On a phase transition in a one-dimensional non-homogeneous model

Bundaru, M; Grünfeld, C. P.

doi:10.1088/0305-4470/32/6/002

Cited by 6 publications

(6 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An infinite-order percolation transition has been seen in models of growing networks [31,32,33,34,35,36,37,38], with exponential scaling in the size of the giant component above the percolation threshold. A prior observation of a finite-temperature, inverted Berezinskii-KosterlitzThouless singularity similar to the one described above was made in Ising models on an inhomogeneous growing network [39] and on a one-dimensional inhomogeneous lattice [40,41,42,43].…”

Section: Introductionmentioning

confidence: 67%

Inverted Berezinskii-Kosterlitz-Thouless singularity and high-temperature algebraic order in an Ising model on a scale-free hierarchical-lattice small-world network

2006

View full text Add to dashboard Cite

We have obtained exact results for the Ising model on a hierarchical lattice incorporating three key features characterizing many real-world networks-a scale-free degree distribution, a high clustering coefficient, and the small-world effect. By varying the probability p of long-range bonds, the entire spectrum from an unclustered, non-small-world network to a highly-clustered, small-world system is studied. Using the self-similar structure of the network, we obtain analytic expressions for the degree distribution P (k) and clustering coefficient C for all p, as well as the average path length ℓ for p = 0 and 1. The ferromagnetic Ising model on this network is studied through an exact renormalization-group transformation of the quenched bond probability distribution, using up to 562,500 renormalized probability bins to represent the distribution. For p < 0.494, we find powerlaw critical behavior of the magnetization and susceptibility, with critical exponents continuously varying with p, and exponential decay of correlations away from Tc. For p ≥ 0.494, in fact where the network exhibits small-world character, the critical behavior radically changes: We find a highly unusual phase transition, namely an inverted Berezinskii-Kosterlitz-Thouless singularity, between a low-temperature phase with non-zero magnetization and finite correlation length and a hightemperature phase with zero magnetization and infinite correlation length, with power-law decay of correlations throughout the phase. Approaching Tc from below, the magnetization and the susceptibility respectively exhibit the singularities of exp(−C/ √ Tc − T ) and exp(D/ √ Tc − T ), with C and D positive constants. With long-range bond strengths decaying with distance, we see a phase transition with power-law critical singularities for all p, and evaluate an unusually narrow critical region and important corrections to power-law behavior that depend on the exponent characterizing the decay of long-range interactions.

show abstract

Section: Introductionmentioning

confidence: 67%

Inverted Berezinskii-Kosterlitz-Thouless singularity and high-temperature algebraic order in an Ising model on a scale-free hierarchical-lattice small-world network

2006

View full text Add to dashboard Cite

show abstract

“…Phys. 64, 193 (1991); [3] M. Bundaru and C.P. Grünfeld, On a phase transition in a one-dimensional non-homogeneous model, J. Phys.…”

mentioning

confidence: 99%

Phase Transition with the Berezinskii-Kosterlitz-Thouless Singularity in the Ising Model on a Growing Network

2005

View full text Add to dashboard Cite

We consider the ferromagnetic Ising model on a highly inhomogeneous network created by a growth process. We find that the phase transition in this system is characterised by the Berezinskii-Kosterlitz-Thouless singularity, although critical fluctuations are absent and the meanfield description is exact. Below this infinite order transition, the magnetization behaves as exp(−const/ √ Tc − T ). We show that the critical point separates the phase with the power-law distribution of the linear response to a local field and the phase where this distribution rapidly decreases. We suggest that this phase transition occurs in a wide range of cooperative models with a strong infinite-range inhomogeneity. Note added.-After this paper had been published, we have learnt that the infinite order phase transition in the effective model we arrived at was discovered by O. Costin, R.D. Costin and C.P. Grünfeld in 1990. This phase transition was considered in the following papers: We would like to note that Costin, Costin and Grünfeld treated this model as a one-dimensional inhomogeneous system. We have arrived at the same model as a one-replica ansatz for a random growing network where expected to find a phase transition of this sort based on earlier results for random networks (see the text). We have also obtained the distribution of the linear response to a local field, which characterises correlations in this system. We thank O. Costin and S. Romano for indicating these publications of 90s. The ferromagnetic Ising model on lattices has an ordinary second-order phase transition [5]. Above the upper critical dimension of the model, the critical fluctuations are absent, and the mean-field description of this transition is exact. In particular, this takes place if couplings between all spins are equal-infinite-range interactions. In this Letter we report the finding of a phase transition with the Berezinskii-Kosterlitz-Thouless (BKT) singularity in the Ising model on a growing network, which is infinitely-dimensional as most of networks. This transition is quite unusual for an infinitely-dimensional system as well for a cooperative model with the order parameter of discrete symmetry.Recall that in "ordinary" continuous phase transitions, pair correlations of an order parameter show a slow, power-law space decay only at the critical point, T c , and decay exponentially both in the low-and high- * Electronic address: michel.bauer@cea.fr † Electronic address: coulomb@dsm-mail.saclay.cea.fr ‡ Electronic address: sdorogov@fis.ua.pt temperature phases. This behavior was observed in the Ising model on equilibrium complex networks [6,7,8,9, 10] (for percolation and for disease spreading on equilibrium networks, see Refs. [11] and [12], respectively). In contrast, the BKT phase transition [13,14] separates the phase with rapidly decreasing correlations and the critical phase with correlations decaying by a power law. The contact of these two phases is characterised by specific dependences. For example, the order parameter behaves as, and th...

show abstract

“…Among issues most recently addressed have been the occurrence of true or algebraic [6,7] order in the geometric or thermal long-range correlations, and the connection between these geometric and thermal characteristics. In Ising magnetic systems on a one-dimensional inhomogeneous lattice [8,9,10,11] and on an inhomogeneous growing network [12], a Berezinskii-Kosterlitz-Thouless (BKT) phase in which the thermal correlations between the spins decay algebraically with distance was found. In growing networks [13,14,15,16,17,18,19,20], geometric algebraic correlations were seen with the exponential (non-power-law) scaling of the size of the giant FIG.…”

mentioning

confidence: 99%

Critical percolation phase and thermal Berezinskii-Kosterlitz-Thouless transition in a scale-free network with short-range and long-range random bonds

2009

View full text Add to dashboard Cite

Percolation in a scale-free hierarchical network is solved exactly by renormalization-group theory in terms of the different probabilities of short-range and long-range bonds. A phase of critical percolation, with algebraic ͓Berezinskii-Kosterlitz-Thouless ͑BKT͔͒ geometric order, occurs in the phase diagram in addition to the ordinary ͑compact͒ percolating phase and the nonpercolating phase. It is found that no connection exists between, on the one hand, the onset of this geometric BKT behavior and, on the other hand, the onsets of the highly clustered small-world character of the network and of the thermal BKT transition of the Ising model on this network. Nevertheless, both geometric and thermal BKT behaviors have inverted characters, occurring where disorder is expected, namely, at low bond probability and high temperature, respectively. This may be a general property of long-range networks. Scale-free networks are of high current interest ͓1-5͔ due to their ubiquitous occurrence in physical, biological, social, and information systems and due to their distinctive geometric and thermal properties. The geometric properties reflect the connectivity of the points of the network. The thermal properties reflect the interactions, along the geometric lines of connectivity, between degrees of freedom located at the points of the network. These interacting degrees of freedom could be voters influencing each other, persons communicating a disease, etc. and can be represented by model systems. Among issues most recently addressed have been the occurrence of true or algebraic ͓6,7͔ order in the geometric or thermal long-range correlations, and the connection between these geometric and thermal characteristics. In Ising magnetic systems on a one-dimensional inhomogeneous lattice ͓8-11͔ and on an inhomogeneous growing network ͓12͔, a Berezinskii-Kosterlitz-Thouless ͑BKT͒ phase in which the thermal correlations between the spins decay algebraically with distance was found. In growing networks ͓13-20͔, geometric algebraic correlations were seen with the exponential ͑non-power-law͒ scaling of the size of the giant component above the percolation threshold. The connection between geometric and thermal properties was investigated with an Ising magnetic system on a hierarchical lattice that can be continuously tuned from non-small-world to highly clustered small-world via increase in the occurrence of quenchedrandom long-range bonds ͓21͔. Whereas in the non-smallworld regime a standard second-order phase transition was found, when the small-world regime is entered, an inverted BKT transition was found, with a high-temperature algebraically ordered phase and a low-temperature phase with true long-range order but delayed short-range order. Algebraic order in the thermal correlations has also been found in a community network ͓22͔. In the current work, the geometric percolation property of the quenched-random long-range bonds is studied, aiming to relate the geometric properties to the algebraic thermal properties. From an exact re...

show abstract

On a phase transition in a one-dimensional non-homogeneous model

Cited by 6 publications

References 3 publications

Inverted Berezinskii-Kosterlitz-Thouless singularity and high-temperature algebraic order in an Ising model on a scale-free hierarchical-lattice small-world network

Inverted Berezinskii-Kosterlitz-Thouless singularity and high-temperature algebraic order in an Ising model on a scale-free hierarchical-lattice small-world network

Phase Transition with the Berezinskii-Kosterlitz-Thouless Singularity in the Ising Model on a Growing Network

Critical percolation phase and thermal Berezinskii-Kosterlitz-Thouless transition in a scale-free network with short-range and long-range random bonds

Contact Info

Product

Resources

About