Advanced Solid State Physics

Strange Metals and Fractionalization

603 Fall 2020


Piers Coleman   Rutgers University

Images Texts
Times of Course
Syllabus outline




Flux attachment and the fractional quantization of Hall Effect in 2D heterojunctions.

Fractionalization and Solitons in the Su Schrieffer Heeger Model

The Kitaev Honeycomb Model, showing the fractionalization of spins into Majorana Fermions

Linear resistivity of LaSrCuO4.  A high temperature superconductor.

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Instructor: Piers Coleman, Room 268
If you have any questions about this course or the homework, please do not hesitate to contact me via email at : coleman@physics.rutgers.edu

Scope of Course. This course will introduce the concepts of fractionalization and strange metals in solid state physics.    There will be a lot of discussion and interaction. We will introduce fractionalization in four classic contexts: the Su Schrieffer Heeger model, the fractional Quantum Hall Effect, the  Kitaev Honeycomb model and the Kondo Lattice. An important part of the  course will be the discussion of key scientific papers, both theory and experiment. The last part of the course will discuss the unsolved problem of strange metals, which  develop in strongly correlated metals near a quantum critical point, discussing current experiment and theory on the topic.

Students with disabilities 

Some useful  reference texts  

"Soliton Excitations in Polyacetylene", W. P. Su, J. R. Schrieffer and A. J. Heeger, Phys. Rev. B 22,  2099 (1980)

 ``Topological Superconductors and Category Theory".  by Titus Neupert and Andrei Bernevig.

 ``Modern Condensed Matter Physics " Chapters 12 and 16. Steve Girvin and Kun Yang, CUP (2019).

``Lectures on the Quantum Hall effect'', David Tong,

``Field Theories of Condensed Matter Physics", Chapter  13, pp 502-512, Eduardo Fradkin, CUP (2013).

 ``Topological Phases and Quantum Computation",  Alexei Kitaev and Chris Laumann, arXiv:0904.2771.

  ``Anyons in an exactly solved model and Beyond",  Alexei Kitaev, Annals of Physics 321 2-111 (2006).

 ``Kitaev quantum spin liquid: concept and materialization",
  Hide Takagi, Tomohiro Takayama, George Jackeli and Giniyat Khaliullin, arXiv 1903.08081,  Nature Review Physics, 1, 264-280 (2019).

 "Heavy Fermions and the Kondo Lattice: a 21st Century Perspective", Piers Coleman, arXiv:1509.05769.

"Topological Approach to Luttinger's Theorem and the Fermi surface of a Kondo Lattice", Masaki Oshikawa, Phys. Rev. Lett., 84, 3370 (2000).

"Extending Luttinger's Theorem to Z2 fractionalized Phases",  Arun Paramekanti and Ashwin Vishwanath, PRB 70, 245118 (2004).

 "Universal T-linear resistivty and Planckian dissipation in overdoped cuprates", A. Legros et al, Nature Physics, 15, 142-147 (2019).

Exercises 603
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 Exercise 1: the SSH Model (due 30th Sept)

 Exercise 2: the FQHE (due 30th Oct)

  Exercise 3: Kitaev Model with Jordan Wigner transformation

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Times: The class will occur online.  To avoid a clash with Many Body Physics 621,  we will be moving the class away from the published time (originally
1.40-3.00 on M and W) to the tentative plan of  Wednesday 10.20-11.40am, Friday 1.40-3.00pm.

Office hour:  To be arranged.

Assessment:   I anticipate four or five take home exercises and one take-home final. I want to encourage an interactive class and will take this into account when grading!

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  Here is the provisional outline.

  • Solitons in the Su Schrieffer Heeger Model: an elementary example of fractionalization
  • The Fractional Quantum Hall Effect (mainly following Girvin and Yang).
  • Spin Fractionalization in the Kitaev Model
  • Fractionalization in Heavy Fermions and the  Kondo Lattice. Oshikawa's Theorem.
  • Strange Metals. 

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 Weds 10.20-12.00am

Friday 1.40-3.00pm

1.Sep 1-4

No Class
No Class

2 Sep 7-11
First Class of the Semester.

Overview of Fractionalization. 
L1 notes

The SSH Model
L2 notes

3 Sep 14-18

SSH Model
L3 notes

The Integer Quantum Hall Effect.
L4 notes

4 Sep 21-25


Integer QHE continued.  FQHE Introduction.

L5 notes


Mechanical momentum and Guiding Centers.
Electrons in the Lowest Landau Level.  Haldane Pseudopotentials.  Laughlin's Plasma Analogy.
L6 Notes

5 Sept 28-Oct 2

The Laughlin wavefunction.
The concept of fractional charge.

L7 notes

The concept of fractional Statistics
Calculation of the statistical angle of Laughlin
quasiparticles.  Evidence for fractional statistics

L8 notes

6 Oct 5-9

Composite Fermions.
L9 notes


Chern Simons Theories
L10 Notes

7. Oct 12-16

Chern Simons Theories

L11 Notes

FQHE wrap up.

Class Discussion on FQHE and Chern Simons Theories

Presentation by Roman Geiko on generalized Chern Simons Theories, his work with Greg Moore.

Short Journal Club discussion on "Floating Topological Phases"
by Devakul, Sondhi, Kivelson and Berg
led by Pavel Volkov

8. Oct 19-23

Kitaev Honeycomb Model

L12 Notes

Kitaev Honeycomb Model

L13 Notes

9. Oct 26-30
Kitaev Honeycomb Model and Kitaev Spin Liquids


Real Kitaev Spin Liquids.
(see L14 Notes)

Introduction: The Kondo Lattice and Heavy Fermions.
(see L15 Notes)

10.  Nov 2-6

The Kondo Lattice: From Anderson to Kondo
L15 Notes

The Kondo Lattice:  Oshikawa's theorem.
L16 Notes

11.  Nov 9-13

The Kondo Lattice: Oshikawa's theorem continued

The Kondo Lattice: Large N Mean Field Approach
L17 Notes

12.  Nov 16-20

Kondo Lattice:  Large N Mean Field Approach

Kondo Lattice:  Heavy Fermion superconductivity
L18 Notes

13.  Nov 23-27

No Class Thanksgiving

No Class Thanksgiving

14.  Nov 30- Dec 4

Strange and Norma Metals contrasted.
The Marginal Fermi liquid Phenomenology

L19 Notes

Strange Metals:  fractionalization picture  in RVB. The Ioffe-Larkin theory.
L20 Notes

15.  Dec 7-11
12noon Weds Class.

Strange metals: Survey of Current Theories

L21 Notes

No Class.

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