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What is the Generic Critical Behaviour for Polymer Collapse in 2D

Adam Nahum

Published Oct 30, 2015• 0 citations
OpenAlex

Abstract

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 $CP^{N-1}$ sigma model in the limit $N \rightarrow 1$ 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 $CP^{N-1}$ 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 $N$. Also we argue that for any $N$ there is a marginal parity-odd operator which is related to the winding angle.

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arXiv:cond-mat/0104033•Physical review. B, Condensed matter•Dec 13, 2001•3 citations
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−1CP^{N-1}CPN−1 sigma model in the limit N→1N \rightarrow 1N→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)O(n)O(n) model at n=0n=0n=0, and different to the case without crossings. We also discuss interesting features of the operator content of the CPN−1CP^{N-1}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 NNN. Also we argue that for any NNN there is a marginal parity-odd operator which 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.
We study the collapse of two-dimensional polymers, via an O(nnn) 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, ΘBNΘ_{\rm BN}ΘBN​ and ΘDSΘ_{\rm DS}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 ΘBNΘ_{\rm BN}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 ΘBNΘ_{\rm BN}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 ΘBNΘ_{\rm BN}ΘBN​ and examine the flow along the multicritical line between ΘBNΘ_{\rm BN}ΘBN​ and ΘDSΘ_{\rm DS}ΘDS​.
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.
A review is given of recent work on the ordinary surface critical behaviour of systems in two dimensions. Several models of interest in statistical mechanics are considered: Potts model, percolation, Ising clusters, ZN-model, O(n) model and polymers. Numerical results for surface exponents, obtained by suitable finite size scaling extrapolations, are discussed in the light of recent advances based on the conformal invariance approach. Surface exponents are often seen as important tests of conformal invariance predictions. In other cases these exponents provide important information for a location of the problem within the classification schemes offered by the conformal approach, and a determination of its universality class. A relevant example of the first aspect is the study of the q-state Potts model with q near 4, for which an analytical study of logarithmic scaling corrections is needed to achieve a successful test. The latter point of view applies, e.g., to the more controversial cases of polymers at the theta point and critical Ising clusters. Emphasis is put on the importance of an integrated study of both bulk and surface properties. Relevant issues, like the possible existence of analytical expressions for the indices in particular model families, or of general relationships between bulk and surface exponents, are critically discussed. The new problem of critical behaviour at fractal boundaries is also considered for random (RW) and self-avoiding walks (SAW). From the numerical analysis of this problem remarkable universalities of the surface exponents seem to emerge, which, in the case of SAW’s, are still far from being understood.
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.
A mapping is developed between the quantum Hall plateau transition and two-dimensional self-interacting lattice polymers. This mapping is exact in the classical percolation limit of the plateau transition, and diffusive behavior at the critical energy is shown to be related to the critical exponents of a class of chiral polymers at the \ensuremath{\theta} point. The exact critical exponents of the chiral polymer model on the honeycomb lattice are found, verifying that this model is in the same universality class as a previously solved model of polymers on the Manhattan lattice. The mapping is obtained by averaging analytically over the local random potentials in a previously studied lattice model for the classical plateau transition. This average generates a weight on chiral polymers associated with the classical localization length exponent $\ensuremath{\nu}=\frac{4}{3}.$ We discuss the differences between the classical and quantum transitions in the context of polymer models and use numerical results on higher-moment scaling laws at the quantum transition to constrain possible polymer descriptions. Some properties of the polymer models are verified by transfer matrix and Monte Carlo studies.