Recursive Lattice Research Articles

Theoretical evidence for the presence of “spin-liquid” like phase in frustrated ferrimagnetic J1 − J2 systems on body-centered cubic lattice

Using the recursive lattice technique the existence of the intermediate phase, which separates two ferrimagnetic phases in the 1/2−σ (σ=1,3/2,2) mixed-spin Ising ferrimagnets with the presence of the next-to-nearest neighbor interaction on the regular body-centered cubic lattice, is predicted. It is shown that, for given spin value of the central sites of the body-centered cubic lattice, this intermediate phase is realized in the corresponding narrow interval of the values of the frustration parameter of the model, that the form of the region, where the intermediate phase exists, depends on the value of the central spin σ (becomes larger with increasing of σ), and that all transitions between various phases of the model always have the second order nature. All possible series of consecutive phase transitions of the model are identified and the properties of the sublattice magnetizations are studied and discussed in detail.

Presence of the weak ferromagnetism in pure antiferromagnetic systems with octahedral structure: Effective field theory analysis

The recently reported (Jurčišinová E. and Jurčišin M., J. Magn. & Magn. Mater., 513 (2020) 167085) possibility of the existence of the weak ferromagnetism with small but nonzero spontaneous magnetization in pure antiferromagnetic systems with the octahedral structure is confirmed using the six-site effective-field theory cluster approximation of the antiferromagnetic spin- Ising model on the octahedral lattice. The sublattice magnetizations, the total spontaneous magnetization, as well as the Néel temperature of the model are found and compared to those obtained in the framework of the recursive lattice approach. The performed analysis allows one to conclude that the existence of the weak ferromagnetism in pure antiferromagnetic systems with the octahedral structure is directly related to their intrinsic geometry. The spontaneous magnetization and the Curie temperature of the ferromagnetic model are also determined and briefly discussed.

Influence of multisite interaction on thermodynamics and ground-state degeneracies of frustrated magnetic systems with pyrochlore structure: An exact theoretical analysis

The antiferromagnetic spin-1∕2 Ising model with the multisite interaction in the presence of the external magnetic field is investigated on the tetrahedron recursive lattice, which represents a suitable and effective approximation of frustrated magnetic materials with pyrochlore structure. It is shown that the model belongs to the rather restricted set of exactly solvable models and exhibits unique solution for all values of the model parameters. The exact expression for the free energy per site is found and the magnetization and entropy properties of the model are investigated in detail. The system of all ground states of the model under the influence of the multisite interaction is identified and their magnetization properties and the values of all residual entropies are found exactly. It is shown that not only the strength but also the sign of the multisite interaction has the significant impact on the properties of the model and that, depending on the value of the multisite interaction in comparison to the absolute value of the antiferromagnetic nearest-neighbor interaction, the model exhibits the existence of three ground states of different orders with the same value of the magnetization equal to zero but with the residual entropies which are considerably different, i.e., with different macroscopic degeneracies.

Grid refinement in the three-dimensional hybrid recursive regularized lattice Boltzmann method for compressible aerodynamics.

Grid refinement techniques are of paramount importance for computational fluid dynamics approaches relying on the use of Cartesian grids. This is especially true of solvers dedicated to aerodynamics, in which the capture of thin shear layers require the use of small cells. In this paper, a three-dimensional grid refinement technique is developed within the framework of hybrid recursive regularized lattice Boltzmann method (HRR-LBM) for compressible high-speed flows, which is an efficient collide-stream-type method on a compact D3Q19 stencil. The proposed method is successfully assessed considering several test cases, namely, an isentropic vortex propagating through transition interface, shock-vortex interaction with intersection between grid refinement interface and shock corrugation, and transonic flows over three-dimensional DLR-M6 wing with seven levels of grid refinement.

Evidence for weak ferromagnetism and metamagnetic phase transitions in frustrated antiferromagnetic systems with octahedral structure

Possible source of the weak ferromagnetism and the metamagnetic phase transitions in the antiferromagnetic systems with octahedral structure, such as perovskites, is identified in the framework of the antiferromagnetic spin-1/2 Ising model on the octahedral recursive lattice. It is shown that the weak ferromagnetism of the studied antiferromagnetic model is directly related to the competition between two qualitatively different kinds of spin loops (frustrated and bipartite) realized within each elementary octahedron. The free energy of the model is found, the magnetization properties of the model are studied, and the phase diagram of the metamagnetic phase transitions in the external magnetic field is determined. The discrete system of the model ground states is found and discussed. The discontinuity of the susceptibility at the metamagnetic phase transitions is studied. The exact expression for the Néel temperature of the antiferromagnetic system is determined and compare to the Curie temperature of the ferromagnetic system.

Ising Model with Nonmagnetic Dilution on Recursive Lattices

Using a method for composing self-consistent equations, we construct a class of approximate solutions of the Ising problem that are a generalization of the Bethe approximation. We show that some of the approximations in this class can be interpreted as exact solutions of the Ising model on recursive lattices. For these recursive lattices, we find exact values of the thresholds of percolation through sites and couplings and show that for the Ising model of a diluted magnet, our method leads to exact values for these thresholds.

Interface tracking characteristics of color-gradient lattice Boltzmann model for immiscible fluids.

We study the interface tracking characteristics of a color-gradient-based lattice Boltzmann model for immiscible flows. Investigation of the local density change in one of the fluid phases, via a Taylor series expansion of the recursive lattice Boltzmann equation, leads to the evolution equation of the order parameter that differentiates the fluids. It turns out that this interface evolution follows a conservative Allen-Cahn equation with a mobility which is independent of the fluid viscosities and surface tension. The mobility of the interface, which solely depends upon lattice speed of sound, can have a crucial effect on the physical dynamics of the interface. Further, we find that, when the equivalent lattice weights inside the segregation operator are modified, the resulting differential operators have a discretization error that is anisotropic to the leading order. As a consequence, the discretization errors in the segregation operator, which ensures a finite interface width, can act as a source of the spurious currents. These findings are supported with the help of numerical simulations.

Ising model with nonmagnetic dilution on recursive lattices

Методом составления самосогласованных уравнений построен класс приближенных решений задачи Изинга, являющихся обобщением приближения Бете. Показано, что некоторые из приближений этого класса можно интерпретировать как точные решения для модели Изинга на рекурсивных решетках. Для этих рекурсивных решеток найдены точные значения порогов протекания по узлам и связям и показано, что для модели Изинга разбавленного магнетика наш метод приводит к точным значениям для этих порогов.

Recursive Least-Squares Lattice Algorithm Combined With Secondary-Path Innovation and Lattice-Order Decision Algorithms for Active Noise Control

This paper proposes a new active noise control (ANC) system based on a recursive least-squares lattice (RLSL) algorithm by designing secondary-path innovation (SPI) and lattice-order decision (LOD) algorithms. The SPI algorithm associated with the power spectral factorization of the secondary path is designed to apply the RLSL algorithm to the ANC system without the filtered-input structure. The SPI algorithm whitens the error microphone signals into the virtual error signals just ahead of the secondary path to construct the virtual desired signals corresponding to the outputs of the lattice filter. The LOD algorithm is designed to operate the ANC system without knowing the tap length of the optimal finite impulse response (FIR) filter. The decision algorithm using the initial and final estimation errors of the lattice filter estimates the tap length of the optimal FIR filter by increasing the lattice order one-by-one until the final estimation error reaches below a threshold value. Even in environments where the tap length of the optimal FIR filter is unknown, the proposed ANC system with low computational complexity shows almost the same performance as the ANC system based on a filtered-input recursive least squares algorithm.

Solid wall and open boundary conditions in hybrid recursive regularized lattice Boltzmann method for compressible flows

Complex geometries and open boundaries have been intensively studied in the nearly incompressible lattice Boltzmann method (LBM) framework. Therefore, only few boundary conditions for the high speed fully compressible LBM have been proposed. This paper deals with the definition of efficient boundary conditions for the compressible LBM methods, with the emphasis put on the newly proposed hybrid recursive regularized D3Q19 LBM (HRR-LBM) with applications to compressible aerodynamics. The straightforward simple extrapolation-based far-field boundary conditions, the characteristic boundary conditions, and the absorbing sponge layer approach are extended and estimated in the HRR-LBM for the choice of open boundaries. Moreover, a cut-cell type approach to handle the immersed solid is proposed to model both slip and no-slip wall boundary conditions with either isothermal or adiabatic behavior. The proposed implementations are assessed considering the simulation of (i) isentropic vortex convection with subsonic to supersonic inflow and outflow conditions, (ii) two-dimensional (2D) compressible mixing layer, (iii) steady inviscid transonic flow over a National Advisory Committee for Aeronautics (NACA) 0012 airfoil, (iv) unsteady viscous transonic flow over a NACA 0012 airfoil, and (v) three-dimensional (3D) transonic flows over a German Aerospace Center (DLR) F6 full aircraft configuration.

Hybrid recursive regularized lattice Boltzmann simulation of humid air with application to meteorological flows.

An extended version of the hybrid recursive regularized lattice-Boltzmann model which incorporates external force is developed to simulate humid air flows with phase change mechanisms under the Boussinesq approximation. Mass and momentum conservation equations are solved by a regularized lattice Boltzmann approach well suited for high Reynolds number flows, whereas the energy and humidity related equations are solved by a finite volume approach. Two options are investigated to account for cloud formation in atmospheric flow simulations. The first option considers a single conservation equation for total water and an appropriate invariant variable of temperature. In the other approach, liquid and vapor are considered via two separated equations, and phase transition is accounted for via a relaxation procedure. The obtained models are then systematically validated on four well-established benchmark problems including a double diffusive Rayleigh Bénard convection of humid air, two- and three-dimensional thermal moist rising bubble under convective atmospheric environment, as well as a shallow cumulus convection in the framework of large-eddy simulation.

Hybrid recursive regularized thermal lattice Boltzmann model for high subsonic compressible flows

A thermal lattice Boltzmann model with a hybrid recursive regularization (HRR) collision operator is developed on standard lattices for simulation of subsonic and sonic compressible flows without shock. The approach is hybrid: mass and momentum conservation equations are solved using a lattice Boltzmann solver, while the energy conservation is solved under entropy form with a finite volume solver. The defect of Galilean invariance related to Mach number is corrected by the third order equilibrium distribution function, supplemented by an additional correcting term and hybrid recursive regularization. The proposed approach is assessed considering the simulation of i) an isentropic vortex convection, ii) a two dimensional acoustic pulse and iii) non-isothermal Gaussian pulse with Ma number in range of 0 to 1. Numerical simulations demonstrate that the flaw in Galilean invariance is effectively eliminated by the compressible HRR model. At last, the compressible laminar flows over flat plate at Ma number of 0.3 and 0.87, Reynolds number of 105 are considered to validate the capture of viscous and diffusive effects.

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.

A Serial Concatenation of Binary-Input Nonbinary-Output Convolutional Code and Recursive Convolutional Lattice Code

The recently proposed recursive convolutional lattice code (RCLC) can form a signal with pseudo-Gaussian constellations, and their parallel concatenation is shown to approach the Shannon limit. A practical limitation is that its input symbol is limited to $L^{2}$ -ary quadrature amplitude modulation (QAM), which has non-power-of-two constellation points when $L$ is chosen from the odd numbers. Therefore, encoding binary information by the RCLC is not straightforward. Furthermore, the information rate is limited to $\log _{2}~L$ bits per complex dimension due to their parallel concatenation. In this paper, we tackle these issues by introducing a serial concatenation of binary-input nonbinary-output convolutional code (CC) and the RCLC, where the outer CC outputs an $L$ -ary symbol that is matched to the input of the inner RCLC. We demonstrate that even with $L=3$ , the proposed approach can achieve 2 bits per complex dimension and still is able to approach the Shannon limit with lower decoding complexity compared with its parallel concatenation counterpart. As is demonstrated through theoretical analysis, the major practical drawback of the constellation generated by the RCLC is its Gaussian-like distribution, which has large peak-to-average power ratio. Therefore, we further introduce an approach to reduce the signal dynamic range for the proposed system. It is shown that a remarkable gain can be achieved in terms of capacity compared with the conventional QAM signals under the constraint of comparable power amplifier efficiency.

Antiferromagnetic geometric frustration under the influence of the next-nearest-neighbor interaction. An exactly solvable model

The influence of the next-nearest-neighbor interaction on the properties of the geometrically frustrated antiferromagnetic systems is investigated in the framework of the exactly solvable antiferromagnetic spin-1∕2 Ising model in the external magnetic field on the square-kagome recursive lattice, where the next-nearest-neighbor interaction is supposed between sites within each elementary square of the lattice. The thermodynamic properties of the model are investigated in detail and it is shown that the competition between the nearest-neighbor antiferromagnetic interaction and the next-nearest-neighbor ferromagnetic interaction changes properties of the single-point ground states but does not change the frustrated character of the basic model. On the other hand, the presence of the antiferromagnetic next-nearest-neighbor interaction leads to the enhancement of the frustration effects with the formation of additional plateau and single-point ground states at low temperatures. Exact expressions for magnetizations and residual entropies of all ground states of the model are found. It is shown that the model exhibits various ground states with the same value of magnetization but different macroscopic degeneracies as well as the ground states with different values of magnetization but the same value of the residual entropy. The specific heat capacity is investigated and it is shown that the model exhibits the Schottky-type anomaly behavior in the vicinity of each single-point ground state value of the magnetic field. The formation of the field-induced double-peak structure of the specific heat capacity at low temperatures is demonstrated and it is shown that its very existence is directly related to the presence of highly macroscopically degenerated single-point ground states in the model.

Highly macroscopically degenerated single-point ground states as source of specific heat capacity anomalies in magnetic frustrated systems

Anomalies of the specific heat capacity are investigated in the framework of the exactly solvable antiferromagnetic spin-1/2 Ising model in the external magnetic field on the geometrically frustrated tetrahedron recursive lattice. It is shown that the Schottky-type anomaly in the behavior of the specific heat capacity is related to the existence of unique highly macroscopically degenerated single-point ground states which are formed on the borders between neighboring plateau-like ground states. It is also shown that the very existence of these single-point ground states with large residual entropies predicts the appearance of another anomaly in the behavior of the specific heat capacity for low temperatures, namely, the field-induced double-peak structure, which exists, and should be observed experimentally, along with the Schottky-type anomaly in various frustrated magnetic system.

Evidence for the ferromagnetic frustration in a classical spin-[formula omitted] system with multisite interaction in external magnetic field: Exact results

We investigate the influence of the multisite interaction among sites within elementary triangles of the kagome-like recursive lattice on the properties of the classical spin-1∕2 ferromagnetic Ising model in the external magnetic field. The exact solution of the model is found and it is shown that the model exhibits a nontrivial structure of the first order as well as second order phase transitions in nonzero external magnetic fields related to the multisite interaction. The equation for the exact determination of the positions of the critical points of the second order phase transitions is derived. The thermodynamic properties of the model are investigated in detail and it is shown that the competition between the ferromagnetic interaction and the multisite interaction leads to the appearance of strong ferromagnetic frustration effects represented by the formation of a nontrivial system of macroscopically degenerated plateau-like and single-point-like ground states. The residual entropies of all ground states are found and the kagome spin-ice-like highly macroscopically degenerated plateau state with nonzero magnetization is identified with the exact residual entropy per site s∕kB=ln(4∕3)∕3≈0.095894. The properties of the specific heat are investigated, its Schottky-type behavior near the single-point ground state values of the magnetic field is identified, the existence of large magnetocaloric effect is discussed, and the existence of the first order phase transitions without the specific heat capacity change is demonstrated.