Hinczewski, Michael; Berker, A. Nihat Inverted Berezinskii-Kosterlitz-Thouless singularity and high-temperature algebraic order in an Ising model on a scale-free hierarchical-lattice small-world network. (English) Zbl 1244.82013 Phys. Rev. E (3) 73, No. 6, Article ID 066126, 22 p. (2006). Summary: We obtain 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 power-law critical behavior of the magnetization and susceptibility, with critical exponents continuously varying with \(p\), and exponential decay of correlations away from \(T_c\). For \(p\geq 0.494\), in fact where the network exhibits small-world character, the critical behavior radically changes: We find a highly unusual phase transition, namely an inverted Berezinskiĭ-Kosterlitz-Thouless singularity, between a low-temperature phase with nonzero magnetization and finite correlation length and a high-temperature phase with zero magnetization and infinite correlation length, with power-law decay of correlations throughout the phase. Approaching \(T_c\) from below, the magnetization and the susceptibility, respectively, exhibit the singularities of \(\exp(-C/\sqrt{T_c-T})\) and \(\exp(D/\sqrt{T_c-T})\), 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. Cited in 16 Documents MSC: 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 82B26 Phase transitions (general) in equilibrium statistical mechanics 05C82 Small world graphs, complex networks (graph-theoretic aspects) Keywords:Ising model; small-world network PDFBibTeX XMLCite \textit{M. Hinczewski} and \textit{A. N. Berker}, Phys. Rev. E (3) 73, No. 6, Article ID 066126, 22 p. (2006; Zbl 1244.82013) Full Text: DOI arXiv References: [1] DOI: 10.1103/RevModPhys.74.47 · Zbl 1205.82086 [2] DOI: 10.1137/S003614450342480 · Zbl 1029.68010 [3] DOI: 10.1080/00018730110112519 [4] DOI: 10.1038/30918 · Zbl 1368.05139 [5] DOI: 10.1126/science.286.5439.509 · Zbl 1226.05223 [6] DOI: 10.1088/0305-4470/33/47/304 · Zbl 0970.82005 [7] DOI: 10.1007/s100510050067 [8] DOI: 10.1103/PhysRevE.64.057104 [9] DOI: 10.1103/PhysRevE.66.018101 [10] DOI: 10.1103/PhysRevE.68.027101 [11] DOI: 10.1103/PhysRevE.64.056135 [12] DOI: 10.1016/S0378-4371(02)00740-9 · Zbl 0995.82016 [13] DOI: 10.1016/S0375-9601(02)01232-X · Zbl 0999.82014 [14] DOI: 10.1103/PhysRevE.66.016104 [15] DOI: 10.1140/epjb/e2002-00220-0 [16] DOI: 10.1103/PhysRevE.67.026123 [17] DOI: 10.1088/0022-3719/12/22/035 [18] DOI: 10.1103/PhysRevB.24.496 [19] DOI: 10.1103/PhysRevB.30.244 [20] DOI: 10.1016/S0378-4371(01)00369-7 · Zbl 0972.57003 [21] DOI: 10.1103/PhysRevE.63.062101 [22] DOI: 10.1103/PhysRevE.65.066122 [23] DOI: 10.1016/S0378-4371(02)00741-0 · Zbl 0995.60103 [24] DOI: 10.1103/PhysRevE.65.056101 [25] DOI: 10.1016/j.physa.2005.08.020 [26] DOI: 10.1103/PhysRevLett.94.018702 [27] DOI: 10.1103/PhysRevE.71.046141 [28] DOI: 10.1103/PhysRevE.71.056131 [29] V. L. Berezinskii, Sov. Phys. JETP 32 pp 493– (1971) ISSN: http://id.crossref.org/issn/0038-5646 [30] DOI: 10.1088/0022-3719/6/7/010 [31] DOI: 10.1103/PhysRevE.64.041902 [32] DOI: 10.1103/PhysRevE.64.066110 [33] DOI: 10.1103/PhysRevE.66.055101 [34] DOI: 10.1088/0305-4470/35/5/305 · Zbl 1032.91709 [35] DOI: 10.1023/A:1022842013935 · Zbl 1014.60093 [36] DOI: 10.1140/epjb/e2003-00290-4 [37] DOI: 10.1002/rsa.20041 · Zbl 1063.05121 [38] DOI: 10.1016/j.physa.2004.05.020 [39] DOI: 10.1103/PhysRevLett.94.200602 [40] DOI: 10.1007/BF01334762 · Zbl 0718.60127 [41] DOI: 10.1007/BF01057873 · Zbl 0952.82506 [42] DOI: 10.1088/0305-4470/32/6/002 · Zbl 0964.82016 [43] DOI: 10.1142/S0217984995001455 [44] A. A. Migdal, Zh. Eksp. Teor. Fiz. 69 pp 1457– (1975) ISSN: http://id.crossref.org/issn/0044-4510 [45] A. A. Migdal, Sov. Phys. JETP 42 pp 743– (1976) ISSN: http://id.crossref.org/issn/0038-5646 [46] DOI: 10.1016/0003-4916(76)90066-X [47] DOI: 10.1103/PhysRevB.29.1315 [48] DOI: 10.1103/PhysRevB.17.3650 [49] DOI: 10.1103/PhysRevB.19.2488 [50] DOI: 10.1103/PhysRevB.29.2630 [51] DOI: 10.1103/PhysRevB.51.8266 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.