Number Of Flux Quanta Research Articles

A topological flux trap: Majorana bound states at screw dislocations

The engineering of non-trivial topology in superconducting heterostructures is a very challenging task. Reducing the number of components in the system would facilitate the creation of the long-sought Majorana bound states. Here, we explore a route toward emergent topology in a trivial superconductor without a need for other proximitized materials. Specifically, we show that a vortex hosting an even number of flux quanta is capable of forming a quasi-one-dimensional topological sub-system that can be mapped to the Kitaev wire, if the vortex is trapped at a screw dislocation. This crystallographic defect breaks inversion symmetry and thereby threads a local spin–orbit coupling through the superconductor. The vortex-dislocation pair in the otherwise trivial bulk can harbor a pair of Majorana bound states located at the two surface terminations. We explain the topological transition in terms of a band inversion in the Caroli-de Gennes-Matricon vortex bound states and discuss favorable material parameters.

Geometric resonance of four-flux composite fermions

Two-dimensional interacting electrons exposed to strong perpendicular magnetic fields generate emergent, exotic quasiparticles phenomenologically distinct from electrons. Specifically, electrons bind with an even number of flux quanta, and transform into composite fermions (CFs). Besides providing an intuitive explanation for the fractional quantum Hall states, CFs also possess Fermi-liquid-like properties, including a well-defined Fermi sea, at and near even-denominator Landau level filling factors such as $\nu=1/2$ or $1/4$. Here, we directly probe the Fermi sea of the rarely studied four-flux CFs near $\nu=1/4$ via geometric resonance experiments. The data reveal some unique characteristics. Unlike in the case of two-flux CFs, the magnetic field positions of the geometric resonance resistance minima for $\nu<1/4$ and $\nu>1/4$ are symmetric with respect to the position of $\nu=1/4$. However, when an in-plane magnetic field is applied, the minima positions become asymmetric, implying a mysterious asymmetry in the CF Fermi sea anisotropy for $\nu<1/4$ and $\nu>1/4$. This asymmetry, which is in stark contrast to the two-flux CFs, suggests that the four-flux CFs on the two sides of $\nu=1/4$ have very different effective masses, possibly because of the proximity of the Wigner crystal formation at small $\nu$.

Integer quantum Hall transition in a fraction of a Landau level

We investigate the quantum Hall problem in the lowest Landau level in two dimensions, in the presence of an arbitrary number of $\delta$-function potentials arranged in different geometric configurations. When the number of delta functions $N_\delta$ is smaller than the number of flux quanta through the system ($N_\phi$), there is a manifold of $(N_\phi-N_\delta)$ degenerate states at the original Landau level energy. We prove that the total Chern number of this set of states is +1 regardless of the number or position of the $\delta$ functions. Furthermore, we find numerically that, upon the addition of disorder, this subspace includes a quantum Hall transition which is (in a well-defined sense) $\textit{quantitatively}$ the same as that for the lowest Landau level without $\delta$-function impurities, but with a reduced number $N_\phi' \equiv N_\phi-N_\delta$ of magnetic flux quanta. We discuss the implications of these results for studies of the integer plateau transitions, as well as for the many-body problem in the presence of electron-electron interactions.

The Integer-Fraction Principle of the Digital Electric Charge for Quarks and Quasiparticles

In the integer-fraction principle of the digital electric charge, individual integral charge and individual fractional charge are the digital representations of the allowance and the disallowance of irreversible kinetic energy, respectively. The disallowance of irreversible kinetic energy for individual fractional charge brings about the confinement of individual fractional charges to restrict irreversible movement resulted from irreversible kinetic energy. Collective fractional charges are confined by the short-distance confinement force field where the sum of the collective fractional charges is integer. As a result, fractional charges are confined and collective. The confinement force field includes gluons in QCD (quantum chromodynamics) for collective fractional charge quarks in hadrons and the magnetic flux quanta for collective fractional charge quasiparticles in the fractional quantum Hall effect (FQHE). The collectivity of fractional charges requires the attachment of energy as flux quanta to bind collective fractional charges. The integer-fraction transformation from integral charges to fractional charges consists of the three steps: 1) the attachment of an even number of flux quanta to individual integral charge fermions to form individual integral charge composite fermions, 2) the attachment of an odd number of flux quanta to individual integral charge composite fermions to form transitional collective integral charge composite bosons, and 3) the conversion of flux quanta into the confinement force field to confine collective fractional charge composite fermions converted from composite bosons. The charges of quarks are fractional, because QCD (the strong force) emerges in the universe that has no irreversible kinetic energy. Kinetic energy emerged in the universe after the emergence of the strong force. The charges of the quasiparticles in the FQHE are fractional because of the confinement by a two-dimensional system, the Landau levels, and an extremely low temperature and the collectivity by high energy magnetic flux quanta. From the integer-fraction transformation from integral charge electrons to fractional charge quarks, the calculated masses of pion, muon and constituent quarks are in excellent agreement with the observed values.

Vortices in a long periodically modulated Josephson contact containing a few magnetic flux quanta

The possibility of existence of vortices containing a few magnetic flux quanta in a long periodically ordered Josephson contact is investigated. The computer algorithm is proposed, which makes it possible to calculate exactly any symmetric current configuration containing vortices of only one direction. The variants of an approximate analytic approach and of an exact computer calculation are considered. The values of the pinning parameter, for which the corresponding solutions exist for the cases of 2, 3, and 4 flux quanta, are calculated. It is shown that vortex energy is a quadratic function of the number of flux quanta in it. The stability of the found configurations to small fluctuations of phase jumps is analyzed. It is shown that at reasonably large values of I, stable solutions of any configuration can exist (in particular, with any number of quanta in a cell). At the beginning of each range of values of I, in which a corresponding solution exists, there is an interval (sometimes very small) in which the solution is unstable.

SUPERCONDUCTING FILM WITH EMBEDDED COBALT NANORODS

Studies of the transport properties of a PbBi superconducting film with an embedded triangular array of cobalt nanorods with a 300 nm period are reported. It is found that the critical current demonstrates strong hysteresis and maxima at magnetic field values corresponding to an integral number of flux quanta per unit cell of the array. The amplitude of the critical current is larger than for a control film without nanorods. The increase varies from several times to several orders of magnitude depending on the magnetic field and temperature.

A Phenomenological Model for the Electromagnetic Origin of Mass in Particles, and Its Quantitative Application to the Electron, the Muon, the Proton, and the Neutron

A simple phenomenological model is developed, which indicates the existence of a direct link between the concept of rest mass of a particle and magnetodynamic energies associated to the formation of the particle. The model is based upon the principles of quantization and conservation of flux, well known for their application in superconductivity. The charge of particles is considered as forming vortices of superconducting currents, which we postulate are created by electromagnetic fluctuations from vacuum (or related processes). A new quantization rule gathers the size, the magnetic moment, and the rest mass of the particle and associates these quantities to the integer number of flux quanta that should be stored in the vortices corresponding to each particle. The model is applied to the electron, the muon, the proton, and the neutron. Quantitative consistency with available experimental data for these subatomic particles is obtained.

Fractional Quantum Hall Effect of Lattice Bosons Near Commensurate Flux

We study interacting bosons on a lattice in a magnetic field. When the number of flux quanta per plaquette is close to a rational fraction, the low-energy physics is mapped to a multispecies continuum model: bosons in the lowest Landau level where each boson is given an internal degree of freedom, or pseudospin. We find that the interaction potential between the bosons involves terms that do not conserve pseudospin, corresponding to umklapp processes, which in some cases can also be seen as BCS-type pairing terms. We argue that in experimentally realistic regimes for bosonic atoms in optical lattices with synthetic magnetic fields, these terms are crucial for determining the nature of allowed ground states. In particular, we show numerically that certain paired wave functions related to the Moore-Read Pfaffian state are stabilized by these terms, whereas certain other wave functions can be destabilized when umklapp processes become strong.

BLACK HOLE STATE COUNTING IN LOOP QUANTUM GRAVITY

The two ways of counting microscopic states of black holes in the U(1) formulation of loop quantum gravity: one counting all allowed spin network labels j, m and the other only m labels, are discussed in some detail. The constraints on m are clarified and the map between the flux quantum numbers and m discussed. Configurations with |m|=j, which are sometimes sought after, are shown to be important only when large areas are involved. The discussion is extended to the SU(2) formulation.

Transference of transport anisotropy to composite fermions

When interacting two-dimensional electrons are placed in a large perpendicular magnetic field, to minimize their energy, they capture an even number of flux quanta and create new particles called composite fermions (CFs). These complex electron-flux-bound states offer an elegant explanation for the fractional quantum Hall effect. Furthermore, thanks to the flux attachment, the effective field vanishes at a half-filled Landau level and CFs exhibit Fermi-liquid-like properties, similar to their zero-field electron counterparts. However, being solely influenced by interactions, CFs should possess no memory whatever of the electron parameters. Here we address a fundamental question: Does an anisotropy of the electron effective mass and Fermi surface (FS) survive composite fermionization? We measure the resistance of CFs in AlAs quantum wells where electrons occupy an elliptical FS with large eccentricity and anisotropic effective mass. Similar to their electron counterparts, CFs also exhibit anisotropic transport, suggesting an anisotropy of CF effective mass and FS.

Control Schemes of Quantum-Based Pico-Newton Force Measurement

We propose control-measurement schemes for flux-quantum-based pico-newton force metrology. A micron-sized superconducting ring placed in magnetic field gradients can serve as a pico-newton force gauge. Constant-force steps can be realized by controlling the flux quanta in the superconductor ring. Two different schemes are proposed to control and measure the number of flux quanta in the ring. They are the fixed bias-flux mode and the fixed bias-current mode. In the fixed bias-flux (bias-current) mode, external bias-flux (transport bias-current) is fixed just below the resistive transition and entry of the flux quanta is monitored by counting the voltage bumps with increasing the bias current (the external flux). At the targeted number of flux quanta, bias current and flux are reset to zero. The number of flux quanta trapped in the ring can be confirmed by counting the voltage bumps with increasing the bias current up to the maximum critical current of the ring. Simultaneous flux control and force measurement is made by measuring force steps in field gradient with counting the number of flux quanta exciting the ring with increasing the bias current.

The influence of time-dependent electric and magnetic fields on the dynamic localization of lattice electrons

In this paper we deal with the derivation of dynamic localization conditions for electrons on the one-dimensional (1D) lattice under the influence of ac electric and magnetic fields of the same frequency. We resort, for convenience, to a tight-binding single-band Hamiltonian. Our emphasis is on a more fundamental theoretical understanding by investigating interplays between such fields and the nearest-neighbor hopping interactions characterizing the Hamiltonian. In general, such conditions get expressed in terms of infinite sums of binary products of Bessel functions of the first kind. These sums are hardly tractable, but we found that selecting in a suitable manner the phases of time-dependent modulations leads to controllable frequency-mixing effects providing appreciable simplifications. Such mixings concern competitions between the number of flux quanta and the quotients of field amplitudes and field frequencies. More exactly, tuning one of the mixed frequencies to zero opens the way to establishing the simplified dynamic localization conditions. By resorting again to the zeros of the Bessel function of zeroth order. This results in quickly tractable relationships between the amplitudes of electric and magnetic fields, the field frequency, and the zeros referred to just above. Pure field limits and superpositions between uniform electric and time-dependent magnetic fields are also discussed. Comments concerning the role of disorder and of the Coulomb interaction are also made.

Cosmic string formation by flux trapping

We study the formation of cosmic strings by confining a stochastic magnetic field into flux tubes in a numerical simulation. We use overdamped evolution in a potential that is minimized when the flux through each face in the simulation lattice is a multiple of the fundamental flux quantum. When the typical number of flux quanta through a correlation-length-sized region is initially about 1, we find a string network similar to that generated by the Kibble-Zurek mechanism. With larger initial flux, the loop distribution and the Brownian shape of the infinite strings remain unchanged, but the fraction of length in infinite strings is increased. A 2D slice of the network exhibits bundles of strings pointing in the same direction, as in earlier 2D simulations. We find, however, that strings belonging to the same bundle do not stay together in 3D for much longer than the correlation length. As the initial flux per correlation length is decreased, there is a point at which infinite strings disappear, as in the Hagedorn transition.

Enhanced vortex trapping by a composite antidot lattice in a superconducting Pb film

Using electrical transport measurements, we study a composite antidot lattice consisting of two interpenetrating antidot square arrays with a different antidot size and the same lattice period. We show that placing an additional small antidot in the unit cell of the array a higher number of flux quanta per unit cell can be trapped inside the antidots, compared to a reference antidot film without the additional small antidots. As a consequence, the field range in which an enhanced critical current is observed is considerably expanded.

CHARGE AND STATISTICS OF QUASIPARTICLES IN FRACTIONAL QUANTUM HALL EFFECT

We have studied here the charge and statistics of quasiparticle excitations in FQH states on the basis of the Berry phase approach incorporating the fact that even number of flux quanta can be gauged away when the Berry phase is removed to the dynamical phase. It is observed that the charge q and statistical parameter θ of a quasiparticle at filling factor ν = n/(2pn+1) are given by q = (n/2pn+1)e and θ = n/(2pn+1), with the fact that the charge of the quasihole is opposite to that of the quasielectron. Using Laughlin wave function for quasiparticles, numerical studies have been done following the work of Kjønsberg and Myrheim1 for FQH states at ν = 1/3 and it is pointed out that as in case of quasiholes, the statistics parameter can be well defined for quasielectrons having the value θ = 1/3.

Direct observation of vortex lattice transitions in mesoscopic superconducting single crystals using STM

Superconductors containing mesoscopic artificially-engineered defects are orders of magnitude more resistant to the destructive effect of the magnetic field. We present scanning tunnelling microscopy study of the vortex lattice configurations in a mesoscopic single crystal superconductor—normal metal heterostructure. We observe co-existence of a strongly interacting multiquanta vortex lattice and interstitial Abrikosov vortices that form a composite magnetic flux distribution which undergoes a series of transitions between different topological configuration states. The vortex configuration states are strongly dependent on the nanoscale architecture of the superconductor and applied magnetic field. Scanning tunnelling spectroscopy images show the evolution of vortex topological states when the number of flux quanta in the system changes.

Theory of magnetic oscillations in Josephson-junction stacks

We consider magnetic oscillations of the critical current in stacks of intrinsic Josephson junctions. Depending on junction lateral size and magnetic field, oscillations may have either the period of half a flux quantum per junction (wide-stack regime) or one flux quantum per junction (narrow-stack regime). For junctions with lateral sizes of the order of several Josephson lengths, the stack crosses over from the wide-stack regime to the narrow-stack regime with increasing magnetic field. This crossover occurs via suppression of the critical-current peaks at the integer-flux-quanta points and enhancement of the critical-current peaks at the half-integer-flux-quanta points. In the narrow-stack regime the lattice structure periodically transforms between rectangular and triangular configurations. The latter configurations is realized only in narrow regions near magnetic-field values corresponding to an integer number of flux quanta per junction.

Direct Observation of Geometrical Phase Transitions in Mesoscopic Superconductors by Scanning Tunneling Microscopy

Using scanning tunneling microscopy, we mapped the distribution of the local density of states in a single crystal superconductor heterostructure with an array of submicron normal metal islands. We observe the coexistence of strongly interacting multiquanta vortex lattice with interstitial Abrikosov vortices. The newly formed composite magnetic flux structure undergoes a series of phase transitions between different topological configuration states. The vortex configuration states are strongly dependent on the number of flux quanta and the nanoscale confinement architecture of the mesoscopic superconductor. Here, we present images of vortex phase transitions due to confinement effects when the number of magnetic flux quanta in the system changes. The vortex dynamics in these systems could serve as a model for behavior of confined many-body systems when the number of particles changes.

Effective Hamiltonian approach for magnetic band structure and additional oscillations in the magnetization originating from the Fermi surface at finite magnetic fields

The one-electron Schr\"odinger equation in a two-dimensional periodic potential and a homogeneous magnetic field $B$ perpendicular to the plane is solved exactly for rational flux quantum numbers per unit cell ${\ensuremath{\Phi}}_{c}∕{\ensuremath{\Phi}}_{0}=p∕q$. For comparison, the spectrum around a certain flux quantum number ${p}_{0}∕{q}_{0}$ has also been obtained by semiclassical quantization of the exact magnetic band structure (MBS) at ${p}_{0}∕{q}_{0}$. To implement and justify this procedure, a generalized effective Hamiltonian theory based on the MBS at finite magnetic fields has been established. The total energy as a function of ${\ensuremath{\Phi}}_{c}∕{\ensuremath{\Phi}}_{0}$ shows a series of kinks, where each kink indicates an insulating state. The kinks of each series converge to a metallic state. The magnetization contains information not only about the band structure (at zero-magnetic field), but also about the magnetic band structures (for finite fields). The period of the oscillations in $M[1∕(B\ensuremath{-}{B}_{0})]$ is determined by the Fermi surface cross sections for the MBS at ${B}_{0}$. The height of the steps in $M(B)$ provides the energy gap in the MBS at $B$. Unlike the standard Lifshitz-Kosevich type approaches, our theoretical de Haas-van Alphen spectra contain the effects of magnetic breakdown, forbidden orbits, and interband coupling implicitly.

On the Principles of Vortex Localization and Motion in Superconductor Thin Films with Artificially Patterned Cavities

In this paper we summarize the results of our recent studies of vortex matter in high-Tc superconductors with antidots, i.e. lithographically patterned holes in superconductor thin films. We analyze theoretically and experimentally the distribution of magnetic flux and effective pinning energy in perforated samples: the equilibrium occupation number (number of flux quanta accomodated by the hole) is evaluated numerically in a range of temperatures and fields for antidot sizes typically used in experiments; magnetic field profiles and activation energy of thermally activated flux creep is obtained from magnetic relaxation measurements by the means of Hall probes array. Possible mechanisms of vortex penetration and propulsion in such structures are discussed.