Generalized Ising Model on a Scale-Free Network: An Interplay of Power Laws.

We consider a recently introduced generalization of the Ising model in which individual spin strength can vary. The model is intended for analysis of ordering in systems comprising agents which, although matching in their binarity (i.e., maintaining the iconic Ising features of '+' or '-', 'up' or 'down', 'yes' or 'no'), differ in their strength. To investigate the interplay between variable properties of nodes and interactions between them, we study the model on a complex network where both the spin strength and degree distributions are governed by power laws. We show that in the annealed network approximation, thermodynamic functions of the model are self-averaging and we obtain an exact solution for the partition function. This allows us derive the leading temperature and field dependencies of thermodynamic functions, their critical behavior, and logarithmic corrections at the interface of different phases. We find the delicate interplay of the two power laws leads to new universality classes.

Narrative structure of A Song of Ice and Fire creates a fictional world with realistic measures of social complexity.

Network science and data analytics are used to quantify static and dynamic structures in George R. R. Martin's epic novels, A Song of Ice and Fire, works noted for their scale and complexity. By tracking the network of character interactions as the story unfolds, it is found that structural properties remain approximately stable and comparable to real-world social networks. Furthermore, the degrees of the most connected characters reflect a cognitive limit on the number of concurrent social connections that humans tend to maintain. We also analyze the distribution of time intervals between significant deaths measured with respect to the in-story timeline. These are consistent with power-law distributions commonly found in interevent times for a range of nonviolent human activities in the real world. We propose that structural features in the narrative that are reflected in our actual social world help readers to follow and to relate to the story, despite its sprawling extent. It is also found that the distribution of intervals between significant deaths in chapters is different to that for the in-story timeline; it is geometric rather than power law. Geometric distributions are memoryless in that the time since the last death does not inform as to the time to the next. This provides measurable support for the widely held view that significant deaths in A Song of Ice and Fire are unpredictable chapter by chapter.

The specific heat and magnetic properties of two species of spin-1/2 ladders with butterfly-shaped unit blocks.

Studies of the structural effects on the thermodynamic properties of two types of Ising-Heisenberg ladders are comprehensively reported. Each structure comprises spin-1/2 particles partitioned into adjoined blocks or cages formed from butterfly-shaped plaquettes and interacting in an Ising manner. We use the transfer-matrix approach to determine the partition functions of these models and numerically investigate their magnetization and specific-heat properties. Both models illustrate all characteristic features of low-temperature magnetization processes and thermodynamics properties such as abrupt variations of the magnetization curves, Schottky-type peak and a double-peak structure of the specific heat. Comparisons between the two ladders reveal some differences in their magnetic and thermodynamic behaviors. For instance, difference in the number of intermediate magnetization plateaus, also difference in the height and temperature-position of the specific heat peaks. We also present that the fluctuations of the specific heat with respect to the magnetic field are in highly accordance with the magnetization plateaus and magnetization jumps. Moreover, we find quite interesting threshold reentrance points within the matrix-plot of the specific heat shown in the (T/J - B/J) plane, at which the temperature-position of the Schottky peak alternatively change. Finally, we prove that the reentrance points can be satisfactorily considered as magnetization plateau witnesses even in the high temperatures.

Use of the Complex Zeros of the Partition Function to Investigate the Critical Behavior of the Generalized Interacting Self-Avoiding Trail Model.

The complex zeros of the canonical (fixed walk-length) partition function are calculated for both the self-avoiding trails model and the vertex-interacting self-avoiding walk model, both in bulk and in the presence of an attractive surface. The finite-size behavior of the zeros is used to estimate the location of phase transitions: the collapse transition in the bulk and the adsorption transition in the presence of a surface. The bulk and surface cross-over exponents, ϕ and ϕ S , are estimated from the scaling behavior of the leading partition function zeros.

Network analysis of the Viking Age in Ireland as portrayed in Cogadh Gaedhel re Gallaibh.

Cogadh Gaedhel re Gallaibh ('The War of the Gaedhil with the Gaill') is a medieval Irish text, telling how an army under the leadership of Brian Boru challenged Viking invaders and their allies in Ireland, culminating with the Battle of Clontarf in 1014. Brian's victory is widely remembered for breaking Viking power in Ireland, although much modern scholarship disputes traditional perceptions. Instead of an international conflict between Irish and Viking, interpretations based on revisionist scholarship consider it a domestic feud or civil war. Counter-revisionists challenge this view and a long-standing and lively debate continues. Here, we introduce quantitative measures to the discussions. We present statistical analyses of network data embedded in the text to position its sets of interactions on a spectrum from the domestic to the international. This delivers a picture that lies between antipodal traditional and revisionist extremes; hostilities recorded in the text are mostly between Irish and Viking-but internal conflict forms a significant proportion of the negative interactions too.

Exact solution of the critical Ising model with special toroidal boundary conditions.

The Ising model in two dimensions with special toroidal boundary conditions is analyzed. These boundary conditions, which we call duality-twisted boundary conditions, may be interpreted as inserting a specific defect line ("seam") in the system, along noncontractible circles of the cylinder, before closing it into a torus. We derive exact expressions for the eigenvalues of a transfer matrix for the critical ferromagnetic Ising model on the M×N square lattice wrapped on the torus with a specific defect line. As a result we have obtained analytically the partition function for the Ising model with such boundary conditions. In the case of infinitely long cylinders of circumference L with duality-twisted boundary conditions we obtain the asymptotic expansion of the free energy and the inverse correlation lengths. We find that the ratio of subdominant finite-size correction terms in the asymptotic expansion of the free energy and the inverse correlation lengths should be universal. We verify such universal behavior in the framework of a perturbating conformal approach by calculating the universal structure constant C_{n1n} for descendent states generated by the operator product expansion of the primary fields. For such states the calculations of an universal structure constants is a difficult task, since it involves knowledge of the four-point correlation function, which in general is not fixed by conformal invariance except for some particular cases, including the Ising model.

Role of Fourier Modes in Finite-Size Scaling above the Upper Critical Dimension.

Renormalization-group theory has stood, for over 40 years, as one of the pillars of modern physics. As such, there should be no remaining doubt regarding its validity. However, finite-size scaling, which derives from it, has long been poorly understood above the upper critical dimension d_{c} in models with free boundary conditions. In addition to its fundamental significance for scaling theories, the issue is important at a practical level because finite-size, statistical-physics systems with free boundaries and above d_{c} are experimentally relevant for long-range interactions. Here, we address the roles played by Fourier modes for such systems and show that the current phenomenological picture is not supported for all thermodynamic observables with either free or periodic boundaries. In particular, the expectation that dangerous irrelevant variables cause Gaussian-fixed-point scaling indices to be replaced by Landau mean-field exponents for all Fourier modes is incorrect. Instead, the Gaussian-fixed-point exponents have a direct physical manifestation for some modes above the upper critical dimension.

Comparison of methods to determine point-to-point resistance in nearly rectangular networks with application to a 'hammock' network.

Considerable progress has recently been made in the development of techniques to exactly determine two-point resistances in networks of various topologies. In particular, two types of method have emerged. One is based on potentials and the evaluation of eigenvalues and eigenvectors of the Laplacian matrix associated with the network or its minors. The second method is based on a recurrence relation associated with the distribution of currents in the network. Here, these methods are compared and used to determine the resistance distances between any two nodes of a network with topology of a hammock.

Dimer model on a triangular lattice.

We analyze the partition function of the dimer model on an M×N triangular lattice wrapped on a torus obtained by Fendley, Moessner, and Sondhi [Phys. Rev. B 66, 214513 (2002)]. From a finite-size analysis we have found that the dimer model on such a lattice can be described by a conformal field theory having a central charge c=-2. The shift exponent for the specific heat is found to depend on the parity of the number of lattice sites N along a given lattice axis: e.g., for odd N we obtain the shift exponent λ=1, while for even N it is infinite (λ=∞). In the former case, therefore, the finite-size specific-heat pseudocritical point is size dependent, while in the latter case it coincides with the critical point of the thermodynamic limit.