Statistical Field Theory - Mathematical Tripos - Paper, Exams of Mathematics

Download Statistical Field Theory - Mathematical Tripos - Paper and more Exams Mathematics in PDF only on Docsity! MATHEMATICAL TRIPOS Part III Tuesday, 5 June, 2012 1:30 pm to 3:30 pm PAPER 49 STATISTICAL FIELD THEORY Attempt no more than TWO questions. There are THREE questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Cover sheet None Treasury Tag Script paper You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. 2 1 Give an account of the Landau-Ginsberg (LG) theory of phase transitions in the context of a scalar field theory which should include a discussion of the following points: (i) the idea of an order parameter; (ii) the distinction between first-order and continuous phase transitions and how their defining properties are explained; (iii) the idea of universality giving an example; (iv) the idea of critical exponents and how they may be derived; (v) the reason why a line of first order transitions must terminate in a critical point associated with a continuous phase transition; (vi) the explanation of the features of a two-dimensional phase diagram containing a tricritical point in which you should identify a line of three-phase coexistence and give a reason for its occurence. Near to a continuous phase transition at T = TC , the critical exponents α, β, γ and δ are defined by Specific heat: CV ∼ |t| −α (h = 0), Magnetization: M ∼ |t|β (t < 0, h = 0), Susceptibility: χ ∼ |t|−γ (h = 0), Magnetization: M ∼ |h|1/δ (t = 0), where t = (T − TC)/TC and h is the applied magnetic field. Calculate α, β, γ and δ for a tricritical point and verify the scaling relations α+ 2β + γ = 2, βδ = β + γ. Part III, Paper 49