Discrete Charge Research Articles

Passing from Discrete to Continuum Models of Electrostatic Energy

We analyze the passage from discrete (i.e., point) charges and the corresponding discrete electrostatic energies to a continuum charge density and the corresponding continuum electrostatic energy in the limit of a large number of point charges. Given a continuous function on a bounded region that represents a continuum charge density, we construct a sequence of point charges and prove that the corresponding discrete electrostatic energies converge to the continuum counterpart. In a more general setting, we consider a given, compactly supported, signed Radon measure in the three-dimensional space representing the distribution of charges. We construct a sequence of point charges that converge to the given signed Radon measure and show that the corresponding discrete energies converge to the continuum one defined by the signed Radon measure. Conversely, for any sequence of point charges that satisfy certain reasonable assumptions on local geometry and excluded volumes, we prove that there exists a subsequence converging to a signed Radon measure and that the corresponding discrete energies converge to the continuum one defined by the limiting signed Radon measure. Tools used in our analysis include the explicit constructions of point charges from a given signed Radon measure as well as approximation properties of signed Radon measures. Finally, we apply our discrete-to-continuum analysis to the minimization of electrostatic energy related to the classical balayage problem in the potential theory. Such minimization can be potentially applied to the modeling of charged molecular systems with heterogeneously distributed charges embedded in a continuum solvent.

Features of Electrostatic Fields and Their Force Action When Using Micro- and Nanosized Inter-Electrode Gaps.

The present work is devoted to assessing the influence of discreteness of electric charge distribution in the double electric layer on the characteristics of the electric fields and their force action in capacitor structures with small interelectrode gaps. Due to the fact that modern technologies often use submicron-sized interelectrode gaps, it is no longer possible to consider the electrodes uniformly charged because of the discreteness of the electric charge. The corresponding development of a mathematical and physical model for the study of a non-uniform electric field is suggested. Numerical calculations are carried out, expressions, criteria, and results that are convenient for practical evaluations are obtained. The physical and mathematical model for force characteristics of a non-uniform electric field is developed. With a sufficiently small size of the interelectrode gap, the integral force effect of discretely distributed charges can be significantly higher than with a uniform distribution of the same charge. At reasonable surface charge densities, these phenomena are usually observed at interelectrode gaps less than tenths of a micrometer.

Effective field theory for fractional quantum Hall systems nearν=5/2

We propose an effective field theory (EFT) of fractional quantum Hall systems near the filling fraction $\nu=5/2$ that flows to pertinent IR candidate phases, including non-abelian Pfaffian, anti-Pfaffian, and particle-hole Pfaffian states (Pf, APf, and PHPf). Our EFT has a 2+1$d$ O(2)$_{2,L}$ Chern-Simons gauge theory coupled to four Majorana fermions by a discrete charge conjugation gauge field, with Gross-Neveu-Yukawa-Higgs terms. Including deformations via a Higgs condensate and fermion mass terms, we can map out a phase diagram with tunable parameters, reproducing the prediction of the recently-proposed percolation picture and its gapless topological quantum phase transitions. Our EFT captures known features of both gapless and gapped sectors of time-reversal-breaking domain walls between Pf and APf phases. Moreover, we find that Pf$\mid$APf domain walls have higher tension than domain walls in the PHPf phase. Then the former, if formed, may transition to the energetically-favored PHPf domain walls; this could, in turn, help further induce a bulk transition to PHPf.

Non-Abelian gauged fracton matter field theory: Sigma models, superfluids, and vortices

By gauging a higher-moment polynomial degree global symmetry and a discrete charge conjugation (i.e., particle-hole) symmetry coupled to matter fields (two symmetries mutually non-commutative), we derive a new class of higher-rank tensor non-abelian gauge field theory with dynamically gauged fractonic matter fields: Non-abelian gauged fractons interact with a hybrid class of higher-rank (symmetric or generic non-symmetric) tensor gauge theory and anti-symmetric tensor topological field theory, generalizing [arXiv:1909.13879, 1911.01804]. We also apply a quantum phase transition similar to that between insulator v.s. superfluid/superconductivity (U(1) symmetry disordered phase described by a topological gauge theory or a disordered Sigma model v.s. U(1) global/gauge symmetry-breaking ordered phase described by a Sigma model with a U(1) target space underlying Goldstone modes): We can regard our tensor gauge theories as disordered phases, and we transient to their new ordered phases by deriving new Sigma models in continuum field theories. While one low energy theory is captured by degrees of freedom of rotor or scalar modes, another side of low energy theory has vortices and superfluids -- we explore non-abelian vortices (two types of vortices mutually interacting non-commutatively beyond an ordinary group structure) and their Cauchy-Riemann relation.

Noise and charge discreteness as ultimate limit for the THz operation of ultra-small electronic devices

To manufacture faster electron devices, the industry has entered into the nanoscale dimensions and Terahertz (THz) working frequencies. The discrete nature of the few electrons present simultaneously in the active region of ultra-small devices generate unavoidable fluctuations of the current at THz frequencies. The consequences of this noise remain unnoticed in the scientific community because its accurate understanding requires dealing with consecutive multi-time quantum measurements. Here, a modeling of the quantum measurement of the current at THz frequencies is introduced in terms of quantum (Bohmian) trajectories. With this new understanding, we develop an analytic model for THz noise as a function of the electron transit time and the sampling integration time, which finally determine the maximum device working frequency for digital applications. The model is confirmed by either semi-classical or full- quantum time-dependent Monte Carlo simulations. All these results show that intrinsic THz noise increases unlimitedly when the volume of the active region decreases. All attempts to minimize the low signal-to-noise ratio of these ultra-small devices to get effective THz working frequencies are incompatible with the basic elements of the scaling strategy. One can develop THz electron devices, but they cannot have ultra-small dimensions. Or, one can fabricate ultra-small electron devices, but they cannot be used for THz working frequencies.

Nonvolatile Multilevel Photomemory Based on Lead-Free Double Perovskite Cs2AgBiBr6 Nanocrystals Wrapped Within SiO2 as a Charge Trapping Layer.

Floating gate transistor photomemory (FGTPM) has been regarded as one of the most prospective nonvolatile photomemory devices because of its compatibility with transistor-based circuits, nondestructive reading, and multilevel storage. Until now, owing to the excellent photoelectric properties, lead-based perovskite nanocrystals (PNCs) have been applied in most of the perovskite-based FGTPM devices and embedded in the polymer matrix as the charge trapping layer. However, the polymer matrix and its solvent would degrade the structure of the PNCs, resulting in the loss of their unique photoresponse ability. In addition, lead-based perovskites have environmental unfriendliness and poor stability. Hence, a novel nonvolatile FGTPM based on oligomeric silica (OS) wrapped lead-free double perovskite Cs2AgBiBr6 NCs was demonstrated for the first time. Acting synchronously as the protection layer for the discrete Cs2AgBiBr6 NCs and charge tunneling layer for the FGTPM device, the OS layer can achieve controllable thickness by adjusting the process parameters, leading to an adjustment of storage properties with a larger memory window (58 V). Owing to the excellent photoresponse ability of the Cs2AgBiBr6@OS composite layer, the FGTPM device exhibited high-performance with repeatable multilevel nonvolatile photomemory and precise photoresponse ability of wavelength/time/power-dependent photoirradiation without extra gate biasing.

Higher-order entanglement and many-body invariants for higher-order topological phases

We discuss how strongly interacting higher-order symmetry protected topological (HOSPT) phases can be characterized from the entanglement perspective: First, we introduce a topological many-body invariant which reveals the non-commutative algebra between flux operator and $C_n$ rotations. We argue that this invariant denotes the angular momentum carried by the instanton which is closely related to the discrete Wen-Zee response and fractional corner charge. Second, we define a new entanglement property, dubbed `higher-order entanglement', to scrutinize and differentiate various higher-order topological phases from a hierarchical sequence of the entanglement structure. We support our claims by numerically studying a super-lattice Bose-Hubbard model that exhibits different HOSPT phases.

Inter-surface effective electrostatic interactions in the presence of surface charge discreteness and solvent granularity

Effects of relative arrangement of discrete surface charges and solvent granularity on the effective potential of mean force (EPMF) are studied. Main conclusions are summarized as follows: (i) With consideration of the solvent granularity the EPMF always becomes more attractive, and one like charge attraction always appears even if the counter-ion is univalent. Moreover, with transition from asymmetrical to symmetrical surface charge distributions of similar surface charge compactness, attraction strength of the EPMF always rises; with increase of the counter-ion valence the EPMF attraction strength rises unless the surface charge configuration is asymmetrical and less compact and the counter-ion high-valence. (ii) Whether the surface charge distribution is asymmetrical or symmetrical, the EPMF tends to become more attractive in the compactness charge configuration than in the non-compactness charge configuration. (iii) The correlation between the EPMF attraction strength and bulk mole concentration can be positive or negative depending on the counter-ion valence, compactness of surface charge distribution, and average surface charge strength. Highly taking value of each of the three quantities helps in inducing one negative correlation and vice versa. (iv) Co-ion size has almost no influence on the EPMF, increasing or decreasing the counter-ion size inhibits or enhances the solvent granularity effect.

Robust spin and charge excitations throughout the high- Tc cuprate phase diagram from incipient Mottness

The generic phase diagram of lightly hole-doped high-$T_c$-cuprates hosts antiferromagnetic insulating phase with well-defined spin-wave excitations. Contrary to the weak-coupling prediction, these modes persist up to the overdoped metallic regime as intense and dispersive paramagnons. Here we report on our study of the low-energy magnetic and charge excitations within the extended Hubbard model at strong-coupling, using a modified $1/N$ expansion method with a variational state serving as the saddle point solution. Despite clear separation of magnetic and Hubbard-$U$ energy scales, we find that incipient Mottness affects qualitatively dispersions and widths of magnetic modes throughout entire phase diagram. The obtained magnetic and charge dynamical structure factors agree semi-quantitatively with recent resonant $X$-ray and neutron scattering data for $\mathrm{La_{2-\mathit{x}}Sr_\mathit{x}CuO_4}$ and $\mathrm{(Bi,Pb)_2(Sr,La)_2CuO_{6+\delta}}$ at all available doping levels. The weak-coupling random-phase-approximation fails already for underdoped samples, pointing to the non-trivial intertwining of distinct energy scales in cuprate superconductors. The existence of a discrete charge mode which splits off the electron-hole continuum is also predicted.

Re-Conceptualizing Burglary: An Analysis of the Use of Burglary as a Criminal Enhancement

Since its inception in the British common law, burglary has transformed from a narrow, discrete charge to an ever expanding and nebulously defined criminal act. Where the original definition included six precise elements, the majority of jurisdictions now define burglary as requiring only the unlawful entrance into a structure with the intent to commit a theft or felony. Additionally, the terms “entrance” and “structure” have expanded to include acts and locations far from those contemplated by the initial definition of the crime. This had led to a growing practice by criminal prosecutors: creatively alleging burglary to effectuate a criminal enhancement based solely on where the underlying crime took place. In this way, burglary is being used as a “location aggravator” rather than an allegation of the traditional standalone offense. While scholars have noted this practice for several decades, there is little scholarship attempting to assess whether this practice results in disproportionality harsh criminal punishments when compared to other location aggravators. This paper attempts to contribute to the debate regarding burglary by directly comparing the increase in criminal punishment that occurs from using burglary as an enhancement or aggravator with those increases in punishment stemming from a similar location aggravator under “Drug Free School Zone” laws. By looking at the legislative framework and caselaw surrounding the use of these enhancements in three different states — New Mexico, California, and New York — this paper will argue that the use of burglary as an enhancement results in disproportionally harsh criminal punishments in relation to the underlying offense when compared to similar location based enhancements.

Pólya–Schur Inequality and the Green Energy of a Discrete Charge

The classical Polya–Schur inequality for the logarithmic energy of a point charge distributed on a circle is generalized to the Green energy with respect to the concentric circular ring.

Weighted Fekete Points on the Real Line and the Unit Circle

Weighted Fekete points are defined as those that maximize the weighted version of the Vandermonde determinant over a fixed set. They can also be viewed as the equilibrium distribution of the unit discrete charges in an external electrostatic field. While these points have many applications, they are very difficult to find explicitly, and are only known in a few (unweighted) classical cases. We give two rare explicit examples of weighted Fekete points. The first one is for the weights $$w(x)=|x-ai|^{-s}$$ on the real line, with $$s\ge 1$$ and $$a\ne 0,$$ while the second is for the weights $$w(z)=1/|z-b|$$ on the unit circle, with $$b\in {{\mathbb {R}}}, b\ne \pm 1.$$ In both cases, we provide solutions of the continuous energy problems with external fields that express the limit versions of considered weighted Fekete points problems.

Harmonic surface mapping algorithm for molecular dynamics simulations of particle systems with planar dielectric interfaces.

We have developed an accurate and efficient method for molecular dynamics simulations of charged particles confined by planar dielectric interfaces. The algorithm combines the image-charge method for near field with the harmonic surface mapping, which converts the contribution of infinite far-field charges into a finite number of charges on an auxiliary spherical surface. We approximate the electrostatic potential of far-field charges via spherical harmonic expansion and determine the coefficients by fitting the Dirichlet-to-Neumann boundary condition, which only requires the potential within the simulation cell. Instead of performing the direct evaluation of spherical harmonic series expansion, we use Green's second identity to transform the series expansion into a spherical integral, which can be accurately represented by discrete charges on the sphere. Therefore, the fast multipole method can be readily employed to sum over all charges within and on the sphere, achieving truly linear O(N) complexity. Our algorithm can be applied to a broad range of charged complex fluids under dielectric confinement.

Wetting Transition of Ionic Substrate by Modulating Surface Charge Distribution.

Surface wettability regulation plays a crucial role in antifouling and related applications. For regulating surface wettability, one of the effective approaches is to modulate the surface charge distribution. Herein, we report a theoretical study for unraveling the mechanistic relation between surface charge distribution and ionic substrate wettability. Specifically, acetonitrile liquids at ambient condition in contact with various ionic substrates are considered. At different surface charge distributions, the interfacial thermodynamic properties are investigated by means of molecular density functional theory. We find that the variation of the spatial interval among the discrete charges strongly alters the substrate-acetonitrile interaction and leads to an oscillation in the interfacial tension, indicating that the substrate can be tuned from a solvophobic one to a solvophilic one. This trend can be further enhanced by increasing the charge quantity. The underlying mechanisms are extensively discussed and expatiated. Our work provides theoretical guidance to engineer and regulate surface wettability.

Building a bigger Hilbert space for superconducting devices, one Bloch state at a time

Superconducting circuits for quantum information processing are often described theoretically in terms of a discrete charge, or equivalently, a compact phase/flux, at each node in the circuit. Here we revisit the consequences of lifting this assumption for transmon and Cooper-pair-box circuits, which are constituted from a Josephson junction and a capacitor, treating both the superconducting phase and charge as noncompact variables. The periodic Josephson potential gives rise to a Bloch band structure, characterised by the Bloch quasicharge. We analyse the possibility of creating superpositions of different quasicharge states by transiently shunting inductive elements across the circuit, and suggest a choice of eigenstates in the lowest Bloch band of the spectrum that may support an inherently robust qubit encoding.

Visualizing molecular orientational ordering and electronic structure in CsnC60 fulleride films

Alkali-doped fullerides exhibit a wealth of unusual phases that remain controversial by nature. Here we report a cryogenic scanning tunneling microscopy study of the sub-molecular structural and electronic properties of expanded fullerene C60 films with various cesium (Cs) doping. By varying the discrete charge states and film thicknesses, we reveal a large tunability of orientational ordering of C60 anions, yet the tunneling conductance spectra are all robustly characteristic of energy gaps, hallmarks of Jahn-Teller instability and electronic correlations. The Fermi level lies halfway within the insulating gap for stoichiometric Cs doping level of n = 1, 2, 3 and 4, apart from which it moves toward band edges with concomitant electronic states within the energy gap. Our findings establish the universality of Jahn-Teller instability, and clarify the relationship among the doping, structural and electronic structures in CsnC60 fullerides.

3-D Simulation of Cavity SGEMP Interference Generated by Pulsed X-Rays

A 3-D electromagnetic particle-in-cell simulation code 3Dcavitysgemp V1.0 is developed to compute the cavity System-Generated Electromagnetic Pulse (SGEMP) interference generated by pulsed X-rays. Discrete charge conservation theorem, conformal mesh technique, and time-biased algorithm are introduced to improve the calculation accuracy. Based on a cylinder cavity model, the 3-D results are verified by previous 2-D simulation and demonstrate the special distributions and resonant frequencies of cavity SGEMP. The peak value of cavity SGEMP main pulse is 72.88 kV/m. When farther away from the charge-emitting face, the field strength becomes smaller quickly and its polarity reverses; when the distance from the symmetry axis increases, the field strength declines relatively slower. Moreover, it is proven that the cavity SGEMP would form a stable resonance and its frequency is dominated by the radius and length of the cavity.

Extremal problems for polynomials with real roots

We consider polynomials of degree d with only real roots and a fixed value of discriminant, and study the problem of minimizing the absolute value of such polynomials at a fixed point off the real line. There are two explicit families of polynomials that turn out to be extremal in terms of this problem. The first family has a particularly simple expression as a linear combination of dth powers of two linear functions. Moreover, if the value of the discriminant is not too small, then the roots of the extremal polynomial and the smallest absolute value in question can be found explicitly. The second family is related to generalized Jacobi (or Gegenbauer) polynomials, which helps us to find the associated discriminants. We also investigate the dual problem of maximizing the value of discriminant, while keeping the absolute value of polynomials at a point away from the real line fixed. Our results are then applied to problems on the largest disks contained in lemniscates, and to the minimum energy problems for discrete charges on the real line.

Structural analysis of confined monovalent salts: Combined effects of steric hindrance, surface charge representation, and dielectric response

The precise ion determination at solid-liquid interfaces and the underlying mechanisms behind this are of paramount relevance in a wide range of technological, environmental and biological applications. In this work, a systematic exploration into the behavior of monovalent salts in an extended nanospace is performed with Monte Carlo simulations in an attempt to provide a more general description. We obtain rather useful information not only on the lateral organization of ions in the Stern layer region but also on their distribution in the direction normal to the interface. Our simulation results show a remarkably consistent picture with the experimentally observed Stern layer structure in the discrete surface charge representation. It is recognized that sensitive to the radii of hydrated counterions and the solution dielectric response, their binding to low-dielectric interface sites forming a set of Bjerrum pairs is crucially driven by electrostatics. Also, we evaluate the validity of the uniform surface charge modeling that underlies previous theories and numerical simulations in predicting the structural properties of monovalent salts under confinement.

Discrete Helmholtz model: a single layer of correlated counter-ions. Metal oxides and silica interfaces, ion-exchange and biological membranes.

The mechanism by which interfaces in solution can be polarised depends on the nature of the charge carriers. In the case of a conductor, the charge carriers are electrons and the polarisation is homogeneous in the plane of the electrode. In the case of an insulator covered by ionic moieties, the polarisation is inhomogeneous and discrete in the plane of the interface. Despite these fundamental differences, these systems are usually treated in the same theoretical framework that relies on the Poisson–Boltzmann equation for the solution side. In this perspective, we show that interfaces polarised by discrete charge distributions are rather ubiquitous and that their associated potential drop significantly differs from those of conductor–electrolyte interfaces. We show that these configurations, spanning liquid–liquid interfaces, charged silica–water interfaces, metal oxide interfaces, supercapacitors, ion-exchange membranes and even biological membranes can be uniformly treated under a common “Discrete Helmholtz” model where the discrete charges are compensated by a single layer of correlated counter-ions, thereby generating a sharp potential drop at the interface.