Fractional vortices

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

The term Fractional vortex is used for two kinds of very different quantum vortices which occur when:

(i) A physical system allows phase windings different from 2\pi \times {\mathit {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 only by an extremely weak link (also called Josephson Junction), such a situation also occurs in some cases in polycrystalline samples on grain boundaries etc. At such superconducting boundaries the phase can have a discontinous jump. Correspondingly a vortex placed onto such a boundary acquires a fractional phase windings hence the term fractional vortex. Similar situation occurs in Spin-1 Bose condensates where, a vortex with \pi phase winding can exist if it is combined with a domain of overturned spins.

(ii) Different situation occurs in uniform multicomponent superconductors which allow stable vortex solution with integer phase winding 2\pi N, where N=\pm 1,\pm 2,... which however carry arbitrarily fractionally quantized magnetic flux.[1]

(i) Vortices with non-integer phase winding

Josephson vortices

Fractional vortices at phase discontinuities

Josephson phase discontinuities may appear in specially designed long Josephson junctions (LJJ). For example, so-called 0-π LJJ have a π discontinuity of the Josephson phase at the point where 0 and π parts join. Physically, such 0-π LJJ can be fabricated using tailored ferromagnetic barrier[2][3] or using d-wave superconductors.[4][5] The Josephson phase discontinuities can also be introduced using artificial tricks, e.g. a pair of tiny current injectors attached to one of the superconducting electrodes of the LJJ.[6][7][8] We denote the value of the phase discontinuity by κ and, without losing generality, assume that 0<κ<2π, because the phase is 2π 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 d\phi (x)/dx localized around discontinuity (0-π boundary). It also results in appearance of a supercurrent \propto \sin \phi (x) circulating around discontinuity. The total magnetic flux Φ, carried by the localized magnetic field, is proportional to the value of the discontinuity κ, namely Φ = (κ/2π)Φ, where Φ0 is a magnetic flux quantum. For π-discontinuity, Φ=Φ0/2 and the vortex of supercurrent is called a semifluxon. When κ≠π, 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).[9]

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,[10][11] which depends on the value of κ.

Splintered vortices (double sine-Gordon solitons)

In context of d-wave superconductivity, a Fractional vortex also known as splintered vortex[12][13] is a vortex of supercurrent carrying unquantized magnetic flux Φ1<Φ0, which depends on parameters of the system. Physically such vortices may appear at the gran boundary between two d-wave superconductors, which often looks like regular or irregular sequence of 0 and π facets. One can also construct an artificial array of short 0 and π facets to achive the same effect. These splintered vortices are solitons. They are able to move and preserve their shape similar to conventional integer Josephson vortices (fluxons). This is opposite to the fractional vortices pinned at phase discontinuity, e.g. semifluxons, that are pinned at the discontinuity and cannot move far from it.

Theoretically, one can describe a grain boundary between d-wave superconductors (or an array of tiny 0 and π facets) by an effective equation for a large-scale phase ψ. Large scale means the scale, which is much larger than the facet size. This equation is double sin-Gordon equation, which in normalized units reads

{\ddot {\psi }}-\psi ''+\sin \psi +g\sin(2\psi )=0

 

 

 

 

(EqDSG )

where g<0 is a dimensionless constant resulting from averaging over tiny facets. The detailed mathematical procedure of averaging is similar to the one done for a parametrically driven pendulum,[14][15] and can be extended to time-dependent phenomena.[16] In essence, (EqDSG) descrived extended φ Josephson junction.

For g<-1 (EqDSG) has two stable equilibrium values (in each 2π interval): ψ=±φ, where φ=cos(-1/g). They corresponding to two energy minima. Correspondingly, there are two fractional vortices (topological solitons): one with the phase ψ(x) going from -φ to +φ, while the other has the phase ψ(x) changing from +φ to -φ+2π. The first vortex has a topological change of 2φ and carries the magnetic flux Φ1=(φ/π)Φ0. The second vortex has a topological change of 2π-2φ and carries the flux Φ2=Φ0-Φ1.

For the first time splintered vortices were observed at the asymmetric 45° grain boundaries between two d-wave superconductors[13] YBa2Cu3O7-δ.

Spin-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 2\pi 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 2\pi .[17]

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

Different kind of "Fractional vortices" appears in the different context in multi-component superconductivity where several independent charged condensates or superconducting components interact with each other electromagnetically. Such a situation occurs for example in the U(1)\times U(1) theories of the projected quantum states of liquid metallic hydrogen, where two order parameters originate from theoretically anticipated coexistence of electronic and protonic Cooper pairs. There topological defects with an 2\pi (i.e. "integer") phase winding only in electronic or only in protonic condensate carries fractionally quantized magnetic flux: a consequence of electromagtnetic interaction with the second condensate. Also these fractional vortices carry a superfluid momentum which does not obey Onsager-Feynman quantization [18] [19] Despite the integer phase winding, the basic properties of this kind of fractional vortices are very different from Abrikosov vortex solutions. For example in contrast to Abrikosov vortex their magnetic field, generically is not exponentially localized in space. Also in some cases the magnetic flux inverts its direction at a certain distance from the vortex center [20]

See also

References

  1. ↑ . Egor Babaev, Phys.Rev.Lett. 89 (2002) 067001.
  2. ↑ M. Weides, M. Kemmler, H. Kohlstedt, R. Waser, D. Koelle, R. Kleiner and E. Goldobin (2006). "0-p Josephson Tunnel Junctions with Ferromagnetic Barrier". Phys. Rev. Lett 97 (24): 247001. arXiv:cond-mat/0605656. Bibcode:2006PhRvL..97x7001W. doi:10.1103/PhysRevLett.97.247001. PMID 17280309. 
  3. ↑ M. L. Della Rocca, M. Aprili, T. Kontos, A. Gomez and P. Spathis (2005). "Ferromagnetic 0-p Junctions as Classical Spins". Phys. Rev. Lett 94 (19): 197003. arXiv:cond-mat/0501459. Bibcode:2005PhRvL..94s7003D. doi:10.1103/PhysRevLett.94.197003. PMID 16090200. 
  4. ↑ Tsuei, C.C.; Kirtley, J.R. (February 2002). "d-Wave pairing symmetry in cuprate superconductors—fundamental implications and potential applications". Physica C: Superconductivity 367 (1-4): 1–8. doi:10.1016/S0921-4534(01)00976-5. 
  5. ↑ Hilgenkamp, Hans; Ariando, ; Smilde, Henk-Jan H.; Blank, Dave H. A.; Rijnders, Guus; Rogalla, Horst; Kirtley, John R.; Tsuei, Chang C. (6 March 2003). "Ordering and manipulation of the magnetic moments in large-scale superconducting π-loop arrays". Nature 422 (6927): 50–53. doi:10.1038/nature01442. 
  6. ↑ A. Ustinov (2002). "Fluxon insertion into annular Josephson junctions". Appl. Phys. Lett. 80 (17): 3153–3155. Bibcode:2002ApPhL..80.3153U. doi:10.1063/1.1474617. 
  7. ↑ B. A. Malomed and A. V. Ustinov (2004). "Creation of classical and quantum fluxons by a current dipole n a long Josephson junction". Phys. Rev. B 69 (6): 064502. arXiv:cond-mat/0310595. Bibcode:2004PhRvB..69f4502M. doi:10.1103/PhysRevB.69.064502. 
  8. ↑ E. Goldobin, A. Sterck, T. Gaber, D. Koelle, R. Kleiner (2004). "Dynamics of semifluxons in Nb long Josephson 0-pi junctions". Phys. Rev. Lett. 92 (5): 057005. arXiv:cond-mat/0311610. Bibcode:2004PhRvL..92e7005G. doi:10.1103/PhysRevLett.92.057005. PMID 14995336. 
  9. ↑ E. Goldobin, D. Koelle, R. Kleiner (2004). "Ground states of one and two fractional vortices in long Josephson 0-\kappa junctions". Phys. Rev. B 70 (17): 174519. arXiv:cond-mat/0405078. Bibcode:2004PhRvB..70q4519G. doi:10.1103/PhysRevB.70.174519. 
  10. ↑ E. Goldobin, H. Susanto, D. Koelle, R. Kleiner, S. A. van Gils (2005). "Oscillatory eigenmodes and stability of one and two arbitrary fractional vortices in long Josephson 0-\kappa junctions". Phys. Rev. B 71 (10): 104518. arXiv:cond-mat/0410340. Bibcode:2005PhRvB..71j4518G. doi:10.1103/PhysRevB.71.104518. 
  11. ↑ K. Buckenmaier, T. Gaber, M. Siegel, D. Koelle, R. Kleiner, E. Goldobin (2007). "Spectroscopy of the Fractional Vortex Eigenfrequency in a Long Josephson 0-\kappa Junction". Phys. Rev. Lett. 98 (11): 117006. arXiv:cond-mat/0610043. Bibcode:2007PhRvL..98k7006B. doi:10.1103/PhysRevLett.98.117006. PMID 17501081. 
  12. ↑ Mints, R. (February 1998). "Self-generated flux in Josephson junctions with alternating critical current density". Physical Review B 57 (6): R3221–R3224. doi:10.1103/PhysRevB.57.R3221. 
  13. ↑ 13.0 13.1 Mints, R.; Papiashvili, Ilya; Kirtley, J.; Hilgenkamp, H.; Hammerl, G.; Mannhart, J. (July 2002). "Observation of Splintered Josephson Vortices at Grain Boundaries in YBa2Cu3O7-δ". Physical Review Letters 89 (6). doi:10.1103/PhysRevLett.89.067004. 
  14. ↑ L. D. Landau and E. M. Lifshitz (1994). Mechnics, Pergamon press, Oxford. 
  15. ↑ V. I. Arnold, V. V Kozlov, and A. I. Neishtandt (1997). Mathematical aspects of classical and celestial mechnics, Springer. 
  16. ↑ M. Moshe and R. G. Mints (2007). "Shapiro steps in Josephson junctions with alternating critical current density". Phys. Rev. B 76 (5): 054518. arXiv:0708.1222. Bibcode:2007PhRvB..76e4518M. doi:10.1103/PhysRevB.76.054518. 
  17. ↑ Dieter Vollhardt , Peter Woelfle The Superfluid Phases Of Helium 3 (1990)
  18. ↑ . Egor Babaev, "Vortices with fractional flux in two-gap superconductors and in extended Faddeev model" Phys.Rev.Lett. 89 (2002) 067001.
  19. ↑ . Egor Babaev, N. W. Ashcroft "Violation of the London Law and Onsager-Feynman quantization in multicomponent superconductors" Nature Physics 3, 530 - 533 (2007).
  20. ↑ 0903.3339. Egor Babaev, Juha Jaykka, Martin Speight, " Magnetic field delocalization and flux inversion in fractional vortices in two-component superconductors" Phys. Rev. Lett 103, 237002 (2009).
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