In the previous chapter, the ferromagnetic Ising model provided a simple example of a phase diagram with an associated fixed point structure of the renormalization group flows. There were stable fixed points corresponding to low and high temperature phases, and a critical fixed point controlling the behaviour of critical Ising Systems. However, more realistic systems often have more complicated phase diagrams, and therefore a richer fixed point structure. In this chapter we study some of these examples, and show how, even with a rather qualitative description of renormalization group flows, it is possible to understand phase diagrams from the renormalization group viewpoint. More importantly, when more than one non-trivial fixed point is present, the question arises as to which is the dominant one in a particular region of the phase space. The renormalization group answers this question through the theory of cross-over behaviour. The existence of such phenomena, whereby different fixed points may influence the properties of the same system on different length scales, is totally absent in mean field treatments.

Ising model with vacancies

As a first example, consider a generalisation of the Ising model in which the spin variables s(r) may take the value 0 as well as ±1. This may be viewed as the classical version of a quantum spin-1 magnet, or as a lattice gas of magnetic particles, with |s(r)| playing the role of the occupation number.

We saw in Chapter 3 that the hamiltonian for a System at a critical point flows under the renormalization group into a critical fixed point. Under a renormalization group transformation, the microscopic length scale is rescaled by a constant factor b, and so the coordinates of a given point, as measured in units of this length scale, transform according to r → b−1r. This is called a scale transformation. Once the flows reach such a fixed point, the parameters of the hamiltonian no longer change, and it is said to be scale invariant. As well as being scale invariant, the fixed point hamiltonian usually possesses other spatial symmetries. For example, if the underlying model is defined on a lattice, so that its hamiltonian is invariant under lattice translations, the corresponding critical fixed point hamiltonian is generally invariant under arbitrary uniform translations. This is because terms which might be added to the hamiltonian which break the symmetry under continuous translations down to its subgroup of lattice translations are irrelevant at such a fixed point. Similarly, if the lattice model is invariant under a sufficiently large subgroup of the rotation group (for example, if the interactions in the x and y directions on a Square lattice are equal), then the fixed point hamiltonian enjoys full rotational invariance. As discussed on p.58, even if the interactions are anisotropic, rotational invariance may often be recovered by a suitable finite relative rescaling of the coordinates.

One of the most striking aspects of critical behaviour is that of the crucial role played by the geometry of the System. Critical exponents depend in a non-trivial manner on the dimensionality d. The very existence of a phase transition depends on the way in which the infinite volume limit is taken, as discussed in Section 4.4. This happens because the critical fluctuations, which determine the universal properties, occur at long wavelengths and are therefore very sensitive to the large scale geometry. By contrast, the non-critical properties of a system are sensitive to fluctuations on the scale of the correlation length and are therefore much less influenced. This line of reasoning also suggests that not all points in a system are equivalent in the way the local degrees of freedom couple to these critical fluctuations. So far, we have considered the behaviour of scaling operators only at points deep inside the bulk of a system. Near a boundary, however, the local environment of a given degree of freedom is different, and we might expect to find different critical properties there. In general, such differences should extend into the bulk only over distances of the order of the bulk correlation length. However, at a continuous bulk phase transition, this distance diverges, and we should expect the influence of boundaries to be more pronounced.

The simplest modification of the bulk geometry to consider is that of a (d − 1)-dimensional hyperplane bounding a semi-infinite d-dimensional system.

Scaling concepts play a central role in the analysis of the ever more complex systems which nowadays are the focus of much attention in the physical sciences. Whether these problems relate to the very large scale structure of the universe, to the complicated forms of everyday macroscopic objects, or to the behaviour of the interactions between the fundamental constituents of matter at very short distances, they all have the common feature of possessing large numbers of degrees of freedom which interact with each other in a complicated and highly non-linear fashion, often according to laws which are only poorly understood. Yet it is often possible to make progress in understanding such problems by isolating a few relevant variables which characterise the behaviour of these systems on a particular length or time scale, and postulating simple scaling relations between them. These may serve to unify sets of experimental and numerical data taken under widely differing conditions, a phenomenon called universality. When there is only a single independent variable, these relations often take the form of power laws, with exponents which do not appear to be simple rational numbers yet are, once again, universal.

The existence of such scaling behaviour may often be explained through a framework of theoretical ideas loosely grouped under the term renormalization. Roughly speaking, this describes how the parameters specifying the system must be adjusted, under putative changes of the underlying dynamics, in such a way as not to modify the measurable properties on the length or time scales of interest.

Before embarking on an exploration of the modern theories of critical behaviour, it is wise to consider first the more traditional approaches, generally lumped together under the heading of mean field theory. Despite the fact that such methods generally fail sufficiently close to the critical point, there are nonetheless several good reasons for their study. Mean field theory is relatively simple to apply in most cases, and often gives a qualitatively correct picture of the phase diagram of a given model. In some cases, where fluctuation effects are suppressed for some physical reason, its predictions are also quantitatively accurate. In a sufficiently large number of spatial dimensions, it gives exact values for the various critical exponents. Moreover, it often serves as an important adjunct to the renormalization group, since the latter by itself may give direct information about the existence and location of phase transitions, but not about the nature of the phases which they separate. For this, further input is necessary, and this often may be provided by applying mean field theory in a region of the phase diagram far from the critical region, where it is applicable.

The mean field free energy

There are as many derivations of the basic equations of mean field theory as there are books written on the subject, all of varying degrees of rigour and completeness.

Take a large piece of material and measure some of its macroscopic properties, for example its density, compressibility or magnetisation. Now divide it into two roughly equal halves, keeping the external variables like pressure and temperature the same. The macroscopic properties of each piece will then be the same as those of the whole. The same holds true if the process is repeated. But eventually, after many iterations, something different must happen, because we know that matter is made up of atoms and molecules whose individual properties are quite different from those of the matter which they constitute. The length scale at which the overall properties of the pieces begin to differ markedly from those of the original gives a measure of what is termed the correlation length of the material. It is the distance over which the fluctuations of the microscopic degrees of freedom (the positions of the atoms and suchlike) are significantly correlated with each other. The fluctuations in two parts of the material much further apart than the correlation length are effectively disconnected from each other. Therefore it makes no appreciable difference to the macroscopic properties if the connection is completely severed.

Usually the correlation length is of the order of a few inter-atomic spacings. This means that we may consider really quite small collections of atoms to get a very good idea of the macroscopic behaviour of the material.

Quenched and annealed disorder

The systems discussed so far have been assumed to be homogeneous. Any real system will inevitably contain impurities. In most circumstances, one tries to eliminate them, but, under well controlled conditions, it is also interesting to study their effect on critical behaviour. In general, one would expect any kind of random in homogeneities to tend to disorder the system, and thus to lower the critical temperature. In fact, under certain circumstances, randomness may completely eliminate the ordered phase. Under other conditions, it is still possible for the system to order, but the universality class of the critical behaviour may be modified.

The first point to be made concerns the important distinction between annealed and quenched disorder. As a concrete example, suppose that we substitute some non-magnetic impurity atoms into a lattice of magnetic ions. The way we might do this is to mix some fraction of impurities into the melt, and let the system crystallise by cooling. If we allow this to happen very slowly, the impurities and the magnetic ions will remain in thermal equilibrium with each other, and the resulting distribution of impurities will be Gibbsian, governed by the final temperature and the various interactions between the different kinds of atom. Such a distribution of impurities is called annealed.

In this chapter the basic concepts of the modern approach to equilibrium critical behaviour, conventionally grouped under the title ‘renormalization group’, are introduced. This terminology is rather unfortunate. The mathematical structure of the procedure, in the sense that it may be said to have any rigorous underpinnings, is certainly not that of a group. Neither is renormalization in quantum field theory an essential element, although it has an intimate connection with some formulations of the renormalization group. In fact, the renormalization group framework may be applied to problems quite unrelated to field theory. The origins of the name may be traced to the particle physics of the 1960s, when it was optimistically hoped that everything in fundamental physics might be explained in terms of symmetries and group theory, rather than dynamics. One of the earliest applications of renormalization group ideas, in fact, was to the rather esoteric subject of the high energy behaviour of renormalized quantum electrodynamics. It took the vision of K. Wilson to realise that these methods had a far wider field of application in the scaling theory of critical phenomena that was being formulated by Fisher, Kadanoff and others in the latter part of the decade. By then, however, the name had become firmly attached to the subject.

Not only are the words ‘renormalization’ and ‘group’ examples of unfortunate terminology, the use of the definite article ‘the’ which usually precedes them is even more confusing.

Bounded and unbounded potentials

A molecular system is one in which the atoms are bound together by chemical bonds between specific pairs of atoms. To study the stability of molecular systems we have to study the stability of these individual chemical bonds. Often the bonds are not represented as a complex of electron orbitals but, rather, represented as attractive potentials between pairs of atoms that are functions only of interatom distance. All realistic models of these potentials are bounded. That is, if the two atoms involved are separated to large enough distances the interaction energy goes to zero, or depending on where one puts the origin in potential, to some finite value. Atoms are assumed to become independent of one another when separated by a large enough distance. In particular a bounded interaction potential does not go to infinity as the separation increases; if that happened atoms would be bound together the way quarks are and could never be found separated.

If one applies standard statistical mechanics to a molecule bound by such bounded potentials one always gets the result that the equilibrium state of the system is the dissociated state. In other words, the molecule is always melted. This is true at all temperatures, even infinitesimal temperatures, and is true for both classical and quantum analysis. In studying melting by statistical mechanics of systems with realistic potentials one is dealing with the contradictory situation of trying to study the melting of a molecule that is already melted.

Dynamics of conformation change

Biological macromolecules often undergo changes in shape carrying out their biological function. For example the action of muscle fibers involves a relative rotation of a section of a macromolecule. An important change in shape in DNA is the B to A conformation change that is caused by changes in the level of hydration of the helix. The A conformation is the low hydration form. In that transition no valence bonds, and likely no H-bonds, are even transiently disrupted. The change is mostly to the helicity of the helix, going from a complete turn in 10 base pairs to one in 11 base pairs. In the process the width of the helix increases, the length decreases, and the base pairs tilt away from the perpendicular to the helix axis (Saenger, 1984). Analysis of this transition has been carried out in the context of a soft mode displacive change transition (Eyster and Prohofsky, 1977; Lindsay et al., 1985). Displacive change is useful for transitions that involve continuous movement of atoms from one conformation to another, i.e. where the transition is second order. The mode that goes soft, in soft mode analysis, is related to the softening of the free energy barrier that separates the free energy minima as a function of order parameter, as discussed in a previous chapter.

Choice of methods

As discussed in earlier chapters, several advantages point to the use of an effective phonon approach to study dynamics in a large nonlinear system such as the DNA double helix. The first advantage is that in the effective phonon formulation unbounded interactions replace bounded interactions and this change allows one to apply statistical mechanical ensemble theory to the system without running into problems associated with infinite thermodynamic integrals. The second advantage is that the selfconsistent phonon approach divides the large nonlinear dynamics problem into two parts, 1) the incorporation of nonlinearities into an effective force constant, and 2) the solution of the large number of degrees of freedom by normal mode means. A third implicit advantage is related to the built in harmonic approximation, i.e. the advantage of not having to solve for the ground state conformation before being able to proceed to the dynamic solutions. All these theoretical and calculational advantages are present in the approach developed.

In addition to the theoretical advantages discussed above there is a very practical reason to formulate the dynamics as an effective phonon problem. Direct experimental observation of the vibrational excitations of double helices by Raman scattering and infrared absorption indicate that they are resonant rather than relaxational. The lines are broad but not so broad as to indicate that one is observing relaxational modes.

Biologically significant DNA

Most biologically significant DNA helices are very long, so long that it is a useful approximation to assume infinite length. Helical lattice dynamics can then be used for analysis without having to resort to large dimensional calculations if the systems have a repeating base sequence symmetry. The studies in earlier chapters dealt only with DNA which had repeating symmetry. One can learn much about the dynamics of native DNA from a study of repeating DNA because the polymer DNAs share much of the dynamics, and often bracket the behavior of native DNAs. For example, the melting temperature of native DNA falls between that of poly(dG)– poly(dC) and poly (dA–dT)–poly(dA–dT). The study of repeating DNA is, however, a study of the material science of the material DNA. It deals with ‘perfect DNA’ and not with the complex broken symmetry material of biological significance.

The departures from symmetry are of great importance as biological information is contained in them. A biologically oriented study then requires methods that can deal with departures from repeating symmetry but can still be applied to very long DNA. In this chapter we develop methods useful with symmetry breaking structures. We get around the difficulties of dealing with large or infinite systems by starting with initial repeat polymer solutions and applying the new methods to achieve appropriate solutions. The approach was initially developed to deal with defects in crystals, the mathematical method is a Green function approach that is detailed in Appendix 3.

Greater helix

The study of the dynamics of a helix with elements, such as a drug, attached to it is easiest when there is an excess of the element so that it forms a repeating pattern of attachment. This allows the use of lattice methods to solve the large dimensional problem associated with the long helix. When only one element (or a few elements) attaches to a long helix the study is best handled by methods developed in Chapters 11 and 13. One of the more extensively studied attaching drugs is daunomycin, probably because it is an important antitumor agent. It is one of a number of drugs that intercalate into the helix, i.e. part of it has a planar structure that enters between base pairs of the helix. The specific example studied is the repeating unit of daunomycin–poly(GCAT)–poly(ATGC) (Chen and Prohofsky, 1994). The calculation determines the probability of the drug dissociating from the DNA helix which can be converted to give the binding constant of the drug to the helix. The choice of the particular DNA sequence was dictated by the availability of X-ray conformational information (Wang et al., 1987). The helix is somewhat distorted by the intercalation and the calculation uses the distorted conformation and the correct daunomycin position. The bases are numbered in the unit cell by calling the first guanine G1 then proceeding down one strand in the 3′ to 5′ direction so that the next cytosine is C2, then comes A3 and T4.