Wavevector-Dependent Susceptibility in Z-Invariant Pentagrid Ising Model

Abstract

We study the q-dependent susceptibility χ(q) of a Z-invariant ferromagnetic Ising model on a Penrose tiling, as first introduced by Korepin using de Bruijn's pentagrid for the rapidity lines. The pair-correlation function for this model can be calculated exactly using the quadratic difference equations from our previous papers. Its Fourier transform χ(q) is studied using a novel way to calculate the joint probability for the pentagrid neighborhoods of the two spins, reducing this calculation to linear programming. Since the lattice is quasiperiodic, we find that χ(q) is aperiodic and has everywhere dense peaks, which are not all visible at very low or high temperatures. More and more peaks become visible as the correlation length increases-that is, as the temperature approaches the critical temperature.

Q-Dependent Susceptibilities in Ferromagnetic Quasiperiodic Z-Invariant Ising Models

“…The number of visible peaks increases as T approaches T c . (1) There are many other quasiperiodic sequences. Still we have examined a variety of cases and believe that the above conclusions are quite generic.…”

“…In our most recent paper, (1) we have studied the q-dependent susceptibility χ(q) for a Z-invariant ferromagnetic Ising model on Penrose tiles. (The χ(q) is in many ways equivalent to the structure function determining diffraction patterns.)…”

“…Since l and m are integers, it is easily seen from (1.1) that the q-dependent susceptibilities for such cases are periodic with periods 2π in q x and q y . When the lattice structure is quasi-periodic, as in the case of the Penrose tiles studied in our previous paper, (1) it is not possible to split the summation in the susceptibility in this way and χ(q) is no longer periodic.…”

Origin of enhanced visible-light photocatalytic activity of transition-metal (Fe, Cr and Co)-doped CeO2: effect of 3d orbital splitting

“…As one metal element (Fe, Cr, or Co) substituting a Ce atom, it produces one or more than one hole below the Fermi Level, thus causing it to act as an acceptor. The positive formation energy indicates that it needs energy (3.57 eV) to create an O vacancy in CeO2 lattice [44,49]. The formation energy is 2.46, -1.09 and 4.39 eV for Fe, Cr and Co doped into CeO2, respectively.…”

“…In this work, we systematically explore the physical mechanism on the enhanced photocatalytic activity of CeO2 doped with transition metal elements, especially the effect of the interaction between transition metal dopant and O vacancy defects, by large-scale ab initio calculations. The various properties, such as relative stability, redox and mechanical properties, of pure CeO2 have been intensively investigated by using first principles calculations [37][38][39][40][41][42][43][44][45][46][47][48][49], which are also widely used to study the properties of doped oxides and/or with O vacancy defects, such as TiO2, Cu2O, and HfO2 [11,13,50,51]. It is revealed that the reduced band gap and strong absorption induced by impurity levels are responsible for the enhanced visible-light photocatalytic activity of CeO2 doped with nonmetal impurity [52].…”