Periodic Chain Research Articles

Explorations for alternating FPU-chains with large mass

We show interaction between high- and low-frequency modes in periodic α-FPU chains with alternating large masses. This high-low frequency interaction is known if the number of particles N is a 4-fold, our treatment discusses the difficult case where the number of particles N=2p involves p prime. A key role is played by identifying symmetric invariant manifolds, thus reducing the dimension of the problems drastically, and a Mathematica programme focused on these systems. We show explicitly high-low frequency interaction for systems with 2pn particles where 2≤p≤47 is prime and n is an arbitrary natural number. In addition we have strong arguments for interactions in arbitrary large chains.

One-dimensional narrow-band conductors

We review one-dimensional lattice models and the corresponding results that describe the low-temperature properties of quasi-one-dimensional lattice systems with long-range interaction. A widely known example is narrow-band low-dimensional conductors with long-range interelectron repulsion. The models deal with particles that live on the one-dimensional host lattice (chain), translation invariant or disordered, and interact via the long-range repulsive potential. The results are presented concerning the translation invariant host chain, in particular: the low-temperature thermodynamics incommensurable ground states and related devil-staircase form of various characteristics as functions of pertinent parameters, the self-localization of particles, a new branch of elementary excitations, etc. In the disordered case, where the sites of the host chain fluctuate randomly around the sites of the periodic chain, the low-temperature thermodynamics and the structure of the ground state are discussed in the framework of a certain model, which we call the cluster model and which seems to be a fairly reasonable approximation for low temperatures and small concentration of particles. Using analytical and numerical tools we analyze the thermodynamics and the ground state of the model. The latter proves to be a sequence of random domains and we study in detail their distribution.

Tunable and high-performance self-powered ultraviolet detectors using leaf-like nanostructural arrays in ternary tin zinc sulfide system

Recently, self-powered ultraviolet (UV) detectors have received increasing attention for developing the future demands for optoelectronic devices due to their distinguished photoresponse properties. In this study, an emerging type of self-powered UV detectors using the ternary SnxZn1-xS alloy thin films (0.20 ≤ x ≤ 0.40 mol) were fabricated using the solvothermal process. The structural study showed the formation of two distinct phases of ZnS and SnS for the samples with high content Sn2+ ions. According to the ideal crystal-growth mechanism proposed by periodic bond chain theory, a morphological evolution from nanorods to nanoflakes and unique aligned three-dimensional leaf-like nanostructures was achieved by enhancing the Sn2+ content up to the maximum value of 0.40 mol. The optical properties featured the less influence of band tailing on the band gap energy at the high concentration of Sn2+ ions, resulting in the widening of the band gap energy up to 4.56 eV thanks to the Burstein-Moss effect. From the photosensing measurements, the Schottky barrier heights of the devices were found to significantly reduce under UVB exposure as the Sn2+ content increases, confirming a high Ion/Ioff ratio (>103) for the devices. The UV detectors exhibited superior self-powered characteristics under UVB illumination, having high photosensitivity and fast photoswitching response times (τr = 1.9 and τd = 2.6 ms) at zero voltage. Interestingly, the photoresponse performance of UV detectors can be adjusted by varying the content of Sn2+ ions, which signifies that tunable self-powered UV detectors can be designed by varying the Sn2+ concentration in the ternary SnxZn1-xS system.

Phantom Bethe excitations and spin helix eigenstates in integrable periodic and open spin chains

We demonstrate the existence of special phantom excitations for open and periodically closed integrable systems at the example of the $XXZ$ Heisenberg spin chain. The phantom excitations do not contribute to the energy of the Bethe state and correspond to special solutions to the Bethe Ansatz equations with infinite "phantom" Bethe roots. The phantom Bethe roots lead to degeneracies between different magnetization sectors in the periodic case and to the appearance of spin helix states (SHS), i.e. periodically modulated states of chiral nature in both open and closed systems. For the periodic chain, phantom Bethe root (PBR) solutions appear for anisotropies $\De=\cosh\eta$ with $\exp(\eta)$ being a root of unity, thus restricting the phenomenon to the critical region $|\De|<1$. For the open chain, PBR solutions appear for any value of anisotropy, both in the critical and in the non-critical region, provided that the boundary fields satisfy a criterion which we derive in this paper. There exist PBR solutions with all Bethe roots being phantom, and PBR solutions that consist of phantom roots as well as regular (finite) roots. Implications of our results for an experiment are discussed.

On Borromean links and related structures.

The creation of knotted, woven and linked molecular structures is an exciting and growing field in synthetic chemistry. Presented here is a description of an extended family of structures related to the classical `Borromean rings', in which no two rings are directly linked. These structures may serve as templates for the designed synthesis of Borromean polycatenanes. Links of n components in which no two are directly linked are termed `n-Borromean' [Liang & Mislow (1994). J. Math. Chem. 16, 27-35]. In the classic Borromean rings the components are three rings (closed loops). More generally, they may be a finite number of periodic objects such as graphs (nets), or sets of strings related by translations as in periodic chain mail. It has been shown [Chamberland & Herman (2015). Math. Intelligencer, 37, 20-25] that the linking patterns can be described by complete directed graphs (known as tournaments) and those up to 13 vertices that are vertex-transitive are enumerated. In turn, these lead to ring-transitive (isonemal) n-Borromean rings. Optimal piecewise-linear embeddings of such structures are given in their highest-symmetry point groups. In particular, isonemal embeddings with rotoinversion symmetry are described for three, five, six, seven, nine, ten, 11, 13 and 14 rings. Piecewise-linear embeddings are also given of isonemal 1- and 2-periodic polycatenanes (chains and chain mail) in their highest-symmetry setting. Also the linking of n-Borromean sets of interleaved honeycomb nets is described.

Linear optical properties of a linear chain of interacting gold nanoparticles

In a Drude-like model for the conduction electrons of metal nanoparticles (MNPs) in a periodic linear chain, considering dipole-dipole interactions of adjacent particles, an analytical expression is derived for each particle permittivity for two different polarizations of incident light: parallel with and perpendicular to the chain line. A numerical analysis is carried out for a chain including 10 identical gold nanoparticles (NPs) for two different sizes of NPs and two different host media: air and glass. It is shown that in the parallel case of polarization, interaction of NPs leads to a substantial increase in the extinction cross section and the redshift of the surface plasmon resonance (SPR) wavelength. In comparison with the linear properties of a single NP, the second and penultimate particles have the most increase in the extinction cross section and SPR wavelength displacement while the first and last particles experience the least variations due to the mutual interactions. For the perpendicular polarization, inversely, the dipolar coupling causes a decrease in extinction cross section of all particles and the blue-shift of SPR wavelength. For the parallel polarization, the absolute values of the real and imaginary parts of complex permittivity of each MNP decrease in comparison with the single particle case while they increase for the perpendicular state of polarization.

Lump chains in the KP‐I equation

AbstractWe construct a broad class of solutions of the Kadomtsev–Petviashvili (KP)‐I equation by using a reduced version of the Grammian form of the ‐function. The basic solution is a linear periodic chain of lumps propagating with distinct group and wave velocities. More generally, our solutions are evolving linear arrangements of lump chains, and can be viewed as the KP‐I analogues of the family of line‐soliton solutions of KP‐II. However, the linear arrangements that we construct for KP‐I are more general, and allow degenerate configurations such as parallel or superimposed lump chains. We also construct solutions describing interactions between lump chains and individual lumps, and discuss the relationship between the solutions obtained using the reduced and regular Grammian forms.

Bandgap and vibration transfer characteristics of scissor-like periodic metamaterials

A periodic mass-spring-truss chain based on a scissor-like structure and inertial amplification is proposed to seek for low-frequency vibration attenuation. The resonant and anti-resonant frequencies of a basic element are obtained analytically to explain the resonance and anti-resonance of the periodic chain. The formation of bandgaps is explored, revealing that the vertical movements of the vertices take the input kinetic energy away from the horizontal movement energy. The cut-off frequencies of three models are derived analytically to compare the effect of the inertia amplification, which can reach more than three times the classical mass-spring chain with a small angle. Also, the effect of stiffness amplification is presented when the spring mounted vertically, unlike the classical inertial amplification systems. Thus, the optimal configuration for low-frequency attenuation is obtained, that is, the spring mounted horizontally and the masses mounted at the upper and lower vertices. The chain with mass on the upper and lower vertices has the lowest cut-off frequency, which is 17.7% lower than that of the classical mass-spring chain. A diatomic chain is studied to find lower and wider bandgaps, where the central frequency is 25% lower than the classical mass-spring diatomic chain. The central frequency and width of the new bandgap can be tunable accordingly. The result shows that small angles or low stiffness ratios result in lower bandgaps. Meanwhile, the large angles, big differences of masses, or high stiffness ratios lead to wider bandgaps.

Correlation-driven phenomena in periodic molecular systems from variational two-electron reduced density matrix theory.

Correlation-driven phenomena in molecular periodic systems are challenging to predict computationally not only because such systems are periodically infinite but also because they are typically strongly correlated. Here, we generalize the variational two-electron reduced density matrix (2-RDM) theory to compute the energies and properties of strongly correlated periodic systems. The 2-RDM of the unit cell is directly computed subject to necessary N-representability conditions such that the unit-cell 2-RDM represents at least one N-electron density matrix. Two canonical but non-trivial systems, periodic metallic hydrogen chains and periodic acenes, are treated to demonstrate the methodology. We show that while single-reference correlation theories do not capture the strong (static) correlation effects in either of these molecular systems, the periodic variational 2-RDM theory predicts the Mott metal-to-insulator transition in the hydrogen chains and the length-dependent polyradical formation in acenes. For both hydrogen chains and acenes, the periodic calculations are compared with previous non-periodic calculations with the results showing a significant change in energies and increase in the electron correlation from the periodic boundary conditions. The 2-RDM theory, which allows for much larger active spaces than are traditionally possible, is applicable to studying correlation-driven phenomena in general periodic molecular solids and materials.

Bulk-edge correspondence in the Haldane phase of the bilinear-biquadratic spin-1 Hamiltonian

The Haldane phase is the prototype of symmetry protected topological (SPT) phases of spin chain systems. It can be protected by several symmetries having in common the degeneracy of the entanglement spectrum (ES). Here we explore in depth this degeneracy for the spin-1 Affleck–Kennedy–Lieb–Tasaki and bilinear-biquadratic Hamiltonians and show the emergence of a bulk-edge correspondence that relates the low energy properties of the entanglement Hamiltonian of a periodic chain and that of the physical Hamiltonian of an open chain. We find that the ES can be described in terms of two spins-1/2 behaving as the effective spins at the end of the open chain. In the case of non-contiguous partitions, we find that the entanglement Hamiltonian is given by the spin-1/2 Heisenberg Hamiltonian, which suggests a relationship between SPT phases and conformal field theory. We finally investigate the string order parameter and the relation with the bulk-edge correspondence.

Geometric lifting of the integrable cellular automata with periodic boundary conditions

Inspired by G Frieden’s recent work on the geometric R-matrix for affine type A crystal associated with rectangular shaped Young tableaux, we propose a method to construct a novel family of discrete integrable systems which can be regarded as a geometric lifting of the generalized periodic box–ball systems. By converting the conventional usage of the matrices for defining the Lax representation of the discrete periodic Toda chain, together with a clever use of the Perron–Frobenious theorem, we give a definition of our systems. It is carried out on the space of real positive dependent variables, without regarding them to be written by subtraction-free rational functions of independent variables but nevertheless with the conserved quantities which can be tropicalized. We prove that, in this setup an equation of an analogue of the ‘carrier’ of the box–ball system for assuring its periodic boundary condition always has a unique solution. As a result, any states in our systems admit a commuting family of time evolutions associated with any rectangular shaped tableaux, in contrast to the case of corresponding generalized periodic box–ball systems where some states did not admit some of such time evolutions.

Topologically induced metastability in a periodic XY chain

Non-trivial topological behavior appears in many different contexts in statistical physics, perhaps the most known one being the Kosterlitz–Thouless phase transition in the two-dimensional XY model. We study the behavior of a simpler, one dimensional, XY chain with periodic boundary and strong interactions, but rather than concentrating on the equilibrium measure, we try to understand its dynamics. The equivalent of the Kosterlitz–Thouless transition in this one-dimensional case happens when the interaction strength scales like the size of system N, yet we show that a sharp transition for the dynamics occurs at the scale of log N. When the interactions are weaker than a certain threshold, topological phases could not be observed over long times, while for interactions that are stronger than that threshold, topological phases become metastable, surviving for diverging time scales.

Observation of an Accidental Bound State in the Continuum in a Chain of Dielectric Disks

Being a general wave phenomenon, bound states in the continuum (BICs) appear in acoustic, hydrodynamic, and photonic systems of various dimensionalities. Here, we report the first experimental observation of an accidental electromagnetic BIC in a one-dimensional periodic chain of coaxial ceramic disks. We show that the accidental BIC manifests itself as a narrow peak in the transmission spectra of the chain placed between two loop antennas. We demonstrate a linear growth of the radiative quality factor of the BICs with the number of disks that is well described with the developed tight-binding model. We estimate the number of disks when the radiation losses become negligible in comparison to material absorption and, therefore, the chain can be considered as practically infinite. The presented analysis is supported by near-field measurements of the BIC profile. The obtained results provide useful guidelines for practical implementations of structures with BICs opening up horizons for the development of radio-frequency and optical metadevices.

Square-root non-Bloch topological insulators in non-Hermitian ring resonators.

We investigate the topological skin effect in a ring resonator array which can be mapped into the square root of a Su-Schrieffer-Heeger (SSH) model with non-Hermitian asymmetric coupling. The asymmetric coupling is realized by integrating the same amount of gain and loss into the two half perimeters of linking rings that effectively couple two adjacent site rings. Such a square-root topological insulator inherits the properties from its parent Hamiltonian, which has the same phase transition points and exhibits non-Bloch features as well. We show the band closing points for open chain are different from that of periodic chain as a result of the skin effect. Moreover, the square-root insulator supports multiple topological edge modes as the number of band gaps is doubled compared to the original Hamiltonian. The full-wave simulations agree well with the theoretical analyses based on a tight-binding model. The study provides a promising approach to investigate the skin effect by utilizing ring resonators and may find potential applications in light trapping, lasers, and filters.

Quantum spin systems and supersymmetric gauge theories. Part I

The relation between supersymmetric gauge theories in four dimensions and quantum spin systems is exploited to find an explicit formula for the Jost function of the N site mathfrak{sl} 2X X X spin chain (for infinite dimensional complex spin representations), as well as the SLN Gaudin system, which reduces, in a limiting case, to that of the N-particle periodic Toda chain. Using the non-perturbative Dyson-Schwinger equations of the supersymmetric gauge theory we establish relations between the spin chain commuting Hamiltonians with the twisted chiral ring of gauge theory. Along the way we explore the chamber dependence of the supersymmetric partition function, also the expectation value of the surface defects, giving new evidence for the AGT conjecture.

Recurrence in generalized semigroup

In [arXiv:1904.12333], we introduced the concept of escaping set in general setting for a topological space and extended the notion of limit set and escaping set for the general semigroup generated by continuous self maps. In this paper we continue with extending the other notions of recurrence for the generalized semigroup analogs to their counterpart in the classical theory of dynamics. We discuss the concept of periodic point, nonwandering point and chain recurrent point in the more general setting of a semigroup and establish the correlation between them.

On Monotonic Pattern in Periodic Boundary Solutions of Cylindrical and Spherical Kortweg–De Vries–Burgers Equations

We studied, for the Kortweg–de Vries–Burgers equations on cylindrical and spherical waves, the development of a regular profile starting from an equilibrium under a periodic perturbation at the boundary. The regular profile at the vicinity of perturbation looks like a periodical chain of shock fronts with decreasing amplitudes. Further on, shock fronts become decaying smooth quasi-periodic oscillations. After the oscillations cease, the wave develops as a monotonic convex wave, terminated by a head shock of a constant height and equal velocity. This velocity depends on integral characteristics of a boundary condition and on spatial dimensions. In this paper the explicit asymptotic formulas for the monotonic part, the head shock and a median of the oscillating part are found.

Topological quantum optical states in quasiperiodic cold atomic chains

Topological quantum optical states in one-dimensional (1D) quasiperiodic cold atomic chains are studied in this work. We propose that by introducing incommensurate modulations on the interatomic distances of 1D periodic atomic chains, the off-diagonal Aubry-Andr\'e-Harper (AAH) model can be mimicked, although the crucial difference is the existence of long-range dipole-dipole interactions. The discrete band structures with respect to the modulation phase, which plays the role of a dimension extension parameter, are calculated for finite chains beyond the nearest-neighbor approximation. It is found that the present system indeed supports nontrivial topological states localized over the boundaries. Despite the presence of long-range dipole-dipole interactions that leads to an asymmetric band structure, it is demonstrated that this system inherits the topological properties of two-dimensional integer quantum Hall systems. The spectral position, for both real and imaginary frequencies, and number of these topologically protected edge states are still governed by the gap-labeling theorem and characterized by the topological invariant, namely, the (first) Chern number, indicating the validity of bulk-boundary correspondence. Due to the fractal spectrum arising from the quasiperiodicity in a substantially wide range of system parameters, our system provides a large number of topological gaps and optical states readily for practical use. It is also revealed that a substantial proportion of the topological edge states are highly subradiant with extremely low decay rates, which therefore offer an appealing route for controlling the emission of external quantum emitters and achieving high-fidelity quantum state storage.

Models for functor categories over self-injective quivers

Let [Formula: see text] be a ring, [Formula: see text] a small [Formula: see text]-preadditive category, and let [Formula: see text] be the category of [Formula: see text]-linear functors from [Formula: see text] to the left [Formula: see text]-module category [Formula: see text]. Given a cotorsion pair [Formula: see text] in [Formula: see text], we construct four cotorsion pairs [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] in [Formula: see text] and investigate when these cotorsion pairs are hereditary and complete. Moreover, under few assumptions, we show that there exist a recollement of homotopy categories induced by these cotorsion pairs. Finally, some applications are given in the category of [Formula: see text]-periodic chain complexes of [Formula: see text]-modules.

Influence of the POuN4-u structural units on the formation energies and transport properties of lithium phosphorus oxynitride: a DFT study.

The potential of mobile applications for digital networking is constantly increasing. A key challenge is to ensure a reliable and long-term power supply. One possible solution is the use of all-solid-state thin-film lithium batteries which use amorphous lithium phosphorus oxynitride (LIPON) as solid electrolyte. It is well known that the electrochemical properties of this material are related to the amorphous state, which correlates with the nitrogen content. Due to the difficulty of calculating amorphous structures using first principles methods, three different LIPON structure models are considered in this study and the influence of the anion POuN4-u sublattice on the Li vacancy and Li interstitial formation as well as on the lithium ion transport is highlighted. While for all three model systems the migration energies of the energetically preferred Li vacancies increase with increasing complexity of the anion POuN4-u sublattice only slightly from 0.38 eV to 0.55 eV, the migration energies for the energetically preferred Li interstitials decrease with increasing complexity of the anion POuN4-u sublattice from 0.68 eV to 0.38 eV. Thus, it was found that the energetically preferred lithium ion (Li vacancy and Li interstitial ion) transport mechanism in LIPON can be explained on the basis of the present POuN4-u structural units. In the presence of isolated PON3x- tetrahedra or periodic PO2N2 chains, the lithium vacancy diffusion dominates, whereas in the presence of periodic POuN4-u planes, the lithium interstitial diffusion becomes dominant.