V. SCALING. A. Length scale - correlation length. h f 1 L a. (50)

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1 V. SCALING A. Length scale - correlation length This broken symmetry business gives rise to a concept pretty much missed until late into the 20th century: length dependence. How big does the system need to be in order for us to truely exhbit the singularities assocciated with a continuous phase transition? The singularity in the magnetization as a function of the magnetic field seems to only be obtained if the system is infinite. When the system is finite we can t observe this singularty, which means that we would see a sudden, but soft, drop in the magnetization as h 0. This drop would sharpen with increasing system size. Now as we approach the transition, the singularity itself - the magnetization jump - becomes smaller. and therefore in order to see this jump, we need to consider bigger and bigger systems. To see this better, let us define h f as the rough symmetry-breaking field, where the magentizationstarts dropping to zero. Clearly, the larger the system, the lower is h f - only at very small SB fields is the system fluctuating and suppressing the long-time average magnetization. So we can guess, that if L is the size of the system, then: h f L a. (50) On the other hand, if we consider h f when the temperature is close to T c, then since even in the infinite system, the average magnetization is tiny, one would need a big h f to stabilize it against finite size fluctuations. So by the same token, we can guess: h f And putting the two observations together, we expect: (T c T ) b (5) h f L a (T c T ) b. (52) Now, suppose we are carrying out the following experiment. We measure the magnetization of several pieces of Ising magnets with different sizes L (but same denisty), but in a constant symmetry breaking field, h 0. For a given temperature T, at what size L, do we start seeing the finite magnetic moments characteristic of the ordered phase? We need to have: h 0 > h f L a (T c T ) b. (53) and therefore: L > (T c T ) = b/a (T c T ) ν (54) We have a special name for this power - ν. This experiment gives an indication of an important length scale that diverges at the critical point. Let s name this mystery length scale, ξ. ξ diverges as we approach the phase transition: ξ t ν (55) where t = (T T c )/T c. form the above discussion, when the size of the system, L > ξ, the singularity of the magnetization is very sharp. But the sharp features of the transition are lost if L < ξ. ξ has an even deeper meaning. It is the correlation length. The rigorous statement associated with the claim above is that the reduced-correlation function decays as: (σ i σ i )(σ j σ j ) e i j /ξ (56) But more important even, is the qualitative meaning. A spin σ i, is sensitive to spins in a sphere of size ξ around it. In order to figure out what it wants to do, it literally seeks the advise of its sphere of interest spins. Pretty much like deciding what stocks to invest in by consulting your best bodies...

2 At the point of the transition, all the spins are so confused, they have to check with everybody in the system, even if it requires asking spins that are miles away. This is the essence of a continuous phase transition. A second qualitative viewpoint on the correlation length, is that it determines the size of droplets that point in the same direction. For instance, above the transition, even though long range order is not established, still there are some flucutations in which a group of spins suddenly form a magnetized droplet. The characteristic size of this droplets is ξ. There is another important conclusion to be made fom this discussion: Whereas in first order phase transitions the system may break down into a coexistence of two phases, second order critical points cannot do that - the size of a droplet diverges at the critical point. A finite size sample will not undergo a sharp transition, but will carry out a croos over between the two phases. 2 B. Critical exponents As it turns out, the critical behavior of systems has many more singularities otehr than that of the correlation length. And eahc singularity is associated with a critical exponent. For teh correlation length, the exponent is ν. Other tabulated singularities:. heat capacity: 2. order parameter: 3. suscptibility: 4. equation of state at t = 0: c t α (57) m t β (58) χ t γ (59) m H /δ (60) the greek letters, here are the critical exponents of the transition. Understanding critical behavior, implies calculating these exponents. VI. UNIVERSALITY, LRO AND MERMIN WAGNER THEOREM, AND EXACT SOLUTION OF THE ISING MODEL A. Role of Models - Universality The greek letters introduced above constitute the critical exponents of a transition. There are other quantities that describe the transition - T c, the speed with which m rises, in fact - the amplitudes in all the above relations. These latter quantities depend very strongly on the microscopic variables of the problem. On the other hand the critical exponents constitute the large-length-scale behavior of the system, and it is independent of the microscopic details. These quantities are therefore called Universal. On the other hand - the quantities that are here are non-universal by definition. As we focus more and more on the large-scale behavior, they fall off, and to us, they are less interesting. For this reason - the independence from microscopic quantities, even the simple bare Ising model is sufficient to descibe the behavior of a critical system that breaks up-down symmetry. This is the essence of Universality. Systems can be categorized into universality classes, which have the same critical behavior (ie critical singularities and exponents), but they differ in the microscopic details of the phases on the two sides of the transition, and in the transition point itself. Systems in the same universality class, share. symmetry,

3 3 2. dimensionality 3. range of interaction (short vs. long with a particular power-law decay) All other details are irrelevant. When I started my graduate studies, many peopled around when discussing physics used the term universal rather freely - they would say - this susceptibility is universal, this is universal, and that is not-universal... and you could tell that they couldn t care less for this last thing. But all this time nobody could tell me what universal actually means. Very quickly i noticed that I don t loose any of the conversation if in my head I translate unviersal, to important. I was quite satisfied with this linguistic short-cut, until it was finally explained to me quite a few years later. Ironically, it was by somebody who wanted to make the point that non-universal aspects fo the system, such as critical temperature, and the amplitude of the diverging quantities is what is actually measured in experiment. I ignored this person. The idea of universality is what makes this quarter worth while - by studying a select group of models, and the canonized methods of scaling and RG, we will be able to understand the behvior of countless systems. B. LRO and QLRO Last class I also emphasized the idea of a broken symmetry and order. How did we qualify order? By defining the order parameter - m = σ (6) when m 0, we have an ordered state. But due to flucutations in finite systems, we know that measuring this equilibrium average might be tricky. An alternative thing we can think of doing, is to ask whether two spin far away from eachother always point in the same direction: if indeed the system is ordered, this correlation will be of the order C i j = S i S j (62) C i j m 2 (63) This we call - Long Range Order. Alternatively order may exist but be fickle, and the correlation will decay very slowly to zero, as a power law: C i j i j φ (64) This is Quasi-long-range order. Long range order is missing if this correlation decays exponentially. In the following, we ll go back to the Ising model and ask whether this LRO we expect at low T can indeed exist. After a qualitative discussion, we will go ahead and just solve the -d Ising exactly. Let s go back to the Ising model: C. Phase transitions at -d? Ĥ = i Jσ i σ j (65) Appealing to your intuition, I claimed that there is a transition temperature above which the Ising model would be disordered and below which it breaks the up down symmetry. But actually is that always true? Suppose it is even in one-d, what do fluctuations do to this symmetry broken state? Consider a very long chain, with all spins pointing up. Disordering such a chain could be most easily done by inserting a domain wall. (DRAW ANOTHER chain). The energy cost of the wall is: 2J

4 The question we need to ask though, is whether the introduction of a domain wall increases or decreases the free energy of the system. ΔF = U T S (66) Since we could insert this domain wall in any of the bonds. This implies an entropy, which is log of the number of possibilities: The free energy of a domain wall, then, is: S = ln L F DW = 2J T ln L (67) Which, as soon as T > 0, becomes. Hence domain walls can always occur - they minimize the free energy. But if domain walls occur freely, then the LRO is completely destroyed. 4 D. 2-d What about 2-d? A domain wall of length L DW now costs roughly U = 2L DW J In addition to being able to occur anywhere along the lattice, which gives a log L contribution, the domain wall can meander as it wishes. roughly speaking it can meander with (z ) L DW possiblities. Therefore the entropy is: S DM ln L + L DW ln(z ) where z is the coordination number of the dual lattice - the number of nearest neighbors in the lattice that the domain wall lives on. This lattice is obtained by drawing lines through the center of the bonds, and perpendicular to them. The log can be neglected compared to the linear term in L DW term. Now the free energy of a domain wall is: which means that below: F L DW (J T ln(z )) (68) T < J/ ln(z ) (69) the Ising model is stable against domain walls. A similar argument applies at higher dimensions. E. Mermin Wagner Thm The above is an illustration of the Mermin-Wagner theorem for the breaking of discrete symmetries - in this case, time reveresal or up-down. Long range order can only obtain at dimensions higher than. We refer to D= as the lower critical dimension of the Ising model. The dimension above which there is no finite temperature symmetry breaking. The Mermin-Wagner theorem implies that this is true for all models with a discrete symmetry - no breaking at finite temperature at D.

5 5 VII. EXACT SOLUTION OF THE ISING MODEL Until now we talked about the general structure of critical phenomena. In particular, we discussed the concept of universality. This concept specifies that critical phenomena is pretty much the same for a variety of models. Therefore, for the rest of the class, we will focus on one particular model - the Ising model, and develop the theory of critical phenomena in this model. We begin with the exact solution of the Ising model, in two setups. First, for free boundaries, and second for periodic boundaries. This way we will gain some intuition and experience as to how interacting models are solved in practice. A. Free boundaries By solution, we mean the partition function and correlations. The partition function is: Z = βj e {S i} L i=0 S is i+ (70) But now, to solve this interacting model, we need a trick. here it is: We can also write as: the last sum is vey easy: = S L =± {S i, i<l} L 2 βj S is i+ e i=0 continue in the same way, and very easily you see that: S L =± e βjs L S L (7) e βjs L S L = e βj + e βj = 2 cosh(βj) (72) Z = (2 cosh(βj)) L (73) real easy, isn t it? Now let me ask you - can you see from here that there is no phase transition? (DISCUSS WITH CLASS) We have to think a bit, but once we have we ralize that in order to have a discontinuity in any order of some thermodynamic derivatice, the partition function need to have a singularity at some point. The partition above, is better behaving the pope himself. The only place where it does something funny is at T = 0, where it diverges. So no criticality, except maybe at T = 0. Above we though of the heat-capacity and susceptibility, for instance. The heat capacity is fairly easy: The entropy: Take another derivative, and you get: F = T ln Z = LT ln(2 cosh(βj)) (74) S = F T = L ln(2 cosh(βj)) L J tanh(βj) (75) T T S ( ) 2 J T = L T cosh 2 βj Near T = 0 we can assume the cosh is just an exponent, and we get: (76) C L ( J T ) 2 e 2βJ (77) Activated decay to zero. The /T 2 looks promising for some divergence, but the exponent just kills it. Also, we know a singularity sohould not be non integrable, otherwise we would have discontinuity in the entropy. so at most: T α with α <. Are we surprised? Not really. The lowest lying excitations are just a domain wall, which cost 2J, and hence the activated form.

6 6 B. Correlation fuctions Unfortunately, we don t know much about correlations from the partiiton function. To calculate the correlations, let me do a trick. The partition function solution is pretty suggestive. It seems as if instead of adding Boltzmann terms associated with each site, we are dealing with eahc bond at a time. Why not then define a bond variable: It is easy to see that: that: η i = S i S i (78) S i = S 0 (S 0 S )(S S 2 )... (S i S i ) = S 0 η η 2... η i (79) and again we use the fact that S 2 i =. Suppose there is one domain wall between 0 and i, this long product will just be. Hence we also call the eta s domain-wall variables. let s call S 0 = η 0 and think about η 0 as a domain wall variable relative to some bounary condition. The partition function is: Z = βj e {S i} L i=0 S is i+ = e βjηi = 2 cosh(βj) (80) so easy! Correlations are also made easy. Consider the correlations between S i and S j wiht j > i. We can write: Now: S i S j = Z {η i} S i S j = e βjηi {η i} j l=i+ j l=i+ η l (8) η l = tanh(βj) i j. (82) This we can write in the form that I showed you earlier - we can write this as an exponential decay with a correlation length: With ξ being: tanh(βj) i j = e i j ln coth(βj) = e i j /ξ (83) ξ = ln coth(βj) = ln( + 2e 2βJ ) 2 e2βj (84) For any finite temperature, we see that ξ is finite, and hence there is no LRO. There is a divergence of ξ as T 0, implying critical behavior there. Usually this divergence is with a power of /t here it is an essential singularity. C. Susceptiblity Even though we didn t manage to solve for the Z with a magnetic field, we can still find the susceptibility. I ll leave it to you to show that: χ = i L h S i = LT L S i S j (85) i,j=0 and to calculate the susceptibility using the above result.