Phase diagram of semiflexible self-avoiding trails with independent crossing and collision weights on the square lattice

The semiflexible self-avoiding trail model places a polymer on a lattice with self-avoidance imposed on edges rather than sites: each lattice edge carries at most one bond, doubly visited sites are weighted separately for crossings (weight $\tau_x$) and collisions (weight $\tau_c$), and each bend at a singly visited site carries a stiffness weight $\omega$. The exact grand-canonical Bethe-lattice solution found non-polymerized, polymerized, dense-polymerized and nematic phases, but on the loopless Bethe lattice the crossing and collision weights enter only through the combination $\tau_x+2\tau_c$, and the dense phases are totally frozen, with the collapse organized by a bicritical point. The authors' subsequent Husimi-lattice solution separated the two weights and showed that the frozen phases, the discontinuous-critical polymerized-nematic transition and the bicritical organization all change once elementary loops are present, although detailed phase diagrams were charted only along two one-parameter weight families. On regular two-dimensional lattices only one-parameter slices are understood: the flexible model with independent weights, the stiff model with equal weights, and the stiff crossings-free model. The open problem is the phase diagram of the generalized model on the square lattice with independent $\tau_x$, $\tau_c$ and $\omega$: whether nematic polymerized phases exist for sufficiently stiff chains, how phase boundaries depend separately on the crossing and collision weights, whether dense phases remain saturated, and how the collapse transition is organized (bicritical, tricritical or critical-end-point structure) with which universality classes when stiffness and crossings coexist.

Background

A polymer in dilute solution collapses from an extended coil to a dense globule when the solvent quality is lowered; the transition point is called the $\Theta$-point. The classical lattice model uses self-avoiding walks (SAW: lattice paths that visit every site at most once) with an attractive energy between monomers on nearest-neighbour sites not joined by a chain bond (the self-attracting self-avoiding walk, SASAW). In two dimensions the tricritical exponents of this collapse transition were computed exactly by Duplantier and Saleur (the DS exponents). An alternative lattice realization moves the excluded-volume constraint from sites to edges: a self-avoiding trail (SAT) visits each lattice edge at most once but may visit a site several times, and the attraction acts between monomers sharing a site. On the square lattice a doubly visited site arises in two topologically distinct ways: in a crossing the trail passes through the site twice along perpendicular straight segments, while in a collision both segments bend at the site, so the trail touches itself and the site carries two bends. Giving crossings and collisions the same Boltzmann weight yields the interacting self-avoiding trail (ISAT); forbidding crossings yields the vertex-interacting self-avoiding walk (VISAW), which is the $n\to 0$ limit of the exactly solved $O(n)$ loop model on the square lattice due to Blöte and Nienhuis (that model sums over configurations of non-intersecting loops, each loop carrying a weight $n$, so that the $n\to 0$ limit suppresses all loops and leaves a single polymer path), whose $\Theta$-exponents (the BN set) differ from the DS set. Which exponent set is generic for two-dimensional polymer collapse was debated for decades; the ISAT collapse point itself is special in that its specific heat diverges strongly. Semiflexibility is introduced by a Boltzmann weight $\omega=e^{-\beta\epsilon_b}$ for each elementary bend of the trail at a singly visited site, with $\omega<1$ corresponding to a positive bending energy $\epsilon_b$ (stiff chains); for the crossings-free semiflexible model on the square lattice, Vernier, Jacobsen and Saleur showed that the two exactly solvable $\Theta$ candidates (BN and DS) are joined by a line of multicritical points, and that the BN point is an unusual conformal field theory with non-compact degrees of freedom. Regular lattices can also host anisotropic collapsed trail phases: on the triangular lattice, the extended ISAT of Doukas, Owczarek and Prellberg has a maximally dense crystalline low-temperature phase, and the ISAT collapse point is a multicritical point where a $\Theta$-like second-order collapse line meets a first-order line. Hierarchical lattices provide exactly solvable mean-field counterparts of these models. The Bethe lattice is the core of an infinite loopless Cayley tree (coordination $q$); the Husimi lattice built with squares is a hierarchical lattice whose elementary plaquettes are squares meeting at most at corners, so elementary loops of length four are present. Both are solved by iteration of exact recursion relations for ratios of partial partition functions of rooted subtrees (partial partition functions fix the configuration of the root edge or square). Such solutions give mean-field-type critical exponents but exact phase-diagram topology, and are used as controlled approximations to regular lattices of the same coordination. Dantas, Oliveira, Stilck and Prellberg solved the general semiflexible trail model in the grand canonical ensemble (bond fugacity $z$, crossing weight $\tau_x$, collision weight $\tau_c$, bend weight $\omega$) on the four-coordinated Bethe lattice, finding four phases: a non-polymerized phase (NP, the empty lattice), a regular polymerized phase (P), a dense polymerized phase (DP, every site doubly visited, every edge occupied) and, for sufficiently stiff chains ($\omega<1/2$), a dense anisotropic nematic phase (AN, all edges of one lattice direction occupied), i.e. a phase with spontaneous bond-orientational order. The NP-P and P-DP critical surfaces meet the NP-DP coexistence surface along a bicritical line (a bicritical point being the common endpoint of two continuous-transition surfaces and a first-order coexistence surface), and the P-AN transition is simultaneously discontinuous and critical: the nematic order parameter jumps, yet the stability limits of both phases coincide with the transition line (a Fisher-Berker-type transition). Because the Bethe lattice has no closed paths, any collision can be deformed into a crossing, so $\tau_x$ and $\tau_c$ enter the solution only through the single combination $\tau=\tau_x+2\tau_c$, and the DP and AN phases are totally frozen (saturated). The flexible ISAT on Bethe and Husimi lattices had earlier been solved by Oliveira and Stilck, with the collapse identified as a bicritical point. The authors' announced follow-up on the Husimi lattice appeared as Oliveira, Dantas, Prellberg and Stilck (2018). It separates $\tau_x$ and $\tau_c$ (the recursion relations contain the combinations $\tau_c+\tau_x$, $\tau_x+2\tau_c$ and $\tau_c$), finds five phases (NP, low-density P1, high-density P2, and two nematic phases AN1 and AN2 for $\omega<1/2$), shows that the dense phases are no longer frozen (P2 is not maximally dense; AN1 is dense with $\rho_z=1/2$ but only partially ordered, its nematic order parameter being $Q=\sqrt{1-4\omega^2}/(1-2\omega^2)<1$; AN2 is not dense and interpolates between P1 and AN1), replaces the discontinuous critical P-AN transition by two continuous transitions, and replaces the bicritical point by a tricritical line (where a continuous transition line becomes first-order) together with a critical end-point line (where a critical line terminates on a coexistence surface), the organization exchanging between the crossings-allowed (sISAT) and crossings-forbidden (sVISAW) families. However, its detailed phase diagrams were charted only along two one-parameter weight families ($\tau_x=\tau^\ast$, $\tau_c=\omega^2\tau^\ast$ and $\tau_x=0$, $\tau_c=\omega^2\tau^\ast$), and it explicitly leaves the square-lattice behaviour open. On the square lattice itself, only one-parameter slices of the full three-weight space are understood: the flexible model with independent crossing and collision weights (Foster 2011; Bedini, Owczarek and Prellberg 2013), the stiff model with equal weights (Bedini, Owczarek and Prellberg 2013), and the stiff crossings-free model (Vernier, Jacobsen and Saleur 2015). Field-theoretically, Nahum (2016) showed via a $CP^{N-1}$ $\sigma$-model (a field theory of complex unit vectors whose $N\to 1$ limit encodes the polymer) that the DS exponents are stable as long as the polymer cannot cross itself, while allowing crossings adds an independent relevant perturbation and moves the flexible collapse to the generic tricritical $O(n=0)$ universality class — but the combined effect of stiffness and crossings on the lattice is untreated. The unresolved target is therefore the behaviour of the generalized model on the square lattice, where the physical question of separate crossing and collision weights, nematic phases and multicritical organization actually lives.

Problem Statement

Determine the equilibrium phase behaviour of the generalized semiflexible self-avoiding trail model on the square lattice as a function of the three independent Boltzmann weights. The model: lattice walks that visit each edge at most once (each site may be visited up to twice on a four-coordinated lattice), with grand-canonical partition function $Y=\sum z^{N}\tau_c^{N_c}\tau_x^{N_x}\omega^{N_b}$, where the sum runs over all configurations of such trails on the lattice (any number of trails, including the empty configuration), $z$ is the bond fugacity, and $N$, $N_c$, $N_x$, $N_b$ count bonds, collisions, crossings, and bends at singly visited sites ($\tau_c$ per collision, $\tau_x$ per crossing, $\omega$ per bend). The question comprises the following linked facets, which jointly make up this phase diagram: (1) Do anisotropic nematic polymerized phases — phases with spontaneous bond-orientational order, measured for instance by the order parameter $Q=|\rho_1-\rho_2|$ built from the bond densities in the two lattice directions — exist on the square lattice for sufficiently stiff chains, as both the Bethe-lattice and Husimi-lattice solutions predict for $\omega<1/2$? Do dense phases remain saturated (all edges occupied, or all edges of one direction occupied) or acquire fluctuating densities, and how do their densities and order parameters behave along the phase boundaries? (2) How do the phase boundaries depend separately on the crossing weight $\tau_x$ and the collision weight $\tau_c$, which on the loopless Bethe lattice are visible only through the combination $\tau_x+2\tau_c$? (3) How is the collapse transition organized on the square lattice — bicritical point (as on the Bethe lattice), tricritical line plus critical end-point line (as on the Husimi lattice), or another scenario — and what are the universality classes (critical exponents) of the continuous transitions when stiffness and crossings are both present? A complete answer supplies the phase diagram in the $(z,\tau_x,\tau_c,\omega)$ space (or an explicit set of canonical fixed-length or temperature-like sections covering that space), identifies the phases and their order parameters and densities, and characterizes the critical behaviour along the transition surfaces, including which transitions are continuous and which are first-order; these parts are jointly required, and no single facet or single fixed-weight section resolves the task on its own.

References

1. W. G. Dantas, Tiago J. Oliveira, Jürgen F. Stilck, Thomas Prellberg, 'Grand-canonical solution of semiflexible self-avoiding trails on the Bethe lattice', Physical Review E 95, 022132 (2017). DOI: 10.1103/PhysRevE.95.022132, https://doi.org/10.1103/PhysRevE.95.022132 (arXiv:1612.07270, https://arxiv.org/abs/1612.07270)
2. Tiago J. Oliveira, Wellington G. Dantas, Thomas Prellberg, Jürgen F. Stilck, 'Solution of semi-flexible self-avoiding trails on a Husimi lattice built with squares', Journal of Physics A: Mathematical and Theoretical 51, 054001 (2018). DOI: 10.1088/1751-8121/aa9e0b, https://doi.org/10.1088/1751-8121/aa9e0b (arXiv:1709.10449, https://arxiv.org/abs/1709.10449)
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