Phase Transitions, Finite-size Scaling and Renormalization Group

These notes were adapted for presentation from E. Stanley, Introduction to Phase Transitions and Critical Phenomena, D. Chandler, Introduction to Statistical Mechanics, and Gould and Tabachnik, Introduction to Computer Simluations.

For an introduction into a thermodynamic version of RNG, rather than real-space percolation example, please see R.J. Creswick, H.A. Farach, C.P. Poole, Jr., Introduction to Renormalization Group Methods in Physics.

Phase Transitions and Correlations

Critical points are temperatures (T), densties (), etc., above which some parameter describing, say, long-range order in a ferromagnetic, or the density change in liquid-gas transition, vanishes. For example, spontaneous Magnetization, M(T), is zero above some critical TC in a M(T) vs. T plot in a ferromagnet. The evidence for such increased correlations was manifest in the so-called critical opalescence observed in carbon dioxide over a hundered years ago by Andrews . As the critical point is approached from above, droplets of fluid acquire a size which is on the order of the wavelenght of light, hence they scatter light and provide a striking example of the critical point which can be observed with the naked eye!

Order Parameter are a quantities which are non-zero below TC and zero above it and which are found to be a common feature associated with critical points in a large variety of physical systems. For example, M(T) is the magnetic order parameter, whereas L - G is the order parameter in the case of liquid-gas transition. In an alloy, it would be the deviations of the sublattice concentrations from the average concentration, i.e. c(r) - c0, which may be measured via scattering experiments using x-ray, electron, or neutrons.

Note that while there is a correspondence between order parameters in different physical systems there is no correspondence of the topology of the individual phase diagrams. For example, for a fluid, P vs. T plot corresponds to a M vs. T in a magnetic system, but the do not look alike. In a magnetic systems H vs. T is a single line separating an "all up spin" from an "all down spin" configuration, and TC is a the point where the system obtains a non-zero M(T). Whereas, for a fluid there is a critical point at the end of the vapor pressure curve where we can continuously transform a gas to a liquid. These are not to be confused with the curves which separate solid-gas (sublimation curve), gas-liquid (vapor pressure curve), and the liquid-solid (fusion curve).

Therefore, such critical behavior can be manifest differently in any particular cut of a (P, , T) diagram or, equivalently, a (H, M, T) diagram for a magnetic system. Certainly, there are other types of global transitions, such as in percolation, which result from a cluster spanning a system and which occurs at some particular critical values of site occupation probability, pC. So such critical behavior, and it associated characteristics, are ubiquitous.

A Correlation Length, , is the distance over which a specific thermodynamic variable in the system are correlated with one another and is relevant in a system near a critical point, as evidenced from the critical opalescence experiments. In the 2-D Ising Model, for example, you can see correlations of spins over larger and larger distances as the TC is approached from above. It becomes larger than the simulation box, L, rapidly near TC. Above TC, such correlations of spins in the Ising model show short-range order (correlations over short distances), whereas below TC the system exhibits long-range order (infinitely-ranged correlations).

Generally speaking, we observed three things near a critical point, which are, in fact, interrelated. There is an

Pair Correlations are clearly important, especially since they are are directly related to thermodynamic quantities, such as KT. With KT0 = <N>kT/V the ideal gas compressibility, this is self-evident when we recall

KT/KT0 = < (N - <N>)2 >/<N> and
< (N - <N>)2 > = dr dr' G(r - r') = V dr G(r)

As T TC, KT means that G(r - r') becomes very long ranged, or G(r - r') 0 slowly as |r - r'| >> 1. Hence, the connection between the density fluctuations, compressibility and density-density correlations. In addition, it is clear that the correlation length diverges ( ) at the critical point in order for the density-density correlation to produce a divergent KT.

The increased correlation length, and associated change in thermodynamic quantities, lead to the concept of scaling in a finite-sized simulation, in order to estimate the proper critical points and critical exponents.

 

Finite-size Scaling

Reduced Coordinates are often convenient to define when discussing scaling of thermodynamic quantities. For example, reduced temperature t = (T - TC)/TC, which is a temperature relative to the critical point. You can define reduced densities. Or, in percolation problems, reduced units would be x = (p - pC)/pC. Its use in what follows should be apparent.

Experimental Evidence for Scaling -- When measured coexistence curves of numerous fluids, e.g., are plotted as T/TC vs. / C, a single curve is found (see, e.g. E. Stanley, pg. 10). A fit to this curve reveals that - C ~ (-t), and =0.33. However, it is generally found that there is a range: 0.33 < < 0.37. For example, in Helium, =0.354. Measurements for other quantities, such as susceptibilities, also lend themselves to this type of scaling analysis.

Whether talking of magnetic systems, fluids, or percolation problems, the order parameters (pair-correlations, correlation lengths, and so on) exhibit this type of scaling behavior. Here we only discuss the named quantities in the magnetic case, (i) Magnetization, M(T), (ii) the isothermal susceptibility, (T), and (iii) the correlation length, (T), and (iv) the zero-field specific heat, C(T). For percolation, they would be the (i) spanning probability, pinf, (ii) the mean cluster size, S(p), and (iii) the correlation length, (p).

The coefficients defining the power law behavior (i.e., , , , etc.) are referred to as critical exponents

 

 

Scaling with Finite System Size

Consider a simulation system with linear dimension L.

Note that for a 2-dimensional system, like the square lattice, =1. Thus, TC(L) should scale linearly with L and the final TC should be (within error bars) close to the exact 2-D square lattice in zero field, 2.269. To determine TC(L), CV(L,T) vs. T is calculated for various L = (4, 8, 16, 32, ...) and the maximum CV(L,T) is used to indicate TC(L). Recall CV(T) = (<E2> - <E>2)/kT2, which reveals the energy fluctuations of the system and the distribution of configurations over energy, i.e. the more possible configurations, the higher the specific heat.

Why not use E vs. T plots to reveal the transition temperature, instead of CV?

 

A Preliminary: Scaling Law for Homogeneous Functions

A function, f(r), is said to scale if for all values of , f( r) = g() f(r).
A function with this property is homogeneous, e.g., f(r)= Br2 f( r)= 2f(r), and g()= 2.
For a homogeneous function, if we know f(r=r0) and we know g(), then we know f(r) everywhere!

The scaling function is not arbitrary. It must be g() = P.
P is the degree of homogeneity.
See Stanley, pg 176 for "proof".

A generalized homogeneous function is given by f(a x, b y) = f(x,y).
There is no P on R.H.S. because you could always re-scale with 1/P and change a'=a/P and b'=b/P, without loss of generality.
Notice this is NOT f( x, y) = P f(x,y), as you might suspect, because functions can be scaled differently in different directions.

 

Static Scaling Hypothesis for Thermodynamic Functions

The (ad hoc) static scaling hypothesis asserts that G(t,H) is a generalized homogeneous function. Hence, G(at t, aH H) = G(t,H).

If G(t,H) is a generalized homogeneous function, then so are all other forms of Free Energy, as they are just Legendre transforms of G.

Relations amongst critical exponents may be obtained by application of thermodynamic relations, e.g., M(t,H) = -dG(t,H)/dH. Taking this derivative, we find (try it!):

aH M(at t, aH H) = M(t,H).

For zero field, M(t,0) = aH-1 M(at t, 0).

From SCALING of the thermodynamic functions, the critical exponents can be determined in terms of ratios of only 2 quantities, aH and at. Hence, by plotting the thermodynamic functions according to the derived scaling relations, both aH and at can be determined.

You can perform the same type of analysis for other Thermodynamic functions or quantities to derive the scaling equalities, which from another analysis are given as inequalities (but you must remember that the scaling hypothesis was, in fact, an ad hoc idea -- but seems to be born out).

 

Renormalization Group

Because the correlation length diverges, and various fluctuation grow (along with the associated pair correlations), as TC is approached from above, it shows that long-range fluctuations control the phase transition and the critical point. This also suggests that the phase transition (or fluctuations) is associated with the degree of connectivity of the system. Recall that there is no connectivity in 1-D and, hence, no phase transition. (You may have connectivity but in combination with frustration due, say, to geometry of lattice and competing interactions which results in no phase transition, as with the triangular lattice and AFM interactions. But this is because there is no unique ground-state, not becuase of non-connectivity.)

The results of scaling, as discussed above, is that the important fluctuations and their range can be scaled up to the appropriate lenght scale, and the local effects (which do not control the phase transition) can be ignored. Such scaling ideas were developed by Leo Kadanoff in the 60's in real space, which provided a very intuitive insight. However, these ideas were still only approximate and not complete. Nonetheless, the ideas of fixed points, Kadanoff transformations, etc. extend directly into many areas of physics, chemistry, and engineering (although perhaps not obvious at first glance).

Based on the work of Kadanoff, Ken G. Wilson in the early 70's proposed a formal mathematical approach (based in reciprocal-space (or k-space)) solvable with large-scale computing, now known as the Renormalization Group (RNG) Theory. For this work he was awarded the Nobel Prize in Physics in 1982. It has less intuitive nature than the so-called Real-space Renormalization Group ideas of Kadanoff, but carries the principals out exactly, in principle. Two good references for an overview of these ideas are in Problems in Physics with Many Scales of Length, Sci. Am. 241, 158 (1979) and Rev. Mod. Phys. 55, 583 (1983) by K.G. Wilson.

A good introduction to the concepts of RNG can be found in D. Chandler's Introduction to Modern Statistical Mechanics, pg. 139. For statistical mechanics problems, the key issues is how to re-scale the partition function such that it (i) retains the same symmetry and ground-state given by the original partition function, (ii) can re-scale the so-called coupling constant, i.e. K = J/kT in the nearest-neighbor model, and (iii) the partition function can be re-cast in terms of partially summed original partition function such that it look the same as original, with different spins and (perhaps) with different coupling constant. If this re-scale is possible, then we can develop a Recursion Relation to compute the partition function from a system with another coupling (e.g., zero, non-interacting system). An iterative solution of this Recursion Relation for the partition function yields "roots" or fixed points of the solution space. The fixed points reveal the possible critical behavior expected.

This development of the RNG ideas and their solution is straightforward, but a little long, in 1-D (see Chandler). Nonetheless, the solution is exact and show there is no critical point in 1-D. In 2-D, which is even more difficult, the model can be also managed by real-space renormalization and the fixed point is found to be KC = J/kTC = 0.50698, whereas Onsager's (or Yang's) exact solution is KC = 0.44069. The specific heat is found to be C ~ |T - TC|- where = 0.131, which is a weak singularity reminiscient of the exact solution C ~ - ln|T - TC|. Notice the results of Real-space RNG are very good, much better than mean-field theory which ignores correlations (or at least the important long-range ones).

 

Important Physics (think Ising for simplicity)

In re-scaling the partition function, there are partial summations performed of spin (in the Ising case). In integrating (or summing) out some spins means we have a Boltzmann samples fluctuations of those spins (see the scaled partition fct. in Chandler's book -- but you at least know in statisitical mechanics the partition function is the Boltzmann weighting and remains so upon re-scaling). Since each spin is indirectly coupled to the non-nearest neighbor through fluctuations, even though remaining spin are not nearest-neighbors after summation, they contain remnant interactions through the fluctuations. And, each new coupling is intrinsic to the degree of connectivity of the lattice. With no connectivity, there can be no phase transition, just a sin 1-D Ising model.

 

Concrete Example (peroclation in real-space)

Scaling of Hamiltonian and Partition Function is not as easy as for percolation in real space. Hence, percolation is best example to reveal the ideas and how a recursion relation is found, giving the fixed point solutions.

Consider an L= 16 x 16 percolation simulation lattice in 2-D. Shaded squares are occupied sites and un-shaded are unoccupied sites. Sites are occupied with probability p = 0.5. Here we will consider a percolation of the lattice (i.e. a connectivity across the lattice) from bottom to top.


For Renormalization one must consider what size of renormalization you are going to perform. Let us consider a b=2 type RNG, where a you half the size of square under consideration each RNG.
As such we must consider all 2 x 2 clusters and how they span (or occupy) in the bottom to top direction to obtain percolation. They each have a Total occupation probability given by the product of the individual site occupations.

The pertinent vertically spanning clusters that renormalizes to a single shaded square (and produced the first RNG image from L=16 to L=8) are obvious, i.e.


First cluster has probability of p4 ,
second set of clusters have 4 p3(1 - p) ,
last set of clusters have 2 p2(1 - p)2 ,
in order to span from bottom to top.
The new (renormalized) cell dimension is obvious reduced by a factor of b. For example, beginning with L=16 and with one renormalization, the new cell is L=8, which is shown in first image.

The renormalization transformation between p' and p must reflect the connectedness, i.e. formation of the spanning path which percolates from top to bottom, and we define a cell to be occupied if it spans the cell vertically. This is how the L=8 square was found. Such an RNG can be performed until we get to L=2, as the final two RNG process are shown in next image.

 

The Renormalization Group Equation

Hence, if sites are occupied with probability p, then the cells are occupied with probability p'. As a result, the RG Equation is p' = R(p). Hence, for the above vertically spanning clusters we have

p' = R(p) = p4 + 4 p3(1 - p) + 2 p2(1 - p)2.

 

The Trivial Fixed Points These two solutions of RG Equation are called Trivial Fixed Points because starting with anything above (or below) the p* percolation threshhold (yet to be found) leads to these two un-interesting values. Recall p=0 is no sites are occupied and p=1 is fully occupied. By the way, it should be clear from the last image for p = 0.5 RNG that there is no percolation because the last renormalization from L=2 to L=1 yields an empty site. What is p*?

 

The Non-Trivial Fixed Point

We want the non-trivial fixed point such that p* = R(p*). In the present case, it is a fourth-degree polynomial, as seen from above. Realize this depends on the size, b, of the renormalization. Solution of this polynomial root equation yields p* = 0 and 1 (the trivial fixed points) and p* = 0.61804 . Try it! We now associate the non-trivial fixed point, p*, with pC the critical point. The best know value for pC = 0.5927. So, with such a simple b=2 RG, we find a very good estimate (4% error) of the critical point. Similar results may be found in the thermodynamic version of RNG.

For an introduction into a thermodynamic version of RNG, where the partition function is renormalized to obtain a scaled interaction (so-called coupling constant) and the RG Equation for the partition function, see R.J. Creswick, H.A. Farach, C.P. Poole, Jr., Introduction to Renormalization Group Methods in Physics. The algebra is straightforward in the nearest-neighbor Ising case for a 1 and 2 Dimension, but grungy.

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November 8, 1999 by Duane Johnson.