PHYS626-0101: Modern Condensed Matter Physics II: Scaling and Renormalization-Fall 2020 jaydsau

PHYS626-0101: Modern Condensed Matter Physics II: Scaling and Renormalization-Fall 2020 jaydsau


An introduction to ideas of scaling and renormalization groups near classical and quantum phase transitions:

 Phys 626 starts with the fascinating phenomenon of scale invariance and universality at phase transitions. This is followed by a discussion of  the renormalization group technique and how it can be used to take advantage of scale invariance to compute properties of many-body systems near these phase transitions. We will also discuss the Berezinski-Kosterlitz-Thouless transition, which was one of the first examples of a topological phase in this context. Finally, we will spend the last part of the course on one of the modern applications of these ideas to the description of quantum phase transitions. This discussion will likely include the renormalization properties of Fermi liquids, something which has no classical analog. 

Some of the topics covered are (suggestions welcome):

  • Phase diagrams
  • Mean-field theory
  • Real space renormalization group (block-spin transformations)
  • Scaling of free-energy
  • Critical exponents
  • Relevant and irrelevant operators in RG
  • Universality classes/fixed points
  • Cross-over between fixed points
  • Finite size scaling
  • Perturbative renormalization group
  •  Wilson Fisher fixed point/epsilon expansion
  • Kosterlitz-Thouless transition
  • Quantum criticality





Required Resources

Course website:

Text book: Scaling and renormalization in Statistical physics, John Cardy, 1st edition, Cambridge University Press 1996.

Other resources: Quantum Phase transitions, Subir Sachdev, Cambridge University Press 1999 - referred to as SS

Review article: Functional renormalization group approach to correlated fermion systems, Metzner et al, Reviews of Modern Physics, vol 84, page 299,2012. (Referred to as FRG)


Lectures on Phase transitions and the renormalization group, Nigel Golden feld, Frontiers in Physics 1992.


















Dr. Jay D. Sau

Class Meets

Wednesday and Friday

10:30 am – 11:45 am


Office Hours

PHY/ATL #4441 (CMTC)

by appointment



Shuyang Wang (email:sywang95)

Suggested Prerequisites

Undergraduate quantum mechanics (PHYS 401/402 or equivalent)

Graduate statistical mechanics (Phys 603 or equivalent)

Course Communication

All updates and information regarding the course will be made using the announcements on ELMS – please make sure your ELMS settings do not delay announcements. I may or may not repeat in class.

Please send any questions or notifications of absences that you need to inform me preferably by email (see above).  



Campus Policies

It is our shared responsibility to know and abide by the University of Maryland’s policies that relate to all courses, which include topics like:

  • Academic integrity
  • Student and instructor conduct
  • Accessibility and accommodations
  • Attendance and excused absences
  • Grades and appeals
  • Copyright and intellectual property

Please visit for the Office of Undergraduate Studies’ full list of campus-wide policies and follow up with me if you have questions.


Activities, Learning Assessments, & Expectations for Students

Lectures: Class time will be occupied by lectures that follow a set of notes that closely follow sections in the textbook. In a few cases I will follow a different textbook, which I will point out. I will post my notes online in this case. In addition to explaining the physical intuition behind concepts I will provide mathematical derivation of some of the more important results where the derivation is instructive. A firm grasp of quantum mechanics at approximately the advanced undergraduate level will be needed to follow parts of the lectures as well do some of the homeworks. The thorough use of quantum mechanics is what distinguishes the graduate solid state course from the undergraduate one.

Participation: The lectures assume that you are keeping track of the material of the previous lecture. This will enhance your learning and participation in the class, which is crucial to the classes success. To ensure a minimal level of participation, I will keep track of your participation through questions you ask or answer. You get full credit for participation if you ask or answer 6 questions in the semester related to the material presented in the lectures. Participation points of 2/lecture will be added to your grades within 24 hours of the lecture you participated in (i.e. asked/answered a relevant question/clarification).  I might forget to credit you for this. IT IS YOUR RESPONSIBILITY TO EMAIL ME IF I FORGET TO ADD THIS WITHIN TWO DAYS.  Late (by more than a few days) may or may not be credited depending on whether I remember.

Homework : Problem sets will be posted as assignments on ELMS. The problems can also be downloaded from the assignments folder. Homework submission should be by file upload. Homeworks turned in after the solutions are posted will not be graded. New assignments will be posted on the course website, along with the homework solutions.  Homework problems are carefully chosen to highlight some of the important topics covered in lecture, complete some of the important steps, as well as important applications of the ideas. It is important that you carefully complete and make sure you understand all of the homework. You are encouraged to work with others on homework, however, it is forbidden to blindly copy another person’s work. There are 6 homework sets and one will be dropped. This will be done by replacing the lowest homework score out of 6 (which will be 0 if you skipped the homework) by the average of the rest.



Grades are not given, but earned.  Your grade is determined by your performance on exams, homeworks and participation in the course and is assigned based on your score according to a curve. Typically I follow a curve scheme where the median score would be graded at B+. The lowest grade would be B- (very few). The top grades will be A+ and As. The number in each category would be roughly equal depends on appropriate breaks in the score distribution. 

Of course, all this being said this rule is subject to change depending on the performance of the class. If some students do extremely poorly (e.g. score well below 40%) I might consider going below B- for the lowest grade.  On the other hand, if everyone does well (i.e. above 90%) I have no hesitation giving the entire class an A. Also, if someone scores above 80 that is a B or better independent of whether the average is above 80.

If earning a particular grade is important to you, please speak with me at the beginning of the semester so that I can offer some helpful suggestions for achieving your goal.

All assessment scores will be posted on the course ELMS page.  If you would like to review any of your grades (including the exams), or have questions about how something was scored, please email me to schedule a time for us to meet in my office.

Late work (as explained in the instruction) will not be accepted for course credit so please plan to have it submitted well before the scheduled deadline.  I am happy to discuss any of your grades with you, and if I have made a mistake I will immediately correct it.  Any formal grade disputes must be submitted in writing and within one week of receiving the grade. 

Learning Assessments


Category Weight

Participation points



Homework (out of 6 assignments)




Course Schedule

Chapters below refer to textbook. SS in chapter refers to Subir Sachdev's book


Week 1

Wednesday, September 2

  • Review of phase transitions in simple system (Section 1.1)
  • Ising ferromagnets
  • anomalous exponent
  • intro to Landau theory

Friday, September 4

  • simple fluids
  • criticality and exponents 

Week 2

Wednesday, September 9

  • Simple models (Section 1.2)
  • XY model for superfluids (Sec 1.2)
  • Historical context (Widom scaling, Kadanoff conjecture, Wilson RG)
  • Formal mean-field theory (Sec 2.4)
  • Mean-field free-energy (Landau theory) (Sec 2.1)

Friday, September 11

  • Mean-field critical exponents (Sec 2.2)
  • Mean-field correlations (Sec 2.3)
  • Corrections to mean-field theory (Sec 2.4)

Wednesday, September 16

  • Renormalization group (Block-spin transformation) (Sec 3.1) 

Friday, September 18

  • Renormalization group (One dimensional Ising model) (Sec 3.2)
  • Higher dimensional Ising models 

Week 4

Wednesday, September 23

  • Renormalization group (General theory) - (Sec 3.3)
  • Scaling of the free-energy - (Sec 3.4)

Friday, September 25

  • Renormalization group (Critical exponents) (Sec 3.5)

Week 5

Wednesday, September 30

  • Irrelevant eigenvalues (Sec 3.6)

Friday, October 2

  • Scaling of correlation functions (Sec 3.7)
  • Scaling operators and scaling dimensions (Sec 3.8)

Week 6

Wednesday, October 7

  • Scaling fields (Sec 3.8)
  • Critical amplitudes (Sec 3.9)

Friday, October 9

  • Anisotropic scaling (Sec 3.10)
  • Phase diagrams/fixed points Blume-Capel model (Sec 4.1)
  • Anisotropic Heisenberg model - Cross-over behavior (Sec. 4.2)

Week 7

Wednesday, October 14

  • Cross-over to long range behavior (Sec 4.3)

Friday, October 16

  • Finite size scaling (Sec 4.4)
  • Perturbative renormalization group (intro) (Chapter 5)

Week 8

Wednesday, October 21

  • Operator product expansion (Sec 5.1)
  • Perturbative renormalization group (formalism) (Sec 5.2)

Friday, October 23

  • Perturbative renormalization group (formalism) (Sec 5.2)
  • Ising model near 4 dimensions (Sec 5.3)

Week 9

Wednesday, October 28

  • Gaussian Fixed point (Sec 5.4)
  • Review of Wick's theorem/field theory (appendix A)
  • Normal ordered operators (Sec 5.5)

Friday, October 30

  • OPE at gaussian fixed point (Sec 5.5)
  • Wilson-Fisher fixed point (Sec 5.5)

Week 10

Wednesday, November 4

  • Irrelevant and redundant operators (Sec 5.5)
  • Log corrections from marginal operators (Sec 5.6)
  • OPE for O(n) model near 4 dimensions (Sec 5.7)

Friday, November 6

  • RG for O(n)(Sec 5.7)
  • Application of Wilson-Fisher (cubic symmetry breaking)(Sec 5.8))
  • Lower critical dimension (Sec 6.1)

Week 11

Wednesday, November 11

  • Villain approximation: from XY model to Coulomb gas (Sec 6.2)
  • Kosterlitz-Thouless transitions (Sec 6.2)

Friday, November 13

  • RG for Coulomb gas (Sec 6.4)
  • Non-linear sigma model, Asym freedom, dim transmutation (Sec 6.5)

Week 12

Wednesday, November 18

  • Quantum criticality (Sec 4.5)
  • Fluctuation dissipation theorem and analytic continuation
  • Ising chain in a transverse field - mapping to Ising model(SS Ch 4)

Friday, November 20

  • Dynamical structure factor
  • Transverse field Ising model - strong coupling (SS 4.1.1)
  • Kramers-Wannier duality

Week 13

Wednesday, November 25

  • No classes - Thanksgiving Recess

Friday, November 27

  • No classes - Thanksgiving Recess

Week 14

Wednesday, December 2

  • Transverse field Ising model - weak coupling (SS 4.1.2)
  • Finite temperature cross-over: quantum criticality (SS 4.5)
  • Emergent relativistic dispersion at criticality (SS 4.5.1)

Friday, December 4

  • Low T dynamical response at weak coupling (SS 4.5.1)

Week 15

Wednesday, December 9

  • Low T  response at strong coupling (SS Sec 4.5.2)
  • Jordan-Wigner transformation of transverse field Ising (SS Sec 4.2)
  • Massless Dirac theory of critical Ising model (SS Sec 4.3)

Friday, December 11

  • High T - quantum critical (SS Sec 4.5.3)
  • Phase transitions in Fermi liquids (?)


Note: This is a tentative schedule, and subject to change as necessary – monitor the course ELMS page for current deadlines.  In the unlikely event of a prolonged university closing, or an extended absence from the university, adjustments to the course schedule, deadlines, and assignments will be made based on the duration of the closing and the specific dates missed.

Course Summary:

Date Details Due