Students · MSc · 2026/27 · P2, 6 Nov – 18 Dec · Fridays · students only

Phase Transitions and Critical Phenomena

Why do magnets, fluids, alloys and superfluids behave identically near a continuous phase transition, and how do we compute that behaviour? A 3-ECTS course on classical critical phenomena: mean-field and Landau–Ginzburg theory, fluctuations and scaling, the renormalisation group, continuous symmetries, topological defects and the Berezinskii–Kosterlitz–Thouless transition, with two simulation portfolios.

Lecturers: Pedro Ribeiro

The course in three parts

Phases, mean field and Landau–Ginzburg

Weeks 1–2. Order parameters, symmetry breaking and universality; the exact Ising chain; Weiss mean field; Landau and Landau–Ginzburg theory; first-order transitions and metastability.

Fluctuations, scaling and the renormalisation group

Weeks 3–5. Gaussian fluctuations and the Ginzburg criterion; scaling and finite-size scaling; the renormalisation group, power counting and the ε-expansion.

Continuous symmetries and topology

Weeks 6–7. Goldstone modes, σ-models and the Mermin–Wagner theorem; topological defects and the Berezinskii–Kosterlitz–Thouless transition.

Schedule

All classes are on Fridays: 14:30–16:30 in room Q4.2 and 16:30–18:00 in room Q4.6.

Week	Friday	14:30–16:30, Q4.2	16:30–18:00, Q4.6
1	6 Nov	L1 Phases, symmetry and universality	L2 The Ising chain and the Weiss mean field
2	13 Nov	L3 Mean field and Landau theory · L4 Landau–Ginzburg theory	Workshop A1 · Tutorial I: trustworthy Monte Carlo (bring a laptop)
3	20 Nov	L5 Gaussian fluctuations and the failure of mean field	L6 Scaling and finite systems
4	27 Nov	L7 The renormalisation group: ideas and structure	Workshop A2 · Tutorial II: finite-size scaling (bring a laptop)
5	4 Dec	L8 The Gaussian fixed point and power counting · L9 Momentum-shell RG of φ⁴ theory	L10 The Wilson–Fisher fixed point and the ε-expansion · Tutorial III demonstration
6	11 Dec	L11 Continuous symmetries and Goldstone modes	L12 Non-linear σ-models and the Mermin–Wagner theorem
7	18 Dec	L13 Topology of ordered phases and defects · L14 The BKT transition	Workshop B · Tutorial IV: vortices and the BKT transition; course synthesis

The written test (60 minutes) is in the last week of classes, and the individual orals in the first week after classes. Dates to be announced.

Assessment

Written test, 60 minutesIn the last week of classes.
Two simulation portfoliosReports written in pairs: A, trustworthy Ising criticality (Workshops A1 and A2); B, XY topology and BKT evidence (Workshop B).
Oral examination, 20 minutesIn the first week after classes, including a short defence of your portfolio work.
Weights35 % written test, 30 % portfolios, 35 % oral.