The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, important steps have been made toward understanding the qualitatively new critical phenomena in complex networks. The results, concepts, and methods of this rapidly developing field are reviewed. Two closely related classes of these critical phenomena are considered, namely, structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. Systems where a network and interacting agents on it influence each other are also discussed. A wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, $k$-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks are mentioned. Strong finite-size effects in these systems and open problems and perspectives are also discussed.

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Critical phenomena in complex networks

http://files.janapv.webnode.cz/200000097-8a03f8afdf/ex_bias_nano.pdf

Exchange bias in nanostructures

R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

Development of theories of inhomogeneous fluids, J.S. Rowlinson statistical mechanical sum rules, J.R. Henderson density functionals in the theory of non-uniform fluids, R. Evans integral equation theories for inhomogeneous fluids, Douglas Henderson inhomogeneous two-dimensional plasmas, B. Jancovici statistical mechanics of electrolytes at interfaces, L. Blum and Douglas Henderson wetting experiments, Carl Franck fluids between walls and in pores, Marcelo Lozada-Cassou freezing, A.D.J. Haymet nucleation, David W. Oxtoby liquid crystals, J.F. Marko nature of microemulsion, K.A. Dawson and Christopher J. Mundy kinetic theory of strongly inhomogeneous fluids, H. Ted Davis.

Fundamentals of Inhomogeneous Fluids

As their name indicates, liquid crystals simultaneously exhibit some characteristics common to both ordinary isotropic liquids and solid crystals. This ambivalence is also found in the effects of surfaces on these systems which lead to a great diversity of phenomena. These phenomena are reviewed focusing on nematic liquid crystals which have the simplest structure among the many existing types and which have been the most extensively studied. Three main kinds of effects can be distinguished. The first concerns the perturbation of the liquid crystalline structure close to the surface. Beyond this transition region, the bulk liquid crystalline structure is recovered with an orientation which is fixed by the surface: this phenomenon of orientation of liquid crystals by surfaces is the so-called anchoring. Finally, close to bulk phase transitions, critical adsorption or wetting can occur at surfaces as is also seen in isotropic systems.

https://iopscience.iop.org/article/10.1088/0034-4885/54/3/002/pdf

Surface effects and anchoring in liquid crystals

We consider the single-spin-flip dynamics of the random-field Ising model on a Bethe lattice at zero temperature in the presence of a uniform external field. We determine the average magnetization as the external field is varied from to by setting up the self-consistent field equations, which we show are exact in this case. The qualitative behaviour of magnetization as a function of the external field unexpectedly depends on the coordination number z of the Bethe lattice. For z = 3, with a Gaussian distribution of the quenched random fields, we find no jump in magnetization for any non-zero strength of disorder. For , for weak disorder the magnetization shows a jump discontinuity as a function of the external uniform field, which disappears for a larger variance of the quenched field. We determine exactly the critical point separating smooth hysteresis curves from those with a jump. We have checked our results by Monte Carlo simulations of the model on three- and four-coordinated random graphs, which for large system sizes give the same results as on the Bethe lattice, but avoid surface effects altogether.

https://www.lassp.cornell.edu/sethna/pubPDF/BetheLatticeHysteresis.pdf

Zero-temperature hysteresis in the random-field Ising model on a Bethe lattice

We consider the single-spin-flip dynamics of the random-field Ising model on a Bethe lattice at zero temperature in the presence of a uniform external field. We determine the average magnetization as the external field is varied from minus infinity to plus infinity by setting up the self-consistent field equations, which we show are exact in this case. We find that for a 3-coordinated Bethe lattice, there is no jump discontinuity in magnetization for arbitrarily small gaussian disorder, but the discontinuity is present for larger coordination numbers. We have checked our results by Monte Carlo simulations employing a technique for simulating classical interacting systems on the Bethe lattice which avoids surface effects altogether.

Zero-temperature Hysteresis in Random-field Ising Model on a Bethe Lattice

Drawing upon a model of Sato and Arrott, it is proposed that the spin-glass behavior observed in iron-rich ${\mathrm{Fe}}_{1\ensuremath{-}x}{\mathrm{Al}}_{x}$ alloys arises by virtue of competition between a nearest-neighbor Fe---Fe ferromagnetic exchange $J$ and a further-neighbor Fe---Al---Fe antiferromagnetic superexchange $\ensuremath{-}\ensuremath{\alpha}J$. The special crystal structure of these alloys makes possible a mapping (decimation) of this model onto a simple-cubic magnet with random exchange. Previous approximate solutions of the simple-cubic model may then be used to construct the ${\mathrm{Fe}}_{1\ensuremath{-}x}{\mathrm{Al}}_{x}$ magnetic phase diagram. For an appropriately chosen ratio $\ensuremath{\alpha}$ (antiferro/ferro), an acceptable fit to the experimental phase diagram is obtained.

Spin-glass behavior in iron-aluminum alloys: A microscopic model

The molecular field theory of nematics is rephrased in density functional language. The free energy in the nematic state is expressed as a functional expansion around the isotropic state in powers of the nematic order parameters Pl and the fractional density change eta at the isotropic-nematic transition. The crucial quantities in the expansion are spherical harmonics of the direct correlation functions in the isotropic state. Sensible approximations to these direct correlation functions lead to other well known formulations of nematic field theory. An exact (in the authors' approximation) result for eta is obtained in terms of the potential parameters Ul and the order parameters Pl at the phase transition. The authors discuss the relationship of their approach to other microscopic theories of nematics. The theory can be extended to discuss inhomogeneous systems.

https://iopscience.iop.org/article/10.1088/0305-4470/16/7/030/pdf

Molecular field theory of nematics: density functional approach. I. Bulk effects

We study hysteresis in the random-field Ising model with an asymmetric distribution of quenched fields, in the limit of low disorder in two and three dimensions. We relate the spin flip process to bootstrap percolation, and show that the characteristic length for self-averaging ${L}^{\ensuremath{\star}}$ increases as $\mathrm{exp}[\mathrm{exp}(J/\ensuremath{\Delta})]$ in 2D, and as $\mathrm{exp}{\mathrm{exp}[\mathrm{exp}(J/\ensuremath{\Delta})]}$ in 3D, for disorder strength $\ensuremath{\Delta}$ much less than the exchange coupling $J$. For system size $1\ensuremath{\ll}Ll{L}^{\ensuremath{\star}}$, the coercive field ${h}_{\mathrm{coer}}$ varies as $2J\ensuremath{-}\ensuremath{\Delta}\mathrm{ln}\mathrm{ln}L$ for the square lattice, and as $2J\ensuremath{-}\ensuremath{\Delta}\mathrm{ln}\mathrm{ln}\mathrm{ln}L$ on the cubic lattice. Its limiting value is 0 for $L\ensuremath{\rightarrow}\ensuremath{\infty}$ for both square and cubic lattices. For lattices with coordination number 3, the limiting magnetization shows no jump, and ${h}_{\mathrm{coer}}$ tends to $J$.

/pdf/hysteresis-in-the-random-field-ising-model-and-bootstrap-2hf4ifvkip.pdf

Prabodh Shukla

Papers

Zero-temperature hysteresis in the random-field Ising model on a Bethe lattice

Zero-temperature Hysteresis in Random-field Ising Model on a Bethe Lattice

Spin-glass behavior in iron-aluminum alloys: A microscopic model

Molecular field theory of nematics: density functional approach. I. Bulk effects

Hysteresis in the random-field Ising model and bootstrap percolation.