Fractional vortices
Encyclopedia
In a standard superconductor, described by a complex field 
(condensates wave function), vortices carry quantized magnetic field: a consequence of 
-invariance of the phase 
of the condensate wave function 
. There a winding of the phase 
by 
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
. 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 
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
, where 
which however carry arbitrarily fractionally quantized magnetic flux.
discontinuities may appear in specially designed long Josephson junction
s (LJJ). For example, so-called 0-
LJJ have a 
discontinuity of the Josephson phase at a point where 0 and 
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 LJJ
.
.
We will denote the value of the phase discontinuity by
and, without losing generality, assume that 
, because the phase is 
periodic.
LJJ reacts to the phase discontinuity by bending the Josephson phase
in the 
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
localized around discontinuity (0-
boundary). It also results in appearance of a supercurrent
circulating around discontinuity. The total magnetic flux 
, carried by the localized magnetic field, is proportional to the value of the discontinuity 
, namely
where
is a magnetic flux quantum
. For
discontinuity, 
and the vortex of supercurrent
is called a semifluxon
. When
, 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)
.
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
.
.
.
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 frequency.
carrying unquantized magnetic flux
, in oppose to conventional Josephson vortex and semifluxon
s. Fractional vortices exist in the so-called 0-π long Josephson junction
s dense chains. Fractional vortices are soliton
s which are able to move and preserve their shape much like conventional Josephson vortices and in opposed to semifluxon
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 difference
between 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
where
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 and
, and can be extended to time dependent phenomena. For
he above equation for the phase, ψ, has two stable equilibrium values 
and 
. There are two fractional vortices which correspond to these two values one carries Φ1=
ψγΦ0/π flux and the other carries Φ2=
Φ0-Φ1 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 superconductor
and 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
and underdamped SIFS-type
.
, along with a 
rotation of spin angle. This is in contrast to 
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 
.
independent charged condensates or superconducting components interact with each other electromagnetically.
Such a situation occurs for example in the
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
(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
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
(condensates wave function), vortices carry quantized magnetic field: a consequence of -invariance of the phase of the condensate wave function . There a winding of the phase by creates a vortex which carries one flux quantum. See Quantum vortexQuantum vortex
In physics, a quantum vortex is a topological defect exhibited in superfluids and superconductors. Superfluids and superconductors are states of matter without friction. They exist only at very low temperatures. The existence of these quantum vortices was independently predicted by Richard Feynman...
.
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
. 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 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
, where which however carry arbitrarily fractionally quantized magnetic flux.Fractional Josephson vortices at phase discontinuities
Josephson phaseJosephson phase
In superconductivity, the Josephson phase is the difference of the phases of the quantum mechanical wave function in two superconducting electrodes forming a Josephson junction....
discontinuities may appear in specially designed long Josephson junction
Long Josephson junction
In superconductivity, a long Josephson junction is a Josephson junction which has one or more dimensions longer than the Josephson penetration depth \lambda_J. This definition is not strict....
s (LJJ). For example, so-called 0-
LJJ have a discontinuity of the Josephson phase at a point where 0 and parts join. Josephson phaseJosephson phase
In superconductivity, the Josephson phase is the difference of the phases of the quantum mechanical wave function in two superconducting electrodes forming a Josephson junction....
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 LJJ
.
.
We will denote the value of the phase discontinuity by
and, without losing generality, assume that , because the phase is periodic.LJJ reacts to the phase discontinuity by bending the Josephson phase
in the Josephson penetration depth
In superconductivity, Josephson penetration depth characterizes the typical length on which an externally-applied magnetic field penetrates into the long Josephson junction...
vicinity of the discontinuity point, so that far away there are no traces of this perturbation. Bending of the Josephson phase
Josephson phase
In superconductivity, the Josephson phase is the difference of the phases of the quantum mechanical wave function in two superconducting electrodes forming a Josephson junction....
inevitably results in appearance of a local magnetic field
localized around discontinuity (0- boundary). It also results in appearance of a supercurrentSupercurrent
A supercurrent is a superconducting current, that is, electric current which flows without dissipation....
circulating around discontinuity. The total magnetic flux , carried by the localized magnetic field, is proportional to the value of the discontinuity , namelywhere
is a magnetic flux quantumMagnetic flux quantum
The magnetic flux quantum Φ0 is the quantum of magnetic flux passing through a superconductor. The phenomenon of flux quantization was discovered B. S. Deaver and W. M. Fairbank and, independently, by R. Doll and M. Nabauer, in 1961...
. For
discontinuity, and the vortex of supercurrentSupercurrent
A supercurrent is a superconducting current, that is, electric current which flows without dissipation....
is called a semifluxon
Semifluxon
In superconductivity, a Semifluxon is a vortex of supercurrent carrying the magnetic flux equal to the half of the magnetic flux quantum. Semifluxons exist in the so-called 0-π long Josephson junctions at the boundary between 0 and π regions. For a shorter junction length In superconductivity, a...
. When
, 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 supercurrentSupercurrent
A supercurrent is a superconducting current, that is, electric current which flows without dissipation....
(clockwise or counterclockwise) circulating around its center (discontinuity point)
.
Semifluxon
Semifluxon
In superconductivity, a Semifluxon is a vortex of supercurrent carrying the magnetic flux equal to the half of the magnetic flux quantum. Semifluxons exist in the so-called 0-π long Josephson junctions at the boundary between 0 and π regions. For a shorter junction length In superconductivity, a...
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
.
.
.
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 frequency.
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 supercurrentSupercurrent
A supercurrent is a superconducting current, that is, electric current which flows without dissipation....
carrying unquantized magnetic flux
Magnetic flux
Magnetic flux , is a measure of the amount of magnetic B field passing through a given surface . The SI unit of magnetic flux is the weber...
, in oppose to conventional Josephson vortex and semifluxon
Semifluxon
In superconductivity, a Semifluxon is a vortex of supercurrent carrying the magnetic flux equal to the half of the magnetic flux quantum. Semifluxons exist in the so-called 0-π long Josephson junctions at the boundary between 0 and π regions. For a shorter junction length In superconductivity, a...
s. Fractional vortices exist in the so-called 0-π long Josephson junction
Long Josephson junction
In superconductivity, a long Josephson junction is a Josephson junction which has one or more dimensions longer than the Josephson penetration depth \lambda_J. This definition is not strict....
s dense chains. Fractional vortices are soliton
Soliton
In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...
s which are able to move and preserve their shape much like conventional Josephson vortices and in opposed to semifluxon
Semifluxon
In superconductivity, a Semifluxon is a vortex of supercurrent carrying the magnetic flux equal to the half of the magnetic flux quantum. Semifluxons exist in the so-called 0-π long Josephson junctions at the boundary between 0 and π regions. For a shorter junction length In superconductivity, a...
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 difference
between the two superconducting banks of the 0-π long Josephson junction
Long Josephson junction
In superconductivity, a long Josephson junction is a Josephson junction which has one or more dimensions longer than the Josephson penetration depth \lambda_J. This definition is not strict....
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
where
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 and, and can be extended to time dependent phenomena. For
he above equation for the phase, ψ, has two stable equilibrium values and . There are two fractional vortices which correspond to these two values one carries Φ1=ψγΦ0/π flux and the other carries Φ2=
Φ0-Φ1 flux where Φ0 is the fundamental unit of magnetic flux quantum
Magnetic flux quantum
The magnetic flux quantum Φ0 is the quantum of magnetic flux passing through a superconductor. The phenomenon of flux quantization was discovered B. S. Deaver and W. M. Fairbank and, independently, by R. Doll and M. Nabauer, in 1961...
.
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 superconductor
and 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
and underdamped SIFS-type
.
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, along with a rotation of spin angle. This is in contrast to 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 .(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 severalindependent charged condensates or superconducting components interact with each other electromagnetically.
Such a situation occurs for example in the
theories of the projected quantum states of liquid metallic hydrogen, where two order parametersoriginate from theoretically anticipated coexistence of electronic and protonic Cooper pairs. There topological defects with
an
(i.e. "integer") phase winding only in electronic or only in protonic condensate carries fractionally quantized magnetic flux: a consequence of electromagtneticinteraction with the second condensate. Also these fractional vortices carry a superfluid momentum which does not obey Onsager-Feynman quantization
Despite the integer phase winding, the basic properties of this kind of fractional vortices are very different from Abrikosov vortex
Abrikosov vortex
In superconductivity, an Abrikosov vortex is a vortex of supercurrent in a type-II superconductor. The supercurrent circulates around the normal core of the vortex. The core has a size \sim\xi — the superconducting coherence length...
solutions. For example in contrast to Abrikosov vortex
Abrikosov vortex
In superconductivity, an Abrikosov vortex is a vortex of supercurrent in a type-II superconductor. The supercurrent circulates around the normal core of the vortex. The core has a size \sim\xi — the superconducting coherence length...
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