Lecture on Branched Polymers and Dimensional Reduction

This is a pedagogical account of the some of the recent results of Brydges and Imbrie, described from the point of view of Grassmann integration. Some simple extensions are pointed out. Historical Introduction The subject of dimensional reduction, in the context considered in this lecture, has a long and rather peculiar history. The strangest aspect of all is that the simplest formulation of the results has been found only very recently. That one physical theory in d space dimensions should be related to another in D = d−2 dimensions was first suggested in the context of the field-theoretic formulation of the critical behaviour of the Ising model in a quenched random magnetic field [1,2]. It was noticed that the most singular Feynman diagrams all had the form of trees before averaging over the quenched randomness. When this was done, it turned out that the diagrams were the same as those for the critical Ising model without any random field, in two fewer dimensions. ∗Address for correspondence 1 Thus, no further work was needed – one should be able to take, for example, the known results for the ordinary Ising model in D dimensions, and find the critical behaviour of the random field Ising model in d = D + 2 dimensions. In particular, since the lower critical dimension (at and below which there is no phase transition) of the ordinary Ising model is D = 1, this should imply that the 3-dimensional random field Ising model has no phase transition. Unfortunately this beautiful idea is wrong. In particular it contradicts a very simple argument of Imry and Ma [1] that the lower critical dimension should be two. However, it took several years of theoretical and experimental confusion and contradictory papers before Imbrie [3] proved that Imry and Ma were right. During this time Parisi and Sourlas [4] came up with a startling explanation of how and why dimensional reduction works: supersymmetry. At that time supersymmetry was thought to be the preserve of particle physics and string theory, so their argument was not widely understood. In their paper was also the germ of the reason why simple dimensional reduction might not work for the random field Ising model: the tree diagrams were seen to be the perturbative solution to the classical field equations in the presence of the random external field. If the solution of these equations was unique and given by the sum of the diagrams, then all should have been well. But it was easy to see that, at low enough temperatures, the solution was not unique, and that the one with the lowest energy was not the perturbative one. Work is still continuing on how, possibly, to rectify this situation, but this seems a very difficult problem. Meanwhile, a few years later, Parisi and Sourlas [5] came up with another ingenious application of their ideas. The problem known under various versions as branched polymers, lattice trees, or lattice animals (to be decribed later) had been attracting some attention. Lubensky and Isaacson [6] had formulated a rather complicated field theory for this problem, had come to the conclusion that the upper critical dimension (above which mean field theory is valid) is d = 8, and had noticed that the first terms in the ǫ-expansion of the critical exponents below this were the same as those of the so-called Yang-Lee edge singularity 2 in D = d − 2 dimensions. This is the problem of an Ising model in a purely imaginary magnetic field, and is described by a simple scalar field theory with a cubic interaction and a purely imaginary coupling. Parisi and Sourlas reformulated the model of Lubensky and Isaacson in a simpler way, and then used their supersymmetry to explain why the dimensional reduction happened. This time, everything was all right. The numerically measured exponents of lattice animals and lattice trees in d = 3 dimensions are given by the exactly known Yang-Lee exponents in D = 1. A non-perturbative proof of dimensional reduction in a supersymmetric field theory was given [7]. It was understood that the main reason for the failure of dimensional reduction in the random field Ising model does not apply here, essentially because the tree diagrams are the branched polymer configurations [8]. But none of these arguments were completely watertight, largely because it seemed necessary to ignore so-called irrelevant terms which spoiled the supersymmetry, but were believed not to affect the critical behaviour. In fact, the Yang-Lee critical theory is nowadays strongly believed [9] to be in the same universality class as an even simpler problem: a classical gas with short-range repulsive interactions in the grand canonical ensemble. As a function of the fugacity z > 0, the grand partition function is sum of positive terms and has no singularity nor any zeroes. But in the complex z plane the closest singularity to the origin lies on the negative axis at z = −zc, and the critical behaviour near this is believed to be independent of the particular details of the interaction (so it is known as the ‘universal repulsive gas singularity’), and it is also believed to be the same as that of the Yang-Lee problem at the critical imaginary magnetic field. Thus the branched polymer problem in d dimensions was believed to be related to the universal repulsive gas singularity in D = d−2 dimensions. But once again these arguments relied on the neglect of irrelevant terms in the appropriate field theories. It took Brydges and Imbrie [10] to realise that most of this theoretical undergrowth could be cut away that field theory is not needed at all! There is a simple and physical model for branched polymers in d dimensions for which dimensional reduction to a repulsive gas is rigorous and exact, not just in the critical region (which in this context means very large 3 polymers) but for all values of the fugacity. The arguments do use supersymmetry, in a very beautiful way, and give compelling evidence of how important it is in theoretical physics to learn how to walk (do problems with finitely many particles) before one tries to run (field theory and beyond.) The aim of these lectures is to present these ideas in as simple a manner as possible, using the ideas of Grassmann integration nowadays familiar to many theoretical physicists. I shall, however, quickly recall all the ingredients necessary for the argument. Classical gas and cluster expansion. Consider a classical gas of point particles, in some large D-dimensional domain Ω, interacting through a rotationally symmetric two-body potential V . The grand partition function is