin@illinois.edu
http://eduardo.physics.illinois.edu/homepage/


Time: 3:30 pm - 5:00 pm, Mondays and Wednesdays
Place: 158 Loomis Laboratory
Call Number: 52571
Credit: 1 unit.
Office Hours: Thursdays 4:00 pm to 5:00 pm, Rm 2119 ESB

TA: TBA
Office: TBA
Phone: TBA
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This course will cover current developments on topological phases in condensed matter physics, including: the theory of the fractional quantum Hall states, spin liquids, topological insulators and superconductors, effective field theories of topological phases, quasiparticles, fractional statistics (Abelian and non-Abelian), experimental detection of quasiparticles, quantum interferometers and the manipulation of quasiparticles, topological phases and topological quantum computing, quantum entanglement at quantum criticality and in topological phases. The course material will consist primarily of a review of recent literature in leading journals. I will also rely heavily on chapters of the second edition of my book "Field Theories of Condensed Matter Systems" (Cambridge University Press, 2013).

Concerning the required background I will assume that the students have taken at least the equivalent of our second graduate course in Quantum Mechanics (Physics 581: Advanced Quantum Mechanics) and at least the first course in Qunatum Field Theory (Physics 582). Although no real expertise field theory will be assumed, students that have taken the Advanced Quantum Field Theory class (Physics 583) and/or the Advanced Condensed Matter Physics class (Physics 561) will profit much more from this class.

There will not be homeworks in this class but all students will be required to write a term paper and make an oral (powerpoint) presentation. To encourage participation, from time to time I may ask students to go over some material in front of the class on some previously agreed material.


Announcements


Course Plan

Ordered Phases in Condensed Matter

The concept of order in condensed matter. Characterization of a phase of matter in terms of order parameters and its correlations. Effective field theory description. Review of classical and quantum phase transitions to an ordered state. Topology and order: topological excitations. Examples: superconductors and superfluids; antiferromagnets.

Characterization of Topological Phases


What is a topological phase. Absence of an order parameter description. Universal properties of topological phases and topological invariants. Chern numbers. Ground state degeneracies and the quantum numbers of the spectrum of excitations. Examples: quantum Hall phases, quantum dimers, topological superconductors and topological insulators.

Frustration, Quantum Disorder, and Gauge Theory


Frustration in classical and quantum spin systems. Quantum disordered states and quantum spin liquids. Dimer models and their phases. Topological phases in dimer models. Gauge theory description and deconfinement. Kitaev' Toric Code. Differences and similarities between one and two dimensional systems.

The Integer Quantum Hall effect


Electrons in two dimensions in large magnetic fields. Single electron picture: Landau quantization and degeneracy of Landau levels. Lattice effects. Quantization of the Hall conductance and plateaus. Chern numbers. Role of disorder and localization. Topological quantization. The plateau transition.

Topological Insulators


Topological classification of band structures. Band crossings and Dirac fermions. Berry's phases and Chern numbers. Spin orbit. The quantized anomalous Hall effect and the spin quantum Hall effect. Weak and strong topological insulators. 3D topological insulators, the Callan-Harvey effect and the parity anomaly in condensed matter. The role of time reversal breaking. Relation with topological superconductors. Examples: graphene, HgTe heterostructures, Bi$_{1-x}$Sb$_x$, Bi$_2$Se$_3$ and other materials. Classification schemes.

The Fractional Quantum Hall Effect


Laughlin, Haldane/Halperin and Jain wavefunctions. Plasma analogy, collective modes and vortices. Fractional charge and statistics. Hierarchical states and the classification of Abelian Fractional quantum Hall states. Non-Abelian fractional quantum Hall phases: Moore-Read and Read-Rezayi states. Classification of non-Abelian states. Pairing and clustering. Topological Superconductors. The concept of non-Abelian fractional statistics. Topological protection and Topological Quantum Computing.

Gauge Theory and the FQHE

The Landau-Ginzburg picture and Chern-Simons gauge theory. Hydrodynamic picture of the FQH fluid. Composite bosons and composite fermions. Collective modes and Raman Scattering experiments. Experimental evidence for composite fermions: focusing experiments, gaps and Shubnikov-de haas experiments. Bilayer systems and partially polarized states. An integer as a fraction: Skyrmions in the integer FQH state and NMR experiments. What happens when the fractional quantum Hall state fails: compressible states, stripe and nematic phases.

Edge States


Wen's theory of the Luttinger liquid of chiral fermions and bosons. Universal effective theories of edge states and relation with conformal field theory. Tunneling signatures and experimental evidence for Luttinger liquids. Noise measurements at point contacts and the measurement of fractional charge. Quantum interferometers and the detection and measurement of fractional statistics, Abelian and non-Abelian.

Quantum Entanglement in Condensed Matter


Scaling of quantum entanglement in condensed matter as a characterization of its phases. Quantum entanglement in a generic phase with finite correlation length: the "area" law. Scaling of quantum entanglement in 1D critical systems. Quantum entanglement and dynamics: quenches. Scaling of the quantum entanglement entropy in topological phases and at quantum phase transitions. Quantum entanglement and the AdS/CFT correspondence.


Term Paper

There will be a Term Paper that will play the role of the final exam. A list of topics for the Term Paper can be found here. The Term paper will be due at 9:00 pm on Wednesday May 11, 2016 (the day before the Final Exam). You must sent me the pdf files of your paper and of your slide presentation by email . The Final exam for Physics 598, will be on Thursday May 12, 2016 in room 3110ESB at 9:00 am . Each student will give a 15-20 minute long computer presentation.




Bibliography




Eduardo Fradkin, "Field Theories of Condensed Matter Physics, Second Edition" (Cambridge University Press, 2013).


Richard Prange and Steven M. Girvin, "The Quantum Hall Effect", Second Edition (Springer-Verlag, 1990)


Michael Stone, "The Quantum Hall Effect", (World Scientific).


B. Andrei Bernevig and Taylor L. Hughes, "Topological Insulators and Superconductors" (Princeton University Press, 2013).


Last updated 4/27/2016