Chiral CP^2 skyrmions in three-band superconductors

9.3 MB·English

by  Julien Garaud

Chiral CP2 skyrmions in three-band superconductors
Julien Garaud,1,2 Johan Carlstr¨om,2 Egor Babaev,1,2 and Martin Speight3
1Department of Physics, University of Massachusetts Amherst, MA 01003 USA
2Department of Theoretical Physics, The Royal Institute of Technology, Stockholm, SE-10691 Sweden
3School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
(Dated: January 15, 2013)
It is shown that under certain conditions, three-component superconductors (and in particular
three-band systems) allow stable topological defects different from vortices. We demonstrate the
existence of these excitations, characterized by a CP2 topological invariant, in models for three-
componentsuperconductorswithbrokentimereversalsymmetry. Wetermthesetopologicaldefects
“chiral GL(3) skyrmions”, where “chiral” refers to the fact that due to broken time reversal sym-
metry, these defects come in inequivalent left- and right-handed versions. In certain cases these
objectsareenergeticallycheaperthanvorticesandshouldbeinducedbyanappliedmagneticfield.
Inothersituationstheseskyrmionsaremetastablestates,whichcanbeproducedbyaquench. Ob-
servation of these defects can signal broken time reversal symmetry in three-band superconductors
3
or in Josephson-coupled bilayers of s and s-wave superconductors.
1 ±
0
2 PACSnumbers: 74.70.Xa74.20.Mn74.20.Rp

n
a I. INTRODUCTION in detail, these topological solitons which we term chi-
J
ral GL(3) skyrmions (chiral skyrmions for short). They

4
are magnetic flux-carrying excitations characterized by
1 Experiments on the recently discovered iron pnictide
a CP2 topological invariant, (by contrast this invariant
superconductors suggest the existence of positive co-
is trivial for ordinary vortices). The topological prop-
] efficient of Josephson coupling between superconduct-
n erties, motivating the denomination skyrmion are rigor-
ing components in two bands (s state) and possibly
o morethantwosuperconductingba±nds.1 Underthesecir- ouslydiscussed. Astheterminologysuggests,thesoliton
c
itself has a given chiral state of the Broken Time Rever-
- cumstances, new physics can appear. That is, frus-
r sal Symmetry. More precisely, different arrangements of
tration of competing interband Josephson couplings in
p
the fractional vortices constituting a skyrmion carrying
u three-component superconductors, can lead to sponta-
s neouslyBrokenTimeReversalSymmetry(BTRS)2,3(an- integerfluxdefinedifferentchiralityoftheskyrmion. Fi-
. nally GL(3) refers to the physical context of the three-
t other scenario for BTRS states in pnictides was dis-
a component Ginzburg–Landau theory. The thermody-
cussed in Refs. 4 and 5). There, the ground state ex-
m
plicitly breaks the discrete U(1)×Z symmetry.6,7 Re- namic and energetic (meta)stability of chiral skyrmions
2
- latedmulticomponentstateswerealsorecentlydiscussed, are discussed, as well as their perturbative stability. In
d
n in connection with other materials.8 If superconductiv- scanning SQUID, scanning Hall or magnetic force mi-
croscopy experiments, chiral GL(3) skyrmions can (un-
o ity in iron pnictides is described by just a two-band s
±
der certain conditions) be distinguished from vortices by
c models, BTRS states can nonetheless be obtained in a
[ their very exotic magnetic field profile. Fig. 1 shows ex-
Josephson-coupled bilayer of s superconductor and or-
±
2 dinary s-wave material.2 Such bilayer systems can be ef- amples of such exotic magnetic field signatures of chiral
skyrmions in three band superconductors with various
v fectively described by a three-component model where
2 the third component is coupled through a “real-space” parameters of the model.
4 inter-layered Josephson coupling. The paper is organized as follows. In Sec. II we in-
3
Duetoanumberofunconventionalphenomena,which troduce a Ginzburg-Landau model for three-component
4
. are not possible in two-band superconductors, the pos- superconductors where phase frustration due to compet-
1 sible experimental realization of three component su- ing Josephson interactions leads to Broken Time Rever-
1
perconductors (either with or without BTRS) recently sal Symmetry states. The structure of the domain walls
2
startedtoattractsubstantialinterest.3,6,7,9–14Thesephe- which are possible due to this new spontaneously broken
1
: nomena include: exotic collective modes which are dif- Z2 symmetryisdiscussedinSec.IIA.Theessentialcon-
v ferent from the Leggett’s mode;7,10,15 the existence of cepts of the topological excitations in multi-band super-
i
X a large disparity in coherence lengths even when inter- conductorsarediscussedinSec.IIC.Afterthat,thenew
component Josephson coupling is very strong, leading kind of topological excitations, chiral GL(3) skyrmions,
r
a to type-1.5 regimes7 (where some coherence lengths are are discussed Sec. IID. The physical properties: (i) en-
smaller and some are larger than the magnetic field pen- ergyofformationofaskyrmionversusvortexlattice,(ii)
etration length16); the possibility of flux-carrying topo- thermodynamical stability of the chiral skyrmions and
logical solitons different from Abrikosov vortices.6 (iii) their perturbative stability are investigated Sec. III.
Thispaperisafollow-uptoRef.6whereweintroduced In the next part, Sec. IV, the very rich interactions be-
new flux-carrying topological solitons. Here we study tween the chiral skyrmions and between skyrmions and
2
Figure 1. (Color online) – Example of unusual observable magnetic field configuration of chiral skyrmions.
vortices are investigated. The model has many inter- In multiband superconductors, a Ginzburg–Landau ex-
esting mathematical aspects as well. Sec. V is devoted pansion of this kind can in certain cases be formally jus-
to the most formal aspects and rigorous justifications of tified microscopically (see e.g. corresponding discussion
the physics and mathematical properties of the three- in two-band case17). In what follows, different physical
component Ginzburg–Landau model and the skyrmionic realizationsofthemodel(2.1)withdifferentbrokensym-
excitations therein. This section aims at a more mathe- metries are considered. Note that in some of the phys-
matical audience. Thus, readers less interested in formal ical realizations of multicomponent GL models, some of
justificationofthephysicscanskipthesediscussions,and the couplings are forbidden (for example on symmetry
go straight after Sec. IV to our conclusions in Sec. VI. grounds). This can occur for intercomponent Joseph-
There we conclude this paper by addressing, in more de- son couplings, in some realizations.18 More terms, con-
tail, the possible experimental signatures of our chiral sistent with symmetries, can be included to extend the
GL(3) skyrmions. GLfunctional. Alternativelyamicroscopicapproachcan
provide a more quantitatively accurate picture at lower
temperatures. However, the properties of the topologi-
II. THE MODEL cal objects which are discussed, should then differ only
quantitatively and not qualitatively in the framework of
e.g. microscopicapproachforasystemwithagivensym-
In this paper we consider various realizations of three-
metry (some examples how phenomenological multiband
component superconductivity described by the following
GLmodelsgivegoodresultsevenatlowtemperaturecan
three-component Ginzburg–Landau (GL) model:
be found in Ref. 17).
Thefieldconfigurationsconsideredinthefollowingare
1 (cid:88)1 1
F = 2(∇×A)2+ 2|Dψa|2+αa|ψa|2+ 2βa|ψa|4 two-dimensional, as well as three dimensional systems
a with translation invariance along the third axis.
(cid:88)
+ γ |ψ |2|ψ |2−η |ψ ||ψ |cos(ϕ −ϕ ). (2.1)
ab a b ab a b b a
a,b>a
A. Broken Time Reversal Symmetry, the U(1)×Z
2
Here D =∇+ieA and ψa =|ψa|eiϕa are complex fields states
representingthesuperconductingcomponents. Thecom-
ponent indices a,b take the values 1,2,3. In the partic- Foragivenparameterset(α ,β ,η ,γ ),theground
a a ab ab
ular case of a three-band superconductor, different su- state is the field configuration which minimizes the po-
perconducting components arise due to Cooper pairing tential energy. The corresponding values of |ψ |’s and
a
in three different bands. The bands are coupled by their ϕ ’s, together with the gauge coupling e determine the
a
interaction withthevectorpotential A andalso through physical length scales of the theory. The particularly in-
potential interactions. The coefficients η are the inter- teresting property of the model (2.1), is that the ground
ab
component Josephson couplings. We also consider the state can be qualitatively different from its two band
more general case which includes bi-quadratic density counterparts. While in two bands systems with Joseph-
interactions with the couplings γ . Here, the London son interactions the phase-locking is trivial (either 0 or
ab
magnetic field penetration length is parametrized by the π), the phase-locking in three bands can be much more
gauge coupling constant e. Functional variation of the involved. Indeed, competition between different phase-
freeenergy(2.1)withrespecttothefieldsgivesGinzburg- locking terms possibly leads to phase frustration. When
Landau equations η > 0, the corresponding Josephson term is minimal
ab
for zero phase difference, while if η < 0 it is minimal
ab
∂V for ϕ ≡ ϕ −ϕ = π. Now if the signs of η ’s are all
DDψ =2 , ∂ (∂ A −∂ A )=J . (2.2) ab b a ab
a ∂ψ∗ i i j j i i positive (we denote it as [+++]), the ground state has
a
ϕ =ϕ =ϕ . Similarlyfor[+−−]couplings,thephase
1 2 3
where V is the collection of all non-gradient terms and locking pattern ϕ =ϕ =ϕ +π. However for [++−]
1 2 3
the supercurrent is defined as or [−−−], the phase locking terms are frustrated. That
is: all three Josephson terms cannot simultaneously at-
(cid:88) (cid:88)
J ≡ J(a) = eIm(ψ∗Dψ ) . (2.3) taintheirminimalvalues. Asaresultgroundstatephase
a a
differencesareneither0norπ. Forexample,considerthe
a=1,2,3 a=1,2,3
3
-2π −π 0 π 2π
case α = −1, β = 1 and η = −1. Symmetry under 2π 0
a a ab
global U(1) phase rotations allows to set ϕ =0 without
1
loss of generality (for the below considerations). There,
two ground states are possible ϕ = 2π/3, ϕ = −2π/3 -2
2 3 π
or ϕ = −2π/3, ϕ = 2π/3. The two ground states are
2 3
eachother’scomplexconjugate. Theactualvaluesofthe
groundstatephasesdependonthepotentialparameters. -4
Note that the free energy is invariant under complex ϕ3 0 F pot
conjugation, (ψ ,ψ ,ψ ) (cid:55)→ (ψ∗,ψ∗,ψ∗), which takes it
1 2 3 1 2 3
-6
to a state with different phase locking. Thus the the-
ory has a spontaneously broken discrete (Z ) symmetry,
2
called Time Reversal Symmetry. That is, the free en- −π
-8
ergyisstillinvariantundercomplexconjugation,butthe
ground state is not. By ‘picking’ one of the two inequiv-
alent phase-locking patterns, the ground state explicitly
-2π -10
breaksthediscreteZ symmetry. Suchstatesaretermed
2 ϕ
2
Broken Time Reversal Symmetry (BTRS) states.
B. Domain walls in BTRS states
BTRS systems have topological excitations related to
Figure 2. (Color online) – Representation of the vacuum
the broken discrete symmetry in the form of domain
submanifold(Top),for(α ,β )=(−1,1)andη =−3. The
walls. The domain walls interpolate between domains a a ab
image shows the potential energy as a function of the phase
of inequivalent ground states. In other words they are
differences: ϕ and ϕ , minimized with respect to all moduli
2 3
walls separating regions of different phase locking. It is degrees of freedom (while ϕ is set to zero by U(1) invari-
1
instructive to display more quantitatively the structure anceassociatedwithsimultaneouschangeofallphases). Red
of the ground state (or “vacuum”) manifold, see Fig. 2. and green dots show inequivalent Z ground states. And the
2
There, the potential energy is minimized with respect lines connecting them represent four different kinds of do-
to the densities |ψ |, for uniform fixed phase difference main wall trajectory over the field manifold. Black dots are
a
configurations. This provides a map of the ground state groundstateslocatedfartherthan2πinthephasedifferences.
The second line, gives a schematic representation of various
manifold. It appears clearly that there are disconnected
Z domainwallsinthree-bandsuperconductorswithdifferent
inequivalent ground states (the red and green dots). In- 2
frustrationsofphaseangles,shownbyarrowsofdifferentcol-
terestingly,thereisnotauniquepathtoconnectinequiv-
ors. The pink line schematically shows the phase difference
alent ground states with inequivalent phase locking, but
between red and green arrow, interpolating between the two
four. The four corresponding domain walls will have dif-
inequivalent ground states.
ferentlinetension(energyperunitlength). Notethat,in-
vestigating the vacuum manifold with fixed ground state
densities |ψa| (at their true ground state value) provides Ourmaininterest,here,isthree-component skyrmionic
a qualitatively similar picture. Namely, this approxima- solutions of the Ginzburg–Landau model. Here
tionpreservesthepositionsoftheminima. However, the skyrmions are topological defects characterized by a
actual values of Fpot are obviously different if |ψa|’s are topological invariant which classifies the maps R2 →
held constant to the ground state, so this approximation CP2. In contrast to the topological invariant character-
does not allow one to calculate the energy of the do- izing vortices (i.e. the winding number which is defined
main walls. In particular the sharp angles appear there as a line integral over a closed path), the topological in-
for strong Josephson couplings, when the ground state dexassociatedwithskyrmionicexcitationsisgivenasan
densities are not fixed. This property is absent when integral over xy-plane :
densities are held to their actual ground state values.
Q(Ψ)=(cid:90) i(cid:15)ji (cid:2)|Ψ|2∂ Ψ†∂ Ψ+Ψ†∂ Ψ∂ Ψ†Ψ(cid:3)d2x,
2π|Ψ|4 i j i j
R2
C. Flux-carrying topological defects in three (2.4)
component Ginzburg–Landau model with Ψ† = (ψ∗,ψ∗,ψ∗). A detailed derivation of this
1 2 3
formulaisgiveninSec.V.Ifwehaveanaxiallysymmetry
As previously stated, three component Ginzburg– vortexwithacorewhereallsuperconductingcondensates
Landau model can exhibit BTRS and domain wall exci- simultaneously vanish, then Q = 0. On the other hand,
tations associated with the broken Z symmetry. There if singularities happen at different locations, then Q(cid:54)=0
2
are also different topological defects, associated with the and the quantization condition Q = B/Φ = N holds
0
other broken symmetries. ( Φ being the flux quantum and N the number of flux
0
4
quanta). This is rigorously discussed in Sec. VB. terms (2.5d) and (2.5b) give trivial contribution to the
free energy, so that the relevant parts now reads
Fractional vortices
1 J2
F = (∇×A)2+
London 2 2e2ρ2
In order to understand the physical properties of the
later introduced chiral skyrmions, it is good to remind + (cid:88) |ψa|2|ψb|2 (cid:18)(∇ϕ )2− 2ηabρ2 cosϕ (cid:19). (2.9)
oneself of the basic features of multi-component super- 2ρ2 ab |ψa||ψb| ab
a,b>a
conductors and their topological excitations. The ele-
mentary vortex excitations in this system are fractional
vortices. They are defined as field configurations with In a [U(1)]3 symmetric model, one fractional vortex
a 2π phase winding only in one phase (e.g. ϕ has giveslogarithmicallydivergentcontributiontotheenergy
1
(cid:72)
∆ϕ ≡ ∇ϕ =2π winding while ∆ϕ =∆ϕ =0). To through the term
1 1 2 3
better illustrate their physical properties, the Ginzburg–
Landau free energy (2.1) can be rewritten as
(cid:90) r (cid:90) 2π |ψ |2|ψ |2 |ψ |2|ψ |2 r
r(cid:48)dr(cid:48) dθ a b (∇ϕ )2 =π a b ln ,
F = 1(∇×A)2+ J2 (2.5a) rc 0 2ρ2 ab ρ2 (2.r1c0)
2 2e2ρ2
r being a sharp cut-off corresponding to the core size of
+(cid:88)1(∇|ψ |)2+α |ψ |2+ βa|ψ |4 (2.5b) acvortex. However a bound state of three such vortices
2 a a a 2 a
(where each phase a=1,2,3 had 2π phase winding) has
a
+ (cid:88) |ψa|2|ψb|2 (cid:18)(∇ϕab)2 − ηabρ2cosϕab(cid:19) (2.5c) fiinngiteinentheergpy.haIsneddeeiffderseuncchesa. bTouhnisdfisntaittee-ehnaesrgnyobwoiunndd-
ρ2 2 |ψ ||ψ |
a b
a,b>a state is a “composite” vortex having one core singular-
+ (cid:88) γ |ψ |2|ψ |2, (2.5d) ity where |ψ1|+|ψ2|+|ψ3| = 0. Around this core all
ab a b
three phases have similar winding ∆ϕ = 2π. A vor-
a,b>a a
tex carrying one quantum Φ of flux is thus a logarith-
0
where ϕ ≡ ϕ − ϕ are the phase differences and mically bound state of fractional vortices. For non-zero
ab b a
ρ2 = (cid:80) |ψ |2. The indices a,b again denote the differ- Josephson coupling, fractional vortices interact linearly,
ent supearcoanducting condensates and take value 1,2,3. so they are bound much more strongly.18 It can be seen
The identity that, for non-zero Josephson coupling, the phase differ-
ence sector (2.5c) or the second line in (2.9) is a sine-
n n Gordon model. There, a given fractional vortex excites
(cid:88)(cid:88)
|ψa|2|ψb|2∇ϕa(∇ϕa−∇ϕb) two Josephson strings (one per phase difference sector).
a=1b=1 Crossections of a string, at a large distance from a vor-
=(cid:88)n (cid:88)n |ψ |2|ψ |2(∇ϕ −∇ϕ )2 , (2.6) tex are sine-Gordon kinks. Such a Josephson string, has
a b a b an energy proportional to its length. Thus for non-zero
a=1b=a+1 Josephson coupling one fractional vortex has linearly di-
verging energy (see App. A for a detailed derivation).
is used to derive this expression. Here, the supercurrent
Note that the Josephson strings are different topological
(2.3) reads, more explicitly
excitations than the domain walls previously discussed.
(cid:88) Having linearly diverging energy, fractional vortices in-
J/e=eρ2A+ |ψ |2∇ϕ . (2.7)
a a teract linearly. As a result an (composite) integer flux
a vortex can be seen as a strongly bound state of three co-
centered fractional vortices. This binding is thus much
Consider now a vortex for which the phase of only one
(cid:72) stronger for non-zero Josephson couplings. Because of
component changes by 2π: ∇ϕ =2π. Such a configu-
a their diverging energies, the fractional vortices are not
ration carries a fraction of flux quantum18
thermodynamically stable in bulk samples:18 A group of
(cid:73) |ψ |21(cid:73) |ψ |2 three different fractional vortices is energetically unsta-
Φa = Ad(cid:96)= ρa2 e ∇ϕa = ρa2 Φ0, (2.8) blewithrespecttocollapseintoanintegerfluxcomposite
σ σ vortex. Note however that under certain conditions, in a
finitesample,theycanbethermodynamicallystablenear
where |ψ | denotes the ground state density of ψ , σ is
a a boundaries19 with strings terminating on a boundary.
a closed curve around the vortex core, and Φ = 2π/e
0
is the flux quantum. For vanishing Josephson interac- Note that in a London limit, magnetic field of frac-
tions,thesymmetryis[U(1)]3 andeachfractionalvortex tional vortices is exponentially localized. However in a
has logarithmically diverging energy.18 This can be seen [U(1)]3 Ginzburg-Landau model, the magnetic field of a
easily in the London limit by setting ψ = const every- fractionalvorticesisinagenerallocalizedonlyaccording
a
whereexceptasharpcutoffinthevortexcore. Therethe to a power law and moreover can invert direction.20
5
Figure3. (Coloronline)–Asinglechargechiralskyrmion,for3mirrorpassivebands(α ,β )=(1,1)andJosephsoncoupling
a a
constants η = −3. Here γ = 0.8 and the gauge coupling constant is e = 0.6. Displayed quantities are the magnetic flux
ab ab
(A) and the sine of phase differences sin(ϕ ) (B), sin(ϕ ) (C). Condensate densities |ψ2|, (D), |ψ2|, (E) and |ψ2|, (F) are
12 13 1 2 3
representedonthesecondline. Thecorrespondingsupercurrentdensities|J |,(G),|J |,(H)and|J |,(I)aredisplayedonthe
1 2 3
third line. To avoid redundant informations, the total energy density is not displayed. It qualitatively follows the magnetic
flux shown in panel (A).
D. Chiral three component Ginzburg–Landau ing it to attain more favorable phase difference values in
skyrmions betweenthesplitfractionalvortices. Asaconsequenceof
these circumstances, the domain wall can trap vortices.
Recall that away from domain walls, fractional vortices
Domain-walls such as those discussed in Sec. IIA can are linearly confined by Josephson terms.
form dynamically in physical systems by a quench. Be-
cause of its line tension, a closed domain wall collapses When the magnetic field penetration length is suffi-
to zero size. From the term (2.5c), in the rewritten ciently large (e small enough), the repulsion between the
Ginzburg–Landau functional, it is clear that in order to fractional vortices confined on the domain wall can be-
decreasetheenergycostassociatedwithagradientinthe come strong enough to overcome the domain wall’s ten-
relative phase ϕ , the densities of the components |ψ |, sion. Itthusresultsinaformationofatopologicalsoliton
ab a
|ψ | should be suppressed on the domain wall. Further- made up of 3N fractional vortices, stabilized by compet-
b
more, on a domain wall, the cosines of phase differences ing forces. Such ‘composite’ topological solitons are thus
cos(ϕ −ϕ ) are energetically unfavorable. Indeed, by made of a closed domain wall along which there are N
b a
definition, it is where they are the farthest from their singularitiesineachcondensate|ψ |. Aroundeachsingu-
a
ground state values. As a result, if an integer composite laritythephaseϕ changesby2π. Thetotalphasewind-
a
(cid:72) (cid:72)
vortexisplacedonthedomainwall,theJosephsonterms ing around the soliton is then ∇ϕ d(cid:96) = ∇ϕ d(cid:96) =
1 2
(cid:72)
shouldtendtosplititintofractionalfluxvortices, allow- ∇ϕ d(cid:96) = 2πN. Therefore it carries N flux quanta.
3
6
Figure 4. (Color online) – A Skyrmion with Q = 6 topological charge (which implies that it carries six flux quanta and
consistsof18fractionalvortices). Displayedquantitiesarethemagneticflux(A)andthesineofphasedifferencessin(ϕ )(B)
12
sin(ϕ ) (C). Condensate densities |ψ2|, (D), |ψ2|, (E) and |ψ2|, (F) are represented on the second line. The corresponding
13 1 2 3
supercurrent densities |J |, (G), |J |, (H) and |J |, (I) are displayed on the third line. Parameters are the same as in Fig. 3.
1 2 3
The CP2 topological invariant (2.4) computed for such ductorwiththreepassivebands(i.e. thequadraticterms
objects is found to be integer, whereas it is zero for or- havepositiveprefactorsα ). Thefactthatthebandsare
a
dinary composite vortices. As a result, the composite passive is not important for the soliton’s existence. It
configuration made out of a domain wall between two consists of three fractional vortices, each one carrying a
Z domainsstabilizedbyrepulsionbetweentrappedvor- fraction|ψ |2/ρ2 ofmagneticfluxwhichaddsuptoaflux
2 a
tices,isinfactadistincttopologicaldefect: ChiralGL(3) quantum Φ . Since the fractional vortices are located
0
skyrmion (chiral skyrmion for short). quite close to each other they cannot be distinguished
in the magnetic field profile in this case. Single charge
It was previously demonstrated that these topological
defectsexistandareindeedatleastmetastable.6 Herewe skyrmionsaremoredifficulttoobtainthanhigher-charge
skyrmions in this model. As will be explained later, in-
furtherinvestigatetheseobjects. Toinvestigatetheexis-
creasing the number of flux quanta N, usually makes
tence and stability of the so-called chiral skyrmions, we
the solution more stable (which contrasts with vortices
use an energy minimization approach, using non-linear
where, in the type-II regime only N =1 vortices are sta-
conjugate gradient algorithm. More details about the
ble). The bi-quadratic density interactions in the model
employednumericalschemesareprovidedinApp.B.The
(2.1) help to stabilize Q = 1 solutions. Single charge
topologicalcharge(2.4)wascomputednumericallyforall
solitons are thus usually supported by bi-quadratic den-
configurations and was found to be integer within small
sity interactions. Clearly, from the density plots (panels
numerical errors, less than 0.1%, thus providing an esti-
(D–F))inFig.3, eachcomponenthasanon-overlapping
mate of the accuracy of our solutions.
zero (the blue spots). A feature which can be observed
Fig. 3 shows a Q = 1 chiral skyrmion in a supercon-
7
Figure 5. (Color online) – A Q=2 quantum soliton in a system with two identical passive bands (α ,β )=(1,1) (a=1,2)
a a
coupledtoathirdactivebandwithsubstantialdisparityinthegroundstatedensities(α ,β )=(−2.75,1). Josephsoncoupling
3 3
constants are η = η = η = −3. The system is in a strongly type-II regime e = 0.08, the solutions here are stable even
12 13 23
in the absence of bi-quadratic density interaction i.e. γ = 0. Displayed quantities are the magnetic flux (A) and the sine
ab
of phase differences sin(ϕ ) (B) sin(ϕ ) (C). Condensate densities |ψ2|, (D), |ψ2|, (E) and |ψ2|, (F) are represented on the
12 13 1 2 3
second line. The corresponding supercurrent densities |J |, (G), |J |, (H) and |J |, (I) are displayed on the third line.
1 2 3
inthisregimeisthestrongdensityovershootoppositeto skyrmion is different from ‘outside’, thus corresponding
the cores (the red spots). to either of the two Z inequivalent ground states. As
2
a result the chiral skyrmions (in contrast to non-chiral)
Higher charge skyrmions are easily formed in many
feature a domain wall separating the regions of different
cases even when there is no bi-quadratic density interac-
BTRSstates. AsdiscussedbelowSec.IVB,thechoiceof
tion. There,thestabilityoftheskyrmionagainstcollapse
one of the Z ground states inside the skyrmion dictates
of the domain wall is supported only by the electromag- 2
aclockwiseversuscounter-clockwisearrangementoffrac-
netic repulsion and Josephson interactions. In different
tional vortices, thus motivating the terminology “chiral”
numerical simulations we quite easily constructed thou-
for these topological defects.
sands of different skyrmionic configurations, forvery dif-
ferentparametersets. Asampleofthevariousskyrmions
Chiralskyrmionsexhibitveryunusualsignaturesofthe
is given in the Figures 3–7. More regimes are given in
magnetic field which can be seen from the panel (A) in
the appendix App. C. For all such configurations the
all of the Figures 3–7 or in Fig. 1. If the bands have
CP2 topological charge (2.4) is integer with very good
similar density, each fractional vortex carries a similar
accuracy ( |Q/N −1|<10−3 ).
fraction of flux quantum. As a result, the magnetic flux
One key feature, in the Figures 3–7, is seen in the is almost uniformly spread along the domain wall, as in
phasedifferencesonpanels(B)and(C). Ineachofthese Fig. 4. On the other hand, when the condensates have
various regimes, the phase locking pattern ‘inside’ the quitedifferentdensities,themagneticfluxiscarriednon-
8
Figure 6. (Color online) – A Q = 5 quantum soliton in a system with two identical passive bands as in Fig. 5 coupled to a
third active band with disparity in the ground state densities (α ,β )=(−1.5,1). Josephson coupling constants are η =−3
3 3 23
and η =η =1. Here e=0.2 and there is no density-density interaction term γ =0. The system is shaped as a pentagon
12 13 ab
deformedbythevorticesofthestrongactivebandcarryinglargerfractionsoffluxquantum. Displayedquantitiesarethesame
as in the previous pictures, e.g. Fig. 5.
uniformly by fractional vortices in different condensates. sity while the small spots are associated with the active
Consequently, the magnetic flux is inhomogeneously dis- band.
tributed along the soliton. This can be seen in Fig. 5
wherethethirdcomponentcarriesagreatfractionofthe
flux. The remaining fraction of flux is spread along the
E. Chiral multi-skyrmions
components having less density. The overall configura-
tion can easily be mistaken for a vortex pair in such a
superconductor. For higher topological charge, the same Besides having non trivial CP2 topological invari-
system exhibits geometric structures (a pentagon as in ant (2.4), the chiral skyrmions in three component
Fig. 6) where the vertices are occupied by the fractional Ginzburg–LandautheorywithBTRShaveagivenchiral-
vortices of the band with bigger density. There again, ity. Namely,thereisadifferencewhetheroneortheother
geometrical arrangement of apparent vortices is a very broken Z2 state is ‘inside’. Here we report bound states
typical signature of the chiral skyrmions. of chiral skyrmions with opposite chirality which can be
called multi-skyrmions. More precisely a bound state of
Among possible observable signatures of chiral askyrmionwithagivenchirality,carryingsometopolog-
skyrmions, is the varying fraction of magnetic flux car- icalchargesayQ andaskyrmionwiththeoppositechi-
1
ried by fractional vortices, as in Fig. 7. There, the mag- rality carrying Q , see Fig. 8. There the inner skyrmion
2
netic field exhibits spots of different magnitude, larger has a smaller charge than the outer one, Q < Q since
1 2
spotsassociatedtothetwosimilarbandswithmoreden- the chiral skyrmion’s size is controlled by the number of
9
Figure 7. (Color online) – A Q=5 quantum soliton in a system with within the same parameter set as in Fig. 6 apart from
(α ,β )=(−0.5,1). Displayed quantities are the same as in the previous pictures, e.g. Fig. 5.
3 3
enclosed quanta. The bigger is the difference between windingandthusthedirectionofcarriedflux. Aswillbe
Q and Q , the weaker is the interaction between the clear from the discussion below, an anti-Skyrmion with
1 2
two chiral skyrmions. Conversely, as Q → Q the chi- the same Z charge as a Skyrmion will also have frac-
1 2 2
ral skyrmions interact progressively more strongly. For tional vortices arranged in a different order.
very close values of Q and Q the chiral skyrmions falls
1 2
into each other’s attractive basins and the domain walls
annihilate. This allows decay to ordinary vortices.
Similarly, there exist also “Russian nesting doll”-
Note that “opposite chirality” should not be confused like multi-skyrmions made of larger number of alternat-
with opposite flux, i.e. these objects have opposite chi- ing skyrmions of opposite chiralities. Such a multiple
rality because they interpolate between two different Z skyrmioncanbeseeninFig.9whichshowstri-ringsolu-
2
ground states. In that respect in the BTRS case, an ad- tions of skyrmion with alternating chiralities. This kind
ditional Z topological charge like those of ordinary do- ofnumericalsolutionisquiteeasilyobtainedgivenagood
2
mainwallscanbeattributedtoskyrmions. Howeverhav- initialguess. Howeverthisconfigurationcanalsosponta-
ing opposite Z topological charges does not mean that neously form from ‘collisional dynamics’ of energy mini-
2
theseobjectsrepresentaskyrmionandananti-skyrmion. mization of an initial configuration of closely spaced or-
This is because they have similar signs of Q and Q dinary vortices. This indicates that formation of multi-
1 2
charges as well as similar signs of the total phase wind- skyrmion solutions does not in general require fine tun-
inginthelocalU(1)sector. Thatis,theycarrymagnetic ing. Instead these solutions have a substantial “attrac-
flux in the same direction. For a given skyrmion one can tive basin” in the GL energy landscape indicating they
construct an anti-skyrmion from similar number of anti- could also be observed in three component superconduc-
vortices. Using anti-vortices changes the overall phase tors with Broken Time Reversal Symmetry.
10
Figure 8. (Color online) – A Q=11 quantum multi-soliton in a system with three identical passive bands as in Fig. 13. The
currentsolitonisnotmadeoutofonebuttwostabilizeddomainwallsthusbeingahomogeneousbi-ringconfiguration. Panels
(B) and (C) clearly display the alternating different ground-states. Since the three bands are identical, the magnetic field
rather homogeneously spreads all along the solitons. Displayed quantities are the same as in the previous pictures, e.g. Fig. 5.
Note that while going counterclockwise along the outter ring, the fractional vortices have order band-”1,2,3”. For the inner
ring they are ordered as band-”1,3,2”. The origin of this is discussed in Sec. IVB
III. PHYSICAL PROPERTIES OF CHIRAL A. Energy of Chiral skyrmions vs vortices
SKYRMIONS
For vanishing bi-quadratic density interaction cou-
plings (i.e. γ = 0), in all the regimes which we in-
It is important to know the energetic properties of ab
vestigated, chiral skyrmions are always more expensive
skyrmionscomparedtoordinaryvortices,aswellastheir
energetically than vortices. However, as suggested in
stability properties. Indeed if skyrmions are thermody-
Ref. 6, bi-quadratic density interaction decreases the en-
namically stable and form as the ground states in mag-
ergy of chiral skyrmions relative to that of vortices. For
netic field, their experimental signatures are straightfor-
sufficiently strong bi-quadratic density interaction chiral
ward to detect. However, if they form as states with
skyrmions are ground state excitations i.e. energetically
higher energy than e.g. a vortex state, they are only
cheaper than vortices and, for certain parameters, ther-
metastable. Whentheyaremetastablestates,skyrmions
modynamically stable.
are protected against decay by an energy barrier. The
height of this barrier depends non-trivially on the pa- The energy properties of the chiral skyrmions are dis-
rameters of the potential and on the number of enclosed played on the left panels of Figures 10-11. There, the
flux quanta. Metastable chiral skyrmions could be pro- energyperfluxquantumofagivenconfigurationisgiven
duced by quenching the system under applied magnetic inunitsofthesinglequantumfluxcarryinggroundstate.
field. In this section, we discuss these aspects. Namely E(N)/[NE(N =1)] is represented as a function