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arXiv:cond-mat/0606138 · Physical Review E

Phase transitions in an Ising model on a Euclidean network

Arnab Chatterjee, Parongama Sen

Published Sep 20, 2006• 23 citations
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Abstract

A one-dimensional network on which there are long-range bonds at lattice distances $l>1$ with the probability $P(l)\ensuremath{\propto}{l}^{\ensuremath{-}\ensuremath{\delta}}$ has been taken under consideration. We investigate the critical behavior of the Ising model on such a network where spins interact with these extra neighbors apart from their nearest neighbors for $0\ensuremath{\leqslant}\ensuremath{\delta}<2$. It is observed that there is a finite temperature phase transition in the entire range. For $0\ensuremath{\leqslant}\ensuremath{\delta}<1$, finite-size scaling behavior of various quantities are consistent with mean-field exponents while for $1\ensuremath{\leqslant}\ensuremath{\delta}\ensuremath{\leqslant}2$, the exponents depend on $\ensuremath{\delta}$. The results are discussed in the context of earlier observations on the topology of the underlying network.

DOI
10.1103/physreve.74.036109

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A one-dimensional network on which there are long-range bonds at lattice distances l>1l>1l>1 with the probability P(l)\ensuremath∝l\ensuremath−\ensuremathδP(l)\ensuremath{\propto}{l}^{\ensuremath{-}\ensuremath{\delta}}P(l)\ensuremath∝l\ensuremath−\ensuremathδ has been taken under consideration. We investigate the critical behavior of the Ising model on such a network where spins interact with these extra neighbors apart from their nearest neighbors for 0\ensuremath⩽\ensuremathδ<20\ensuremath{\leqslant}\ensuremath{\delta}<20\ensuremath⩽\ensuremathδ<2. It is observed that there is a finite temperature phase transition in the entire range. For 0\ensuremath⩽\ensuremathδ<10\ensuremath{\leqslant}\ensuremath{\delta}<10\ensuremath⩽\ensuremathδ<1, finite-size scaling behavior of various quantities are consistent with mean-field exponents while for 1\ensuremath⩽\ensuremathδ\ensuremath⩽21\ensuremath{\leqslant}\ensuremath{\delta}\ensuremath{\leqslant}21\ensuremath⩽\ensuremathδ\ensuremath⩽2, the exponents depend on \ensuremathδ\ensuremath{\delta}\ensuremathδ. The results are discussed in the context of earlier observations on the topology of the underlying network.
The critical behavior of the Ising model on a one-dimensional network, which has long-range connections at distances l>1 with the probability theta(l) approximately l(-m), is studied by using Monte Carlo simulations. Through analyzing the Ising model on networks with different m values, this paper discusses the impact of the global correlation, which decays with the increase of m, on the phase transition of the Ising model. Adding the analysis of the finite-size scaling of the order parameter [M], it is observed that in the whole range of 0<m<2, a finite-temperature transition exists, and the critical exponents show consistence with mean-field values, which indicates a mean-field nature of the phase transition.
The phase transition in the Ising model and the percolation transition in the lattice percolation model have many common characteristics which have motivated researchers to explore whether the former is a percolation transition of a correlated percolation model. Previous attempts to draw such a connection have been either unsuccessful or unsatisfactory. Considering each lattice site with an Ising spin occupied and the nearest-neighbor (NN) coupling between occupied sites as a bond with a bond probability ppp depending on the NN coupling constant JJJ and the temperature TTT, we formally show that the partition function of the Ising model is the generating function of bond-correlated percolation model (BCPM) with a bond probability $p=1\ensuremath{-}\mathrm{exp}(\frac{\ensuremath{-}2J}{\mathrm{kT}})$. The BCPM has the Ising critical temperature and exponents, including $\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}},\ensuremath{\eta},\ensuremath{\beta},\ensuremath{\alpha},{\ensuremath{\alpha}}^{\ensuremath{'}}$, and $\ensuremath{\gamma}$ (perhaps also ${\ensuremath{\gamma}}^{\ensuremath{'}}$). From the connection between the Ising model and the BCPM, we also derive and hence give a geometrical meaning of the finite-size scaling and broadening at first-order phase transitions of the Ising model. Our approach may easily be extended to many spin models and give geometrical meaning to other properties of spin models.
We investigate the zero-temperature and finite-temperature phase transitions of quantum Ising and quantum rotor models. We here assume a long-range (falling off as ${1/r}^{d+\ensuremath{\sigma}},$ where r is the distance between two spins/rotors in units of lattice spacing) ferromagnetic interaction among the spins or rotors. We find that the long-range behavior of the interaction drastically modifies the universal critical behavior of the system. The corresponding upper critical dimension and the hyperscaling relation and exponents associated with the quantum transition are modified and, as expected, they attain values of short-range system when $\ensuremath{\sigma}=2.$ The dynamical exponent varies continuously as the parameter \ensuremath{\sigma} and is unity for $\ensuremath{\sigma}=2.$ The one-dimensional long-range quantum Ising system shows a phase transition at T=0T=0T=0 for all values of \ensuremath{\sigma}. The most interesting observation is that the phase diagram for $\ensuremath{\sigma}=d=1$ shows a line of Kosterlitz-Thouless transition at finite temperature even though the T=0T=0T=0 transition is a simple order-disorder transition. These finite temperature transitions are studied near the phase boundary using renormalisation group equations and a region with diverging susceptibility is located. We have also studied one-dimensional quantum rotor model which exhibits a rich and interesting transition behavior depending upon the parameter \ensuremath{\sigma}. We explore the phase diagram extending the short-range quantum nonlinear \ensuremath{\sigma} model renormalisation group equations to the present case.
The study of the Ising model from a percolation perspective has played a significant role in the modern theory of critical phenomena. We consider the celebrated square-lattice Ising model and construct percolation clusters by placing bonds, with probability p, between any pair of parallel spins within an extended range beyond nearest neighbors. At the Ising criticality, we observe two percolation transitions as p increases: starting from a disordered phase with only small clusters, the percolation system enters into a stable critical phase that persists over a wide range p_{c_{1}}<p<p_{c_{2}}, and then develops a long-ranged percolation order with giant clusters for both up and down spins. At p_{c1} and for the stable critical phase, the critical behaviors agree well with those for the Fortuin-Kasteleyn random clusters and the spin domains of the Ising model, respectively. At p_{c2}, the fractal dimension of clusters and the scaling exponent along p direction are estimated as y_{h2}=1.9580(6) and y_{p2}=0.552(9), of which the exact values remain unknown. These findings reveal interesting geometric properties of the two-dimensional Ising model that has been studied for more than 100 years.
Critical exponents for a one-dimensional general continuous Ising model with long-range ferromagnetic interactions decaying as 1/r1+σ are calculated using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between −1 and 1. The critical point behavior is investigated. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents; the singularities in the specific heat and susceptibility depend on σ. For σ=1, there are extremely weak finite size effects: the first-order and the second-order cumulants of the order parameter yield ν=2.42(1). The susceptibility exponent γ=2.259(9). Results for σ=0.7 will be shown and discussed.