Infinite Ladder Research Articles

Power dissipation in fractal AC circuits

We extend Feynman’s analysis of an infinite ladder circuit to fractal circuits, providing examples in which fractal circuits constructed with purely imaginary impedances can have characteristic impedances with positive real part. Using (weak) self-similarity of our fractal structures, we provide algorithms for studying the equilibrium distribution of energy on these circuits. This extends the analysis of self-similar resistance networks introduced by Fukushima, Kigami, Kusuoka, and more recently studied by Strichartz et al.

Power dissipation in fractal Feynman-Sierpinski AC circuits

This paper studies the concept of power dissipation in infinite graphs and fractals associated with passive linear networks consisting of non-dissipative elements. In particular, we analyze the so-called Feynman-Sierpinski ladder, a fractal AC circuit motivated by Feynman’s infinite ladder, that exhibits power dissipation and wave propagation for some frequencies. Power dissipation in this circuit is obtained as a limit of quadratic forms, and the corresponding power dissipation measure associated with harmonic potentials is constructed. The latter measure is proved to be continuous and singular with respect to an appropriate Hausdorff measure defined on the fractal dust of nodes of the network.

Trapped modes in thin and infinite ladder like domains. Part 1: Existence results

The present paper deals with the wave propagation in a particular two dimensional structure, obtained from a localized perturbation of a reference periodic medium. This reference medium is a ladder like domain, namely a thin periodic structure (the thickness being characterized by a small parameter $\epsilon > 0$) whose limit (as $\epsilon$ tends to 0) is a periodic graph. The localized perturbation consists in changing the geometry of the reference medium by modifying the thickness of one rung of the ladder. Considering the scalar Helmholtz equation with Neumann boundary conditions in this domain, we wonder whether such a geometrical perturbation is able to produce localized eigenmodes. To address this question, we use a standard approach of asymptotic analysis that consists of three main steps. We first find the formal limit of the eigenvalue problem as the $\epsilon$ tends to 0. In the present case, it corresponds to an eigenvalue problem for a second order differential operator defined along the periodic graph. Then, we proceed to an explicit calculation of the spectrum of the limit operator. Finally, we prove that the spectrum of the initial operator is close to the spectrum of the limit operator. In particular, we prove the existence of localized modes provided that the geometrical perturbation consists in diminishing the width of one rung of the periodic thin structure. Moreover, in that case, it is possible to create as many eigenvalues as one wants, provided that e is small enough. Numerical experiments illustrate the theoretical results.

Two extended Wells–Dawson arsenomolybdate architectures directed by Na(i) and/or Cu(i) organic complex linkers

Two extended Wells–Dawson arsenomolybdates, (imi)2[{CuI(imi)2}2{Na(imi)2}{AsIIIAs2VMo18O62}]·2H2O (1) and {CuI0.5(trz)}6 [{CuI0.5(trz)}6(As2Mo18O62)] (2), (imi = iminazole; trz = 1,2,3-triazole), are hydrothermally synthesized and extensively characterized. Compounds 1 and 2 contain a fully oxidized Dawson {As2Mo18O62}6− (abbreviated as {As2Mo18}) anion as the parent cluster unit. Compound 1 is a symmetric mono-arsenate capped anion cluster modified by two half occupancy {AsO3} units, which represents a new capping mode for Dawson clusters. The two adjacent mono-arsenate capped structures are linked by two {Na(imi)2} bridges to form a dimeric cluster, which, as a tetra-dentate ligand, further connects with adjacent dimers by four {Cu(imi)2} linkers to form an infinite ladder shaped chain. In compound 2, each classical Dawson cluster links six adjacent analogs by six {Cu0.5(trz)2} fragments forming an unprecedented 3D network via a unique belt-to-belt staggered pattern. This structure can be simplified as a 2,6-connected 3-D framework with {812·123}{8}3 topology. Compounds 1 and 2 represent the first 1-D and 3-D assemblies of Dawson-type arsenomolybdates, respectively. They display unique electrocatalytic behavior for the oxidation of ascorbic acid (AA) and reduction of NO2− as well as good photocatalytic properties for the degradation of the difficult-to-degrade dye AP under UV light.

On the impedance of infinite LC ladder networks

The subject of electrical impedance is on the syllabi of most undergraduate courses in physics and electrical engineering. For example, Richard Feynman in his famous undergraduate text Lectures on Physics shows how to calculate the impedance of an infinite LC ladder. However, the formula he obtains has no useful physical interpretation if considered in the steady state frequency domain. In fact the value of this impedance becomes infinite unless one assumes that the energy flow along the infinite LC ladder is spatially uniform and in one direction only. This ad-hoc assumption, which renders the solution non-causal, is entirely unnecessary if the problem is considered in the time domain. It is important for students to appreciate that the concept of impedance works well only in dissipative circuits where the effects of transients are largely short lived. The purpose of this paper is to show that the same problem treated in the time domain by the Laplace transform method provides a qualitatively different and more satisfying explanation. We show that the current response of an infinite LC ladder, which is in the zero state before a causal harmonic driving voltage is applied, contains a significant non-harmonic component. This component, which is present in addition to the forced harmonic waveform, decays only very slowly and extracts an infinite amount of energy from the source.

Four new coordination polymers based on carboxyphenyl-substituted dipyrazinylpyridine ligand: Syntheses, structures, magnetic and luminescence properties

Four new coordination polymers, namely, [Co(L)2]n (1) and [Co(L)(tp)0.5]n·nH2O (2), [Zn(L)2]n·0.5nH2O (3) and [Zn(L)(suc)0.5]n·nH2O (4) (HL = 4-(3-carboxyphenyl)-2,6-di(pyrazinyl)pyridine, H2tp = benzene-1,4-dicarboxylic acid, H2suc = succinic acid), were constructed by changing metal ion and introducing auxiliary ligand under hydrothermal condition. X-ray single diffraction studies reveal that compounds 1 and 3 have similar structures and crystallized in the monoclinic system with P21/c space group, which show infinite zigzag chains and are further extended into a 2D supramolecular layer by inter-molecular hydrogen bonds. Compounds 2 and 4 display infinite ladder chains and are further extended into a 3D supramolecular network by inter-molecular hydrogen bonds. Additionally, magnetic and photoluminescence properties of 1–4 have been investigated.

Non-relativistic scale anomalies

We extend the cohomological analysis in arXiv:1410.5831 of anisotropic Lifshitz scale anomalies. We consider non-relativistic theories with a dynamical critical exponent z = 2 with or without non-relativistic boosts and a particle number symmetry. We distinguish between cases depending on whether the time direction does or does not induce a foliation structure. We analyse both 1 + 1 and 2 + 1 spacetime dimensions. In 1 + 1 dimensions we find no scale anomalies with Galilean boost symmetries. The anomalies in 2 + 1 dimensions with Galilean boosts and a foliation structure are all B-type and are identical to the Lifshitz case in the purely spatial sector. With Galilean boosts and without a foliation structure we find also an A-type scale anomaly. There is an infinite ladder of B-type anomalies in the absence of a foliation structure with or without Galilean boosts. We discuss the relation between the existence of a foliation structure and the causality of the field theory.

Effect of the particle-hole channel on BCS-Bose-Einstein condensation crossover in atomic Fermi gases.

BCS–Bose-Einstein condensation (BEC) crossover is effected by increasing pairing strength between fermions from weak to strong in the particle-particle channel, and has attracted a lot of attention since the experimental realization of quantum degenerate atomic Fermi gases. Here we study the effect of the (often dropped) particle-hole channel on the zero T gap Δ(0), superfluid transition temperature Tc, the pseudogap at Tc, and the mean-field ratio 2Δ(0)/, from BCS through BEC regimes, using a pairing fluctuation theory which includes self-consistently the contributions of finite-momentum pairs and features a pseudogap in single particle excitation spectrum. Summing over the infinite particle-hole ladder diagrams, we find a complex dynamical structure for the particle-hole susceptibility χph, and conclude that neglecting the self-energy feedback causes a serious over-estimate of χph. While our result in the BCS limit agrees with Gor’kov et al., the particle-hole channel effect becomes more complex and pronounced in the crossover regime, where χph is reduced by both a smaller Fermi surface and a big (pseudo)gap. Deep in the BEC regime, the particle-hole channel contributions drop to zero. We predict a density dependence of the magnetic field at the Feshbach resonance, which can be used to quantify χph and test different theories.

Solvable and algebraic systems on infinite ladder

We consider two solvable models with equal entropy on the infinite ladder graph Z×{1,2}: the uniform spanning forest (USF), the abelian sandpile (ASM). We show that the symbolic models (abelian sandpile and spanning forest) are equal entropy symbolic covers of a certain algebraic dynamical system. In the past results of this nature have been established for sandpile models on lattices Zd. But we present a first example in case of spanning trees.

A new crystalline framework formed from 1,4-bis(4-pyridylmethyl)piperazine and Cd(NO3)2: Interpenetrating molecular ladders from T-shaped building blocks

Coordination of a pyridine-based bridging ligand, 4-bpmp, with cadmium nitrate afforded an infinite ladder-type complex {[Cd(4-bpmp)2(H2O)Cl2]}n (A) (4-bpmp = 1,4-bis(4-pyridylmethyl)piperazine) containing T-shaped building blocks. IR spectra, elemental analysis, and X-ray single-crystal diffraction were carried out to determine the composition and crystal structure of complex A. Inclined interpenetration of infinite ladders was observed in the solid structure. The combined work demonstrates the ability of bipyridyl coordination polymer leads to a novel metal complex with impressive structural motif.

Second order transport coefficient from the chiral anomaly at weak coupling: Diagrammatic resummation

We compute one of the second order transport coefficients arising from the chiral anomaly in a high temperature weakly coupled regime of quark-gluon plasma. This transport coefficient is responsible for the CP-odd current that is proportional to the time derivative of the magnetic field, and can be considered as a first correction to the chiral magnetic conductivity at finite, small frequency. We observe that this transport coefficient has a non-analytic dependence on the coupling as $\sim 1/(g^4 \log(1/g))$ at weak coupling regime, which necessitates a re-summation of infinite ladder diagrams with leading pinch singularities to get a correct leading log result: a feature quite similar to that one finds in the computation of electric conductivity. We formulate and solve the relevant CP-odd Schwinger-Dyson equation in real-time perturbation theory that reduces to a coupled set of second order differential equations at leading log order. Our result for this second order transport coefficient indicates that chiral magnetic current has some resistance to the time change of magnetic field, which may be called "chiral induction effect". We also discuss the case of color current induced by color magnetic field.

Diverse architectures and luminescence properties of two novel copper(i) coordination polymers assembled from 2,6-bis[3-(pyrid-4-yl)-1,2,4-triazolyl]pyridine ligands

Two attractive Cu(I) coordination polymers with unique structures, namely, {[Cu8(bptp)4]·6H2O} (1) and {[Cu5(bptp)2(CN)]·2H2O} (2), have been successfully synthesized based on a V-shaped multidentate N-containing ligand 2,6-bis[3-(pyrid-4-yl)-1,2,4-triazolyl]pyridine (H2bptp). The structural analysis reveals that complex 1 displays a Cu8 cluster-based infinite 1D ladder chain, in which each H2bptp ligand links four or five Cu(I) ions through nitrogen atoms of pyridine/triazole groups. Complex 2 features an unusual (3,4,5)-connected 3D topological network with the Schlafli symbol (42·54·63·7)(42·63·7)(5·6·8)(5·62·7·82), which has not been reported before. The structural and dimensional diversity of the two complexes indicates that H2bptp exhibits strong coordination ability and diverse coordination modes. It is worth mentioning that the Cu⋯Cu distances in 1 and 2 are shorter than the sum of the van der Waals radii of copper(I) (2.80 A), implying the metal–metal bonding interactions. What is more, the solid-state luminescence bands of 1 and 2 have also been investigated between 298 and 10 K. Interestingly, the low-energy emission bands of 1 and 2 exhibit yellow/orange-red luminescence, and their intensities increase gradually upon cooling.

Collective excitations of a harmonically trapped, two-dimensional, spin-polarized dipolar Fermi gas in the hydrodynamic regime

The collective excitations of a zero-temperature, spin-polarized, harmonically trapped, two-dimensional dipolar Fermi gas are examined within the Thomas-Fermi--von Weizs\"acker hydrodynamic theory. We focus on repulsive interactions and investigate the dependence of the excitation frequencies on the strength of the dipolar interaction and particle number. We find that the mode spectrum can be classified according to bulk modes, whose frequencies are shifted upward as the interaction strength is increased, and an infinite ladder of surface modes, whose frequencies are independent of the interactions in the large particle limit. We argue quite generally that it is the local character of the two-dimensional energy density that is responsible for the insensitivity of surface excitations to the dipolar interaction strength and not the precise form of the equation of state. This property will not be found for the collective excitations of harmonically trapped, dipolar Fermi gases in one and three dimensions, where the energy density is manifestly nonlocal.

Solution of the antiferromagnetic Ising model with multisite interaction on a zigzag ladder.

We consider the antiferromagnetic spin-1/2 Ising model with multisite interaction in an external magnetic field on an infinite zigzag ladder. The model is solved exactly by using the transfer matrix method. Using the exact expression for the total magnetization per site, the magnetic properties of the model are investigated in detail. The model exhibits the formation of magnetization plateaus for low temperatures, and it is shown that their properties depend strongly on the strength of the multisite interaction. All possible ground states of the model are found and discussed. The existence of nontrivial singular ground states is proven and exact explicit expressions for them are found. The macroscopic degeneracy of the ground states is investigated and discussed.

A fractal-like resistive network

The equivalent resistance of a fractal-like network is calculated by means of approaches similar to those employed in defining the equivalent resistance of an infinite ladder. Starting from an elementary triangular circuit, a fractal-like network, named after Saggese, is developed. The equivalent resistance of finite approximations of this network is measured, and the didactical implications of the model are highlighted.

A new europium fluorous metal–organic framework with pentafluorobenzoate and 1,10-phenanthroline ligands: Synthesis, structure and luminescent properties

A new europium fluorous metal–organic framework, [Eu2(OOCC6F5)6(C12H8N2)2(H2O)2] (OOCC6F5=pentafluorobenzoate, C12H8N2=1,10-phenanthroline monohydrate) (1) has been successfully synthesized and structurally characterized by single crystal X-ray diffraction. The X-ray diffraction revealed that an infinite one-dimensional ladder chain was created by offset face-to-face π–π stacking interactions between adjacent 1,10-phenanthroline rings. Further inspection of the crystal packing demonstrated that each chain interacts with the other chains through intermolecular CH···FC interactions to form a three-dimensional supramolecular architecture. Luminescent properties have also been studied. Compound 1 exhibits characteristic lanthanide-centered luminescence with a moderate quantum efficiency of 22.5%, suggesting effective energy transfer happened in this Eu(III) compound.

Bounded solutions in a T-shaped waveguide and the spectral properties of the Dirichlet ladder

The Dirichlet problem is considered on the junction of thin quantum waveguides (of thickness h ≪ 1) in the shape of an infinite two-dimensional ladder. Passage to the limit as h → +0 is discussed. It is shown that the asymptotically correct transmission conditions at nodes of the corresponding one-dimensional quantum graph are Dirichlet conditions rather than the conventional Kirchhoff transmission conditions. The result is obtained by analyzing bounded solutions of a problem in the T-shaped waveguide that the boundary layer phenomenon.

Hamiltonians with two-ladder spectra and solutions to the Painlevé IV equation

Some supersymmetric partners of the harmonic oscillator Hamiltonian possess third-order differential ladder operators and, thus, they realize the second-order polynomial Heisenberg algebras. The corresponding spectra consist of an infinite ladder of energy levels plus a finite one that can be placed at any position with respect to the infinite ladder. Both of them have equally spaced energy levels. Departing from these Hamiltonians several families of real and complex solutions to the Painlevé IV equation with real parameters will be generated.

Flexible Bis(imidazole) Mediated Assembly of Silver(I)–Organic Frameworks with Ethynide and Trifluoroacetate Ligands

A new family of 10 organosilver(I) complexes, namely, [(AgC≡CPh)2(AgO2CCF3)4(cis-1,2-bix)(CH3OH)]·2CH3OH (1,2-bix = 1,2-bis(imidazol-1-ylmethyl)benzene, 1A), [(AgC≡CC6H4Me-2)2(AgO2CCF3)4(cis-1,3-bix)]·5.25H2O (1,3-bix = 1,3-bis(imidazol-1-ylmethyl)benzene, 1B), [(AgC≡CPh)3(AgO2CCF3)6(trans-1,4-bix)(CH3OH)3]·2CH3OH (1,4-bix = 1,4-bis(imidazol-1-ylmethyl)benzene, 2A), [(AgC≡CC6H4Me-4)3(AgO2CCF3)6(trans-1,4-bix)(CH3OH)3]·2CH3OH (2B), [(AgC≡CPh)(AgO2CCF3)3(trans-1,3-bix)(H2O)]·0.5H2O (3A), [(AgC≡CC6H4Me-4)(AgO2CCF3)3(trans-1,3-bix)(H2O)] (3B), [(AgC≡CC6H4Me-4)(AgO2CCF3)3(bbi)(H2O)] (bbi = 1,1′-(1,4-butanediyl)bis(imidazole), 3C), [(AgC≡CC6H4tBu-4)(AgO2CCF3)3(trans-1,4-bix)] (3D), [(AgC≡CPh)2(AgO2CCF3)4(bbi)(H2O)]·2H2O (4), and [(AgC≡CtBu)2(AgO2CCF3)4(bbi)] (5), have been synthesized employing a variable combination of ethynide, trifluoroacetate, and bridging bis(imidazole) ligands. In both 1A and 1B, the flexible 1,2-bix or 1,3-bix ligands in cis-conformations are alternatively attached on both sides of an infinite cationic silver(I) chain, and neighboring organosilver(I) chains are further interconnected by intermolecular hydrogen bonds to generate a three-dimensional supramolecular architecture. In structural analogues 2A and 2B, pairs of silver(I) chains are linked via the trans-1,4-bix ligands to yield infinite ladders, which are further joined together by hydrogen bonds to afford a three-dimensional supramolecular network. In the isostructural series 3A–3D, centrosymmetric (PhC2)2Ag8 aggregates are connected by trifluoroacetate groups to give an infinite chain, and such chains are then bridged by bis(imidazole) ligands to generate a coordination layer. Furthermore, adjacent layers are linked by hydrogen bonds to form a three-dimensional supramolecular network. In 4, infinite silver(I)–phenylethynide chains constructed by Ag6 aggregates are bridged by bbi ligands in a gauche–trans–gauche conformation to yield a three-dimensional coordination framework. In contrast, infinite silver(I)-tert-butylethynide chains based on Ag8 and Ag6 aggregates in 5 are linked by bbi ligands exhibiting different trans–trans–trans and gauche–trans–gauche conformations to generate a three-dimensional coordination network. Finally, the roles played by various bis(imidazole) ligands and ethynides bearing different substituents in the framework assembly are discussed in detail.

Improving frequency selection of driven pulses using derivative-based transition suppression

Many techniques in quantum control rely on frequency separation as a means for suppressing unwanted couplings. In its simplest form, the mechanism relies on the low bandwidth of control pulses of long duration. Here we perform a higher-order quantum-mechanical treatment that allows for higher precision and shorter times. In particular, we identify three kinds of off-resonant effects: i) simultaneous unwanted driven couplings (e.g. due to drive crosstalk), ii) additional (initially undriven) transitions such as those in an infinite ladder system, and iii) sideband frequencies of the driving waveform such as we find in corrections to the rotating wave approximation. With a framework that is applicable to all three cases, in addition to the known adiabatic error responsible for a shift of the energy levels we typically see in the spectroscopy of such systems, we derive error terms in a controlled expansion corresponding to higher order adiabatic effects and diabatic excitations. We show, by also expanding the driving waveform in a basis of different order derivatives of a trial function (typically a Gaussian) these different error terms can be corrected for in a systematic way hence strongly improving quantum control of systems with dense spectra.