Narrow Results

The structural phase transition in complex network models is known to yield knowledge relevant for several problems of practical application. Examples include resilience of artificial networks, dynamics of epidemic spreading, among others. The model of random graphs is probably the simplest model of complex networks, and the solution of this model falls into the universality class mean field percolation. In this work, we concentrate on a similar problem, namely, the structural phase transition in the ensemble of random acyclic graphs. It should be noted that several approaches to the problem of random graphs, such as generating functions or the Molloy–Reed criterion, rely on the fact that before the critical point, cycles should not be present in random graphs. In this way, up to the critical point our solution should produce results that are equivalent to these other methods. Our approach takes advantage of the fact that acyclic graphs allow for an exact combinatorial enumeration of the whole ensemble, what leads to an exact expression for the entropy of this system. With this definition of entropy we can determine the onset of the critical transition as well as the critical exponents associated with the transition. Our results are illustrated with Monte-Carlo results and are discussed within the context of general random graphs, as well as in comparison with another model of acyclic graphs.

The asymptotic iteration method is used to calculate the eigenenergies for the asymmetrical quantum anharmonic oscillator potentials , with (α = 2) for quartic, and (α = 3) for sextic asymmetrical quantum anharmonic oscillators. An adjustable parameter β is introduced in the method to improve its rate of convergence. Comparing the present results with the exact numerical values, and with the numerical results of the earlier works, it is found that asymptotically, this method gives accurate results over the full range of parameter values A_j.

Development of holomorphy-based methods in super-Yang–Mills theories started in the early 1980s and lead to a number of breakthrough results. I review some results in which I participated. The discovery of Seiberg’s duality and the Seiberg–Witten solution of ${𝒩}=2$ Yang–Mills were the milestones in the long journey of which, I assume, much will be said in other talks. I will focus on the discovery (2003) of non-Abelian vortex strings with various degrees of supersymmetry, supported in some four-dimensional Yang–Mills theories and some intriguing implications of this discovery. One of the recent results is the observation of a soliton string in the bulk ${𝒩}=2$ theory with the $\mathrm{\text{U}}(2)$ gauge group and four flavors, which can become critical in a certain limit. This is the case of a “reverse holography,” with a very transparent physical meaning.

We consider close-packed dimers, or perfect matchings, on two-dimensional regular lattices. We review known results and derive new expressions for the free energy, entropy, and the molecular freedom of dimers for a number of lattices including the simple-quartic, honeycomb, triangular, kagome, 3-12 and its dual, and 4-8 and its dual Union Jack lattices. The occurrence and nature of phase transitions are also elucidated and discussed in each case.

We calculate zeros of the q-state Potts model partition function Z(G_Λ,q,v) for large q, where v is the temperature variable and G_Λ is a section of a lattice Λ with coordination number κ_Λ and various boundary conditions. Lattice types studied include square, triangular, honeycomb, and kagomé. We show that for large q these zeros take on approximately circular patterns in the complex x_Λ plane, where x_Λ=v/q^2/κ_Λ. This generalizes a known result for the square lattice to the other lattices considered.

The mixed-spin Ising model on a decorated square lattice with two different decorating spins of integer magnitudes S_B = 1 and S_C = 2 placed on horizontal and vertical bonds of the lattice, respectively, is examined within an exact analytical approach based on the generalized decoration–iteration mapping transformation. Besides the ground-state analysis, finite-temperature properties of the system are investigated in detail. The most interesting numerical result to emerge from our study relates to a striking critical behavior of the spontaneously ordered "quasi-1D" spin system. It was found that this quite remarkable spontaneous order arises when one sublattice of the decorating spins (either S_B or S_C) tends toward their "nonmagnetic" spin state S = 0, and the system becomes disordered only upon further single-ion anisotropy strengthening. In particular, the effect of single-ion anisotropy upon the temperature dependence of the total and sublattice magnetization is investigated.

In this paper we discuss the properties of the reduced density matrix of quantum many body systems with permutational symmetry and present basic quantification of the entanglement in terms of the von Neumann (VNE), Renyi and Tsallis entropies. In particular, we show, on the specific example of the spin 1/2 Heisenberg model, how the RDM acquires a block diagonal form with respect to the quantum number k fixing the polarization in the subsystem conservation of S_z and with respect to the irreducible representations of the S_n group. Analytical expression for the RDM elements and for the RDM spectrum are derived for states of arbitrary permutational symmetry and for arbitrary polarizations. The temperature dependence and scaling of the VNE across a finite temperature phase transition is discussed and the RDM moments and the Rényi and Tsallis entropies calculated both for symmetric ground states of the Heisenberg chain and for maximally mixed states.

In this paper, inspired by the pseudo-fractal networks (PFN) and the delayed pseudo-fractal networks (DPFN), we present a novel delayed pseudo-fractal networks model, denoted by NDPFN. Different from the generation algorithm of those two networks, every edge of the novel model has a time-delay to generate new nodes after producing one node. We derive exactly the main structural properties of the novel networks: degree distribution, clustering coefficient, diameter and average path length. Analytical results show that the novel networks have small-world effect and scale-free topology. Comparing topological parameters of these three networks, we find that the degree exponent of the novel networks is the largest while the clustering coefficient and the average path length are the smallest. It means that this kind of delay could weaken the heterogeneity and the small-world features of the network. Particularly, the delay effect in the NDPFN is contrary to that in the DPFN, which illustrates the variety of delay method could produce different effects on the network structure. These present findings may be helpful for a deeper understanding of the time-delay influence on the network topology.

In this paper we discuss the properties of the reduced density matrix of quantum many body systems with permutational symmetry and present basic quantification of the entanglement in terms of the von Neumann (VNE), Renyi and Tsallis entropies. In particular, we show, on the specific example of the spin 1/2 Heisenberg model, how the RDM acquires a block diagonal form with respect to the quantum number k fixing the polarization in the subsystem conservation of S_z and with respect to the irreducible representations of the S_n group. Analytical expression for the RDM elements and for the RDM spectrum are derived for states of arbitrary permutational symmetry and for arbitrary polarizations. The temperature dependence and scaling of the VNE across a finite temperature phase transition is discussed and the RDM moments and the Rényi and Tsallis entropies calculated both for symmetric ground states of the Heisenberg chain and for maximally mixed states.