The nature of the θ point for a polymer in two dimensions has long been debated, with a variety of candidates put forward for the critical exponents. This includes those derived by Duplantier and Saleur for an exactly solvable model. We use a representation of the problem via the CP^{N-1}σ model in the limit N→1 to determine the stability of this critical point. First we prove that the Duplantier-Saleur (DS) critical exponents are robust, so long as the polymer does not cross itself: They can arise in a generic lattice model and do not require fine-tuning. This resolves a longstanding theoretical question. We also address an apparent paradox: Two different lattice models, apparently both in the DS universality class, show different numbers of relevant perturbations, apparently leading to contradictory conclusions about the stability of the DS exponents. We explain this in terms of subtle differences between the two models, one of which is fine-tuned (and not strictly in the DS universality class). Next we allow the polymer to cross itself, as appropriate, e.g., to the quasi-two-dimensional case. This introduces an additional independent relevant perturbation, so we do not expect the DS exponents to apply. The exponents in the case with crossings will be those of the generic tricritical O(n) model at n=0 and different from the case without crossings. We also discuss interesting features of the operator content of the CP^{N-1} model. Simple geometrical arguments show that two operators in this field theory, with very different symmetry properties, have the same scaling dimension for any value of N (or, equivalently, any value of the loop fugacity). Also we argue that for any value of N the CP^{N-1} model has a marginal odd-parity operator that is related to the winding angle.
The nature of the θ point for a polymer in two dimensions has long been debated, with a variety of candidates put forward for the critical exponents. This includes those derived by Duplantier and Saleur for an exactly solvable model. We use a representation of the problem via the CP^{N-1}σ model in the limit N→1 to determine the stability of this critical point. First we prove that the Duplantier-Saleur (DS) critical exponents are robust, so long as the polymer does not cross itself: They can arise in a generic lattice model and do not require fine-tuning. This resolves a longstanding theoretical question. We also address an apparent paradox: Two different lattice models, apparently both in the DS universality class, show different numbers of relevant perturbations, apparently leading to contradictory conclusions about the stability of the DS exponents. We explain this in terms of subtle differences between the two models, one of which is fine-tuned (and not strictly in the DS universality class). Next we allow the polymer to cross itself, as appropriate, e.g., to the quasi-two-dimensional case. This introduces an additional independent relevant perturbation, so we do not expect the DS exponents to apply. The exponents in the case with crossings will be those of the generic tricritical O(n) model at n=0 and different from the case without crossings. We also discuss interesting features of the operator content of the CP^{N-1} model. Simple geometrical arguments show that two operators in this field theory, with very different symmetry properties, have the same scaling dimension for any value of N (or, equivalently, any value of the loop fugacity). Also we argue that for any value of N the CP^{N-1} model has a marginal odd-parity operator that is related to the winding angle.
The nature of the theta point for a polymer in two dimensions has long been debated, with a variety of candidates put forward for the critical exponents. This includes those derived by Duplantier and Saleur (DS) for an exactly solvable model. We use a representation of the problem in terms of the CPN−1 sigma model in the limit N→1 to determine the stability of this critical point. First we prove that the DS critical exponents are robust so long as the polymer does not cross itself: they can arise in a generic lattice model, and do not require fine tuning. However there is an apparent paradox: two different lattice models, apparently both in the DS universality class, show different numbers of relevant perturbations, apparently leading to contradictory conclusions about the stability of the DS exponents. We explain this in terms of subtle differences between the two models, one of which is fine-tuned. Next, we allow polymer crossings, as appropriate e.g. to the quasi-2D case. This introduces an additional independent relevant perturbation, so we do not expect the DS exponents to apply: the exponents in the case with crossings will be those of the generic tricritical O(n) model at n=0, and different to the case without crossings. We also discuss interesting features of the operator content of the CPN−1 model. Simple geometrical arguments show that two operators in this field theory, with very different symmetry properties, have the same scaling dimension for any value of N. Also we argue that for any N there is a marginal parity-odd operator which is related to the winding angle.
We study the collapse of two-dimensional polymers, via an O(n) model on the square lattice that allows for dilution, bending rigidity and short-range monomer attractions. This model contains two candidates for the theta point, ΘBN and ΘDS, both exactly solvable. The relative stability of these points, and the question of which one describes the `generic' theta point, have been the source of a long-standing debate. Moreover, the analytically predicted exponents of ΘBN have never been convincingly observed in numerical simulations. In the present paper, we shed a new light on this confusing situation. We show in particular that the continuum limit of ΘBN is an unusual conformal field theory, made in fact of a simple dense polymer decorated with {\sl non-compact degrees of freedom}. This implies in particular that the critical exponents take continuous rather than discrete values, and that corrections to scaling lead to an unusual integral form. Furthermore, discrete states may emerge from the continuum, but the latter are only normalizable---and hence observable---for appropriate values of the model's parameters. We check these findings numerically. We also probe the non-compact degrees of freedom in various ways, and establish that they are related to fluctuations of the density of monomers. Finally, we construct a field theoretic model of the vicinity of ΘBN and examine the flow along the multicritical line between ΘBN and ΘDS.
Determining whether an additional local symmetry affects the universality class of a sta-tistical model is an important issue in the theory of critical phenomena. A basic example is provided by the RP N −1 model, in which N -component spin variables at each lattice site interact through an Hamiltonian invariant under global O(N ) rotations and local spin re-versals. The local symmetry makes the difference with the usual O(N ) model and amounts to the head-tail symmetry characteristic of liquid crystals. In three dimensions, the weak first order transition observed in numerical simulations of the ferromagnetic model is con-sistent with the mean field scenario. On the other hand, in the two-dimensional case –the one we focus on in this thesis – fluctuations are stronger and minimize the reliability of mean field predictions, as illustrated by the phase transition of the three-state Potts model, which becomes continuous on planar lattices. For the RP N −1 model, the absence of spontaneous breaking of continuous symmetry in two dimensions generically suggests that criticality is limited to zero temperature, and Monte Carlo studies for T → 0 showed a fast growth of the correlation length which made particularly hard to reach the asymp-totic limit and draw conclusions. On the other hand, the possibility of finite temperature topological transitions similar to the Berezinskii-Kosterlitz-Thouless (BKT) one – which should definitely occur for RP 1 ∼ O(2) – and mediated by ”disclination” defects has also been debated in numerical studies. While two-dimensional criticality has allowed for an impressive amount of exact solutions thanks to lattice integrability and conformal field theory, models with local symmetries traditionally remained outside the range of application of these methods. In this thesis, however, we show how the renormalization group fixed points of the RP N −1 model and of its complex generalization, the CP N −1 model, can be accessed in an exact way. This is achieved in the scale invariant scattering framework which, as we review in the introductory part of the thesis, implements in the basis of particle excitations the infinite-dimensional conformal symmetry characteristic of critical points in two dimensions and has provided in the last few years new results for pure and disordered systems. In the last part of the thesis, we will exploit the generality of the scale invariant scattering method to progress with another long standing problem of two-dimensional criticality, namely that of spin clusters in Potts correlated percolation. It has been known for long time that the problem can be addressed considering two coupled Potts models, but the need to consider the number of states of one of these models as a continuous variable has severely limited the analytical or numerical study of the critical points. Also here, the ability of the scattering method to enforce conformal invariance for the internal symmetry characteristic of the universality class will allow us to obtain exact equations for the critical points and to determine their solutions in the relevant limits.
We report a numerical investigation of the tricritical behaviour of the dilute equilibrium polymerization model with annealed dilution on the square lattice using transfer matrix techniques. In general, the tricritical exponent ν t is found to be compatible with the value for the tricritical n = 1 model. In the limit where the dilute model is equivalent to the interacting polymer model (whose tricritical point is known as the Θ-point), we observe a crossover of ν t to the tricritical n → 0 model value.
The universal behavior of two-dimensional loop models can change dramatically when loops are allowed to cross. We study models with crossings both analytically and with extensive Monte Carlo simulations. Our main focus (the ``completely packed loop model with crossings'') is a simple generalization of well-known models that shows an interesting phase diagram with continuous phase transitions of a new kind. These separate the unusual ``Goldstone'' phase observed previously from phases with short loops. Using mappings to Z2 lattice gauge theory, we show that the continuum description of the model is a replica limit of the $\ensuremath{\sigma}$ model on real projective space (${\mathbb{RP}}^{n\ensuremath{-}1}$). This field theory sustains Z2 point defects, which proliferate at the transition. In addition to studying the new critical points, we characterize the universal properties of the Goldstone phase in detail, comparing renormalization group (RG) calculations with numerical data on systems of linear size up to L=106 at loop fugacity n=1. (Very large sizes are necessary because of the logarithmic form of correlation functions and other observables.) The model is relevant to polymers on the verge of collapse, and a particular point in parameter space maps to self-avoiding trails at their $\ensuremath{\Theta}$ point; we use the RG treatment of a perturbed $\ensuremath{\sigma}$ model to resolve some perplexing features in the previous literature on trails. Finally, one of the phase transitions considered here is a close analog of those in disordered electronic systems---specifically, Anderson metal-insulator transitions---and provides a simpler context in which to study the properties of these poorly understood (central-charge-zero) critical points.
We explore the phase diagram of an O (n) model on the honeycomb lattice with vacancies, using finite-size scaling and transfer-matrix methods. We make use of the loop representation of the O (n) model, so that n is not restricted to positive integers. For low activities of the vacancies, we observe critical points of the known universality class. At high activities the transition becomes first order. For n=0 the model includes an exactly known theta point, used to describe a collapsing polymer in two dimensions. When we vary n from 0 to 1, we observe a tricritical point which interpolates between the universality classes of the theta point and the Ising tricritical point.