MIT OpenCourseWare http://ocw.mit.edu 8.334 Statistical Mechanics II: Statistical Physics of Fields Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.III. The Scaling Hypothesis III.A The Homogeneity Assumption In the previous chapters, the singular behavior in the vicinity of a continuous transi-tion was characterized by a set of critical exponents {α, β, γ, δ, ν, η, · · ·}. The saddle–point estimates of these exponents were found to be unreliable due to t he importance of fluctua-tions. Since the various thermodynamic quantities are related, these exponents can not be independent of each other. The goal of this chapter is to discover the relationships between them, and to find the minimum number of independent exponents needed to describe the critical point. The non-analytical structure is a coexistence line for t < 0 and h = 0, that terminates at the critical point t = h = 0. The various exponents describe the leading singular be-havior of a thermodynamic quantity Q(t, h), in the vicinity of this point. A basic quantity in the canonical ensemble is the free energy, which in the saddle–point approximation is given by  t2    − for h = 0, t < 0 f(t, h) = min tm 2 + um 4 − h.m ∝  u. (III.1) 2  h4/3 m   − for h 6 0, t = 0= u1/3 The singularities in the free energy can in fact be described by a single homogeneous function† in t and h, as f(t, h) = |t|2 gf  h/|t|Δ . (III.2) The function gf only depends on the combinati on x ≡ h/|t|Δ, where Δ is known as the gap exponent. The asymptotic behavior of gf is easily obtained by comparing eqs.(III.1) a nd (III.2). The h = 0 limit is recovered if limx→0 gf(x) ∼ 1/u, while to get the proper power of h, we must set limx→∞ gf(x) ∼ x4/3/u1/3 . T he latter implies f ∼ |t|2h4/3/(u1/3|t|4Δ/3). † In general, a function f(x1, x2, · · ·) is homogeneous if f (bp1 x1, bp2 x2, · · ·) = bpf f(x1, x2, · · ·), for any rescaling factor b. Wit h the proper choice of b one o f the arguments can be removed, leading to a scaling forms used in this section. 35Since f can have no t dependence along t = 0, the gap exponent (corresponding to Eq.(III.1) has the value 3 Δ = . (III.3) 2 The assumption of homogeneity is that, on going beyond the saddle–po int approxima-tion, the singular form of the free energy (and any other thermodynamic quantity) retains the homogeneous form fsing.(t, h) = |t|2−α gf  h/|t|Δ . (III.4) The actual exp o nents α and Δ depend on the critical point being considered. The depen-dence on t is chosen to reproduce the heat capacity singularity at h = 0. The singular part of the energy is o bta ined from (say for t > 0) 1−α ′ Esing. ∼ ∂f ∼ (2 − α)t gf  h/|t|Δ − Δht1−α−Δ gf  h/|t|Δ ∂t    Δh     (III.5) ∼ t1−α (2 − α)gf h/|t|Δ− tΔ gf ′ h/|t|Δ≡ t1−α gE h/|t|Δ. Thus the derivative of o ne homogeneous function is another. Similarly, the second deriva-tive takes the form (again for t > 0) Csing. ∼ − ∂2f ∼ t−α gC  h/|t|Δ , (III.6) ∂t2 reproducing the scaling Csing. ∼ t−α, as h → 0. It may appear that we have the freedom to p ostulate a more general form C±(t, h) = |t|−α±g±  h/|t|Δ±  , (III.7) with different functions and exponents for t > 0 and t < 0, that match at t = 0. However, this is ruled out by the condition that the free energy is analytic everywhere ex cept on the coexistence line for h = 0 and t < 0 , as proven as follows: Consider a point at t = 0 and finite h. By assumption, the function C is perfectly analytic in the vicinity of this point, expandable in a taylor series, C  t ≪ hΔ = A(h) + tB(h) + O(t2). (III.8) Furthermore, the same expansion should be obtained from both C+ and C−. But eq.(III.7) leads to the expansions,   p±  q±  C± = |t|−α± A± |t|h Δ± + B± |t|h Δ± + · · · , (III.9) 36( where {p±, q±} are the leading powers in asymptotic expansions of g± for large arguments, and {A±, B±} are the corresponding pre-factors. Matching to the taylo r series in eq.(III.8) requires p±Δ± = −α± and q±Δ± = −(1 + α±), and leads to C±  t ≪ hΔ = A±h−α±/Δ± + B±h−(1+α±)/Δ±|t| + · · · . (III.10) Continuity at t = 0 now forces α+/Δ+ = α−/Δ−, and (1 + α+)/Δ+ = (1 + α−)/Δ−, which in turn implies α+ = α− ≡ α . (III.11) Δ+ = Δ− ≡ Δ Despite using |t| in the postulated scaling form, we can still ensure the analyticity of the function at t = 0 for finite h by appropriate choice of parameters, e.g. by setting B− = −B+ to match E q.(III.10) to the analytic form in Eq.(III.8). Having established this result, we can be somewhat careless henceforth in replacing |t| in the scaling equations with t. Naturall y these arguments apply to any quantity Q(t, h). Starting from the free energy in eq.(III.4), we can compute the singular parts of other quantities of interest: • The magnetization is obtained from m(t, h) ∼ ∂f ∼ |t|2−α−Δ gm  h/|t|Δ . (III.12) ∂h In the l imit x → 0, gm(x) is a constant, and m(t, h = 0) ∼ |t|2−α−Δ , =⇒ β = 2 − α − Δ. (III.13) On the other hand, if x → ∞, gm(x) ∼ xp, and m(t = 0, h) ∼ |t|2−α−Δ  |th |Δ p . (III.14) Since this limit i s independent of t, we must have pΔ = 2 − α − Δ. Hence m(t, h = 0) ∼ h(2−α−Δ)/Δ , =⇒ δ = Δ/(2 − α − Δ) = Δ/β. (III.15) • Simil arly, the susceptibility is computed as χ(t, h) ∼ ∂m ∂h ∼ |t|2−α−2Δ gχ(h/|t|Δ), ⇒ χ(t, h = 0) ∼ |t|2−α−2Δ , ⇒ γ = 2Δ − 2 + α. (III.16) 37Thus, the consequences of the homogeneity assumption are: (1) The singular parts of all criti cal quantities Q (t, h), are homogeneous, with the same exponents above and