Chain Of Infinite Length Research Articles

Rise and Fall, and Slow Rise Again, of Operator Entanglement under Dephasing.

The operator space entanglement entropy, or simply "operator entanglement" (OE), is an indicator of the complexity of quantum operators and of their approximability by matrix product operators (MPOs). We study the OE of the density matrix of 1D many-body models undergoing dissipative evolution. It is expected that, after an initial linear growth reminiscent of unitary quench dynamics, the OE should be suppressed by dissipative processes as the system evolves to a simple stationary state. Surprisingly, we find that this scenario breaks down for one of the most fundamental dissipative mechanisms: dephasing. Under dephasing, after the initial "rise and fall," the OE can rise again, increasing logarithmically at long times. Using a combination of MPO simulations for chains of infinite length and analytical arguments valid for strong dephasing, we demonstrate that this growth is inherent to a U(1) conservation law. We argue that in an XXZ spin model and a Bose-Hubbard model the OE grows universally as 1/4log_{2}t at long times and as 1/2log_{2}t for a Fermi-Hubbard model. We trace this behavior back to anomalous classical diffusion processes.

Evaluación teórica de las propiedades electrónicas y estructurales que afectan a la conductividad eléctrica en copolímeros de furanotiofeno

Desde el punto de vista científico y tecnológico ha habido un gran interés en el uso de monosustituyentes de furano y tiofeno como polímeros conductores, debido a sus múltiples aplicaciones como OLED, amplificadores ópticos, nanotecnología, entre otros. Por ello, el propósito de este trabajo fue estudiar los aspectos teóricos que afectan las propiedades electroconductoras de este tipo de moléculas. Se determinaron teóricamente los aspectos estructurales y electrónicos que influyeron en la conductividad de copolímeros de furano–tiofeno monosustituidos, al utilizar grupos carboxilos, metilos, hidroxilos, ciano y fluoruros como sustituyentes en el carbono C3 y C10 de cada heterociclo. La diferencia de energía entre el LUMO y el HOMO (band gap, Eg) y el potencial de ionización (PI) fue calculada a partir de las geometrías optimizadas en DFT para el estado neutro, anión y catión. Los PI y la Eg de los copolímeros fueron obtenidos mediante la extrapolación de los valores del oligómero a (1/N) y de una cadena de longitud infinita (1/N=0), obteniéndose una correlación lineal (R=0,99), la cual se mantiene a lo largo de todos los modelos de ajuste de cada copolímero analizado en el estudio.

Survival of time-evolved quantum correlations depending on whether quenching is across a critical point in anXYspin chain

The time-dynamics of quantum correlations in the quantum transverse anisotropic XY spin chain of infinite length is studied at zero as well as finite temperatures. The evolution occurs due to the instantaneous quenching of the coupling constant between the nearest-neighbor spins of the model, which is either performed within the same phase or across the quantum phase transition point connecting the order-disorder phases of the model. We characterize the time-evolved quantum correlations, entanglement and quantum discord, which exhibit varying behavior depending on the initial state and the quenching scheme. We show that the system is endowed with enhanced bipartite quantum correlations compared to that of the initial state, when quenched from ordered to the deep disordered phase. However, bipartite quantum correlations are almost washed out when the system is quenched from disordered to the ordered phase with the initial state being at the zero-temperature. Moreover, we identify the condition for the occurrence of enhanced bipartite correlations when the system is quenched within the same phase. Finally, we investigate the bipartite quantum correlations when the initial state is a thermal equilibrium state with finite temperature which reveals the effects of thermal fluctuation on the phenomena observed at zero-temperature.

Corrigendum to “Polynomial Cunningham chains” [J. Number Theory 131 (11) (2011) 2100–2106

A sequence of polynomials f 1 ( x ) , f 2 ( x ) , … in Z [ x ] , such that f 1 ( x ) has positive leading coefficient, each f i ( x ) is irreducible in Q [ x ] and f i ( x ) = x f i − 1 ( x ) + ϵ for all i , is defined to be a polynomial Cunningham chain of the first or second kind, depending on whether ϵ = 1 or −1 respectively. If k is the least positive integer such that f k + 1 ( x ) is reducible in Q [ x ] , then we say the chain has length k . In the previous article, “Polynomial Cunningham chains”, a proof is given that there exist infinitely many polynomial Cunningham chains of length k of either kind. It is also deduced from this result that there exist infinitely polynomial Cunningham chains of infinite length of either kind. However, the proof contains an error. In this article we give correct proofs of these statements.

Polynomial Cunningham chains

A sequence of prime numbers p 1 , p 2 , p 3 , … , such that p i = 2 p i − 1 + ϵ for all i, is called a Cunningham chain of the first or second kind, depending on whether ϵ = 1 or −1 respectively. If k is the smallest positive integer such that 2 p k + ϵ is composite, then we say the chain has length k. It is conjectured that there are infinitely many Cunningham chains of length k for every positive integer k. A sequence of polynomials f 1 ( x ) , f 2 ( x ) , … in Z [ x ] , such that f 1 ( x ) has positive leading coefficient, each f i ( x ) is irreducible in Q [ x ] and f i ( x ) = x f i − 1 ( x ) + ϵ for all i, is defined to be a polynomial Cunningham chain of the first or second kind, depending on whether ϵ = 1 or −1 respectively. If k is the least positive integer such that f k + 1 ( x ) is reducible in Q [ x ] , then we say the chain has length k. In this article, for polynomial Cunningham chains of both kinds, we prove that there are infinitely many chains of length k and, unlike the situation in the integers, that there are infinitely many chains of infinite length, by explicitly giving infinitely many polynomials f 1 ( x ) , such that f k + 1 ( x ) is the only term in the sequence { f i ( x ) } i = 1 ∞ that is reducible.

Peierls instability and ferrimagnetic ordering of quasi-one-dimensional carbon chains generated in H-passivated graphene

Using first-principle density-functional calculations, we investigate the competition between Peierls instability and spin ordering in a zigzag C chain generated in an H-passivated graphene. We find that such a quasi-one-dimensional (1D) C chain of infinite length stabilizes a Peierls instability showing bond alternation with a band-gap opening. As the chain length becomes finite, the Peierls distortion is enhanced at even-numbered chains, while a ferrimagnetic spin ordering with the atomic structure of a topological soliton is stabilized at odd-numbered chains. These results indicate the existence of complex interplay between electron-lattice and electron-electron interactions with respect to the parity of the quasi-1D C chains.

The effect of spin dilution on magnetism of the linear chain system β-Cu2−x Zn x V2O7

We have measured the magnetic susceptibility (χ) and heat capacity (C p) of β-Cu2−x Zn x V2O7 (x = 0, 0.05, 0.1, 0.15, 0.2, 0.3, 2) in the temperature range 2–300 K. A one-dimensional alternating exchange Heisenberg antiferromagnetism (HAF) is observed in all compositions with chains of infinite length. The intra-chain exchange remains uniform and decreases marginally with dilution of the magnetic state. A cooperative ordering is seen in the magnetic chains for all Zn concentrations (x ≤ 0.3). The temperature of occurrence of this transition decreases with increasing Zn concentration. Though the conventional spin-wave theory has been used here to describe the properties of the ordered phase, the presence of some contributions like the lattice heat capacity in C p and the Curie-Weiss term in susceptibility introduces some uncertainties in the estimation of the proportions contributed by the spin system. Therefore, the nature of the ordered phase could not be ascertained unambiguously.

Dispersion relations and wave operators in self-similar quasicontinuous linear chains

We construct self-similar functions and linear operators to deduce a self-similar variant of the Laplacian operator and of the D'Alembertian wave operator. The exigence of self-similarity as a symmetry property requires the introduction of nonlocal particle-particle interactions. We derive a self-similar linear wave operator describing the dynamics of a quasicontinuous linear chain of infinite length with a spatially self-similar distribution of nonlocal interparticle springs. The self-similarity of the nonlocal harmonic particle-particle interactions results in a dispersion relation of the form of a Weierstrass-Mandelbrot function that exhibits self-similar and fractal features. We also derive a continuum approximation, which relates the self-similar Laplacian to fractional integrals, and yields in the low-frequency regime a power-law frequency-dependence of the oscillator density.

Maximally Reliable Markov Chains Under Energy Constraints

Signal-to-noise ratios in physical systems can be significantly degraded if the outputs of the systems are highly variable. Biological processes for which highly stereotyped signal generations are necessary features appear to have reduced their signal variabilities by employing multiple processing steps. To better understand why this multistep cascade structure might be desirable, we prove that the reliability of a signal generated by a multistate system with no memory (i.e., a Markov chain) is maximal if and only if the system topology is such that the process steps irreversibly through each state, with transition rates chosen such that an equal fraction of the total signal is generated in each state. Furthermore, our result indicates that by increasing the number of states, it is possible to arbitrarily increase the reliability of the system. In a physical system, however, an energy cost is associated with maintaining irreversible transitions, and this cost increases with the number of such transitions (i.e., the number of states). Thus, an infinite-length chain, which would be perfectly reliable, is infeasible. To model the effects of energy demands on the maximally reliable solution, we numerically optimize the topology under two distinct energy functions that penalize either irreversible transitions or incommunicability between states, respectively. In both cases, the solutions are essentially irreversible linear chains, but with upper bounds on the number of states set by the amount of available energy. We therefore conclude that a physical system for which signal reliability is important should employ a linear architecture, with the number of states (and thus the reliability) determined by the intrinsic energy constraints of the system.

Resonant filtering of compositional waves in multicellular networks

Theoretically predicting the stationary and traveling compositional wave patterns observed over trivial multicellular topologies is common place in biology. Systematic methods are needed to find the relationships between the forms of such waves and the underlying network topologies. Here, we introduce one such method based on the Interface Response Theory (IRT) that combines models of intracellular biochemical reactions with those of intercellular networks to analytically track time variations of concentration profiles across nontrivial cellular network topologies. Cellular chains of infinite length are shown to sustain traveling planar waves under certain conditions among the coefficients of the intracellular chemical reactions. A non-trivial cellular network composed of a finite length side chain attached to the infinite backbone leads to the formation of a standing wave pattern upstream from the site of attachment. Downstream, the wave proceeds although a limited number of frequency bands are filtered out of the original content with that number being equal to the number of cells in the side chain.

Crystal Structures of [Bu2Sn(O2PPh2)2], [Ph2Sn(O2PPh2)2], and [PhClSn(O2PPh2)OMe]2. Raman Spectra of [Ph2Sn(O2PPh2)2] and [PhClSn(O2PPh2)OMe

[(n-Bu)2Sn(O2PPh2)2] (1), and [Ph2Sn(O2PPh2)2] (2) have been synthesized by the reactions of R2SnCl2 (R=n-Bu, Ph) with HO2PPh2 in Methanol. From the reaction of Ph2SnCl2 with diphenylphosphinic acid a third product [PhClSn(O2PPh2)OMe]2 (3) could be isolated. X-ray diffraction studies show 1 to crystallize in the monoclinic space group P21/c with a = 1303.7(1) pm, b = 2286.9(2) pm, c = 1063.1(1) pm, β = 94.383(6)°, and Z = 4. 2 crystallizes triclinic in the space group , the cell parameters being a = 1293.2(2) pm, b = 1478.5(4) pm, c = 1507.2(3) pm, α = 98.86(3)°, β = 109.63(2)°, γ = 114.88(2)°, and Z = 2. Both compounds form arrays of eight-membered rings (SnOPO)2 linked at the tin atoms to form chains of infinite length. The dimer 3 consists of a like ring, in which the tin atoms are bridged by methoxo groups. It crystallizes triclinic in space group with a = 946.4(1) pm, b = 963.7(1) pm, c = 1174.2(1) pm, α = 82.495(6)°, β = 66.451(6)°, γ = 74.922(6)°, and Z = 1 for the dimer. The Raman spectra of 2 and 3 are given and discussed.

Numerical evidence of Luttinger liquid behavior of potassium doped trans-polyacetylene chain of infinite length

We report on the first direct numerical evidence of a Tomonaga–Luttinger liquid behavior in potassium doped trans-polyacetylene system and a possible deviation from Tomonaga–Luttinger liquid state at very high doping concentration. Using density functional theoretical calculation, an analysis of density of states near the Fermi energy reveals a power law scaling factor of Tomonaga–Luttinger liquid at low dopant concentration in the metallic regime. As soon as the doping level reaches 0.0816 e/C, a drastic change in phase with power law scaling factor as low as 0.0023 has been realized.

Synthesis and characterization of perfuoropolyethers from perfluoromethylvinylether

The synthesis of a novel perfluoropolyethers (PFPEs) family has been carried out by the low temperature photooxidation of perfluoromethylvinylether. The macromolecular chains are terpolymeric in nature, being constituted by random sequences of the following groups: the regular –CF 2CF(OCF 3)O–, the orthoestereal –CF(OCF 3)O–, and the acetalic –CF 2O–. The presence of an ethereal oxygen in the short side chains represents the main structural difference with other PFPE families studied up to date [G. Marchionni, G. Ajroldi, G. Pezzin, in: S.L. Aggarwal, S. Russo (Eds.), Comprehensive Polymer Science, 2nd suppl., Pergamon Press, London, 1996.] Several fractions have been obtained, in the present work, by a single sample of the new PFPE, and some of their most important physical and molecular properties have been determined by a variety of different techniques. The properties change regularly with chain length, as expected. The group contribution analysis previously applied by our group to other families of PFPE has been found to be applicable also to chains of infinite length of the present family, which contains oxygen in the side chain.

Strongly coupled Ising chain under a weak random field

We present here an analytical method based on the transfer matrix M to study quenched disorder in one-dimensional spin systems in the limit of strong couplings and weak disorder. The procedure is formulated for the random-field Ising chain of finite length L, and its properties, represented as functions of M, satisfy a differential equation of the Fokker-Planck type. This equation describes “evolution” with L, and a central-limit theorem of a novel kind provides the equation and its solutions with universal character. We obtain analytical expressions for the moments of the magnetization of the infinite length chain and study the approach to the infinite coupling limit. We find that the random-field free energy f H and the Edwards-Anderson order parameter m 2 satisfy a simple relation. We discuss our results in connection to previous work by Luck and Nieuwenhuizen.

Electronic structure and electronic response of a model magnetic superlattice

We explore the electronic structure and wave-vector-dependent magnetic susceptibility as a function of temperature, of a model magnetic superlattice fabricated from one-dimensional Hubbard chains. Each chain, when infinite in length, exhibits a spin-density-wave instability. Within mean-field theory and a random-phase-approximation description of the magnetic susceptibility, we explore the nature of the magnetic order present when a superlattice is fabricated from such chains. Attention is directed to the approach to the limit wherein each unit cell contains many atoms, and the magnetic structure evolves from one unique to the superlattice (in the limit where each unit cell contains few atoms), to that realized for chains of infinite length (the limit where each unit cell contains many atoms).

Molecular dynamics simulation of diffusion of small molecules in polymers. II. Effect of free volume distribution

The influence of the free volume distribution on the diffusion of small molecules in polymers is studied by means of molecular dynamics simulations. A model is used which is composed of 10 O2 molecules, and 600 CH2 groups each belonging to an infinite length chain or its periodic images. A fully vibrational treatment is applied to the chains with a torsional potential and nonbonded interactions. Four model systems, each consisting of chains having the same molecular parameters except for the equilibrium skeletal bond angle, are compared at 300 K and at densities giving the same free volume fraction. The self-diffusion coefficient of oxygen strongly correlates with the average interchain separation; a tight chain packing reduces the diffusion even when the total free volume content is kept constant. The importance of the free volume distribution for diffusion was demonstrated by the good correlation obtainable in terms of the effective free volume available between the nearest neighbor chains.

Molecular dynamics simulations of diffusion of small molecules in polymers: Effect of chain length

In order to investigate the influence of chain length on the diffusion of small molecules in polymers, molecular dynamics simulations have been carried out for two model systems. One is the short chain model which is composed of 30 n-alkane-like chains each having 20 segments. The other is the infinite length chain model, used as a model for polyethylene, which is composed of a noncyclic chain of 600 segments with no free ends. Self-diffusion coefficients of the small molecules in the two model systems are compared. It was found that the chains of shorter length facilitate faster diffusion of the small molecules as well as faster relaxation of internal rotations of the chains even for a fixed total amount of free volume. However, the self-diffusion coefficient of small molecules in the infinite length chain model was still larger than that obtained from experiment. Possible factors which give rise to the effect of chain length are discussed.

A jump motion of small molecules in glassy polymers: A molecular dynamics simulation

Motions of small molecules in an amorphous polymer below its glass transition temperature Tg have been studied using molecular dynamics simulation. An infinite length chain model having 600 interacting sites, each representing a CH2 unit, is used. The system, which is composed of the infinite length chain and ten spherical molecules representing oxygen, is examined at 30 deg below Tg. Although most of the molecules are trapped by cages created by the chain, a molecule reveals clear jump motions. During the jump, no significant change is found in the local free volume as well as in the nonbonded energy acting on the molecule. Potential energy surfaces for the molecules clearly show the cooperative process of jump and chain motions: a generation of a vacant cage and a channel which connects two cages. It is found that the molecule passes through the channel without significant energy barrier and concentration of free volume.

A diffusion equation approach to spin diffusion in biomolecules

A theoretical description of 1H- 1H dipolar nuclear spin relaxation in a multispin system has been worked out by forming a diffusion equation for a one-dimensional chain of equidistant spins. The spin-diffusion equation is formed from first principles by assuming nearest neighbor interactions for a molecule undergoing isotropic random reorientation. This equation describes diffusion only in the long correlation limit (for ( ωτ c > 1.118) and is solved for driven NOE experiments, for spins in a chain of infinite length (0 < × < ∞), or for spins in a chain of finite length (0 < × < L). The solutions are obtained using the method of the Laplace transform for specified initial and boundary conditions. The observed selectivity of the NOE transfer in driven NOE experiments on a biomolecule which has a correlation factor ωτ c ∼ 3 is indeed in conformity with the predictions obtained from the spin-diffusion equation.

Computer simulation of polyethylene crystals

The computer programs described in Part 1 [9] have been used to investigate the structure and energy of stacking-faults and coherent twin boundaries in model polyethylene crystals. They are particularly relevant to the deformation behaviour of polyethylene, and it is explained how simulation studies can supplement experimental observations. Generalized translation faults have been simulated on the low-index planes of orthorhombic and monoclinic crystals constructed from rigid molecular chains of infinite length, and the stable faults have been identified. In the former structure, the only faults of significance are on the {110} planes, and three different configurations exist, each of energy ∼ 10 mJ m−2. In the monoclinic phase, two stable faults occur on each of the planes (100) and (010), and in both cases the most stable fault has an energy ∼ 3 mJ m−2. The possible implications of these findings for chain-axis and transverse slip in the two crystal structures are discussed. The structures of the coherent boundaries of {110} and {310} twins in orthorhombic and (100) and (010) twins in monoclinic crystals have been investigated, and in all cases several stable configurations are possible. The {310} interfaces have high energy (≳ 30 mJ m−2), whereas the others are similar in energy to the stacking faults. The results indicate the need to simulate incoherent boundaries. All the stable structures observed are found to have a simple geometrical explanation, and are not sensitive to the interatomic potentials used.