# Fractional vortices

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Fractional vortices

In a standard superconductor, described by a complex field $|Psi|e^\left\{iphi\right\}$ (condensates wave function), vortices carry quantized magnetic field: a consequence of $2pi$-invariance of the phase $phi$ of the condensate wave function $|Psi|e^\left\{iphi\right\}$. There a winding of the phase $\left\{phi\right\}$by $2pi$ creates a vortex which carries one flux quantum. See Quantum vortex.

The term Fractional vortex is used for various very different quantum vortices or topological defects in very different contexts when:

(i) A physical system allows phase windings different from $2pi imes integer$. I.e. non-integer or fractional phase winding. Quantum mechanics prohibits it in a uniform ordinary superconductor. But it becomes possible in an inhomogeneous system for example if a vortex is placed on a boundary between two superconductors which are connected by a weak link (Josephson Junction), such a situation also occurs in some cases in polycrystalline samples on grain boundaries etc. It results in vortices having fractional phase windings and thus fractional flux). Similar situation occurs in Spin-1 Bose condensates for a vortex with $pi$ phase winding combined with a domain of overturned spins.

(ii) Different situation occurs in uniform multicomponent superconductors which allow stable vortex solution with a phase winding $2pi$ which however carry arbitrarily fractionally quantized magnetic flux [ [http://link.aps.org/abstract/PRL/v89/e067001] . Egor Babaev, Phys.Rev.Lett. 89 (2002) 067001.] .

=Technical explanation=

(i) Vortices with non-integer phase winding

Josephson vortices

Fractional Josephson vortices at phase discontinuities

(not yet finished)

Josephson phase discontinuities may appear in specially designed long Josephson junctions (LJJ). For example, so-called 0-$pi$ LJJ have a $pi$ discontinuity of the Josephson phase at a point where 0 and $pi$ parts join. Josephson phase discontinuities can also be introduces using artificial tricks, e.g. a pair of tiny current injectors attached to one of the superconducting electrodes of the LJJcite journal
author = A. Ustinov
year = 2002
title = Fluxon insertion into annular Josephson junctions
journal = Appl. Phys. Lett.
volume = 80
pages = 3153--3155
doi = 10.1063/1.1474617
] cite journal
author = B. A. Malomed and A. V. Ustinov
year = 2004
title = Creation of classical and quantum fluxons by a current dipole n a long Josephson junction
journal = Phys. Rev. B
volume = 69
pages = 064502
doi = 10.1103/PhysRevB.69.064502
] .cite journal
author = E. Goldobin, A. Sterck, T. Gaber, D. Koelle, R. Kleiner
year = 2004
title = Dynamics of semifluxons in Nb long Josephson 0-$pi$ junctions
journal = Phys. Rev. Lett.
volume = 92
pages = 057005
doi = 10.1103/PhysRevLett.92.057005
] .We will denote the value of the phase discontinuity by $kappa$ and, without losing generality, assume that

LJJ reacts to the phase discontinuity by bending the Josephson phase $phi\left(x\right)$ in the $lambda_J$ vicinity of the discontinuity point, so that far away there are no traces of this perturbation. Bending of the Josephson phase inevitably results in appearance of a local magnetic field $propto dphi\left(x\right)/dx$ localized around discontinuity (0-$pi$ boundary). It also results in appearance of a supercurrent $proptosinphi\left(x\right)$ circulating around discontinuity. The total magnetic flux $Phi$, carried by the localized magnetic field, is proportional to the value of the discontinuity $kappa$, namely$Phi = Phi_0 frac\left\{kappa\right\}\left\{2pi\right\},$where $Phi_0$ is a magnetic flux quantum. For $pi$ discontinuity, $Phi=Phi_0/2$ and the vortex of supercurrent is called a semifluxon. When $kappa eqpi$, one speaks about arbitrary fractional Josephson vortices. This type of vortices are pinned at the phase discontinuity point, but may have two polarities, positive and negative, distinguished by the direction of the fractional flux and direction of the supercurrent (clockwise or counterclockwise) circulating around its center (discontinuity point)cite journal
author = E. Goldobin, D. Koelle, R. Kleiner
year = 2004
title = Ground states of one and two fractional vortices in long Josephson 0-$kappa$ junctions
journal = Phys. Rev. B
volume = 70
pages = 174519
doi = 10.1103/PhysRevB.70.174519
] .

Semifluxon is a particular case of such a fractional vortex pinned at the phase discontinuity point.

Although, such fractional Josephson vortices are pinned, they, if perturbed, may perform a small oscillations around the phase discontinuity point with the eigenfrequency, which depends on the value of $kappa$cite journal
author = E. Goldobin, H. Susanto, D. Koelle, R. Kleiner, S. A. van Gils
year = 2005
title = Oscillatory eigenmodes and stability of one and two arbitrary fractional vortices in long Josephson 0-$kappa$ junctions
journal = Phys. Rev. B
volume = 71
pages = 104518
doi = 10.1103/PhysRevB.71.104518
] .cite journal
author = K. Buckenmaier, T. Gaber, M. Siegel, D. Koelle, R. Kleiner, E. Goldobin
year = 2007
title = Spectroscopy of the Fractional Vortex Eigenfrequency in a Long Josephson 0-$kappa$ Junction
journal = Phys. Rev. Lett.
volume = 98
pages = 117006
doi = 10.1103/PhysRevLett.98.117006
] ..

This type of fractional Josephson vortices may find applications in classical and quantum information storage and processing as well as to build tunable band gap materials for the frequency range of the order of the Josephson plasma frequencycite journal
author = H. Susanto, E. Goldobin, D. Koelle, R. Kleiner, S. A. van Gils
year = 2005
title = Controllable plasma energy bands in a one-dimensional crystal of fractional Josephson vortices
journal = Phys. Rev. B
volume = 71
pages = 174510
doi = 10.1103/PhysRevB.71.174510
] .

Vortices on grain boundaries in d-wave superconductors and Josephson Junctions

In context of d-wave superconductivity, a Fractional vortex known also as splinter vortex is a vortex of supercurrent carrying unquantized magnetic flux, in oppose to conventional Josephson vortex and semifluxons. Fractional vortices exist in the so-called 0-π long Josephson junctions dense chains. Fractional vortices are solitons which are able to move and preserve their shape much like conventional Josephson vortices and in opposed to semifluxons which are attached to the boundary between 0 and π regions.

Theoretically one can obtain an effective double sin-Gordon equation for the phase differencebetween the two superconducting banks of the 0-π long Josephson junctions dense chains. This is done by taking the asymptotic expansion of the phase difference equation of motion to the second order which results in $au^2ddot\left\{psi\right\}-Lambda^2psi"+sinpsi-gammasin2psi=0$

where $gamma$ is a dimensionless constant defined by the junction's properties. The detailed mathematical procedure is similar to the one done for a parametrically driven pendulum, see for example cite book
author = L. D. Landau and E. M. Lifshitz
title = "Mechnics", Pergamon press, Oxford
year = 1994
] and cite book
author = V. I. Arnold, V. V Kozlov, and A. I. Neishtandt
title = "Mathematical aspects of classical and celestial mechnics", Springer
year = 1997
] , and can be extended to time dependent phenomenacite journal
author = M. Moshe and R. G. Mints
year = 2007
title = "Shapiro steps in Josephson junctions with alternating critical current density"
journal = Phys. Rev. B
volume = 76
pages = 054518
doi = 10.1103/PhysRevB.76.054518
] . For $gamma>1$ he above equation for the phase, ψ, has two stable equilibrium values $psi_\left\{gamma\right\}=cos\left(1/gamma\right)$ and $-psi_\left\{gamma\right\}$. There are two fractional vortices which correspond to these two values one carries Φ1γΦ0/π flux and the other carries Φ201 flux where Φ0 is the fundamental unit of magnetic flux quantum.

For the first time fractional vortices were observed using d-wave superconductors at asymmetric 45° grain boundaries YBa2Cu3O7-δ . In these systems the phase shift of π takes place inside the d-wave superconductorand not at the barrier. Due to the advent of controlled coupling by proper chosen ferromagnetic thicknesses,0–π JJs have also recently been realized in low-Tc SFS-like systems cite journal
author = M. L. Della Rocca, M. Aprili, T. Kontos, A. Gomez and P. Spathis
year = 2005
title = Ferromagnetic 0-π Junctions as Classical Spins
journal = Phys. Rev. Lett
volume = 94
pages = 197003
doi = 10.1103/PhysRevLett.94.197003
] and underdamped SIFS-typecite journal
author = M. Weides, M. Kemmler, H. Kohlstedt, R. Waser, D. Koelle, R. Kleiner and E. Goldobin
year = 2006
title = 0-π Josephson Tunnel Junctions with Ferromagnetic Barrier
journal = Phys. Rev. Lett
volume = 97
pages = 247001
doi = 10.1103/PhysRevLett.97.247001
] .

pin-triplet Superfluidity

In certain states of spin-1 superfluids or Bose condensates condensate's wavefunction is invariant if to change a superfluid phase by $pi$, along with a $pi$ rotation of spin angle. This is in contrast to $2pi$ invariance of condensate wavefunction in a spin-0 superfluid. A vortex resulting from such phase windings is called fractional or half-quantum vortex, in contrast to one-quantum vortex where a phase changes by $2pi$ [Dieter Vollhardt , Peter Woelfle The Superfluid Phases Of Helium 3 (1990)] .

(ii) Vortices with integer phase winding and fractional flux in multicomponent superconductivity

The term "Fractional vortex" appears also in context of multicomponent superconductivity of e.g. in the theories of the projected quantum states of liquid metallic hydrogen, where two order parametersoriginate from theoretically anticipated coexistence of electronic and protonic superconductivity. There a topological defects with an $2pi$ (i.e. "integer") phase winding only in electronic or only in protonic condensate carries fractionally quantized magnetic flux. Also it carriers a superfluid momentum which does not obey Onsager-Feynman quantisation and is called "fractional flux vortex" [ [http://arxiv.org/pdf/cond-mat/0111192] . Egor Babaev, "Vortices with fractional flux in two-gap superconductors and in extended Faddeev model" Phys.Rev.Lett. 89 (2002) 067001.] [ [http://www.nature.com/nphys/journal/v3/n8/full/nphys646.html] . Egor Babaev, N. W. Ashcroft "Violation of the London Law and Onsager-Feynman quantization in multicomponent superconductors" Nature Physics 3, 530 - 533 (2007).]

*Josephson junction
*Pi Josephson junction
*Semifluxon
*Quantum vortex

=References=

*cite journal
author = Mints, R. G. and Papiashvili, Ilya and Kirtley, J. R. and Hilgenkamp, H. and Hammerl, G. and Mannhart, J.
year = 2002
title = Observation of Splintered Josephson Vortices at Grain Boundaries in YBa2Cu3O7-δ
journal = Phys. Rev. Lett"'.
volume = 89
pages = 067004
doi = 10.1103/PhysRevLett.89.067004

*cite journal
author = Mints, R. G.
year = 1998
title = Self-generated flux in Josephson junctions with alternating critical current density"'
journal = Phys. Rev. B
volume = 57
pages = R3221
doi = 10.1103/PhysRevB.57.R3221

*cite journal
author = C. C. Tsuei and J. R. Kirtley
title = d-Wave pairing symmetry in cuprate superconductors --- fundamental implications and potential applications
journal = Physica C
year = 2002
volume = 367
pages = 1
doi = 10.1016/S0921-4534(01)00976-5

and

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