Quantum Spins and Quasiperiodicity: a real space renormalization group approach

We study the antiferromagnetic spin-1/2 Heisenberg model on a two-dimensional bipartite quasiperiodic structure, the octagonal tiling -- the aperiodic equivalent of the square lattice for periodic systems. An approximate block spin renormalization scheme is described for this problem. The ground state energy and local staggered magnetizations for this system are calculated, and compared with the results of a recent Quantum Monte Carlo calculation for the tiling. It is conjectured that the ground state energy is exactly equal to that of the quantum antiferromagnet on the square lattice.Comment: To appear in Physical Review Letter

Surface Properties of Aperiodic Ising Quantum Chains

We consider Ising quantum chains with quenched aperiodic disorder of the coupling constants given through general substitution rules. The critical scaling behaviour of several bulk and surface quantities is obtained by exact real space renormalization.Comment: 4 pages, RevTex, reference update

Invaded cluster algorithm for critical properties of periodic and aperiodic planar Ising models

We demonstrate that the invaded cluster algorithm, recently introduced by Machta et al, is a fast and reliable tool for determining the critical temperature and the magnetic critical exponent of periodic and aperiodic ferromagnetic Ising models in two dimensions. The algorithm is shown to reproduce the known values of the critical temperature on various periodic and quasiperiodic graphs with an accuracy of more than three significant digits. On two quasiperiodic graphs which were not investigated in this respect before, the twelvefold symmetric square-triangle tiling and the tenfold symmetric T\"ubingen triangle tiling, we determine the critical temperature. Furthermore, a generalization of the algorithm to non-identical coupling strengths is presented and applied to a class of Ising models on the Labyrinth tiling. For generic cases in which the heuristic Harris-Luck criterion predicts deviations from the Onsager universality class, we find a magnetic critical exponent different from the Onsager value. But also notable exceptions to the criterion are found which consist not only of the exactly solvable cases, in agreement with a recent exact result, but also of the self-dual ones and maybe more.Comment: 15 pages, 5 figures; v2: Fig. 5b replaced, minor change

Strong disorder renormalization group study of aperiodic quantum Ising chains

We employ an adaptation of a strong-disorder renormalization-group technique in order to analyze the ferro-paramagnetic quantum phase transition of Ising chains with aperiodic but deterministic couplings under the action of a transverse field. In the presence of marginal or relevant geometric fluctuations induced by aperiodicity, for which the critical behavior is expected to depart from the Onsager universality class, we derive analytical and asymptotically exact expressions for various critical exponents (including the correlation-length and the magnetization exponents, which are not easily obtainable by other methods), and shed light onto the nature of the ground state structures in the neighborhood of the critical point. The main results obtained by this approach are confirmed by finite-size scaling analyses of numerical calculations based on the free-fermion method

Three modes of adaptive speciation in spatially structured populations

Adaptive speciation with gene flow via the evolution of assortative mating has classically been studied in one of two different scenarios. First, speciation can occur if frequency-dependent competition in sympatry induces disruptive selection, leading to indirect selection for mating with similar phenotypes. Second, if a subpopulation is locally adapted to a specific environment, there is indirect selection against hybridizing with maladapted immigrants. While both of these mechanisms have been modeled many times, the literature lacks models that allow direct comparisons between them. Here, we incorporate both frequency-dependent competition and local adaptation into a single model, and investigate whether and how they interact in driving speciation. We report two main results. First, we show that, individually, the two mechanisms operate under separate conditions, hardly influencing each other when one of them alone is sufficient to drive speciation. Second, we also find that the two mechanisms can operate together, leading to a third speciation mode, in which speciation is initiated by selection against maladapted migrants, but completed by within-deme competition in a distinct second phase. While this third mode bears some similarity to classical reinforcement, it happens considerably faster, and both newly formed species go on to coexist in sympatry. KEYWRODS: parapatric speciation, adaptive speciation, assortative mating, frequency dependent selection, reinforement, local adaptatio

An Analytically Tractable Model for Competitive Speciation

Several recent models have shown that frequency-dependent disruptive selection created by intraspecific competition can lead to the evolution of assortative mating 4 and, thus, to competitive sympatric speciation. However, since most results from these 5 models rely on limited numerical analyses, their generality has been subject to considerable debate. Here, we consider one of the standard models, the so-called Roughgarden model, with a simplified genetics where the selected trait is determined by a single diallelic locus. This model is sufficiently complex to maintain key properties of the general multilocus case, but still simple enough to allow for a comprehensive analytical treatment. By means of invasion fitness analysis, we describe the impact of all model parameters on the evolution of assortative mating. Depending on (1) the strength and (2) shape of stabilizing selection, (3) the strength and (4) shape of pairwise competition, (5) the shape of the mating function, and (6) the type of assortative mating, which may or may not lead to sexual selection, we find five different evolutionary regimes. In one of these regimes, the evolution of complete reproductive isolation is possible through arbitrarily small steps in the strength of assortative mating. Our approach provides a mechanistic understanding of several phenomena that have been found in previous models. The results demonstrate how, even in a simple model of competitive speciation, results depend in a complex way on ecological and genetic parameters

Entanglement entropy of aperiodic quantum spin chains

We study the entanglement entropy of blocks of contiguous spins in non-periodic (quasi-periodic or more generally aperiodic) critical Heisenberg, XX and quantum Ising spin chains, e.g. in Fibonacci chains. For marginal and relevant aperiodic modulations, the entanglement entropy is found to be a logarithmic function of the block size with log-periodic oscillations. The effective central charge, c_eff, defined through the constant in front of the logarithm may depend on the ratio of couplings and can even exceed the corresponding value in the homogeneous system. In the strong modulation limit, the ground state is constructed by a renormalization group method and the limiting value of c_eff is exactly calculated. Keeping the ratio of the block size and the system size constant, the entanglement entropy exhibits a scaling property, however, the corresponding scaling function may be nonanalytic.Comment: 6 pages, 2 figure

Field behavior of an Ising model with aperiodic interactions

We derive exact renormalization-group recursion relations for an Ising model, in the presence of external fields, with ferromagnetic nearest-neighbor interactions on Migdal-Kadanoff hierarchical lattices. We consider layered distributions of aperiodic exchange interactions, according to a class of two-letter substitutional sequences. For irrelevant geometric fluctuations, the recursion relations in parameter space display a nontrivial uniform fixed point of hyperbolic character that governs the universal critical behavior. For relevant fluctuations, in agreement with previous work, this fixed point becomes fully unstable, and there appears a two-cycle attractor associated with a new critical universality class.Comment: 9 pages, 1 figure (included). Accepted for publication in Int. J. Mod. Phys.

Weighted Dirac combs with pure point diffraction

A class of translation bounded complex measures, which have the form of weighted Dirac combs, on locally compact Abelian groups is investigated. Given such a Dirac comb, we are interested in its diffraction spectrum which emerges as the Fourier transform of the autocorrelation measure. We present a sufficient set of conditions to ensure that the diffraction measure is a pure point measure. Simultaneously, we establish a natural link to the theory of the cut and project formalism and to the theory of almost periodic measures. Our conditions are general enough to cover the known theory of model sets, but also to include examples such as the visible lattice points.Comment: 44 pages; several corrections and improvement

Anderson Localization, Non-linearity and Stable Genetic Diversity

In many models of genotypic evolution, the vector of genotype populations satisfies a system of linear ordinary differential equations. This system of equations models a competition between differential replication rates (fitness) and mutation. Mutation operates as a generalized diffusion process on genotype space. In the large time asymptotics, the replication term tends to produce a single dominant quasispecies, unless the mutation rate is too high, in which case the populations of different genotypes becomes de-localized. We introduce a more macroscopic picture of genotypic evolution wherein a random replication term in the linear model displays features analogous to Anderson localization. When coupled with non-linearities that limit the population of any given genotype, we obtain a model whose large time asymptotics display stable genotypic diversityComment: 25 pages, 8 Figure