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Register a new userThe elastic backbone phase transition in the Ising model
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Added by
Morteza Nattagh Najafi
Publication date
2020
fields
Physics
and research's language is
English
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The two-dimensional (zero magnetic field) Ising model is known to undergo a second order para-ferromagnetic phase transition, which is accompanied by a correlated percolation transition for the Fortuin-Kasteleyn (FK) clusters. In this paper we uncover that there exists also a second temperature $T_{text{eb}}<T_c$ at which the elastic backbone of FK clusters undergoes a second order phase transition to a dense phase. The corresponding universality class, which is characterized by determining various percolation exponents, is shown to be completely different from directed percolation, proposing a new anisotropic universality class with $beta=0.54pm 0.02$, $ u_{||}=1.86pm 0.01$, $ u_{perp}=1.21pm 0.04$ and $d_f=1.53pm 0.03$. All tested hyper-scaling relations are shown to be valid.
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Phase transition of the Ising model is investigated on a planar lattice that has a fractal structure. On the lattice, the number of bonds that cross the border of a finite area is doubled when the linear size of the area is extended by a factor of four. The free energy and the spontaneous magnetization of the system are obtained by means of the higher-order tensor renormalization group method. The system exhibits the order-disorder phase transition, where the critical indices are different from that of the square-lattice Ising model. An exponential decay is observed in the density matrix spectrum even at the critical point. It is possible to interpret that the system is less entangled because of the fractal geometry.
We study the phase transition of the Ising model in networks with core-periphery structures. By Monte Carlo simulations, we show that prior to the order-disorder phase transition the system organizes into an inhomogeneous intermediate phase in which core nodes are much more ordered than peripheral nodes. Interestingly, the susceptibility shows double peaks at two distinct temperatures. We find that, if the connections between core and periphery increase linearly with network size, the first peak does not exhibit any size-dependent effect, and the second one diverges in the limit of infinite network size. Otherwise, if the connections between core and periphery scale sub-linearly with the network size, both peaks of the susceptibility diverge as power laws in the thermodynamic limit. This suggests the appearance of a double transition phenomenon in the Ising model for the latter case. Moreover, we develop a mean-field theory that agrees well with the simulations.
We study the collective behavior of an Ising system on a small-world network with the interaction $J(r) propto r^{-alpha}$, where $r$ represents the Euclidean distance between two nodes. In the case of $alpha = 0$ corresponding to the uniform interaction, the system is known to possess a phase transition of the mean-field nature, while the system with the short-range interaction $(alphatoinfty)$ does not exhibit long-range order at any finite temperature. Monte Carlo simulations are performed at various values of $alpha$, and the critical value $alpha_c$ beyond which the long-range order does not emerge is estimated to be zero. Thus concluded is the absence of a phase transition in the system with the algebraically decaying interaction $r^{-alpha}$ for any nonzero positive value of $alpha$.
The continuous ferromagnetic-paramagnetic phase transition in the two-dimensional Ising model has already been excessively studied by conventional canonical statistical analysis in the past. We use the recently developed generalized microcanonical inflection-point analysis method to investigate the least-sensitive inflection points of the microcanonical entropy and its derivatives to identify transition signals. Surprisingly, this method reveals that there are potentially two additional transitions for the Ising system besides the critical transition.
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