Husimi Lattice Research Articles

Exact calculation of the elastic antiferromagnetic Ising model under static external force on Husimi lattice

We investigate an elastic Ising model with a modified Lennard-Jones potential on the Husimi recursive lattice, where the model can be solved exactly. An external force field to either stretch and press the lattice is applied to approach the elasticity in Ising model in a static way. As expected, both stretching and compressing induce less stability with lower T c, nevertheless due to the asymmetric feature of the LJ well, trivial but realistic destabilization is observed in the stretch case implying the drawing crystallization, and interesting phenomenon of supercooled state and Kauzmann Paradox addressed by zero entropy at finite temperature is observed in the compression case on the left side of the LJ potential well.

Entropy of fully packed rigid rods on generalized Husimi trees: A route to the square-lattice limit.

Although hard rigid rods (k-mers) defined on the square lattice have been widely studied in the literature, their entropy per site, s(k), in the full-packing limit is only known exactly for dimers (k=2) and numerically for trimers (k=3). Here, we investigate this entropy for rods with k≤7, by defining and solving them on Husimi lattices built with diagonal and regular square-lattice clusters of effective lateral size L, where L defines the level of approximation to the square lattice. Due to an L-parity effect, by increasing L we obtain two systematic sequences of values for the entropies s_{L}(k) for each type of cluster, whose extrapolations to L→∞ provide estimates of these entropies for the square lattice. For dimers, our estimates for s(2) differ from the exact result by only 0.03%, while that for s(3) differs from best available estimates by 3%. In this paper, we also obtain a new estimate for s(4). For larger k, we find that the extrapolated results from the Husimi tree calculations do not lie between the lower and upper bounds established in the literature for s(k). In fact, we observe that, to obtain reliable estimates for these entropies, we should deal with levels L that increase with k. However, it is very challenging computationally to advance to solve the problem for large values of L and for large rods. In addition, the exact calculations on the generalized Husimi trees provide strong evidence for the fully packed phase to be disordered for k≥4, in contrast to the results for the Bethe lattice wherein it is nematic.

Phase diagram and critical properties of a two-dimensional associating lattice gas.

We revisit the associating lattice gas (ALG) introduced by Henriques etal. [Phys. Rev. E 71, 031504 (2005)PLEEE81539-375510.1103/PhysRevE.71.031504] in its symmetric version. In this model, defined on the triangular lattice, interaction between molecules occupying nearest-neighbor sites depends on their relative orientation, mimicking the formation of hydrogen bonds in network-forming fluids. Although all previous studies of this model agree that it has a disordered fluid (DF), a low-density liquid (LDL), and a high-density liquid (HDL) phase, quite different forms have been reported for its phase diagram. Here, we present a thorough investigation of its phase behavior using both transfer matrix calculations and Monte Carlo (MC) simulations, along with finite-size scaling extrapolations. Results in striking agreement are found using these methods. The critical point associated with the DF-HDL transition at full occupancy, identified by Furlan et al. [Phys. Rev. E 100, 022109 (2019)2470-004510.1103/PhysRevE.100.022109] is shown to be one terminus of a critical line separating these phases. In opposition to previous simulation studies, we find that the transition between the DF and LDL phases is always discontinuous, similar to the LDL-HDL transition. The associated coexistence lines meet at the point where the DF-HDL critical line ends, making it a critical-end-point. Overall, the form of the phase diagram observed in our simulations is very similar to that found in the exact solution of the model on a Husimi lattice. Our results confirm that, despite the existence of some waterlike anomalies in this model, it is unable to reproduce key features of the phase behavior of liquid water.

Binary lattice-gases of particles with soft exclusion: exact phase diagrams for tree-like lattices

We study equilibrium properties of binary lattice-gases comprising A and B particles, which undergo continuous exchanges with their respective reservoirs, maintained at chemical potentials μ A = μ B = μ. The particles interact via on-site hard-core exclusion and also between the nearest-neighbours: there are a soft repulsion between AB pairs and also interactions of arbitrary strength J, positive or negative, for AA and BB pairs. For tree-like Bethe and Husimi lattices we determine the full phase diagram of such a ternary mixture of particles and voids. We show that for J being above a lattice-dependent threshold value, the critical behaviour is similar: the system undergoes a transition at μ = μ c from a phase with equal mean densities of species into a phase with a spontaneously broken symmetry, in which the mean densities are no longer equal. Depending on the value of J, this transition can be either continuous or of the first order. For sufficiently large negative J, the behaviour on the two lattices becomes markedly different: on the Bethe lattice there exist two separate phases with different kinds of structural order, which are absent on the Husimi lattice, due to stronger frustration effects.

A Comparison of Thermal and Magnetic Behaviors in Triangular, Square and Pentagonal Husimi-Like Structures: Monte Carlo Simulations

In this work, Monte Carlo simulations have been performed to investigate the magnetic properties and thermal behavior of different Husimi-like structures: triangular, square and pentagonal. In fact, the Blume-Capel model has been adopted to study such Husimi lattices in the framework of the Ising model spin-1. Furthermore, we have examined the computed blocking temperatures. Also, the behavior of each Husimi geometry has been studied and discussed. Besides, the effects of the magnetization versus the crystal field, the exchange coupling interaction and the external magnetic field have been investigated. Such comparison of triangular, square and pentagonal Husimi-like structures allowed us to deduce interesting properties related to the geometry of each structure.

Husimi-lattice solutions and the coherent-anomaly-method analysis for hard-square lattice gases.

Although lattice gases composed of particles preventing up to their kth nearest neighbors from being occupied (the kNN models) have been widely investigated in the literature, the location and the universality class of the fluid-columnar transition in the 2NN model on the square lattice are still a topic of debate. Here, we present grand-canonical solutions of this model on Husimi lattices built with diagonal square lattices, with 2L(L+1) sites, for L⩽7. The systematic sequence of mean-field solutions confirms the existence of a continuous transition in this system, and extrapolations of the critical chemical potential μ_{2,c}(L) and particle density ρ_{2,c}(L) to L→∞ yield estimates of these quantities in close agreement with previous results for the 2NN model on the square lattice. To confirm the reliability of this approach, we employ it also for the 1NN model, where very accurate estimates for the critical parameters μ_{1,c} and ρ_{1,c}-for the fluid-solid transition in this model on the square lattice-are found from extrapolations of data for L⩽6. The nonclassical critical exponents for these transitions are investigated through the coherent anomaly method (CAM), which in the 1NN case yields β and ν differing by at most 6% from the expected Ising exponents. For the 2NN model, the CAM analysis is somewhat inconclusive, because the exponents sensibly depend on the value of μ_{2,c} used to calculate them. Notwithstanding, our results suggest that β and ν are considerably larger than the Ashkin-Teller exponents reported in numerical studies of the 2NN system.

Nature of intermediate magnetization plateaus of a spin-1/2 Ising-Heisenberg model on a triangulated Husimi lattice resembling a triangulated kagome lattice.

The spin-1/2 Ising-Heisenberg model on a triangulated Husimi lattice is exactly solved in a magnetic field within the framework of the generalized star-triangle transformation and the method of exact recursion relations. The generalized star-triangle transformation establishes an exact mapping correspondence with the effective spin-1/2 Ising model on a triangular Husimi lattice with a temperature-dependent field, pair and triplet interactions, which is subsequently rigorously treated by making use of exact recursion relations. The ground-state phase diagram of a spin-1/2 Ising-Heisenberg model on a triangulated Husimi lattice, which bears a close resemblance with a triangulated kagomé lattice, involves, in total, two classical and three quantum ground states manifested in respective low-temperature magnetization curves as intermediate plateaus at 1/9, 1/3, and 5/9 of the saturation magnetization. It is verified that the fractional magnetization plateaus of quantum nature have character of either dimerized or trimerized ground states. A low-temperature magnetization curve of the spin-1/2 Ising-Heisenberg model on a triangulated Husimi lattice resembling a triangulated kagome lattice may exhibit either no intermediate plateau, a single 1/3 plateau, a single 5/9 plateau, or a sequence of 1/9, 1/3, and 5/9 plateaus depending on a character and relative size of two considered coupling constants.

Fluid-fluid demixing and density anomaly in a ternary mixture of hard spheres.

We report the grand-canonical solution of a ternary mixture of discrete hard spheres defined on a Husimi lattice built with cubes, which provides a mean-field approximation for this system on the cubic lattice. The mixture is composed of pointlike particles (0NN) and particles which exclude up to their first (1NN) and second neighbors (2NN), with activities z_{0},z_{1}, and z_{2}, respectively. Our solution reveals a very rich thermodynamic behavior, with two solid phases associated with the ordering of 1NN (S1) or 2NN particles (S2), and two fluid phases, one being regular (RF) and the other characterized by a dominance of 0NN particles (F0 phase). However, in most part of the phase diagram these fluid (F) phases are indistinguishable. Discontinuous transitions are observed between all the four phases, yielding several coexistence surfaces in the system, among which a fluid-fluid and a solid-solid demixing surface. The former one is limited by a line of critical points and a line of triple points (where the phases RF-F0-S2 coexist), both meeting at a special point, after which the fluid-fluid coexistence becomes metastable. Another line of triple points is found, connecting the F-S1,F-S2, and S1-S2 coexistence surfaces. A critical F-S1 surface is also observed meeting the F-S1 coexistence one at a line of tricritical points. Furthermore, a thermodynamic anomaly characterized by minima in isobaric curves of the total density of particles is found, yielding three surfaces of minimal density in the activity space, depending on which activity is kept fixed during its calculation.

Unsaturated Bipartite Entanglement of a Spin-1/2 Ising-Heisenberg Model on a Triangulated Husimi Lattice

A bipartite entanglement between two nearest-neighbor Heisenberg spins of a spin-1/2 Ising-Heisenberg model on a triangulated Husimi lattice is quantified using a concurrence. It is shown that the concurrence equals zero in a classical ferromagnetic and a quantum disordered phase, while it becomes sizable though unsaturated in a quantum ferromagnetic phase. A thermally-assisted reentrance of the concurrence is found above a classical ferromagnetic phase, whereas a quantum ferromagnetic phase displays a striking cusp of the concurrence at a critical temperature.

A significant equilibrium distinguishing DNA/RNA-like phases in 2T SAW model on Husimi tetrahedron lattice

A modified two-tolerate (2T) SAW polymer model is carried on the Husimi tetrahedron lattice to describe the DNA/RNA-like biomacromolecules. With a simplified assemble of configurations to catch the zipping/unzipping feature of intra-pair-able polymer chains, the thermodynamics are exactly derived via the recursive property of Husimi lattice. The probability(density) of monomers in single and paired strands, and solvent are calculated from recursive partial partition functions to characterize the phase diagrams. The model is found to be a practical setup fitting both DNA- and RNA-like polymers, with rich phase diagrams presenting the DNA/RNA-like regions and the unwinding/polymerase transitions. An intriguing equilibrium is found to separate DNA- and RNA-like regions with a particular parameters setup, where the single and double strands co-exist in the system with constant densities regardless of temperature.

Thermodynamic behavior of binary mixtures of hard spheres: Semianalytical solutions on a Husimi lattice built with cubes.

We study binary mixtures of hard particles, which exclude up to their kth nearest neighbors (kNN) on the simple cubic lattice and have activities z_{k}. In the first model analyzed, point particles (0NN) are mixed with 1NN ones. The grand-canonical solution of this model on a Husimi lattice built with cubes unveils a phase diagram with a fluid and a solid phase separated by a continuous and a discontinuous transition line which meet at a tricritical point. A density anomaly, characterized by minima in isobaric curves of the total density of particles against z_{0} (or z_{1}), is also observed in this system. Overall, this scenario is identical to the one previously found for this model when defined on the square lattice. The second model investigated consists of the mixture of 1NN particles with 2NN ones. In this case, a very rich phase behavior is found in its Husimi lattice solution, with two solid phases-one associated with the ordering of 1NN particles (S1) and the other with the ordering of 2NN ones (S2)-beyond the fluid (F) phase. While the transitions between F-S2 and S1-S2 phases are always discontinuous, the F-S1 transition is continuous (discontinuous) for small (large) z_{2}. The critical and coexistence F-S1 lines meet at a tricritical point. Moreover, the coexistence F-S1,F-S2, and S1-S2 lines meet at a triple point. Density anomalies are absent in this case.

Order-disorder transition in a two-dimensional associating lattice gas.

We study an associating lattice gas (ALG) using Monte Carlo simulation on the triangular lattice and semianalytical solutions on Husimi lattices. In this model, the molecules have an orientational degree of freedom and the interactions depend on the relative orientations of nearest-neighbor molecules, mimicking the formation of hydrogen bonds. We focus on the transition between the high-density liquid (HDL) phase and the isotropic phase in the limit of full occupancy, corresponding to chemical potential μ→∞, which has not yet been studied systematically. Simulations yield a continuous phase transition at τ_{c}=k_{B}T_{c}/γ=0.4763(1) (where -γ is the bond energy) between the low-temperature HDL phase, with a nonvanishing mean orientation of the molecules, and the high-temperature isotropic phase. Results for critical exponents and the Binder cumulant indicate that the transition belongs to the three-state Potts model universality class, even though the ALG Hamiltonian does not have the full permutation symmetry of the Potts model. In contrast with simulation, the Husimi lattice analyses furnish a discontinuous phase transition, characterized by a discontinuity of the nematic order parameter. The transition temperatures (τ_{c}=0.51403 and 0.51207 for trees built with triangles and hexagons, respectively) are slightly higher than that found via simulation. Since the Husimi lattice studies show that the ALG phase diagram features a discontinuous isotropic-HDL line for finite μ, three possible scenarios arise for the triangular lattice. The first is that in the limit μ→∞ the first-order line ends in a critical point; the second is a change in the nature of the transition at some finite chemical potential; the third is that the entire line is one of continuous phase transitions. Results from other ALG models and the fact that mean-field approximations show a discontinuous phase transition for the three-state Potts model (known to possess a continuous transition) lends some weight to the third alternative.

Three stable phases and thermodynamic anomaly in a binary mixture of hard particles.

While the realistic modeling of the thermodynamic behavior of fluids usually demands elaborated atomistic models, much has been learned from simplified ones. Here, we investigate a model where pointlike particles (with activity z0) are mixed with molecules that exclude their first and second neighbors (i.e., cubes of lateral size λ=3a, with activity z2), both placed on the sites of a simple cubic lattice with parameter a. Only hard-core interactions exist among the particles so that the model is athermal. Despite its simplicity, the grand-canonical solution of this model on a Husimi lattice built with cubes revels a fluid-fluid demixing, yielding a phase diagram with two fluid phases (one of them dominated by small particles-F0) and a solidlike phase coexisting at a triple-point. Moreover, the fluid-fluid coexistence line ends at a critical point. An anomaly in the total density (ρT) of particles is also found, which is hallmarked by minima in the isobaric curves of ρT vs z0 (or z2). Interestingly, the line of minimum density crosses the phase diagram starting inside the region where both fluid phases are stable, passing through the F0 one and ending deep inside its metastable region, in a point where the spinodals of both fluid phases cross each other.

Phase transitions of antiferromagnetic Ising spins on the zigzag surface of an asymmetrical Husimi lattice.

An asymmetrical two-dimensional Ising model with a zigzag surface, created by diagonally cutting a regular square lattice, has been developed to investigate the thermodynamics and phase transitions on surface by the methodology of recursive lattice, which we have previously applied to study polymers near a surface. The model retains the advantages of simple formulation and exact calculation of the conventional Bethe-like lattices. An antiferromagnetic Ising model is solved on the surface of this lattice to evaluate thermal properties such as free energy, energy density and entropy, from which we have successfully identified a first-order order–disorder transition other than the spontaneous magnetization, and a secondary transition on the supercooled state indicated by the Kauzmann paradox.

A Husimi rhombus lattice with random angles to present the configurational stochasticity in exact thermodynamic calculation

Abstract The square unit in Husimi lattice is generalized to be rhombus with random angles. The independence feature of unit cells in recursive lattice makes the random conformation possible, which is unfeasible in conventional lattices. Since the randomness of conformations in real system is naturally introduced into the model, this new lattice can describe the off-crystal metastable states without artificial randomness. With reasonable simplification, a coefficient A ( θ ) is formulated to present the effect of angle in the rhombus unit. A “visit and count” recursive calculation is developed to numerically approach the thermodynamics. The critical temperature T c of spontaneous magnetization is lowered with the presence of angle randomness, implying a less stable system. Besides consistent results to the regular lattice, the random-angled model features a distribution of solutions and thermal fluctuation with exact calculation. The Kauzmann paradox and ideal glass transition due to the off-crystal metastable portion are identified by the appearance of negative entropy. The effects of energy parameters on thermodynamics and the ground states are investigated.

Relevance of recursive lattice approximations for description of frustrated magnetic systems: Star kagome-like recursive lattice approximation

We introduce and investigate in detail thermodynamic properties of the antiferromagnetic spin-1∕2 Ising model with the multisite interaction in the presence of the external magnetic field on the star kagome-like recursive lattice that takes into account not only basic triangular structure of the regular kagome lattice responsible for the geometric frustration but also a typical star-like structure of the kagome lattice consisting of six site-connected triangles. It is shown that the model exhibits single solution for arbitrary values of the parameters of the model, which is driven by a sixth-order polynomial equation. The free energy per site of the model is found and the magnetization, the entropy, and the specific heat capacity properties of the model are investigated and compared to those obtained in the framework of the model on the kagome-like Husimi lattice. The system of all ground states of the model is determined and exact expressions for their magnetization properties as well as for their residual entropies are derived. It is shown that all results obtained in the framework of the model on the star kagome-like recursive lattice are unprecedentedly close to those obtained on the much simpler kagome-like Husimi lattice. Besides, in the case when an exact result of the model on the real kagome lattice exists it is very close to the corresponding results obtained on the recursive lattices. These quite surprising but nontrivial facts allow one to make important assumptions about the properties of the model on the real two-dimensional kagome lattice even in the nonzero external magnetic field. First of all, one can suppose that various thermodynamic properties of the model on the kagome lattice must be qualitatively the same as the corresponding properties obtained in the framework of the model on various kagome-like recursive lattices. Moreover, one can also suppose that even the quantitative results obtained for the model on various kagome-like recursive lattices must be very close (it seems that some of them are even identical) to those that should be valid on the regular kagome lattice, although the exact solution of the model on the kagome lattice does not exist yet.

Antiferromagnetic Ising model with frustrations on Husimi lattice of polygons with odd number of sides

Abstract Antiferromagnetic Ising model on Husimi lattice of polygons with odd number of sides are exactly solved to investigate the frustration of spins. Recursive lattices of pentagon, heptagon and nonagon were studied. The spontaneous magnetizations of these systems can easily be located from the solution and entropy curve. The solutions present N branches from 0 to 1 below TC, and the asymmetric arrangement of solutions confirms the frustration, which also can be identified by the non-zero residual entropy at T = 0. As expected, the TC increases with the number of sides, while ST=0 decreases as a measure of the degree of frustration.

Multipeak low-temperature behavior of specific heat capacity in frustrated magnetic systems: An exact theoretical analysis.

We investigate in detail the process of formation of the multipeak low-temperature structure in the behavior of the specific heat capacity in frustrated magnetic systems in the framework of the exactly solvable antiferromagnetic spin-1/2 Ising model with the multisite interaction in the presence of the external magnetic field on the kagome-like Husimi lattice. The behavior of the entropy of the model is studied and exact values of the residual entropies of all ground states are found. It is shown that the multipeak structure in the behavior of the specific heat capacity is related to the formation of the multilevel hierarchical ordering in the system of all ground states of the model. Direct relation between the maximal number of peaks in the specific heat capacity behavior and the number of independent interactions in studied frustrated magnetic system is identified. The mechanism of the formation of the multipeak structure in the specific heat capacity is described and studied in detail, and it is generalized to frustrated magnetic systems with arbitrary numbers of independent interactions.

Magnetic Properties of Simplest Pure Husimi Lattice: a Monte Carlo Study

The magnetic properties of a Husimi lattice with antiferromagnetic spins-|S| = 1/2, 1, 3/2, and 2 Ising model with three sites in the elementary polygon (p = 3) and the coordination number z = 4 have been investigated using a Monte Carlo study. The reduced critical temperature is deduced for different values of magnetic external field, for a fixed value of crystal field and spin moments. The magnetization versus the crystal field is determined for different values of temperatures and for different values of external magnetic field and spin moments. The hysteresis curves of magnetization with the magnetic field have been established for different values of reduced temperatures, reduced crystal field, and spin moments.

Solution of semi-flexible self-avoiding trails on a Husimi lattice built with squares

We study a model of semi-flexible self-avoiding trails, where the lattice paths are constrained to visit each lattice edge at most once, with configurations weighted by the number of collisions, crossings and bends, on a Husimi lattice built with squares. We find a rich phase diagram with five phases: a non-polymerised phase (NP), low density (P1) and high density (P2) polymerised phases, and, for sufficiently large stiffness, two additional anisotropic (nematic) (AN1 and AN2) polymerised phases within the P1 phase. Moreover, the AN1 phase which shows a broken symmetry with a preferential direction, is separated from the P1 phase by the other nematic AN2 phase. Although this scenario is similar to what was found in our previous calculation on the Bethe lattice, where the AN–P1 transition was discontinuous and critical, the presence of the additional nematic phase between them introduces a qualitative difference. Other details of the phase diagram are that a line of tri-critical points may separate the P1–P2 transition surface into a continuous and a discontinuous portion, and that the same may happen at the NP–P1 transition surface, details of which depend on whether crossings are allowed or forbidden. A critical end-point line is also found in the phase diagram.