Exact partition function zeros and the collapse transition of a two-dimensional lattice polymer

Universality class of the special adsorption point of two-dimensional lattice polymers

In recent work [Rodrigues et al., Phys. Rev. E 100, 022121 (2019)10.1103/PhysRevE.100.022121], evidence was found that the surface adsorption transition of interacting self-avoiding trails (ISATs) placed on the square lattice displays a nonuniversal behavior at the special adsorption point (SAP) where the collapsing polymers adsorb. In fact, different surface exponents ϕ^{(s)} and 1/δ^{(s)} were found at the SAP depending on whether the surface orientation is horizontal (HS) or diagonal (DS). Here, we revisit these systems and study other ones, through extensive Monte Carlo simulations, considering much longer trails than previous works. Importantly, we demonstrate that the different exponents observed in the reference above are due to the presence of a surface-attached-globule (SAG) phase in the DS system, which changes the multicritical nature of the SAP and is absent in the HS case. By considering a modified horizontal surface (mHS), on which the trails are forbidden from having two consecutive steps, resembling the DS situation, a stable SAG phase is found in the phase diagram, and both DS and mHS systems present similar 1/δ^{(s)} exponents at the SAP, namely, 1/δ^{(s)}≈0.44, whereas 1/δ^{(s)}≈0.34 in the HS case. Intriguingly, while ϕ^{(s)}≈1/δ^{(s)} is found for the DS and HS scenarios, as expected, in the mHS case ϕ^{(s)} is about 10% smaller than 1/δ^{(s)}. These results strongly indicate that at least two universality classes exist for the SAPs of adsorbing ISATs on the square lattice.

Partition-function-zero analysis of polymer adsorption for a continuum chain model

Polymer chains undergoing adsorption are expected to show universal critical behavior which may be investigated using partition function zeros. The focus of this work is the adsorption transition for a continuum chain, allowing for investigation of a continuous range of the attractive interaction and comparison with recent high-precision lattice model studies. The partition function (Fisher) zeros for a tangent-hard-sphere N-mer chain (monomer diameter σ) tethered to a flat wall with an attractive square-well potential (range λσ, depth ε) have been computed for chains up to N=1280 with 0.01≤λ≤2.0. In the complex-Boltzmann-factor plane these zeros are concentrated in an annular region, centered on the origin and open about the real axis. With increasing N, the leading zeros, w_{1}(N), approach the positive real axis as described by the asymptotic scaling law w_{1}(N)-y_{c}∼N^{-ϕ}, where y_{c}=e^{ε/k_{B}T_{c}} is the critical point and T_{c} is the critical temperature. In this work, we study the polymer adsorption transition by analyzing the trajectory of the leading zeros as they approach y_{c} in the complex plane. We use finite-size scaling (including corrections to scaling) to determine the critical point and the scaling exponent ϕ as well as the approach angle θ_{c}, between the real axis and the leading-zero trajectory. Variation of the interaction range λ moves the critical point, such that T_{c} decreases with λ, while the results for ϕ and θ_{c} are approximately independent of λ. Our values of ϕ=0.479(9) and θ_{c}=56.8(1.4)^{∘} are in agreement with the best lattice model results for polymer adsorption, further demonstrating the universality of these constants across both lattice and continuum models.

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.

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.

Large scale behavior of a two-dimensional model of anisotropic branched polymers

We study critical properties of anisotropic branched polymers modeled by semi-directed lattice animals on a triangular lattice. Using the exact transfer-matrix approach on strips of quite large widths and phenomenological renormalization group analysis, we obtained pretty good estimates of various critical exponents in the whole high-temperature region, including the point of collapse transition. Our numerical results suggest that this collapse transition belongs to the universality class of directed percolation.

Exact partition function zeros of the Wako-Saitô-Muñoz-Eaton β hairpin model

I compute exact partition function zeros of β hairpins, using both analytic and numerical methods, extending previous work [J. Lee, Phys. Rev. Lett. 110, 248101 (2013)] where only a restricted class of hairpins was considered. The zeros of β hairpins with an odd number of peptide bonds are computed and the difference of the distribution of zeros from those for an even number of peptide bonds is explained in terms of additional entropy of liberating the extra bond at the turn region. Upon the introduction of a hydrophobic core in the central region of the hairpin, the zeros are distributed uniformly on two concentric circles corresponding to the hydrophobic collapse and the transition to the fully folded conformation. One of the circles dissolves as the core moves toward the turn or the tip region, which is explained in terms of the similarity of the intermediate state with the folded or unfolded states. The exact partition function zeros for a hairpin with a more complex structure of native contacts, the 16 C-terminal residues of streptococcal protein G B1, are numerically computed and their loci are closely approximated by concentric circles.

Partition function zeros and phase transitions for a square-well polymer chain

The zeros of the canonical partition functions for flexible square-well polymer chains have been approximately computed for chains up to length 256 for a range of square-well diameters. We have previously shown that such chain molecules can undergo a coil-globule and globule-crystal transition as well as a direct coil-crystal transition. Here we show that each of these transitions has a well-defined signature in the complex-plane map of the partition function zeros. The freezing transitions are characterized by nearly circular rings of uniformly spaced roots, indicative of a discontinuous transition. The collapse transition is signaled by the appearance of an elliptical horseshoe segment of roots that pinches down towards the positive real axis and defines a boundary to a root-free region of the complex plane. With increasing chain length, the root density on the circular ring and in the space adjacent to the elliptical boundary increases and the leading roots move towards the positive real axis. For finite-length chains, transition temperatures can be obtained by locating the intersection of the ellipse and/or circle of roots with the positive real axis. A finite-size scaling analysis is used to obtain transition temperatures in the long-chain (thermodynamic) limit. The collapse transition is characterized by crossover and specific-heat exponents of φ≈0.76(2) and α≈0.66(2), respectively, consistent with a second-order phase transition.

Exact partition function zeros of the Wako-Saitô-Muñoz-Eaton protein model

I compute exact partition function zeros of the Wako-Saitô-Muñoz-Eaton model for various secondary structural elements and for two proteins, 1BBL and 1I6C, by using both analytic and numerical methods. Two-state and barrierless downhill folding transitions can be distinguished by a gap in the distribution of zeros at the positive real axis.

Partition function zeros of a square-lattice homopolymer with nearest- and next-nearest-neighbor interactions

We study distributions of the partition function zeros in the complex temperature plane for a square-lattice homopolymer with nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions. The dependence of distributions on the ratio of NN and NNN interaction strengths R is examined. The finite-size scaling of the zeros is performed to obtain the crossover exponent, which is shown to be independent of R within error bars, suggesting that all of these models belong to the same universality class. The transition temperatures are also computed by the zeros to obtain the phase diagram, and the results confirm that the model with stronger NNN interaction exhibits stronger effects of cooperativity.

Exact partition function zeros of a polymer on a simple cubic lattice

We study conformational transitions of a polymer on a simple-cubic lattice by calculating the zeros of the exact partition function, up to chain length 24. In the complex temperature plane, two loci of the partition function zeros are found for longer chains, suggesting the existence of both the coil-globule collapse transition and the melting-freezing transition. The locus corresponding to coil-globule transition clearly approaches the real axis as the chain length increases, and the transition temperature could be estimated by finite-size scaling. The form of the logarithmic correction to the scaling of the partition function zeros could also be obtained. The other locus does not show clear scaling behavior, but a supplementary analysis of the specific heat reveals a first-order-like pseudotransition.