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2D Ising model: Logarithmic scaling behavior of the finite-size extensions of the CRMES defined with the help of specific heat maxima according to the successive minimal approximations (Eq. (5)) and the analogous extensions of the CRMES defined with the help of the susceptibility maxima (Eq. (16)).
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![FIG. 1: 2D Ising model (L = 10 − 100 [14], L = 120, 140, ..., 200 new...](profile/Ioannis-Hadjiagapiou/publication/7291159/figure/fig1/AS:394573399511046@1471085037406/2D-Ising-model-L-10-100-14-L-120-140-200-new-data-Behavior-of-scaling_Q320.jpg)
Dominant energy subspaces of statistical systems are defined with the help of restrictive conditions on various characteristics of the energy distribution, such as the probability density and the fourth order Binder's cumulant. Our analysis generalizes the ideas of the critical minimum energy subspace (CRMES) technique, applied previously to study...
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The free fermion nature of interacting spins in one dimensional (1D) spin chains still lacks a rigorous study. In this letter we show that the length-$1$ spin strings significantly dominate critical properties of spinons, magnons and free fermions in the 1D antiferromagnetic spin-1/2 chain. Using the Bethe ansatz solution we analytically calculate...
Citations
... In this paper we obtain the density of states via the Wang-Landau (WL) Monte Carlo algorithm [54,55] with high precision, which makes it possible to calculate various physical quantities at any temperature. Exact enumeration provides the exact density of states [41] but requires enormous computational time, making it unsuitable to investigate the properties of the specific heat for R = 1/2 on a large finite lattice. ...
The Ising model with nearest-neighbor and next-nearest-neighbor interactions of the coupling constants J1 and J2, respectively, is investigated on a square lattice. For J1=2 and J2=1, the model becomes frustrated because ground states are infinitely degenerate. We obtain the density of states by using the Wang-Landau Monte Carlo method and calculate the specific heat. We find two separate peaks in the specific heat: a sharp peak related to the critical behavior and a round peak related to the specific heat of a disordered system such as spin glass. As the system size increases, the sharp-peak temperature decreases towards zero, and the maximum height of the sharp peak increases logarithmically, supporting that the spatial correlation length diverges exponentially at zero temperature. The partition-function zeros calculated by the density of states also suggest the zero-temperature phase transition.
... Note that in the flat histogram technique we have to choose an energy range (E min , E max ) [58,59]. We divide this energy range to R subintervals with overlaps between them to have a smooth matching between two consecutive subintervals. ...
We study the nature of the smectic–isotropic phase transition using a mobile 6-state Potts model. Each Potts state represents a molecular orientation. We show that with the choice of an appropriate microscopic Hamiltonian describing the interaction between individual molecules modeled by a mobile 6-state Potts spins, we observe the smectic phase dynamically formed when we cool the molecules from the isotropic phase to low temperatures (T). In order to elucidate the order of the transition and the low-T properties, we use the high-performance Wang–Landau flat energy-histogram technique. We show that the smectic phase goes to the liquid (isotropic) phase by melting/evaporating layer by layer starting from the film surface with increasing T. At a higher T, the whole remaining layers become orientationally disordered. The melting of each layer is characterized by a peak of the specific heat. Such a succession of partial transitions cannot be seen by the Metropolis algorithm. The successive layer meltings/evaporations at low T are found to have a first-order character by examining the energy histogram. These results are in agreement with experiments performed on some smectic liquid crystals.
... Binder cumulant is a quantity widely used in the analysis of phase transition systems [20]. In our system we have analyzed the fourth order Binder cumulant in the context of the protein coding analogy according to the Rett syndrome RNA sequences employed to construct random fractals. ...
We have proposed a random walk model to model the post-transcription process. We have chosen mature microRNA sequences related to individuals with Rett syndrome. We have analyzed the micro-RNA sequences by transcribing the nucleotides into random fractals. We have reported
... Binder cumulant is a quantity widely used in the analysis of phase transition systems [20]. ...
... One basic ingredient of this implementation is a suitable restriction of the energy subspace for the implementation of the WL algorithm. This was originally termed as the critical minimum energy subspace restriction [55] and it can be carried out in many alternative ways, the simplest being that of observing the finite-size behavior of the tails of the energy probability density function of the system [55]. ...
... One basic ingredient of this implementation is a suitable restriction of the energy subspace for the implementation of the WL algorithm. This was originally termed as the critical minimum energy subspace restriction [55] and it can be carried out in many alternative ways, the simplest being that of observing the finite-size behavior of the tails of the energy probability density function of the system [55]. ...
... is the density of states (DOS) and we follow the usual modification factor adjustment f j +1 = f j and f 1 = e [54,55]. The preliminary stage may consist of the levels: j = 1, . . . ...
The effects of bond randomness on the universality aspects of a two-dimensional (d=2) Blume-Capel model embedded in the triangular lattice are discussed. The system is studied numerically in both its first- and second-order phase-transition regimes by a comprehensive finite-size scaling analysis for a particularly suitable value of the disorder strength. We find that our data for the second-order phase transition, emerging under random bonds from the second-order regime of the pure model, are compatible with the universality class of the two-dimensional (2D) random Ising model. Furthermore, we find evidence that, the second-order transition emerging under bond randomness from the first-order regime of the pure model, belongs again to the same universality class. Although the first finding reinforces the scenario of strong universality in the 2D Ising model with quenched disorder, the second is in difference from the critical behavior, emerging under randomness, in the cases of the ex-first-order transitions of the Potts model. Finally, our results verify previous renormalization-group calculations on the Blume-Capel model with disorder in the crystal-field coupling.
... The study of frustrated magnetic systems always arouses the interest [1] [2] [3] [4] [5] [6] [7]. It is known that the nature of the ordering at low temperatures regime of these materials depends on the concentration of the magnetic species and on the relative magnitude of competing energy scales [8] [9]. ...
... The study of frustrated magnetic systems always arouses the interest1234567. It is known that the nature of the ordering at low temperatures regime of these materials depends on the concentration of the magnetic species and on the relative magnitude of competing energy scales [8, 9]. ...
... In fact nowadays everyone agrees that the transition line from the disordered to the ordered p(2 × 2) phase is continuous when the ratio between the second " J SN " and the first " J FN " neighboring interactions R = J SN /J FN < 0.5. The situation R > 0.5 where the degenerate p(2 × 1)/ p(1 × 2) ordered phase holds is still under debate, particularly the extent of interaction rate for which the transition is first order [1, 2,14151617 19]. In both cases, the characterization of phase transition is done by means of order parameters that quantify the breakdown of lattice occupation symmetry. ...
When interfaces between ordered domains are ordered clusters, frustration disappears. A phase with mixed ordered structures emerges but no length scale can be associated to. We show that sum of densities of each structure plays the role of order parameter. For, we consider a regular half full lattice with repulsive interaction extended to second neighboring particles. Ordered structures are p(2X2) when ratio between second and first neighboring interaction energies R=J SN /J FN <0.5 and degenerate p(2X1)/p(1X2) for R>0.5. The ground states coexist with another named p(4X2)/p(2X4) at central bi-critical point: a state to cross when passing between non-frustrated ordered phases.
... The study of frustrated magnetic systems always arouses the interest [1][2][3][4][5][6][7]. It is known that the nature of the ordering at low temperatures regime of these materials depends on the concentration of the magnetic species and on the relative magnitude of competing energy scales [8,9]. ...
... In fact nowadays everyone agrees that the transition line from the disordered to the ordered p(2 × 2) phase is continuous when the ratio between the second "J SN " and the first "J FN " neighboring interactions R = J SN /J FN < 0.5. The situation R > 0.5 where the degenerate p(2 × 1)/ p(1 × 2) ordered phase holds is still under debate, particularly the extent of interaction rate for which the transition is first order [1,2,[14][15][16][17]19]. In both cases, the characterization of phase transition is done by means of order parameters that quantify the breakdown of lattice occupation symmetry. ...
We deal with a 2D half occupied square lattice with repulsive interactions between first and second neighboring particles. Despite the intensive studies of the present model the central point of the phase diagram for which the ratio of the two interaction strengths R=JSN/JFN=0.5 is still open. In the present paper we show, using standard Monte Carlo calculations, that the situation corresponds to a phase of mixed ordered structures quantified by an “algebraic” order parameter defined as the sum of densities of the existing ordered clusters. The introduced grandeur also characterizes the transitions towards the known pure ordered phases for the other values of R as mentioned by the agreement of our results with those of the literature. The computation of the Cowley short range order parameter against R suggests that the central point is bicritical and is a state to cross when passing between the two pure phases.
... The study of frustrated magnetic systems always arouses the interest [1][2][3][4][5][6][7]. It is known that the nature of the ordering at low temperatures regime of these materials depends on the concentration of the magnetic species and on the relative magnitude of competing energy scales [8,9]. ...
... In fact nowadays everyone agrees that the transition line from the disordered to the ordered p(2 × 2) phase is continuous when the ratio between the second "J SN " and the first "J FN " neighboring interactions R = J SN /J FN < 0.5. The situation R > 0.5 where the degenerate p(2 × 1)/ p(1 × 2) ordered phase holds is still under debate, particularly the extent of interaction rate for which the transition is first order [1,2,[14][15][16][17]19]. In both cases, the characterization of phase transition is done by means of order parameters that quantify the breakdown of lattice occupation symmetry. ...
We deal with a 2D half occupied square lattice with repulsive interactions between first and second neighboring particles. Despite the intensive studies of the present model the central point of the phase diagram for which the ratio of the two interaction strengths R = J SN /J FN = 0.5 is still open. In the present paper we show, using standard Monte Carlo calculations, that the situation corresponds to a phase of mixed ordered structures quantified by an "algebraic" order parameter defined as the sum of densities of the existing ordered clusters. The introduced grandeur also characterizes the transitions towards the known pure ordered phases for the other values of R as mentioned by the agreement of our results with those of the literature. The computation of the Cowley short range order parameter against R suggests that the central point is bicritical and is a state to cross when passing between the two pure phases.
We study the question of universality in the two-dimensional spin-1 Baxter-Wu model in the presence of a crystal field Δ. We employ extensive numerical simulations of two types, providing us with complementary results: Wang-Landau sampling at fixed values of Δ and a parallelized variant of the multicanonical approach performed at constant temperature T. A detailed finite-size scaling analysis in the regime of second-order phase transitions in the (Δ,T) phase diagram indicates that the transition belongs to the universality class of the four-state Potts model. Previous controversies with respect to the nature of the transition are discussed and attributed to the presence of strong finite-size effects, especially as one approaches the pentacritical point of the model.
In this work, a variant of the Wang and Landau algorithm for calculation of the configurational energy density of states is proposed. The algorithm was developed for the purpose of using first-principles simulations, such as density functional theory, to calculate the partition function of disordered sublattices in crystal materials. The expensive calculations of first-principles methods make a parallel algorithm necessary for a practical computation of the configurational energy density of states within a supercell approximation of a solid-state material. The algorithm developed in this work is tested with the two-dimensional (2d) Ising model to bench mark the algorithm and to help provide insight for implementation to a materials science application. Tests with the 2d Ising model revealed that the algorithm has good performance compared to the original Wang and Landau algorithm and the 1/t algorithm, in particular the short iteration performance. A proof of convergence is presented within an adiabatic assumption, and the analysis is able to correctly predict the time dependence of the modification factor to the density of states. The algorithm was then applied to the lithium and lanthanum sublatticeofthesolid-statelithiumionconductorLi0.5La0.5TiO3.Thiswasdonetohelpunderstandthedisordered nature of the lithium and lanthanum. The results find, overall, that the algorithm performs very well for the 2d Ising model and that the results for Li0.5La0.5TiO3 are consistent with experiment while providing additional insight into the lithium and lanthanum ordering in the material. The primary result is that the lithium and lanthanumbecomemoremixedbetweenlayersalongthecaxisforincreasingtemperature.Inpart,thesimulation ofthedisorderedLi0.5La0.5TiO3 systemservesasabenchmarkforwhatsizesystemsarecurrentlyandinthenear future practical to calculate with density functional theory methods.
