Square-lattice phase diagram of semi-flexible self-avoiding trails with independent crossing and collision weights

The semi-flexible interacting self-avoiding trail (sISAT) model places a lattice polymer on a square lattice with a bond activity $z$, a bending weight $\omega$, and separate statistical weights $\tau_x$ and $\tau_c$ for crossings and collisions of chain segments at doubly visited sites. Its exact grand-canonical solution on the four-coordinated Bethe lattice found non-polymerized, polymerized, maximally dense polymerized, and anisotropic nematic phases, with the two doubly-visited-site weights entering only through the combination $\tau = \tau_x + 2\tau_c$, totally frozen dense phases, and an unusual Fisher-Berker-type discontinuous critical transition ($\beta = 0$, $\gamma = 1$) between the polymerized and nematic phases. The follow-up solution on the Husimi lattice built with squares lifted all of these tree-topology artifacts - separating $\tau_x$ from $\tau_c$, unfreezing the dense phases, and replacing the discontinuous-critical transition by two nearby continuous transitions - and conjectured that on the square lattice a single nematic phase with continuously varying densities appears inside the polymerized phase with a continuous transition. Whether this is what happens in two dimensions is unknown: existing square-lattice studies cover only restricted sub-families (equal crossing and collision weights in canonical single-chain simulations, which find no nematic phase; crossing-free trails with nearest-neighbour attraction, which do show an anisotropic $\beta$-sheet phase). The square-lattice phase diagram of sISAT with independent $\tau_x$ and $\tau_c$, the nature of its polymerized-nematic sector, and the persistence of the opposite ISAT-versus-VISAW stiffness dependence of the collapse all remain open.

Background

A linear polymer in dilute solution can be modeled on a lattice by a self-avoiding walk (SAW), a path visiting each lattice site at most once. An alternative that allows the chain to be locally twice as dense is the self-avoiding trail (SAT), a path that uses each lattice edge (bond) at most once; on the square lattice, whose coordination number is four, a site can then be visited twice, and the two chain segments meeting there either collide (the two bonds at the site are perpendicular, so two monomers share the site with bends) or cross (both segments continue straight through). Interacting versions weight each doubly visited site by an attraction, driving a collapse transition from an extended coil to a dense phase; the collapse of interacting self-avoiding trails (ISAT) in two dimensions has a universality behavior of its own, studied numerically by Foster (2009), and is in general distinct from the theta-point collapse of the interacting self-avoiding walk (ISAW). If crossings are forbidden ($\tau_x = 0$) the model becomes the vertex-interacting self-avoiding walk (VISAW), which is integrable in two dimensions, and whose semi-flexible version on the square lattice was analyzed by Vernier, Jacobsen and Saleur. Semi-flexibility is included by a weight $\omega = \exp(-\beta\epsilon_b)$ per elementary bend of the chain, so that $\omega < 1$ favors stiff chains.

Dantas, Oliveira, Stilck and Prellberg defined the general semi-flexible interacting self-avoiding trail (sISAT) model, with grand-canonical partition function

$$Y = \sum z^{N}\, \tau_c^{N_c}\, \tau_x^{N_x}\, \omega^{N_b},$$

where $N$ is the number of bonds, $N_c$ the number of collisions, $N_x$ the number of crossings and $N_b$ the number of bends at singly visited sites, and solved it exactly on the four-coordinated Bethe lattice, an infinite treelike lattice without closed loops whose solution is the Bethe approximation to the square lattice. Four phases appear: a non-polymerized phase NP (vanishing bond density), a polymerized phase P (finite density of bonds in both lattice directions), a maximally dense phase DP (all lattice edges occupied, so every site is doubly visited), and an anisotropic nematic phase AN in which all edges of one lattice direction are occupied and every site is visited once - orientational order of stiff chains. Two features are artifacts of the tree topology: the crossing and collision weights enter only through the combination $\tau = \tau_x + 2\tau_c$, and the DP and AN phases are totally frozen, with no local fluctuations. The polymerized-nematic transition is a discontinuous critical transition of the type analyzed by Fisher and Berker - simultaneously first-order-like, with a discontinuous nematic order parameter ($\beta = 0$), and critical, with a diverging susceptibility ($\gamma = 1$). In the rigid-rod limit $\omega = 0$ the model is also solved exactly on the square lattice by transfer matrix, with the same phase diagram as on the Bethe lattice.

The announced follow-up - a solution on the Husimi lattice built with squares, a hierarchical lattice obtained by decorating the sites of a Bethe lattice with squares so that elementary closed loops of length four are present - was carried out by Oliveira, Dantas, Prellberg and Stilck. On the Husimi lattice the model has five phases (NP, a low-density polymerized phase P1, a high-density polymerized phase P2, and two nematic phases AN1 and AN2); $\tau_x$ and $\tau_c$ now act separately, P2 is no longer maximally dense, AN1 is no longer totally ordered, AN2 has continuously varying densities, the Bethe bicritical point becomes a tricritical point joined to a critical end point, and the discontinuous-critical P-AN transition is replaced by two nearby continuous transitions bordering AN2. Since the Husimi lattice is a better approximation to regular lattices than the Bethe lattice, these authors conjectured that on the square lattice a single nematic phase with properties similar to AN2 appears inside P1, with a continuous transition between them - but this remains a conjecture.

What is actually known in two dimensions covers only sub-families of the weight space. Bedini, Owczarek and Prellberg simulated the canonical single-chain model with equal crossing and collision weights (the ISAT line) up to length $n = 512$ (and $n = 1024$ at fixed stiffness) with flatPERM, a chain-growth Monte Carlo algorithm that samples the full space of lengths and energies with a flat histogram; they found a single maximally dense low-temperature phase for all finite stiffness, no nematic phase, and an ISAT-like collapse, with bimodal energy distributions at large stiffness that they attribute to a crossover toward a first-order transition at infinite stiffness. For crossing-free trails (called grooves) with nearest-neighbour attraction and stiffness on the square lattice, the same authors found three low-temperature phases, including an anisotropic $\beta$-sheet phase of aligned straight segments - so orientational ordering does occur in two dimensions within that sub-family. The semi-flexible VISAW phase diagram on the square lattice (Vernier, Jacobsen and Saleur) is, according to the Husimi-lattice authors, closer to the Bethe than to the Husimi result. No published study treats the square-lattice sISAT model with independent crossing and collision weights.

Problem Statement

Determine the equilibrium phase behavior of the semi-flexible interacting self-avoiding trail (sISAT) model on the square lattice - the finite-dimensional counterpart of the exactly solved Bethe- and Husimi-lattice models - with independent crossing weight $\tau_x$, collision weight $\tau_c$, and bending weight $\omega$ (grand-canonical bond activity $z$, or the equivalent canonical single-chain formulation). The single overarching objective is a square-lattice phase diagram with identified phases, transition loci, orders, and critical exponents, which must in particular settle:

(a) whether anisotropic nematic phases appear inside the polymerized phase in two dimensions - testing the Husimi-lattice conjecture of a single nematic phase with continuously varying densities and a continuous polymerized-nematic transition - or whether the negative result of the equal-weight canonical single-chain simulations is generic;

(b) whether the polymerized-nematic transition, if present, is a Fisher-Berker-type discontinuous critical transition (as on the Bethe lattice, with $\beta = 0$, $\gamma = 1$) or is replaced by continuous transition(s) (as on the Husimi lattice);

(c) whether $\tau_x$ and $\tau_c$ have distinguishable effects on the two-dimensional phase topology and criticality, as they do on the Husimi lattice, or enter only in the combination $\tau = \tau_x + 2\tau_c$ as they do on the Bethe lattice;

(d) whether the maximally dense and nematic phases remain frozen, become partially ordered, or acquire continuously varying densities in two dimensions, and whether the opposite stiffness dependence of the collapse found on the Bethe lattice for the crossing-allowed (ISAT-like) and crossing-free (VISAW-like) sub-families persists in two dimensions.

A complete answer may be an exact solution, a transfer-matrix finite-size-scaling study, or a simulation study with finite-size-scaling analysis, provided it covers both the crossing-allowed and crossing-free sectors as well as the region where $\tau_x$ and $\tau_c$ differ, and is consistent with the exactly known limits: the $\omega = 0$ rod-limit solution on the square lattice, the classical ISAT line, and the VISAW line $\tau_x = 0$.

References

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