Physics 7240: ADVANCED STATISTICAL MECHANICS Fall 2017 Tuesday and Thursday 2:00-3:15pm, Duane, G2B41 Instructor: Prof. Leo Radzihovsky Office Hours: Tuesday 3:15-4:30pm (or by appointment), Gamow Tower Leo: Rm F623, phone: 492-5436, email: [email protected] Content: The course will cover selected advanced topics in statistical mechan- ics (see course outline) Pre/Corequisites: Introductory Statistical Mechanics, Quantum Mechanics, Electricity and Magnetism, and interest and desire to learn Text: In-class notes Additional suggested • Pathria: Statistical Mechanics, Butterworth-Heinemann reading/references: (1996). • Mehran Kardar: Statistical Physics of Particles, Cambridge University Press (2007). • Mehran Kardar: Statistical Physics of Fields, Cambridge Uni- versity Press (2007) • J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. J. Newman : The Theory of Critical Phenomena, Oxford (1995). • John Cardy: Scaling and Renormalization in Statistical Physics, Cambridge Lecture Notes in Physics • P. M. Chaikin and T. C. Lubensky: Principles of Condensed Matter Physics, Cambridge (1995). • P. G. de Gennes and J. Prost: The Physics of Liquid Crystals, Oxford (1993). • P. G. de Gennes: Superconductivity of Metals and Alloys, Addison-Wesley (1989). • J. Zinn-Justin: Quantum Field Theory and Critical Phenom- ena, Oxford (1989). • Renormalization Group review articles: – M. E. Fisher, Rev. Mod. Phys. 42, 597 (1974). – K. G. Wilson and J. Kogut, Phys. Rep. 12 C, 77 (1974). Homework: There will be periodic problem set assignments, approximately three weeks apart Term paper and in- You will be required to research an advanced (not necessarily original class seminar: or unpublished) topic in statistical mechanics, and write a review term paper on it, due on the day of the scheduled final. Approximate Homework: 80% Grading: Research paper: 20% COURSE OUTLINE 1. Introduction and review (a) Review of basic statistical mechanics (b) Noninteracting systems (ideal gas and Langevin paramagnetism) (c) Phase transitions in real systems (L-G, BEC, PM-FM) (d) Statistical field theory (Gaussian integrals) 2. Magnetism (a) Magnetic interaction (dipolar and exchange) and phenomenology (b) Heisenberg, XY and Ising models (c) Exact solution of the 1d Ising model (transfer matrix) (d) Exact solution of the 2d Ising model (1+1d transverse-field Ising model, Jordan- Wigner, Bogoluibov-de Gennes) (e) Mean-field (d = ∞) theory (f) Landau theory (g) Exact solution of N → ∞ O(N) model in d dimensions 3. Renormalization Group (RG) (a) What’s missing from Landau theory ? (fluctuations and Ginzburg criterion) (b) RG motivation, transformation, and phenomenology (c) Real space RG for 1d Ising model (Nelson and Fisher) (d) Momentum shell RG (Ising model φ4 theory) (e) -expansion (f) Physical observables via scaling out of critical region and matching (g) “Dangerously” irrelevant operators (h) O(N) model (i) Nonlinear σ-model via = d − 2-expansion 4. Superfluids (a) Phenomenology for He4 (experiments, 2nd sound, phonon-roton) (b) Quantum many-body Hamiltonian (c) Review of BEC (d) Bogolubov theory (e) Landau theory (XY model) via coarse-graining (f) Kosterlitz and Thouless transition in films (g) Coulomb gas, duality and sine-Gordon model (h) Roughening transition and floating solids 5. Superconductors (a) Phenomenology (b) Ginzburg-Landau theory (c) Vortex states and dissipation in type II superconductors 6. Duality in Physics (a) Classical 2d XY to sine-Gordon model (b) Classical 3d XY to superconductor (c) Quantum 2+1d superfluid to superconductor 7. Liquid Crystals (a) Phenomenology (b) Isotropic-Nematic transition (c) Nematic-Smectic-A transition (de Gennes model) (d) Smectic anomalous elasticity (Grinstein and Pelcovits) (e) Cholesterics (chiral nematics and their smectic elasticity) (f) Critical states of soft matter 8. Heterogeneous systems (a) Quenched versus annealed disorder (b) Disorder in ordered states: random field, random bond, Imry-Ma-Larkin stability analysis (c) Disorder near a continuous phase transitions: random field, random bond, Harris and Imry-Wortis criteria (d) Physical systems i. charge-density wave ii. vortex lattice iii. polymerized membranes iv. liquid crystals in aerogel v. liquid crystal with a “dirty” substrate (e) Methods: replica trick and correlators (f) RG in heterogeneous field theories (random bond and field) (g) 2d random field finite temparature RG (Cardy-Ostlund) (h) Zero temperature functional RG (D.S. Fisher) 9. Equilibrium and Non-equilibrium Dynamics (a) Brownian motion (Einstein, Smoluchowski, Langevin theories) (b) Fokker-Planck equation (c) Fluctuation-dissipation theorem (d) Stochastic quantization via Martin-Siggia-Rose and DeDominicis-Janssen (MSR DJ) formulations (e) Dynamical RG (via S.K. Ma, MSR, DJ).