- Fractional vortices
In a standard superconductor, described by a complex field $|Psi|e^\{iphi\}$ (condensates wave function), vortices carry quantized magnetic field: a consequence of $2pi$-invariance of the phase $phi$ of the condensate wave function $|Psi|e^\{iphi\}$. There a winding of the phase $\{phi\}$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 designedlong Josephson junction s (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 $02pi\; math>,\; because\; the\; phase\; is$ 2pi$periodic.$LJJ reacts to the phase discontinuity by bending the Josephson phase $phi(x)$ 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 $propto\; dphi(x)/dx$ localized around discontinuity (0-$pi$ boundary). It also results in appearance of asupercurrent $proptosinphi(x)$ circulating around discontinuity. The total magnetic flux $Phi$, carried by the localized magnetic field, is proportional to the value of the discontinuity $kappa$, namely$Phi\; =\; Phi\_0\; frac\{kappa\}\{2pi\},$where $Phi\_0$ is amagnetic flux quantum . For $pi$ discontinuity, $Phi=Phi\_0/2$ and the vortex ofsupercurrent is called asemifluxon . 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 thesupercurrent (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 theJosephson plasma frequency cite 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 ofsupercurrent carrying unquantizedmagnetic flux , in oppose to conventional Josephson vortex andsemifluxon s. Fractional vortices exist in the so-called 0-πlong Josephson junction s dense chains. Fractional vortices aresoliton s which are able to move and preserve their shape much like conventional Josephson vortices and in opposed tosemifluxon s 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 junction s dense chains. This is done by taking the asymptotic expansion of the phase difference equation of motion to the second order which results in $au^2ddot\{psi\}-Lambda^2psi"+sinpsi-gammasin2psi=0$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 $psi\_\{gamma\}=cos(1/gamma)$ and $-psi\_\{gamma\}$. There are two fractional vortices which correspond to these two values one carries Φ_{1}=ψ_{γ}Φ_{0}/π flux and the other carries Φ_{2}=Φ_{0}-Φ_{1}flux where Φ_{0}is the fundamental unit ofmagnetic flux quantum .For the first time fractional vortices were observed using d-wave superconductors at asymmetric 45° grain boundaries YBa

_{2}Cu_{3}O_{7-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-T_{c}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).=See also=

*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 YBa**_{2}Cu_{3}O_{7-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)00976-5and

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