Critical behavior of magnetic polymers on the three-dimensional Sierpinski Gasket.

We present the (numerically) exact phase diagram of a magnetic polymer on the Sierpinsky gasket embedded in three dimensions using the renormalization group method. We report distinct phases of the magnetic polymer, including paramagnetic swollen, ferromagnetic swollen, paramagnetic collapsed, and ferromagnetic collapsed states. By evaluating critical exponents associated with phase transitions, we located the phase boundaries between different phases. If the model is extended to include a four-site interaction which disfavors configurations with a single spin of a given type, we find a rich variety of critical behaviors. Notably, we uncovered a phenomenon of reentrance, where the system transitions from a collapsed (paramagnetic) state to a swollen (paramagnetic) state followed by another collapse (paramagnetic) and ultimately reaching a ferromagnetic collapsed state. These findings shed new light on the complex behavior of (lattice) magnetic polymers.

Critical behavior of magnetic polymers in two and three dimensions.

We explore the critical behavior of two- and three-dimensional lattice models of polymers in dilute solution where the monomers carry a magnetic moment which interacts ferromagnetically with near-neighbor monomers. Specifically, the model explored consists of a self-avoiding walk on a square or cubic lattice with Ising spins on the visited sites. In three dimensions we confirm and extend previous numerical work, showing clearly the first-order character of both the magnetic transition and the polymer collapse, which happen together. We present results in two dimensions, where the transition is seen to be continuous. Finite-size scaling is used to extract estimates for the critical exponents and the transition temperature in the absence of an external magnetic field.

Efimov-DNA phase diagram: Three stranded DNA on a cubic lattice.

We define a generalized model for three-stranded DNA consisting of two chains of one type and a third chain of a different type. The DNA strands are modeled by random walks on the three-dimensional cubic lattice with different interactions between two chains of the same type and two chains of different types. This model may be thought of as a classical analog of the quantum three-body problem. In the quantum situation, it is known that three identical quantum particles will form a triplet with an infinite tower of bound states at the point where any pair of particles would have zero binding energy. The phase diagram is mapped out, and the different phase transitions are examined using finite-size scaling. We look particularly at the scaling of the DNA model at the equivalent Efimov point for chains up to 10 000 steps in length. We find clear evidence of several bound states in the finite-size scaling. We compare these states with the expected Efimov behavior.

Polymer in wedge-shaped confinement: Effect on the θ temperature.

The equilibrium properties of a finite-length linear polymer chain confined in an infinite wedge composed of two perfectly reflecting hard walls meeting at a variable apex angle (α) are presented. One end of the polymer is anchored a distance y from the apex on the conical axis of symmetry, while the other end is free. We report here, the nonmonotonic behavior of θ temperature as a function of y for a finite-length chain. Data collapse for different chain lengths indicates that such behavior will exist for all finite lengths. We delineate the origin of such nonmonotonic behavior, which may have potential applications in understanding the cellular process occurring in nanoconfined geometries.

Use of the Complex Zeros of the Partition Function to Investigate the Critical Behavior of the Generalized Interacting Self-Avoiding Trail Model.

The complex zeros of the canonical (fixed walk-length) partition function are calculated for both the self-avoiding trails model and the vertex-interacting self-avoiding walk model, both in bulk and in the presence of an attractive surface. The finite-size behavior of the zeros is used to estimate the location of phase transitions: the collapse transition in the bulk and the adsorption transition in the presence of a surface. The bulk and surface cross-over exponents, ϕ and ϕ S , are estimated from the scaling behavior of the leading partition function zeros.