Many Body 621

 Spring 2022



Piers Coleman   Rutgers University

Images Texts
Times of Course
Syllabus outline


  Phase diagram of Magic Angle


Magic Angle Graphene

Illustrating the Effective Action in Path Integral


Relationship between Meissner Effect and Phase Rigidity of a Superconductor.

Gap Structure of a d-wave superconductor
115 Superconductor

Phase Diagram of the Kondo Effect

CeCoIn5: a 115 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 continues from Many Body 620, and will introduce many body physics needed to understand current research activities in quantum condensed matter, including finite temperature methods, response functions,  path integrals, conventional and unconventional superconductivity, strongly correlated electron systems.   I will also review essential material and offer additiional tuition to cater to those who were unable to take 620 last semester.  Please ask Shirley Hinds for a special permission to register. There will be a lot of discussion and interaction. Please register as soon as possible.

Students with disabilities 

Introduction to Many-Body Physics

The  reference texts will be
    ``Introduction to Many-Body Physics'', Piers Coleman, (CUP, Jan 2016). Chapters 12-18.

      Here are some additional useful references:

      • Condensed Matter Field Theory by Alexander Altland and Ben Simons.(CUP, 2006)
        An excellent introduction to Field Theory applied in condensed matter physics. 
      • Advanced Solid State Physics by Philip Phillips, second edition (CUP, 2012).
      • Basic Notions in Condensed Matter Physics by P. W. Anderson, Benjamin Cummings 1984. A classic reference. Many of us still turn to this book for inspiration, and philosophy. It also has a fine selection of important reprints at the back.
      • Quantum Field Theory in Condensed Matter Phyiscs,  A. M. Tsvelik, Cambridge University Press, 2nd edition (2003).
      • R. Shankar, Rev Mod Phys 66 129 (1994). An amazingly self-contained review of the renormalization group and functional integral techniques written by one of the best expositors of condensed matter physics.
      • ``Field Theories of Condensed Matter Physics'' by E. Fradkin. (Frontiers in Physics, Addison Wesley). Interesting material on the fractional statistics and the fractional quantum Hall effect.

      Traditional Many Body Theory and Greens Functions

      • ``Methods of Quantum Field Theory in Statistical Physics'' by Abrikosov, Gorkov and Dzyalozinskii. (Dover Paperback) - Classic text from the sixties, known usually as AGD.
      • ``A guide to Feynman Diagrams in the Many-Body problem by R. D. Mattuck. A light introduction to the subject. Reprinted by Dover.
      • ``Greens functions for Solid State Physics'' S.Doniach and E. H. Sondheimer. Not as thorough as AGD, but less threatening and somehow more manageable. Frontiers in Physics series no 44.
      • ``Quantum Many Particle Systems'' by J. W. Negele and H. Orland. Alas all the good physics is in the unsolved exercises! However, it is the only one of this set to touch on the subject of functional integrals.

      Further references:

      • The Theory of Quantum Liquids by D. Pines and P. Nozieres. Excellent introduction to Fermi liquid theory that avoids the use of field theory.
      • Statistical Physics, vol II by Lifshitz and Pitaevskii. Pergammon. Marvellous book on applications of many body physics, mainly to condensed matter physics.

      Some Online references (Check it out- this is a great link).

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Exercises 621
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Exercise 1. (Path integrals)    Solution to Exercise 1.
Exercise 2.      Solution to Exercise 2
Exercise 3. (conductivity and BCS theory)


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Times: 12.10 Weds and 2 pm Friday in Serin 287. NOTE: We will start on Weds, Jan 19th online - there will be no classes week 2 and I hope we will return to face-to-face classes in February.  Occasionally, to make up for my travel, we will hold an additional  class at a time to be decided.

Office hour:  Time to be decided.  Tel x 9033.

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.

  • Finite Temperature approach to Many Body Systems reviewed.
  • Response Functions
  • Broken Symmetry
  • Disordered systems
  • Functional Integral Approach to interacting electron systems
  • Superconductivity,  particularly Anisotropic pairing and superfluid He-3.
  • Heavy Fermion Materials

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         Schedule:  UNDER CONSTRUCTION


Extra class
Time: 3pm Mondays

Weds 12.10-1.30 SRN 330

Friday 2.00-3.20pm SRN 330
Jan 17-21

 19 Jan

Introduction to Course.



 21 Jan

  Path integrals and
finite temperature many body physics

  Notes L1

1 Jan 24-28

      No classes week 2

  No classes week 2

2 Jan 31-Feb 4

    2nd Feb SRN 287

    Path integrals Bosons

    Notes L2

5th Feb Serin 287

  Path Integrals: Many Bosons

  Notes L3

3 Feb 7-11

   9 Feb SRN 330

   Sources in Gaussian Ints
   BECs and Superfluids.

   Notes L4
  12 Feb

  Spectrum of a superfluid and
  Derivation of Critical Velocity

  Notes L5

4 Feb 14-18

  16 Feb
   Fermions and Grassman          Numbers: Grassman Calculus

    Notes L6
  19 Feb
   Fermion Path Integral

   Notes L7

5 Feb 21-25
21 Feb
Makeup Class                      
3pm Serin 330

Gaussian Path integral for Fermions

Notes L8
  23 Feb

  Integrating out Fermions.

  Hubbard Stratonovich  I: Attractive Interaction. Effective Interaction

Notes L9
  25 Feb

  Hubbard Stratonovich for Real and Repulsive Interactions

Electrons with a Coulomb Interaction
 Notes L10

6 Feb 28-Mar 4
28 Feb
Makeup Class
3pm Serin 330

Feynman Rules for Electrons in a potential Field.

RPA as a large N approximation.
 Notes L11

2nd March

  RPA Approach to the interacting Electron Plasma.

Notes L12
4th March


Notes L13

7. Mar 7 -11

9th March


Bardeen Pines

Notes L13
11th March

Fluctuations and
Dissipation I.  Response Functions.

Notes L14

8. Mar 14-18

Spring Break

9. Mar 21-25

11th March

Fluctuations and Dissipation II.
The relation
between noise, response
and imaginary time Greens

Imaginary time response: Ohm's law.

10.  Mar 28-Apr 1

No Classes this week

No Classes This Week

11. Apr 4-8

Ohm's law continued
Ohm's law.

12. Apr 11-15

Anderson's Pseudospin Picture

13. Apr 18-22
Path Integral Formulation of BCS Theory

Notes L20
Nambu Spinors

Superfluid Stiffness

14. Apr 25-29

15.  May 2-6

Last Day of Classes

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