Spring 2018

Lecture 1

Topics discussed: QFT as a language for stat-phys, cond-mat, particle physics (and quantum gravity). QFT vs "cutoff physics". Correlation length and mass gap. Emergence of QFT description at continuous phase transitions, thermodynamic and quantum. Effective Lorentz invariance at cond-mat phase transitions.

Reading assignments:

  • Inspirational reading:
    • John McGreevy "Whence QFT?", Section 0.1
    • Polyakov "Gauge fields and strings". Start on p.1 and stop as soon as you run out of energy to follow his stream of consciousness
  • Preparation for the future lectures:

Lecture 2

Topics discussed: Quantum-classical correspondence. RG philosophy. Various types of real-space RG (block-spin, Migdal-Kadanoff bond-moving approximation, majority rule, decimation). Review of attempts at systematic improvement: Wilson's Rev.Mod.Phys. decimation calculation with O(100) couplings, functional RG in Local Potential Approximation. Tensor Network Renormalization. RG flow diagrams with several couplings. RG and 1st order phase transitions (3-state Potts model in 3d as an example).

Reading assignment: Cardy "Scaling and renormalization in statistical physics", section 4.1 for RG flows in the Ising model with vacancies. This is one of my favorite books, check it out!

Lecture 3

Topics discussed: RG flow structure near fixed point. Critical, tricritical, and self-organized ("dead-end") behavior. Classification of couplings by symmetry. Universality. What is universal? Reasons for universality and possible exceptions. Gaussian FP. Free massless scalar in d>2. Classification of perturbing operators. RG definitions of: QFT, UV-complete QFT, EFT.

Exercise: Consider the tricritical point for 3d Ising model, which as we said is just the Gaussian free scalar. Ising model is on the cubic lattice, while the Gaussian fixed point is rotationally invariant. Can you use RG to argue that the rotational invariance must emerge at large distances? What is the leading dependence of corrections which break rotational invariance on the distance? Harder: same question for the critical point.

Lecture 4

Topics discussed: (\phi^4)_4 beta-function in 5 minutes. Examples of UV-complete theories (YM in 4d, NLSM in 2d, Gross-Neveu in 2d). Further examples of Gaussian FP: theory of elasticity vs Maxwell; fermionic gaussian FPs. The simplest fermionic gaussian FP - symplectic fermions. Their role in abelian sandpile model and self-organized criticality. Start to discuss (phi^4)_2 as the simplest example of strongly coupled QFT. A digression about integrable theories and the Coleman-Mandula theorem.

Lecture 5

Topics discussed: This was a lecture about (\phi^4)_2 theory. We discussed simple perturbative calculations (like the correction to the physical particle mass), and why this expansion is divergent. We then discussed how we know that the path integral of the theory actually makes sense. This required reviewing how mathematicians view path integral in quantum mechanics (Wiener measure), and how this generalizes to field theory (measure on the space of distributions, integrability of the interaction). We then discussed Monte Carlo approach to computing observables in this theory. Finally, we discussed Rayleigh-Ritz method in quantum mechanics and its generalization to field theory - the Truncated Spectrum Approach or Hamiltonian Truncation. We discussed the results of applying this method to (\phi^4)_2 theory, in particular quantum phase transition to a Z2 spontaneously broken phase above a critical coupling. Existence of phase transition was demonstrated using Chang duality.

Literature: Hamiltonian Truncation material discussed today, while elementary, is of current research interest. If you are curious, see my papers 1412.3460, 1512.00493, 1706.09929 and 1512.06901 by other authors. The original paper on Chang duality (1976)

Lecture 6

Topics discussed: Effective Field Theory. Example 1: scalar with shift symmetry. Structure of effective Lagrangian. Role of symmetry in protecting the mass. Aside: naturalness problem of the Standard Model. UV completion. Why need EFT? What is computable in EFT? Nonanalytic parts of the scattering amplitude. The convenience of dim.reg. in EFT calculations. Example 2: Nonlinear sigma model. SO(N)/SO(N-1) as an example.

Lecture 7

1) Coleman-Mermin-Wagner theorem. NLSMs in 2d. 2) Weakly broken global symmetries, pseudogoldstones. Pi-meson masses in 2-flavor QCD. Spurion analysis. 3) EFTs with electromagnetic field. QED at energies below electron mass: Euler-Heisenberg effective lagrangian. Critical electric field and the Schwinger effect. Critical electric charge. 4) EFT for neutral particles interacting with EM field. Rayleigh scattering.

Literature: See 2017 lecture notes L5.pdf and L7.pdf for pi-meson masses and EM field EFTs. For spurions, see e.g. section 5.3 of Georgi "Weak interactions" (although not much more of an explanation there than what I gave).