PHYSICAL REVIEW A VOLUME 38, NUMBER 11 DECEMBER 1, 1988 Critical behavior of branched polymers of even functionality near d =4 P. D. Gujrati Department of Physics, Department of Polymer Science, and Institute of Polymer Science, Uni Uersity of Akron, Akron, Ohio 44325 (Received 15 July 1988) We present an exact new model of branched polymers of even functionality. Surprisingly, our model belongs to the universality class of the O(n) model where n is the activity for loops: Branch- ings do not affect the universality class and the upper critical dimension is 4. However, in the dilute limit, the model undergoes a first-order transition. These results are contrary to current belief. I. INTRODUCTION The critical behavior of long linear polymers in solu- tion is well understood, largely as a result of the n =0 magnetic analogy. ' The magnetic system exhibits a critical behavior at zero magnetic field (H =0) and this critical behavior describes the long-range properties of a dilute solution polymer chains. Therefore the statistics of linear polymers can be studied by studying the magnet- ic analogy in the formal limit n =0. As H~0, one ob- tains a dilute limit of linear polymers in the preceding analogy. Therefore the critical behavior in the magnetic system is related to the critical behavior of a single linear chain. In other words, a single linear polymer exhibits critical properties as first noted by de Gennes. ' As we will see below, this observation is going to be a very impor- tant theme of our work, and is going to have profound impact on our thinking about the role of branching in determining physical properties. The statistics of branched polymers are much more complex than those of linear polymers because of poly- functionality. However, major progress was made by Lu- bensky and Isaacson ' who proposed a field theory to al- low for branching by introducing polyfunction potentials in terms of an nm component field. As n~0, the field theory gives rise to a collection of polymer configurations that allow for branchings (even and odd). Moreover, loops are also allowed. The theory proposed by Lubensky and Isaacson is very rich in structure, which also makes it remarkably corn- plex to study. Even the corresponding lattice version of the theory is highly intricate. By a very complicated analysis, which is not easy to follow, they arrived at vari- ous important and convincing conclusions, the most im- portant of which is the identification of the upper critical dimension d„=8 for branched polymers in good solvents. They also argued that d„=6 for branched polymers in 0 solvents. This suggests that the introduction of even a slight amount ofhranching in linear chains, which is usu- ally very common in real systems, should alter the critical behavior from the O(0) theory, for which d„=4. This is quite remarkable. The existence of d„=6 is not hard to understand, in view of the presence of a cubic term in the theory. However, the peculiarity of the n =0 limit shows that the first infrared singularities in the one-loop dia- grams appear for d =8, suggesting d„=8. Parisi and Sourlas have argued that this shift is peculiar to the n =0 limit and is due to a hidden supersymmetry. Due to this hidden supersymmetry, the critical behavior of the D-dimensional polymer problem is identified with that of a d=D — 2 — dimensional pure magnetic system. This identification again is based on the presence of the cubic term as argued by Parisi and Sourlas. It should be stated that the authors of Refs. 6 — 9 consider the dilute limit (of a single branched polymer) in identifying d„. Therefore they implicitly assume that the dilute limit of branched polymers exhibits a critical phenomenon, analogous to the situation in linear polymers. As we will demonstrate here, this is incorrect. Our aim in the present paper is to propose a new and exact model for a grand canonic-al ensemble of branched polymers, which we believe is much more transparent to understand. The model is restrictive in that we can dis- tinguish various branchings by their parity only, i.e. , by whether they are euen or odd. However, we cannot dis- tinguish various branchings of the same parity. This is due to a rich symmetry in our model (see below). The symmetry in the original model of Lubensky and Isaac- son is not as rich. We will show that our model without loops belongs to the same universality class as the linear self-avoiding walks. If loops are present, then the critical behavior is similar to that of linear polymers with rings, i.e. , similar to that of the 0 (n) model, where n is the ac- tivity for loops. In both cases, d„=4. This is not hard to understand, because of the rich symmetry of our model: It contains no odd terms (except the linear term) in field variables. To be more precise, it has the symmetry P„XZ2, where P„ is the Potts permutation symmetry and Z2 represents the Ising symmetry. Therefore branchings do not necessarily change the upper critical dimension d„. We also consider the dilute limit, where the model exhibits a first-order transition and not a con- tinuous transition. Therefore branched polymers in the dilute limit do not become critical as is usually believed. We accomplish our aim by proposing a magnetic mod- el on a lattice in the limit n =0. We find that the mag- netic field plays the role of the activity for odd vertices. Therefore, as this field vanishes, all odd vertices are suppressed and we have branched polymers with only eUen vertices. A somewhat different model for branched polymers was also considered by Lubensky et al. ' How- ever, as we will see, we do not agree with their analysis. 38 1988 The American Physical Society