Non-universal equilibrium crystal shape results from sticky steps

The anisotropic surface free energy, Andreev surface free energy, and equilibrium crystal shape (ECS) z=z(x,y) are calculated numerically using a transfer matrix approach with the density matrix renormalization group (DMRG) method. The adopted surface model is a restricted solid-on-solid (RSOS) model with "sticky" steps, i.e., steps with a point-contact type attraction between them (p-RSOS model). By analyzing the results, we obtain a first-order shape transition on the ECS profile around the (111) facet; and on the curved surface near the (001) facet edge, we obtain shape exponents having values different from those of the universal Gruber-Mullins-Pokrovsky-Talapov (GMPT) class. In order to elucidate the origin of the non-universal shape exponents, we calculate the slope dependence of the mean step height of "step droplets" (bound states of steps) using the Monte Carlo method, where p=(dz/dx, dz/dy)$, and represents the thermal averag |p| dependence of , we derive a |p|-expanded expression for the non-universal surface free energy f_{eff}(p), which contains quadratic terms with respect to |p|. The first-order shape transition and the non-universal shape exponents obtained by the DMRG calculations are reproduced thermodynamically from the non-universal surface free energy f_{eff}(p).Comment: 31 pages, 21 figure

Vicinal Surface with Langmuir Adsorption: A Decorated Restricted Solid-on-solid Model

We study the vicinal surface of the restricted solid-on-solid model coupled with the Langmuir adsorbates which we regard as two-dimensional lattice gas without lateral interaction. The effect of the vapor pressure of the adsorbates in the environmental phase is taken into consideration through the chemical potential. We calculate the surface free energy $f$, the adsorption coverage $\Theta$, the step tension $\gamma$, and the step stiffness $\tilde{\gamma}$ by the transfer matrix method combined with the density-matrix algorithm. Detailed step-density-dependence of $f$ and $\Theta$ is obtained. We draw the roughening transition curve in the plane of the temperature and the chemical potential of adsorbates. We find the multi-reentrant roughening transition accompanying the inverse roughening phenomena. We also find quasi-reentrant behavior in the step tension.Comment: 7 pages, 12 figures (png format), RevTeX 3.1, submitted to Phys. Rev.

Non semi-simple sl(2) quantum invariants, spin case

Invariants of 3-manifolds from a non semi-simple category of modules over a version of quantum sl(2) were obtained by the last three authors in [arXiv:1404.7289]. In their construction the quantum parameter $q$ is a root of unity of order $2r$ where $r>1$ is odd or congruent to $2$ modulo $4$. In this paper we consider the remaining cases where $r$ is congruent to zero modulo $4$ and produce invariants of $3$-manifolds with colored links, equipped with generalized spin structure. For a given $3$-manifold $M$, the relevant generalized spin structures are (non canonically) parametrized by $H^1(M;\mathbb C/2\mathbb Z)$.Comment: 13 pages, 16 figure

Flexible construction of hierarchical scale-free networks with general exponent

Extensive studies have been done to understand the principles behind architectures of real networks. Recently, evidences for hierarchical organization in many real networks have also been reported. Here, we present a new hierarchical model which reproduces the main experimental properties observed in real networks: scale-free of degree distribution $P(k)$ (frequency of the nodes that are connected to $k$ other nodes decays as a power-law $P(k)\sim k^{-\gamma}$) and power-law scaling of the clustering coefficient $C(k)\sim k^{-1}$. The major novelties of our model can be summarized as follows: {\it (a)} The model generates networks with scale-free distribution for the degree of nodes with general exponent $\gamma > 2$, and arbitrarily close to any specified value, being able to reproduce most of the observed hierarchical scale-free topologies. In contrast, previous models can not obtain values of $\gamma > 2.58$. {\it (b)} Our model has structural flexibility because {\it (i)} it can incorporate various types of basic building blocks (e.g., triangles, tetrahedrons and, in general, fully connected clusters of $n$ nodes) and {\it (ii)} it allows a large variety of configurations (i.e., the model can use more than $n-1$ copies of basic blocks of $n$ nodes). The structural features of our proposed model might lead to a better understanding of architectures of biological and non-biological networks.Comment: RevTeX, 5 pages, 4 figure

Product Wave Function Renormalization Group: construction from the matrix product point of view

We present a construction of a matrix product state (MPS) that approximates the largest-eigenvalue eigenvector of a transfer matrix T, for the purpose of rapidly performing the infinite system density matrix renormalization group (DMRG) method applied to two-dimensional classical lattice models. We use the fact that the largest-eigenvalue eigenvector of T can be approximated by a state vector created from the upper or lower half of a finite size cluster. Decomposition of the obtained state vector into the MPS gives a way of extending the MPS, at the system size increment process in the infinite system DMRG algorithm. As a result, we successfully give the physical interpretation of the product wave function renormalization group (PWFRG) method, and obtain its appropriate initial condition.Comment: 8 pages, 8 figure

Primordial Gravitational Waves Enhancement

We reconsider the enhancement of primordial gravitational waves that arises from a quantum gravitational model of inflation. A distinctive feature of this model is that the end of inflation witnesses a brief phase during which the Hubble parameter oscillates in sign, changing the usual Hubble friction to anti-friction. An earlier analysis of this model was based on numerically evolving the graviton mode functions after guessing their initial conditions near the end of inflation. The current study is based on an equation which directly evolves the normalized square of the magnitude. We are also able to make a very reliable estimate for the initial condition using a rapidly converging expansion for the sub-horizon regime. Results are obtained for the energy density per logarithmic wave number as a fraction of the critical density. These results exhibit how the enhanced signal depends upon the number of oscillatory periods; they also show the resonant effects associated with particular wave numbers.Comment: 25 pages, 14 figure

Colored Vertex Models, Colored IRF Models and Invariants of Trivalent Colored Graphs

We present formulas for the Clebsch-Gordan coefficients and the Racah coefficients for the root of unity representations ($N$-dimensional representations with $q^{2N}=1$) of $U_q(sl(2))$. We discuss colored vertex models and colored IRF (Interaction Round a Face) models from the color representations of $U_q(sl(2))$. We construct invariants of trivalent colored oriented framed graphs from color representations of $U_q(sl(2))$.Comment: 39 pages, January 199

Interacting Boson Theory of the Magnetization Process of the Spin-1/2 Ferromagnetic-Antiferromagnetic Alternating Heisenberg Chain

The low temperature magnetization process of the ferromagnetic-antiferromagnetic Heisenberg chain is studied using the interacting boson approximation. In the low field regime and near the saturation field, the spin wave excitations are approximated by the $\delta$ function boson gas for which the Bethe ansatz solution is available. The finite temperature properties are calculated by solving the integral equation numerically. The comparison is made with Monte Carlo calculation and the limit of the applicability of the present approximation is discussed.Comment: 4 pages, 7 figure

Self-Consistent Tensor Product Variational Approximation for 3D Classical Models

We propose a numerical variational method for three-dimensional (3D) classical lattice models. We construct the variational state as a product of local tensors, and improve it by use of the corner transfer matrix renormalization group (CTMRG), which is a variant of the density matrix renormalization group (DMRG) applied to 2D classical systems. Numerical efficiency of this approximation is investigated through trial applications to the 3D Ising model and the 3D 3-state Potts model.Comment: 12 pages, 6 figure