0.1. CORRELATION LENGTH

Transcription

1 0.1. CORRELAION LENGH Correlation length Correlation length: intuitive picture Roughly speaking, the correlation length ξ of a spatial configuration is the representative size of the patterns. Look at the 2-Ising configuration as a function of temperature. high increasing ξ c low At the order-disorder phase transition point = critical point c, the correlation length diverges. hus, fro a finite visual window, you cannot see the whole picture: small ponds are in islets that are in lakes on islands in bigger lakes on larger islands... hus, from a finite window almost always you see significantly more blue or more yellow; if you see colors always evenly, you are not at the criticality. It is not very trivial to make a critical equilibrium configuration, even if you can tune the temperature accurately at c, because big patterns cannot evolve very quickly; since the pattern size increases indefinitely, the relaxation time also diverges at c, so everything becomes sluggish (critical slowing down) Correlation length: mathematical expression We define the spin-spin correlation function: C(r) = hs(0)s(r)i, (0.1.1)

2 2 where we have assumed that we are above c so s = m (magnetization per spin) is zero and the pattern is, on the average, translationally symmetric. Our guess is that this function decays exponentially for sufficiently large r = r. herefore, the correlation length is defined as log C(r) ξ = lim. (0.1.2) r r We must demonstrate that this is well defined. hat is, the limit exists Correlation length exists An important fact about ferromagnetic order (order that favors similar properties, say up spin, to propagate spatially) is that spins are positively correlated: if you hold a spin at the origin upward, the chance for a spin at r to be up increases (or at leas it never decreases). Generally, for ferromagnetic Ising models the following inequality holds (Griffiths second inequality) s n s m s n s m, (0.1.3) where s n = i s(r i) n i (recall Hadamard s notation) for any positive integers n i. Intuitively, this is very plausible, since the spin-spin correlation must be positive. In particular, we have s(0)s(x) 2 s(x + y) s(0)s(x) s(x)s(x + y) (0.1.4) or, since s 2 = 1 (spins are +1 or 1), s(0)s(x + y) s(0)s(x) s(0)s(y). (0.1.5) We have used the translational symmetry. Define a(x) = log C(x). (0.1.6) hen, we are interested in the limit lim r a(r)/r. (0.1.5) implies for positive integers a(n + m) a(n)a(m). (0.1.7) hat is, {a(n)} are subadditive. hen, Fekete s lemma immediately implies a(n) lim n n = inf n a(n) n. (0.1.8)

3 0.1. CORRELAION LENGH 3 herefore, we have only to say that the RHS is finite. o this end we must say that C(n) does not decay too quickly. he decay rate should not be faster than the 1-Ising along the line. Suppose the spin at the origin is up. hen, the average value of the spin s(1) in its nearest neighbor is s(1) eβj e βj = tanh βj, (0.1.9) e βj + e βj because ignoring the spin chain beyond 1 would reduce the extent of order (notice a(n) that up spin s(2) helps s(1) to be up). herefore, inf n tanh βj. hus, ξ exists n for > c Fekete s lemma If f is subadditive (i.e., f(n+m) f(m)+f(n) for any n, m N), then lim n f(n)/n = inf m f(m)/m. [Demo] Recall lim inf n f(n)/n = lim m inf n>m f(n). Obviously, lim inf f(n)/n inf f(m)/m. Let n = s + km. hen, for any m > 0 f(n) n f(s + km) = n f(s) + kf(m) s + km f(m) m. (0.1.10) herefore, lim sup f(n)/n inf f(m)/m (the largest of LHS must not be larger than the smallest of RHS). hat is, lim inf n f(n)/n = lim sup n f(n)/n = lim n f(n)/n = inf n f(n)/n Correlation function is bounded by exponential factor (0.1.8) implies log C(n) ξ (0.1.11) n We have the following important inequality: s(0)s(r e r /ξ. (0.1.12) What is known about correlation for Ising models? For d-ising model, the following is known or expected:

4 4 If > c it is shown that 1 s(0)s(r) r (d 1)/2 e r/ξ. (0.1.13) If < c, we have spontaneous magnetization for d 2, so We expect s(0)s(r) m 2. (0.1.14) s(0)s(r s(0) s(r) his is shown only for sufficiently small. 1 r (d 1)/2 e r/ξ. (0.1.15) How correlation behaves at criticality As we see from (0.1.14) (or more intuitively from the figure in 0.1.1), ξ diverges at = c. herefore, the correlation function cannot decay exponentially. You might expect a much slower decay something like a stretched exponential decay (say, e a r ). In equilibrium statistical mechanics with short-range interactions this never happens. We start from the following inequality for ferromagnetic Ising models: 1 φ(0)φ(r) s B φ(0)φ(s) φ(s)φ(r). (0.1.16) Here, B can be any set of lattice points such that removal of all the points in B destroys all the paths along the lattice bonds connecting the origin 0 and r. You may imagine B to be a d 1-sphere for d-ising model. If the system is translationally symmetric, then from (0.1.16) we get φ(s)φ(r) = φ(0)φ(r s) s 1 B φ(0)φ(s 1 ) φ(s 1 )φ(r s) (0.1.17) All the moments of spins are positive. 2 herefore, introducing the relation (0.1.17) into (0.1.16), we obtain φ(0)φ(r) s B φ(0)φ(s) ( ) φ(0)φ(s 1 ) φ(s 1 )φ(r s). (0.1.18) s 1 B 1 For ferromagnetic Ising models, see B. Simon, Correlation Inequalities and the decay of correlations in ferromagnets, Commun. Math. Phys., 77, 111 (1980). 2 his is one of Griffiths inequalities (the first inequality): for any finitely many positive integers a i i sai i 0. A simple proof may be found in J. Glimm and A. Jaffe, Quantum Physics, a functional integral point of view, Second Edition (Springer, 1987) Sect. 4.1.

5 0.1. CORRELAION LENGH 5 o be sure let us reiterate this procedure once more: φ(0)φ(r) s B φ(0)φ(s) { ( )} φ(0)φ(s 1 ) φ(0)φ(s 2 ) φ(s 2 )φ(r s s 1 ). s 1 B (0.1.19) If r is sufficiently far away from the origin and the above procedure may be repeated q times, then replacing the term φ(s q )φ(r s s 1 s q 1 ) that is expected to appear after these iterations with its maximum value C, we obtain [ q φ(0)φ(r) φ(0)φ(s) ] C. (0.1.20) s B Here, q may be chosen to be a number proportional to r if r is far away from the origin. If we can make s B φ(0)φ(s) < 1, the correlation function decays exponentially. hat is, the summability of the correlation function on a large sphere B is the key for exponential decay. Suppose that the correlation function behaves as r µ (d 1 < µ). hen, φ(0)φ(s) S d 1 r d 1 r µ r d 1 µ. (0.1.21) s B herefore, if d 1 µ < 0 or µ > d 1, the decay must be exponential. hat is, if the correlation decay faster than 1/r d 1, it must decay exponentially. On the other hand, as can be seen from the decay of the fundamental solution to the Laplace equation, even without any fluctuation, correlation decays as 1/r d 2. If there are fluctuations, intuitively speaking at least for the ferromagnetic case, the correlation is expected to decay faster than without fluctuations. herefore, when ξ diverges, the correlation function must decay algebraically and faster than 1/r d 2. herefore, we may conclude that s 2 B φ(0)φ(r) 1 r d 2+η (0.1.22) is the general expression for the order parameter correlation function at the critical point. Here, η ( (0, 1]) is one of the critical indices we will encounter in the next section. Notice that if the correlation decays exponentially, scaling (shrinking space) makes the correlation length smaller (scaled down), so if you would not see any correlation the fluctuation pattern from distance. In contrast, if the decay is algebraic as (0.1.22) scaling r r/l (l > 1) preserves the same algebraic decay.

6 6 0.2 fluid Universality Universality It is natural to guess that near the critical point, large, even semimacroscopic, fluctuations dominate the equilibrium behavior of the systems, so microscopic details (materials scientific details) may well be washed away. hus, Ising models with different coupling constants or on different lattices exhibits certain common features. his may even be true for totally different systems (such as magnets, binary mixture, etc.). We expected this from the phase diagrams at least qualitatively. his is called universality. he universality is quantitative: watch Dr Ashton s wonderful video, increasing observation distance magnet We could say the universality near the critical point is due to statistical selfsimilarity of the fluctuations See the following figure and watch another Dr. Ashton s video successive magnifications hus, absolute scales become unimportant, and microscopic details irrelevant.

7 0.2. UNIVERSALIY 7 Only global features such as spatial dimensionality, interaction decay (slow algebraic or not), spin dimensionality, etc., become decisive Critical indices Since fluctuations become large near the critical point the fluctuation response relation tells us that various susceptibilities diverge. ypical examples are magnetic susceptibility χ (proportional to the second moment of magnetization) and specific heat C (proportional to the energy fluctuation). It is believed that they diverge as follows at criticality (see the figure below): Here, χ τ γ (h = 0), C B τ α (h = 0). (0.2.1) τ = c c. (0.2.2) he magnetization m becomes nonzero below c. As is lowered m continuously increases from zero, but it is not differentiable at = c as m ( τ) β (h = 0, τ < 0). (0.2.3) m χ C c c he positive (or non-negative) numbers α, β and γ are called critical indices. However, remember that mathematically we do not know whether such divergence actually occur, although empirically these behaviors look realistic. he universality means that these exponents are the same for a class of materials. However, the numerical factors in front of the powers of τ are not at all universal. hat is, these factors are sensitive to materialistic details. Such quantities are, from the purely theoretical point of view, of secondary importance. he actual value of c is also a typical such non-universal quantity extremely sensitive to microscopic details. hus, to try to compute these nonuniversal quantities is a fetish endeavor. c

8 Critical index (in)equalities Representative values of critical indices may be found in the following table. As empirically confirmed, these exponents also apply to critical fluids and binary mixtures. Ising critical exponents. Exponents 2-space 3-space d( 4)-space α 0 (log) (jump) β 1/ /2 γ ν /2 Notice that these values are universal constants just as π in the sense that their values are determined by the structure of the universe. here are relations among critical indices such as Rushbrooke s equality: α + 2β + γ = 2. hermodynamic stability of the system tells us (as noted and demonstrated in33.3-4) Rushbrooke s inequality: α + 2β + γ 2. Needless to say these relations are under the assumption the critical indices exist. As we will see soon, only two indices are independent Kadanoff construction Let us observe the spin configuration of a spin system slightly off the critical point. Suppose we step back from the system and observe it from the distance l times as large as the original distance, while keeping our eyesight. hen, the apparent pattern size we observe is scaled by l, so the correlation length looks as ξ/l. If we regard the size of our minimum resolution as a new effective spin (which consists of several original spins, so we will call it a block spin), the apparent pattern we observe looks like that of the original system away from the critical point. Kadanoff interpreted this as follows. If we shrink the system while keeping the smallest length scale by coarse-graining the pattern appropriately, the configuration corresponds to that of the state of the system further away from the critical point τ = 0 and h = 0. herefore, if the shrinking rate is l (that is, ξ ξ/l), the apparent

9 0.2. UNIVERSALIY 9 pattern is the same as the original system with τ τl y 1 and h hl y 2, where y 1 and y 2 are positive numbers. Kadanoff s idea 3 is apparently vindicated by simulation. Watch the following video by Dr Ashton: Kadanoff s idea (Kadanoff construction) may be formulated as follows. A ξ shrinking ξ B minimum discernible volume after shrinking shrinking copy ξ original minimum discernible volume At c the system is invariant under shrinking and coarse-graining, but if ξ is finite, then ξ ξ/l Critical exponent for susceptibility How does the equation of state m = M(τ, h) change? Magnetic energy stored in the minimum discernible block is hm in the original system. After shrinking, there are l d original spins in the block spin, so the magnetic energy per block spin reads h m. Due to the extensivity of the energy, we expect We get an identity called the scaling relation: h m = l d hm. (0.2.4) m = l d (h /h)m = l y 2 d M(τ, h ) = l y 2 d M(τl y 1, hl y 2 ). (0.2.5) 3 which must be inspired by Widom s homogeneity hypothesis that thermodynamic quantities are (generalized) homogeneous functions of τ nd h.

10 10 l can be chosen as any positive number. Here, note that m is nonzero only if τ < 0, so let us demand τl y 1 = 1. hen, using the definition of the critical exponent β 0.2.2, we get m(τ, 0) = τ (d y 2)/y 1 m( 1, 0) β = d y 2. (0.2.6) y 1 Differentiating m wrt h we can calculate the susceptibility. χ = m h herefore, setting τ l y 1 = 1, we get = l 2y2 d M h (τl y 1, hl y 2 ) = l 2y2 d χ(τl y 1, hl y 2 ) (0.2.7) τ γ = 2y 2 d y 1. (0.2.8) Critical exponent for specific heat he (singular part of the) free energy: f s = F s (τ, h) per minimum discernible volume unit scales as f s = F s (τ, h) = l d F s (τl y 1, hl y 2 ). (0.2.9) because f s = l d f s due to the extensivity. If we differentiate (0.2.9) with h, we get (0.2.5). Differentiating (0.2.9) twice with respect to τ (that is, ), we obtain herefore, hus, we can obtain Rushbrooke s equality: C(τ, h) = l 2y 1 d C(τl y 1, hl y 2 ). (0.2.10) α = 2y 1 d y 1. (0.2.11) α + 2β + γ = 2y 1 d + 2 d y 2 + 2y 2 d = 2. (0.2.12) y 1 y 1 y What does renormalization do?

11 0.2. UNIVERSALIY 11 Magnet H to perfect up phase A c to = 0 limit A' B C to high temperature limit fixed point b a actual n > 1 n = 0 n >> 1 renormalization transformation c Fluid V Gas Solid to perfect down phase A triple point B supercritical fluid C CP A' Liquid