THE INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS ...

FRACTIONAL VORTICITY AND FRACTIONAL FLUX 191
Using the closed loop of the corresponding Alice string, one can produce
the baryonic charge (or other charge) from the vacuum by creating the baryon–
antibaryon pair and forcing the antibaryon to move through the loop. In this
way the antibaryon transforms to the baryon, and one gains the double baryonic
charge from the vacuum. In this process the loop of the Alice string acquires
the opposite charge distributed along the string – the so-called Cheshire charge
(Alford et al. 1990).
15.3.3 Fractional flux in chiral superconductor
In 3 He-A the fractional vorticity is still to be observed. However, its discussion extended
to unconventional superconductivity led to predictions of half-quantum
vortices in superconductors (Geshkenbein et al. 1987); and finally such a vortex
was discovered by Kirtley et al. (1996). It was topologically pinned by the
intersection line of three grain boundary planes in a thin film of a cuprate superconductor,
YBa2Cu3O7−δ (see Sec. 15.3.4).
In unconventional superconductors, the U(1)Q gauge symmetry is broken
together with some symmetry of the underlying crystal, which is why the crystalline
structure of the superconductor becomes important. Let us first start with
the axial or chiral superconductor, whose order parameter structure is similar
to that in 3 He-A. The possible candidate is the superconductivity in Sr2RuO4,
whose crystal structure has tetragonal symmetry. In the simplest representation,
the 3 He-A order parameter – the off-diagonal element in eqn (7.57) – adapted
to the crystals with tetragonal symmetry has the following form:
∆(p) =∆0 ( ˆ d · σ) (sin p · a + i sin p · b) e iθ . (15.20)
Here θ is the phase of the order parameter (here we use θ instead of Φ to distinguish
the order parameter phase from the magnetic flux Φ); a and b are the
elementary vectors of the crystal lattice within the layer. When |p · a|/¯h ≪ 1
and |p · b|/¯h ≪ 1, the order parameter acquires the familiar form applicable to
liquids with triplet p-wave pairing: ∆(p) =eµip i σµ, with eµi ∝ ˆ dµ(âi + i ˆ bi).
Vortices with fractional winding number n1 can be constructed in two ways.
The traditional way discussed for liquid 3 He-A is applicable to superconductors
when the ˆ d-vector is not strongly fixed by the crystal fields, and is flexible enough.
Then one obtains the analog of n1 =1/2 vortex in eqn (15.18): after circling
around this Alice string, ˆ d →− ˆ d, while the phase of the order parameter θ →
θ +π. According to eqn (15.17) such a vortex traps the magnetic flux Φ = hc/4e,
which is one-half of the flux trapped by a conventional Abrikosov vortex having
n1 =1.
In another scenario in Fig. 15.3, the crystalline properties of the chiral superconductor
are exploited. Twisting the crystal axes a and b in the closed wire of a
tetragonal superconductor, one obtains an analog of the Möbius strip geometry
(Volovik 2000b). The closed loop traps the fractional flux, if it is twisted by an
angle π/2 before gluing the ends. Since the local orientation of the crystal lattice
continuously changes by π/2 around the loop, axes a and b transform to each