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Register a new userContinuous Percolation Phase Transitions of Two-dimensional Lattice Networks under a Generalized Achlioptas Process
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The percolation phase transitions of two-dimensional lattice networks under a generalized Achlioptas process (GAP) are investigated. During the GAP, two edges are chosen randomly from the lattice and the edge with minimum product of the two connecting cluster sizes is taken as the next occupied bond with a probability $p$. At $p=0.5$, the GAP becomes the random growth model and leads to the minority product rule at $p=1$. Using the finite-size scaling analysis, we find that the percolation phase transitions of these systems with $0.5 le p le 1$ are always continuous and their critical exponents depend on $p$. Therefore, the universality class of the critical phenomena in two-dimensional lattice networks under the GAP is related to the probability parameter $p$ in addition.
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Using the finite-size scaling, we have investigated the percolation phase transitions of evolving random networks under a generalized Achlioptas process (GAP). During this GAP, the edge with minimum product of two connecting cluster sizes is taken with a probability $p$ from two randomly chosen edges. This model becomes the ErdH os-Renyi network at $p=0.5$ and the random network under the Achlioptas process at $p=1$. Using both the fixed point of $s_2/s_1$ and the straight line of $ln s_1$, where $s_1$ and $s_2$ are the reduced sizes of the largest and the second largest cluster, we demonstrate that the phase transitions of this model are continuous for $0.5 le p le 1$. From the slopes of $ln s_1$ and $ln (s_2/s_1)$ at the critical point we get the critical exponents $beta$ and $ u$, which depend on $p$. Therefore the universality class of this model should be characterized by $p$ also.
We consider quantum Heisenberg ferro- and antiferromagnets on the square lattice with exchange anisotropy of easy-plane or easy-axis type. The thermodynamics and the critical behaviour of the models are studied by the pure-quantum self-consistent harmonic approximation, in order to evaluate the spin and anisotropy dependence of the critical temperatures. Results for thermodynamic quantities are reported and comparison with experimental and numerical simulation data is made. The obtained results allow us to draw a general picture of the subject and, in particular, to estimate the value of the critical temperature for any model belonging to the considered class.
A random growth lattice filling model of percolation with touch and stop growth rule is developed and studied numerically on a two dimensional square lattice. Nucleation centers are continuously added one at a time to the empty sites and the clusters are grown from these nucleation centers with a tunable growth probability g. As the growth probability g is varied from 0 to 1 two distinct regimes are found to occur. For gle 0.5, the model exhibits continuous percolation transitions as ordinary percolation whereas for gge 0.8 the model exhibits discontinuous percolation transitions. The discontinuous transition is characterized by discontinuous jump in the order parameter, compact spanning cluster and absence of power law scaling of cluster size distribution. Instead of a sharp tricritical point, a tricritical region is found to occur for 0.5 < g < 0.8 within which the values of the critical exponents change continuously till the crossover from continuous to discontinuous transition is completed.
In this paper, we apply machine learning methods to study phase transitions in certain statistical mechanical models on the two dimensional lattices, whose transitions involve non-local or topological properties, including site and bond percolations, the XY model and the generalized XY model. We find that using just one hidden layer in a fully-connected neural network, the percolation transition can be learned and the data collapse by using the average output layer gives correct estimate of the critical exponent $ u$. We also study the Berezinskii-Kosterlitz-Thouless transition, which involves binding and unbinding of topological defects---vortices and anti-vortices, in the classical XY model. The generalized XY model contains richer phases, such as the nematic phase, the paramagnetic and the quasi-long-range ferromagnetic phases, and we also apply machine learning method to it. We obtain a consistent phase diagram from the network trained with only data along the temperature axis at two particular parameter $Delta$ values, where $Delta$ is the relative weight of pure XY coupling. Besides using the spin configurations (either angles or spin components) as the input information in a convolutional neural network, we devise a feature engineering approach using the histograms of the spin orientations in order to train the network to learn the three phases in the generalized XY model and demonstrate that it indeed works. The trained network by using system size $Ltimes L$ can be used to the phase diagram for other sizes ($Ltimes L$, where $L e L$) without any further training.
We investigate the generalized p-spin models that contain arbitrary diagonal operators U with no reflection symmetry. We derive general equations that give an opportunity to uncover the behavior of the system near the glass transition at different (continuous) p. The quadrupole glass with J=1 is considered as an illustrating example. It is shown that the crossover from continuous to discontinuous glass transition to one-step replica breaking solution takes place at p=3.3 for this model. For p <2+Delta p, where Delta p= 0.5 is a finite value, stable 1RSB-solution disappears. This behaviour is strongly different from that of the p-spin Ising glass model.
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