Intervortex Forces Research Articles

Abrikosov-Nielsen-Olesen strings from the Coleman-Weinberg potential

We study properties of Abrikosov-Nielsen-Olesen (ANO) strings with the Coleman-Weinberg (CW) potential, which we call CW-ANO strings. While the scale-invariant scalar potential has a topologically trivial vacuum admitting no strings at the classical level, quantum correction allows topologically nontrivial vacua and stable string solutions. We find that the system of the CW potential exhibits significant difference from that of the conventional Abelian-Higgs model with the quadratic-quartic potential. While a single-winding string is qualitatively similar in both systems, and the static intervortex force between two strings at large distance is attractive/repulsive in the type-I/II regime for both, that between two CW-ANO strings exhibits a nontrivial structure. It develops an energy barrier between them at intermediate distance, implying that the string with winding number $n>1$ can constitute a metastable bound state even in the type-II regime. We name such a superconductor type-$\overline{1.5}$. We also discuss implications to high-energy physics and cosmology.

Pinning effects in a two-dimensional cluster glass

We study numerically the glass formation and depinning transition of a system of two-dimensional cluster-forming monodisperse particles in presence of pinning disorder. The pairwise interaction potential is nonmonotonic, and is motivated by the intervortex forces in type-$1.5$ superconductors. Such systems can form cluster glasses due to the intervortex interactions following a thermal quench, without underlying disorder. We study the effects of vortex pinning in these systems. We find that a small density of pinning centers of moderate depth has limited effect on vortex glass formation, i.e., formation of vortex glasses is dominated by intervortex interactions. At higher densities pinning can significantly affect glass formation. The cluster glass depinning, under a constant driving force, is found to be plastic, with features distinct from non-cluster-forming systems such as clusters merging and breaking. We find that in general vortices with cluster-forming interaction forces can exhibit stronger pinning effects than regular vortices.

Dynamics of global and local vortices with orientational moduli

The dynamics of both global and local vortices with non-Abelian orientational moduli is investigated in detail. Head-on collisions of these vortices are numerically simulated for parallel, anti-parallel and orthogonal internal orientations where we find interesting dynamics of the orientational moduli. A detailed study of the inter-vortex force is provided and a phase diagram separating Abelian and non-Abelian vortex types is constructed. Some results on scatterings with non-zero impact parameter and multi-vortex collisions are included.

Vortices with magnetic field inversion in noncentrosymmetric superconductors

Superconducting materials with noncentrosymmetric lattices lacking space inversion symmetry exhibit a variety of interesting parity-breaking phenomena, including the magneto-electric effect, spin-polarized currents, helical states, and the unusual Josephson effect. We demonstrate, within a Ginzburg-Landau framework describing noncentrosymmetric superconductors with $O$ point group symmetry, that vortices can exhibit an inversion of the magnetic field at a certain distance from the vortex core. In stark contrast to conventional superconducting vortices, the magnetic-field reversal in the parity-broken superconductor leads to non-monotonic intervortex forces, and, as a consequence, to the exotic properties of the vortex matter such as the formation of vortex bound states, vortex clusters, and the appearance of metastable vortex/anti-vortex bound states.

Properties of dirty two-band superconductors with repulsive interband interaction: Normal modes, length scales, vortices, and magnetic response

Disorder in two-band superconductors with repulsive interband interaction induces a frustrated competition between the phase-locking preferences of the various potential and kinetic terms. This frustrated interaction can result in the formation of an $s+is$ superconducting state, that breaks the time-reversal symmetry. In this paper we study the normal modes and their associated coherence lengths in such materials. We especially focus on the consequences of the soft modes stemming from the frustration and time-reversal-symmetry breakdown. We find that two-bands superconductors with such impurity-induced frustrated interactions display a rich spectrum of physical properties that are absent in their clean counterparts. It features a mixing of Leggett's and Anderson-Higgs modes, and a soft mode with diverging coherence length at the impurity-induced second order phase transition from $s_{\pm}/s_{++}$ states to the $s+is$ state. Such a soft mode generically results in long-range attractive intervortex forces that can trigger the formation of vortex clusters. We find that, if such clusters are formed, their size and internal flux density have a characteristic temperature dependence that could be probed in muon-spin-rotation experiments. We also comment on the appearance of spontaneous magnetic fields due to spatially varying impurities.

Glass Transitions in Monodisperse Cluster-Forming Ensembles: Vortex Matter in Type-1.5 Superconductors.

At low enough temperatures and high densities, the equilibrium configuration of an ensemble of ultrasoft particles is a self-assembled, ordered, cluster crystal. In the present Letter, we explore the out-of-equilibrium dynamics for a two-dimensional realization, which is relevant to superconducting materials with multiscale intervortex forces. We find that, for small temperatures following a quench, the suppression of the thermally activated particle hopping hinders the ordering. This results in a glass transition for a monodispersed ensemble, for which we derive a microscopic explanation in terms of an "effective polydispersity" induced by multiscale interactions. This demonstrates that a vortex glass can form in clean systems of thin films of "type-1.5" superconductors. An additional setup to study this physics can be layered superconducting systems, where the shape of the effective vortex-vortex interactions can be engineered.

Short-range intervortex interaction and interacting dynamics of half-quantized vortices in two-component Bose-Einstein condensates

We study the interaction and dynamics of two half-quantized vortices in two-component Bose- Einstein condensates. Using the Pade approximation for the vortex core profile, we calculate the intervortex potential, whose asymptotic form for a large distance has been derived by Eto et al. [Phys. Rev. A, 83, 063603 (2011)]. Through numerical simulations of the two-dimensional Gross-Pitaevskii equations, we reveal different kinds of dynamical trajectories of the vortices depending on the combinations of signs of circulations and the intercomponent density coupling. Under the adiabatic limit, we derive the equations of motion for the vortex coordinates, in which the motion is caused by the balance between Magnus force and the intervortex forces. The initial velocity of the vortex motion can be explained quantitatively by this point vortex approximation, but under- standing the long-time behavior of the dynamics needs more consideration beyond our model.

Critical current oscillations in the intrinsic hybrid vortex state of SmFeAs(O,F).

In layered superconductors the order parameter may be modulated within the unit cell, leading to nontrivial modifications of the vortex core if the interlayer coherence length ξ(c)(T) is comparable to the interlayer spacing. In the iron pnictide SmFeAs(O,F) (T(c)≈50 K) this occurs below a crossover temperature T(⋆)≈41 K, which separates two regimes of vortices: anisotropic Abrikosov-like at high and Josephson-like at low temperatures. Yet in the transition region around T(⋆), hybrid vortices between these two characteristics appear. Only in this region around T(⋆) and for magnetic fields well aligned with the FeAs layers, we observe oscillations of the c-axis critical current j(c)(H) periodic in 1/sqrt[H] due to a delicate balance of intervortex forces and interaction with the layered potential. j(c)(H) shows pronounced maxima when a hexagonal vortex lattice is commensurate with the underlying crystal structure. The narrow temperature window in which oscillations are observed suggests a significant suppression of the order parameter between the superconducting layers in SmFeAs(O,F), despite its low coherence length anisotropy (γ(ξ)≈3-5).

Three and four-body intervortex forces in the Ginzburg–Landau models of single- and multicomponent superconductivity

A systematic numerical study of non-pairwise vortex interaction forces in the Ginzburg–Landau model for single-and multicomponent superconductivity is presented. The interactions are obtained by highly accurate numerical free energy minimization. In particular a three-body interaction is defined as the difference between the total interaction and sum of pairwise interactions in a system of three vortices and such interactions are studied for single and two-component type-1, type-2, and type-1.5 superconductors. In the investigated regimes, the three-body interaction is found to be short-range repulsive but long-range attractive in the type-1 case, zero in the critical κ (Bogomoln’y) case, attractive in the type-2 case and repulsive in the type-1.5 case. Some systems of four vortices are also studied and results indicate that four-body forces are of substantially less significance than the three-body interactions.

Current–Voltage Measurements on a Bi2Sr2CaCu2O x Whisker and Investigation of Its Vortex Dynamics in Self Field

We report current–voltage measurements carried out on a high quality Bi2Sr2CaCu2O x whisker in a large temperature range below the critical temperature. When a perpendicular field is applied, the current–voltage curves are typical of the flux flow regime at low fields and resemble flux creep characteristics at high fields. This change, related to what is described as the critical current peak effect is associated to a transition between different vortex phases. In this contribution, we focus on the dynamics of the vortices in self field. We show that the critical current in the flux flow regime can be calculated straightforwardly taking into account inter-vortex forces only, supposing that free vortices run between pinned vortices.

Critical current reduction in coated conductors when in-plane fields are applied

We report measurements of the critical current of two commercial coated conductors at low temperatures when in-plane magnetic fields up to 15 T are applied. The comparison between the two samples suggests that the presence of a magnetic substrate may have a detrimental effect, at least at low and medium fields. Nevertheless, the critical current density at 30 K and 15 T is in the range of that measured in self field at 77 K. We ascribe its reduction in presence of the field to inter-vortex forces.

Vortex coalescence and type-1.5 superconductivity in Sr2RuO4

Recently vortex coalescence was reported in superconducting Sr${}_{2}$RuO${}_{4}$ by several experimental groups for fields applied along the $c$ axis. We argue that Sr${}_{2}$RuO${}_{4}$ is a type-1.5 superconductor with long-range attractive, short-range repulsive intervortex interaction. The type-1.5 behavior stems from an interplay of the two orbital degrees of freedom describing this chiral superconductor together with the multiband nature of the superconductivity. These multiple degrees of freedom give rise to multiple coherence lengths, some larger and some smaller than the magnetic field penetration length, resulting in nonmonotonic intervortex forces.

Current–Voltage Measurements and Vortex Dynamics in a Bi2Sr2CaCu2O x Whisker: Observation of the Peak Effect and Calculation of the Critical Current in the Flux Flow Regime with and Without Applied Field

In this article, we report current–voltage measurements carried out on a Bi2Sr2CaCu2Ox whisker in a large temperature range below the critical temperature with and without applied magnetic field. We examine the critical current peak effect and the vortex dynamics at low field. The critical current peak effect consists of the initial increase of the critical current that subsequently decreases as the applied field is increased. For current–voltage measurements, this effect is associated with a change in the current–voltage curves that are typical of the flux flow regime at low fields and resemble flux creep characteristics for higher fields. As a general rule, our observations are consistent with the models that link the peak effect to vortex phase transitions. We calculate the critical current in self-field in the flux flow regime taking into account intervortex forces. We suppose that most vortices are pinned by defects while mobile vortices move through plastic channels between the strongly pinned vortex regions. When an external field is applied, we suggest that the increase in the critical current that is observed is linked to oscillations of the pinned vortices.

Effective dynamics of multi-vortices in an external potential for the Ginzburg–Landau gradient flow

We study the dynamics of multi-vortex configurations to the Ginzburg–Landau dissipative/gradient flow equations with external potential in . We show that for initial data close to the widely spaced multi-vortex configurations, the effective dynamics of the vortex centres are governed by the inter-vortex forces and by external potential forces for weak and strong external potentials, respectively.

Schrödinger–Chern–Simons vortex dynamics

We study the motion of vortices in the planar Ginzburg–Landau model with Schrödinger–Chern–Simons dynamics. We compare the moduli space approximation with the results of numerical simulations of the full field theory and find that there is agreement if the coupling constant is very close to the critical value separating type I from type II superconductors. However, there are significant qualitative differences even for modest deviations from the critically coupled regime. Radiation effects produce forces which are of the same order of magnitude as the intervortex force and therefore have a significant impact on vortex motion. We conclude that the moduli space approximation does not provide a good description of the dynamics in this regime.

Vortices and type-I superconductivity in neutron stars

In a recent paper by Link, it was pointed out that the standard picture of the neutron star core, composed of a mixture of a neutron superfluid and a proton type-II superconductor, is inconsistent with observations of long period precession in isolated pulsars. In the following we will show that intervortex force between the magnetic flux tubes may be attractive, resulting in a type-I (rather than type-II) superconductor. In this case the magnetic field cannot exist in the form of magnetic flux tubes, supporting Link's observation. This behavior of the system is due to the strong interaction between the proton-neutron Cooper pairs, which was previously ignored. We also calculate the critical magnetic fields ${H}_{c}$ and ${H}_{c2}$ for type-I∕II superconductors. These results also support our claim of type-I superconductivity in the cores of neutron stars.

Simple frictional model for interacting vortices in an isotropic type-II superconductor

In a simple model for an isotropic superconducting thin-plate sample, repulsive intervortex forces are balanced by surface forces produced by an external field ( H) and by frictional forces representing the effects of pinning and unpinning on displaced vortices. For the field-cooled state, the uniform vortex density plus the empirical fact that the average flux density ( B ) and H are essentially equal yield an operational inverse-square dependence of the intervortex force on the intervortex spacing. For the zero-field-cooled state, expressions are then derived for the hysteretic changes of B with H and for the associated flux-density profiles across the sample thickness, where the vortices are considered to interact individually or collectively (as bundles) with the pinning centers. For various values of a vortex-bundling parameter and for a scaling parameter linked to the saturation remanence, the calculated hysteretic properties are compared to experimental data on polycrystalline YBa 2Cu 3O 7 and (Ba, K)BiO 3. For both compounds, the comparison appears to favor a non-zero but small degree of vortex bundling, for which the predicted properties resemble those of the simple critical-state model of Bean (where vortex bundling is not considered).

Schematic frictional model for interacting vortices in an isotropic superconducting plate (abstract)

In the simple model proposed, repulsive intervortex forces are balanced by containing forces produced by the external field (H) and by frictional forces representing the effects of pinning on displaced vortices. For the field-cooled (FC) state, whose vortex density is presumably uniform, the empirical fact that the average flux density (B̄) in nearly equal to H yields an operational inverse-square dependence of the intervortex force on the intervortex spacing. For both the FC and zero-field-cooled (ZFC) states, expressions are derived for B̄ vs H (including the remanences at H=0) and for the profiles of B across the sample thickness. Calculations of these properties are compared with experiment and with the macroscopically related critical-state model, revealing again that the pinning forces are strongly dependent on H. The frictional interacting-vortex model is also used in deriving the critical current as a transport property of the FC and ZFC states.

Linear and nonlinear AC response of Abrikosov vortices in layered superconductors adiabatic approximation

Linear and nonlinear AC responses for a magnetically coupled stack of two-dimensional vortices (2DV's) forming an Abrikosov vortex line perpendicular to the layers of a layered superconducting film are calculated. By solving the Fokker-Planck equations for the distribution functions of 2DV positions in the adiabatic approximation and including randomly distributed pinning centers, a nonlinear differential equation for the continuous shift function S( z, t) is derived. Josephson coupling only changes coefficients in this equation. The equation for S( z, t) is solved perturbatively for the linear AC response and for the third-order mixing current at frequency 2ω 1-ω 2. This pseudoharmonic mixing probes the nonlinearity of the intervortex force and its interplay with pinning. Explicit analytical results are given for the surface impedance Z s and the mixing current δ j( 3). The cases of thin and thick films with respect to the AC penetration length are considered in detail.

AC response of Abrikosov vortices in layered superconductors

In a layered high-T c superconductor an Abrikosov vortex line perpendicular to the layers can be considered as a stack of coupled two-dimensional «pancake» vortices (2DV) with cores residing inside the superconducting layers. When the in-plane penetration length λ ab =λ is smaller than the Josephson length λ J =λ c d/λ (d: layer period) the interaction between 2DV is via a magnetic pair force which is complicated but exactly known. We calculate linear and nonlinear ac responses for this model including randomly distributed pinning centers. In detail, the thermally averaged shift S n (t) of a 2DV in layer n under the influence of a finite external ac current in the frequency range up to 1 GHz is computed from coupled Fokker-Planck equations for the distribution functions of 2DV positions. In the adiabatic approximation, the Fokker-Planck equations can be cast into a nonlinear differential equation for a continuous shift function S(z,t) using a gradient expansion. The constitutive equation for S(z,t) is solved perturbatively for the linear ac response and for the mixing current at frequency 2ω 1 -ω 2 from two external currents at frequencies ω 1 and ω 2 ≃2ω 1 due to a third order process. This pseudo-harmonic mixing probes the nonlinearity of the intervortex force and its interplay with pinning. Explicit results for the film geometry are given for the surface impedance Z s and the mixing current amplitude δj (3) as functions of external frequencies, crossover frequency ω c ∞n i dω D (measuring the pinning), and diffusion frequency ω D of free 2DV