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Register a new userPerturbing the Shortest Path on a Critical Directed Square Lattice
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We investigate the behaviour of the shortest path on a directed two-dimensional square lattice for bond percolation at the critical probability $p_c$ . We observe that flipping an edge lying on the shortest path has a non-local effect in the form of power-law distributions for both the differences in shortest path lengths and for the minimal enclosed areas. Using maximum likelihood estimation and extrapolation we find the exponents $alpha = 1.36 pm 0.01$ for the path length differences and $beta = 1.186 pm 0.001$ for the enclosed areas.
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We investigate the effect of sequentiallydisrupting the shortest path of percolation clusters at criticality by comparing it with the shortest alternative path. We measure the difference in length and the enclosed area between the two paths. The sequential approach allows to study spatial correlations. We find the lengths of the segments of successively constant differences in length to be uncorrelated. Simultaneously, we study the distance between red bonds. We find the probability distributions for the enclosed areas A, the differences in length $Delta l$, and the lengths between the redbonds $l_r$ to follow power law distributions. Using maximum likelihood estimation and extrapolation we find the exponents $beta$ = 1.38 $pm$ 0.03 for $Delta l$, $alpha$ = 1.186 $pm$ 0.008 for A and $delta$ = 1.64 $pm$ 0.025 for thedistribution of $l_r$.
The critical properties of the stochastic susceptible-exposed-infected model on a square lattice is studied by numerical simulations and by the use of scaling relations. In the presence of an infected individual, a susceptible becomes either infected or exposed. Once infected or exposed, the individual remains forever in this state. The stationary properties are shown to be the same as those of isotropic percolation so that the critical behavior puts the model into the universality class of dynamic percolation.
The edge-cubic spin model on square lattice is studied via Monte Carlo simulation with cluster algorithm. By cooling the system, we found two successive symmetry breakings, i.e., the breakdown of $O_h$ into the group of $C_{3h}$ which then freezes into ground state configuration. To characterize the existing phase transitions, we consider the magnetization and the population number as order parameters. We observe that the magnetization is good at probing the high temperature transition but fails in the analysis of the low temperature transition. In contrast the population number performs well in probing the low- and the high-$T$ transitions. We plot the temperature dependence of the moment and correlation ratios of the order parameters and obtain the high- and low-$T$ transitions at $T_h = 0.602(1)$ and $T_l=0.5422(2)$ respectively, with the corresponding exponents of correlation length $ u_h=1.50(1)$ and $ u_l=0.833(1)$. By using correlation ratio and size dependence of correlation function we estimate the decay exponent for the high-$T$ transition as $eta_h=0.260(1)$. For the low-$T$ transition, $eta_l = 0.267(1)$ is extracted from the finite size scaling of susceptibility. The universality class of the low-$T$ critical point is the same as the 3-state Potts model.
The pure-quantum self-consistent harmonic approximation, a semiclassical method based on the path-integral formulation of quantum statistical mechanics, is applied to the study of the thermodynamic behaviour of the quantum Heisenberg antiferromagnet on the square lattice (QHAF). Results for various properties are obtained for different values of the spin and successfully compared with experimental data.
We investigate the role of a transverse field on the Ising square antiferromagnet with first-($J_1$) and second-($J_2$) neighbor interactions. Using a cluster mean-field approach, we provide a telltale characterization of the frustration effects on the phase boundaries and entropy accumulation process emerging from the interplay between quantum and thermal fluctuations. We found that the paramagnetic (PM) and antiferromagnetic phases are separated by continuous phase transitions. On the other hand, continuous and discontinuous phase transitions, as well as tricriticality, are observed in the phase boundaries between PM and superantiferromagnetic phases. A rich scenario arises when a discontinuous phase transition occurs in the classical limit while quantum fluctuations recover criticality. We also find that the entropy accumulation process predicted to occur at temperatures close to the quantum critical point can be enhanced by frustration. Our results provide a description for the phase boundaries and entropy behavior that can help to identify the ratio $J_2/J_1$ in possible experimental realizations of the quantum $J_1$-$J_2$ Ising antiferromagnet.
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