TENTAMEN I STATISTISK MEKANIK

PART B: problems (5 points/problem). Motivate all steps in detail.6. Starting from the scaling form for the singular part of the free energyf s (t, h) ∼ l −d f s (l yt t, l y hh),where t = (T − T c )/T c is the reduced temperature, show the Rushbrook scaling relationα + 2β + γ = 2.7. Consider a system with quenched disorder, such as the presence of impurities atrandom sites in a crystal lattice. The pure system undergoes a continuous phase transitionat T c . The effect of the disorder may be viewed as changing the nearest-neighbourexchange interaction from site to site. Therefore, the system can be modelled as a nearestneighbourmodel with a fluctuating coupling constant. Derive the Harris criterion, statingthat the disorder is irrelevant at the fixed point of the pure system (no disorder) if dν > 2.8. Calculate the magnetization for the one-dimensional Ising model in a magnetic fieldin the Bethe approximation. Compare with the exact resultm =sinh βh√sinh 2 βh + e −4βJ .Definitions of the critical exponents:c(t, h = 0) = −T ∂2 f∂T 2 ∼ |t|−αm(t, h = 0) = − ∂f∂h ∼ (−t)βχ(t, h = 0) = ∂m∂h ∼ |t|−γLYCKA TILL! / GOOD LUCK!2