Semi-Compact Skyrmion-like Structures

3.5 MB·

by  D. Bazeia

Semi-Compact Skyrmion-like Structures
D. Bazeia and E.I.B. Rodrigues
Departamento de F´ısica, Universidade Federal da Para´ıba, 58051-970 Joa˜o Pessoa, PB, Brazil
(Dated: January 25, 2017)
We study three distinct types of planar, spherically symmetric and localized structures, one of
them having non-topological behavior and the two others being of topological nature. The non-
topologicalstructureshaveenergydensitylocalizedinacompactregionintheplane,butareunstable
against spherically symmetric fluctuations. The topological structures are stable and behave as
vortices and skyrmions at larger distances, but they engender interesting compact behavior as one
approaches their inner cores. They are semi-compact skyrmion-like spin textures generated from
models that allow to control the internal behavior of such topological structures.
PACSnumbers: 75.70.Kw,75.50.-y,11.27.+d,05.45.Yv
7
I. INTRODUCTION interest to high energy physics. We then introduce mod-
1
els that provide semi-compact solutions, which are new
0
2 The study of skyrmions first appeared in the context isolatedstructuresthatengenderintegerandhalf-integer
n of particle physics [1] and have been extended to various skyrmion numbers, for which we are able to control the
spin textures at their inner core.
a other areas of nonlinear science, in particular as local-
J ized structures in a diversity of magnetic materials; see The study deals with three distinct models, one
3 e.g., Refs. [2–10]. In this work we focus mainly on mag- describing non-topological solutions, with topological
2 netic skyrmions, dealing with some analytical tools to charge Q=0, and the two others being able to generate
describe the formation of skyrmions in magnetic mate- topologicalstructureswithintegerandhalf-integertopo-
l] rials. The subject concerns several mechanisms working logical charge. In the case of solutions with vanishing
al together [11] and describes properties such as topologi- topological charge, we show that they are linearly un-
h calchargeorwindingnumber, vorticityandhelicity[12]. stable against radial deformations, so they are of limited
- In general, the topological charge of a skyrmion may be interest. Butwedealwiththembecauseonecandevelop
s
e integer, Q = ±1, half-integer, Q = ±1/2 (half-skymion mechanisms to make them stable, although this is out of
m or vortex) [9, 13–17], and null Q = 0, the later being the scope of the current work. The solutions with inte-
. non-topological, considered in Ref. [18]. Here one notes ger and half-integer topological charge are linearly sta-
at that skyrmions with high-topological-number (Q ≥ 2) ble against radial deformations, so we focus mainly on
m has also been suggested; see, e.g., the recent work [19]. them, showing how to control their inner core, without
modifyingtheirasymptoticbehavior. Wecallthemsemi-
- We will deal with skyrmion-like structures, paying
d closer attention to compact solutions, as one explains compact structures, and they have an important differ-
n ence,whencomparedtothecompactstructuresdiscussed
below. Compact skyrmions have been studied before in
o inRef.[21]. Thekeypointhereisthatthesemi-compact
[20], in the context of the baby Skyrme model with a
c
solutions that we introduce have asymptotic behavior
[ specific potential, and also in [21], to describe magnetic
spin textures that appear in magnetic materials. The similar to the standard spin textures, but they behave
1 differently as one approaches the core of the structure.
study of compact skyrmions in magnetic materials is of
v We understand that this is important behavior, because
current interest, since the recent advances on miniatur-
5
ithelpusshowhowtocontroltheinnercoreofskyrmions.
4 izationisleadingustomanipulatemagneticmaterialsat
We organize the work as follows: in the next section
6 constrained geometries and this may modify the profile
6 andpropertiesofthemagneticspintexturesthatinhabits we follow Refs. [23, 24] and discuss three distinct models
0 the material. An example of this was identified experi- describing a single real scalar field in two spatial dimen-
. sions, as suggested in Ref. [25]. We describe explicit
1 mentallyin[22], showinghowadomainwallmaychange
solutions, and study their stability. In Section III, we
0 profileasitisshrankinsideaconstrainedregion. Forthis
7 reason, we think it is worth investigating the possibility deal with the topological charge density and study the
1 of shrinking vortices and skyrmions to a compact region topology and the skyrmion number. We end the work in
: Sec. IV,whereweincludeourcommentsandconclusions.
v in the plane.
i Inthecurrentwork,wefollowthelinesofRefs.[23–25]
X
and use scalar fields to model spin textures for which we
r can control their spin arrangements. The investigation II. ANALYTICAL PROCEDURE
a
presented in [21] has found periodic solutions described
intermsoftheJacobianellipticfunctions,whicharethen We start focusing on the localized spin textures that
worked out to give rise to compact structures. Here we appearinmagneticmaterials. Wesupposethatthemate-
followanotherroute,inspiredbytherecentinvestigations rial is homogeneous along the zˆdirection, with the mag-
[26,27]oncompactandasymmetricstructuresofcurrent netization M being in general a three-component vector
2
with unit modulus that depends on the planar coordi- and the energy for static solution φ(r) is given by
nates, such that M=M(x,y) and M·M=1.
(cid:90) ∞
To describe skyrmions, we introduce the skyrmion E =2π rdrρ(r), (8)
number, which is a conserved topological quantity, given 0
by with ρ(r) being the energy density, such that
1 (cid:90) ∞ 1(cid:18)dφ(cid:19)2 1
Q= 4π dxdy M·∂xM×∂yM. (1) ρ(r)= 2 dr + 2r2P(φ). (9)
−∞
Thepointhereisthatwecandescribeexplicitmodels,
In this work we concentrate on helical excitations, with
using distinct functions for the Polynomial P(φ). We
the magnetization M = M(r) only depending on the
do this for three distinct models in the next subsections.
radial coordinate, being now a two-dimensional vector,
Before doing that, however, we note that for φ → φ ,
orthogonal to the radial direction. In cylindrical coordi- v
with φ being constant and uniform, the energy density
nates, the helical excitations impose that M·rˆ = 0, so v
vanishesifφ isaminimumofP(φ)suchthatP(φ )=0
one allows that the magnetization has the form v v
[23]. In this case, φ is a ground state and will guide us
v
M(r)=θˆcosΘ(r)+zˆsinΘ(r), (2) toward the construction of models.
Anotherissueisthatthemodeloneusesengenderscale
With this decomposition, we can then describe Θ(r) as invariance; see, for instance, the equation of motion (7).
the only degree of freedom to be used to model the mag- Aninterestingroutetodescribemagneticskyrmionsisto
netic excitation in the magnetic material. If we now use break scale invariance, and this can be achieved via the
Eq.(1)withthe(r,θ)variables,theabovemagnetization introduction of Dzyaloshinskii-Moriya interactions (see,
can be used to obtain the skyrmion number. It leads to e.g., [19] and references therein); another route is to de-
scribeskyrmionsonalattice(see,e.g.,[28]andreferences
1 1 therein). In this work we will keep using the model (5)
Q= sinΘ(0)− sinΘ(∞). (3)
2 2 to describe the magnetization since it will lead to ana-
lytical solutions. In this case, although we cannot use
This result shows that the topological profile of the so- results to describe quantitative properties, we can still
lution Θ(r) is related to its value at the origin, and the usethemqualitativelyand,moreimportantly,wecanex-
asymptotic behavior for r →∞. plore the topological properties of the solutions, because
In order to model skyrmion-like solutions analytically, thetopologyfollowsfromtheirasymptoticbehavior,and
we take advantage of the recent study [23, 24] and con- may appear despite the breaking of the scale invariance.
sider
π
Θ(r)= φ(r)+δ, (4) A. Non-topological solutions
2
where δ is a constant phase, which we use to control the The first model that we describe is governed by the
value of the magnetization at the center of the magnetic polynomial P(φ), given by
structure. Also, we suppose that the scalar field φ is ho-
mogeneousanddimensionlessquantitywhichisdescribed P(φ)=n2φ2(φ−2/n−1). (10)
by the planar system investigated in [25]. In this case,
Here n = 3,5,7,... is an odd integer. This model fol-
the Lagrange density L has the form
lows the suggestion of Ref. [26], concerning the presence
of one-dimensional, compact lump-like structures there
1 1
L= φ˙2− ∇φ·∇φ−U(φ), (5) studied. We depict this polynomial P(φ) in Fig. 1 for
2 2
three distinct values of n, to show how it behaves as one
where dot stands for time derivative, and ∇ represents changessuchparameter. Weseethatthepolynomialhas
the gradient in the (x,y) or (r,θ) plane. We search for a local minimum at φ0 =0, and two zeroes at φ± =±1,
time independent and spherically symmetric configura- irrespective of the value of n. We can write the equation
tion, φ=φ(r), and consider U =U(r;φ) in the form of motion as
d2φ dφ
1 r2 +r +n2φ−n(n−1)φ1−2/n =0. (11)
U(r,φ)= P(φ), (6) dr2 dr
2r2
One then searches for solutions related to the ground
with P(φ) an even polynomial which contains non- state φ = 0. The investigation has led us with the in-
0
gradient terms in φ. In this case, the field equation be- teresting family of analytical solutions
comes

cosn(ln(r)), e−π/2 ≤r ≤eπ/2

d2φ dφ 1dP φ (r)= . (12)
r2 +r − =0, (7) n
dr2 dr 2 dφ  0, r <e−π/2 or r >eπ/2
3
Thereisanothersolution,withtheminussign,butitbe-
havessimilarly. Theprofileoftheabovesolutionsarede-
picted in Fig. 2, for three values of n. We note that as n
increases, the solution becomes more and more localized
0
inside a compact region. We also note that the solution
L
goestozeroattheoriginandasymptotically,forr →∞, HP
and this induces its non-topological profile. This is an
example of a compact solution that will be mapped into
a compact skyrmion with vanishing topological charge,
with vanishing skyrmion number.
We use the above solution to see that the energy den-
sity is given by -1 0 1
ρ(r)= n2cos2n(ln(r))(cid:0)cos−2(ln(r))−1(cid:1), (13) FIG. 1: The polynomial (10), depicted for n = 3,5, and 7,
r2
with dot-dashed (red), dashed (blue) and solid (green) lines,
respectively.
intheintervale−π/2 ≤r ≤eπ/2; also, itvanishesoutside
this compact region.
We can integrate the energy density to get the total
1
energy in the form
nπ3/2Γ(n−1/2)
E = , (14)
n Γ(n)
Lr
H
where Γ(z) is the Gamma function. In this case, one can
use Eqs. (3) and (4) to calculate the skyrmion number,
giving Q=0.
0
B. Topological solutions
r
ThesecondmodelisdescribedbythepolynomialP(φ) FIG. 2: The solution (12) for the model (10), depicted for
that has the form n = 3,5 and 7, with dot-dashed (red), dashed (blue) and
solid (green) lines, respectively.
1
P(φ)= φ2ln2(φ2), (15)
(1−s)2
where the parameter s is now in the interval s ∈ [0,1).
This model has three minima, two at φ = ±1 and one
±
at φ =0. It was introduced in [27], and here we depict
0 L
the above polynomial in Fig. 3, for three values of s, to HP
see how it changes as one varies s.
Wecanwritetheequationofmotionforstaticsolutions
as
d2φ dφ φln(φ2)(2+ln(φ2))
r2 +r − =0, (16)
dr2 dr (1−s)2 -1 0 1
andnowonesearchesforsolutionsconnectingtheground
states φ = 0 and φ = 1; the equation can be solved FIG. 3: The polynomial (15), depicted for s = 0.3,0.6 and
0 +
0.9, with dot-dashed (red), dashed (blue) and solid (green)
analytically to give
lines, respectively.
φ (r)=e−r−2/(1−s), (17)
s
There is another solution, with the minus sign, which approachesthecenterofthesolution,indicatingitssemi-
behaves similarly. We depict the solution (17) in Fig. 4, compact behavior. It vanishes for r going to zero, and
forsomevaluesofs. Wenotethatthesolutionissharper approaches the minimum φ =1 as r increases to larger
+
for larger values of s. Also, it behaves standardly for andlargervalues,andthisprofileinducesthetopological
very larger values of r, but it decays very strongly as r behavior which we will show below.
4
1
Lr L
H HP
0 0
r -1 0 1
FIG. 4: The solution (17) for the model (15), depicted for
s = 0.3,0.6 and 0.9, with dot-dashed (red), dashed (blue) FIG. 5: The polynomial (21), depicted for s = 0, and for
and solid (green) lines, respectively. n=1,3andforn→∞,withdot-dashed(red),dashed(blue)
and solid (green) lines, respectively.
The corresponding energy density is given by
1
2
ρ (r)= r−2(3−s)/(1−s)e−2r−2/(1−s), (18)
s (1−s)2
and the total energy has the form
HLr 0
π
E = . (19)
s 2(1−s)
We can calculate the mean matter radius introduced
in [23]. In this case we get -1
(cid:18) (cid:19) r
3+s
r¯s =212(1−s)Γ 2 . (20) FIG. 6: The solution (23) for the model (21) in the limit
n → ∞, depicted for s = 0.3,0.6 and 0.9, with dot-dashed
It decreases as s increases, and it is always smaller than (red), dashed (blue) and solid (green) lines, respectively.
it is for the standard vortex-like structure investigated
in [23]; in particular, for s = 0 it is r¯= 1.2533, smaller
thaninthestandardcaseof[23],whichgivesr¯=1.3561.
The equation of motion in this case becomes
Moreover, one can use Eqs. (3) and (4) to calculate the
skyrmion number in this case; the result is Q=1/2. d2φ dφ (1+φ)(1−φn)2
r2 +r − +
dr2 dr (1−s)2
C. Other topological solutions nφn−1(1+φ)2(1−φn)
+ =0, (22)
(1−s)2
The third model is described by
and here one searches for solutions that connect the two
1 minima, φ = ±1. The above equation can be solved
P(φ)= (1+φ)2(1−φn)2 (21) ±
(1−s)2 analytically in the limit n→∞ to give,
where s ∈ [0,1), and n = 1,3,5,···. This polynomial  1, 0<r ≤1;
hastwominimaatφ =±1,irrespectiveofthevaluesof 
±
φ (r)= (23)
s and n, and a local maximum in between [0,1], whose s 2−r1/(1−s)
valuedependsonn,andincreasesasnincreasestolarger  , r >1.
r1/(1−s)
and larger values. We depict (21) in Fig. 5 for s = 0
and for three distinct values of n, to illustrate how the It is depicted in Fig. 6 for three distinct values of s. The
polynomial appears as n increases to very larger values. energy density is
We note that for n = 1 we get to polynomial already
studiedin[23],sowedonotconsiderithere. Thismodel  0, 0<r ≤1;
first appeared in [27], and there it was used to describe 
ρ (r)= (24)
asymmetric structures capable of generating braneworld s 4
scenario having asymmetric behavior.  (1−s)2 r−2(2−s)/(1−s), r >1.
5
which leads to the total energy
4π
E = . (25)
s (1−s)
In this model, the mean matter radius is
Lr
2 Hq
r¯ = . (26)
s 1+s
It is always greater than it is in the standard case; in
particular, for s=0 is has the highest value, r¯=2.0000;
this is to be contrasted with the value r¯ = 1.1781 that
one gets in the standard case described in [23]. In this r
case, if one uses Eqs. (3) and (4) it is easy to calculate
FIG. 7: The non-topological charge density (37)-(38), de-
the skyrmion number, giving Q=1.
pictedforn=3,5and7,withdot-dashed(red),dashed(blue)
and solid (green) lines, respectively.
D. Stability
We now study stability of the solutions found above.
We consider stability against spherically symmetric de-
formations, taking φ = φ(r)+(cid:15)η(r), with (cid:15) being very
small real and constant parameter. This allows that we
Lr
write the total energy in the form Hq
E =E +(cid:15)E +(cid:15)2E +··· (27)
(cid:15) 0 1 2
where E ,n = 1,2,... is the contribution to the energy
n
at order n in the small parameter (cid:15). We can use the
equation of motion to show that E = 0 and we obtain
1 r
that
(cid:18)(cid:90) 1 P(cid:48)(φ)(cid:19) FIG.8: Thetopologicalchargedensity(39)and(40),depicted
η(r)=Aexp dr . (28) for s=0.3,0.6 and 0.9, with dot-dashed (red), dashed (blue)
r2 φ(cid:48)(r)
and solid (green) lines, respectively.
This is the zero mode, where A is constant of normaliza-
tion, P(cid:48)(φ)=dP(φ)/dφ and φ(cid:48)(r)=dφ (r)/dr.
s We see that it is positive, E >0 for any s∈[0,1). This
Weconsiderthemodel(10),andfrom(28)itispossible 4
proves that the solution φ(r) which we obtained in (17)
to write
is stable against spherically symmetric fluctuations.
η(r)=A cosn−1(ln(r))sin(ln(r)) (29) The last case is the model (21). It has analytical so-
n
lution in the limit n→∞, given by Eq. (23). We follow
intheintervale−π/2 ≤r ≤eπ/2; also, itvanishesoutside as we did before, to show that the solution is also stable
this compact interval. Here, An is constant of normal- against radial fluctuations. The zero mode is
ization. We can prove that E = 0, E = 0 and E has
2 3 4 η(r)=Ar−1/(1−s), r >1 (33)
the form
π3/2(2+n)Γ(n−5/2) where A is normalization constant. The energy is such
E4 =− 8n2Γ(n−2) , (30) that E1 =0, and E2 has the form
π
We see that E4 is negative, and this proves that the E2 = 1−s (34)
spherically symmetric solution (12) is unstable against
spherically symmetric fluctuations. which is positive, indicating stability.
The next case is the model (15), and now the zero
mode is given by
III. SKYRMION NUMBER
η(r)=A r−2/(1−s)e−r−2/(1−s), (31)
s
Letusnowdealwiththemagnetization,asintroduced
where A is constant of normalization. We can prove
s inthebeginningofthepreviousSection. Itispossibleto
that E =0, E =0 and E has the form
2 3 4 use (1) to get the topological charge density in the form
3π r
E4 = 16(1−s). (32) q(r)= 2M·∂xM×∂yM, (35)
6
(a)
Lr
Hq
(b)
r
FIG.9: Thetopologicalchargedensity(41)and(42),depicted
for s=0.3,0.6 and 0.9, with dot-dashed (red), dashed (blue) (c)
and solid (green) lines, respectively.
whereMisthemagnetization. Aftertaking(2),(3),and
(4) one gets
π ∂φ(r) FIG. 10: (Color online) The non-topological structure with
q(r)=− cosΘ(r) . (36)
skyrmion number Q = 0 which is controlled by the model
4 ∂r
(10),depictedforn=3,5and7inthe(a),(b)and(c)panels,
We now apply this procedure to the models investi- respectively.
gated above. For the model (10) we have that
(cid:16)π (cid:17) 1
q(r)=q (r)sin cosn(ln(r)) , (37)
0 2
where we used δ =π/2 and q (r) is given by
0
πncosn−1(ln(r))sin(ln(r))
q (r)= (38)
0 4r
0
which we depict in Fig. 7 for three distinct values of n. (a) (b) (c)
For the model (15) we use δ =π/2 and get
FIG. 11: (Color online) Alternative way to depict the non-
(cid:16)π (cid:17)
q(r)=q (r)sin e−r−2/(1−s) , (39) topologicalstructurecontrolledbythemodel(10),forn=3,5
0 2 and 7, in the (a), (b) and (c) panels, respectively.
where
πr−(3−s)/(1−s)e−r−2/(1−s) where q(r) is given by eq. (36). We use this and
q (r)= (40)
0 2(1−s) the eqs. (37) and (38) to see that Q = 0 in this case.
The skyrmion number for the model (10) vanishes, so
which we depict in Fig. 8 for three distinct values of s. themodelsupportsanon-topologicalcompactstructure.
For the model (21) the topological charge density be- The Fig. 10 describes an array of vectors in the plane,
comes, using δ =0, which represents the helical magnetic excitation with
π (cid:18)2−r1/(1−s)(cid:19) vanishing skyrmion number. Alternatively, we can de-
q(r)=q (r)cos , (41) pict the magnetic structures using a continuum of color,
0 2 r1/(1−s)
takingblueforthemagnetizationpointingupwardinthe
zˆdirection, yellowforthemagnetizationvanishingalong
where
the zˆ direction, and red for the magnetization pointing
πr−(2−s)/(1−s) downward in the zˆ direction. We use this to depict in
q (r)= , (42)
0 2(1−s) Fig. 11 the non-topological solution with the same val-
ues of s considered in Fig. 10.
which is depicted in Fig. 9 for three distinct values of s.
Wenowconsider(36)anduse(39)and(40)togetthat
To get the skyrmion number or the topological charge
Q = 1/2. In this case we are dealing with a skyrmion
we recall that
with skyrmion number 1/2 which represents a topologi-
(cid:90) ∞ calstructureofthevortextype. Wedepictsomeofthese
Q= drq(r), (43)
vortex structures in Fig. 12. Also, in Fig. 13 we depict
0
7
(a) (a)
(b) (b)
(c) (c)
FIG. 12: (Color online) The topological structure with FIG. 14: (Color online) The topological structure with
skyrmion number Q = 1/2 which is controlled by the model skyrmion number Q = 1 which is controlled by the model
(15), depicted for s=0.3,0.6 and 0.9 in the (a), (b) and (c) (21), depicted for s=0.3,0.6 and 0.9 in the (a), (b) and (c)
panels, respectively. panels, respectively.
1 1
0
0 -1
(a) (b) (c) (a) (b) (c)
FIG. 13: (Color online) Alternative way to depict the topo- FIG. 15: (Color online) Alternative way to depict the topo-
logicalstructurecontrolledbythemodel(15),fors=0.3,0.6 logicalstructurecontrolledbythemodel(21),fors=0.3,0.6
and 0.9, in the (a), (b) and for (c) panels, respectively. and 0.9, in the (a), (b) and for (c) panels, respectively.
similar to the case of the vortex-like configurations.
the same figures, but now using a continuum of colors,
as we did before for the non-topological structure. We
note that the structures behave standardly asymptoti-
cally,however,thefieldconfigurationvanishesrapidly,as IV. ENDING COMMENTS
r diminishes toward the origin, so it represents a semi-
compact vortex-like structure. We described three types of planar, localized and
The last model is controlled by (21), so we use (41) spherically symmetric structures, one of them having
and (42) to get that Q = 1. The solutions then repre- non-topologicalbehavior,andtheotherbeingoftopolog-
senttopologicalstructuresoftheskyrmiontypewithunit ical nature. The non-topological structure can be used
skyrmion number. We depict some of them in Figs. 14 to describe compact structure, having energy density lo-
and15,usinganarrayofvectorsandacontinuumofcol- calized in a compact region in the plane. It is, however,
ors, respectively. We note that for Q=1, the skyrmion- unstable against spherically symmetric fluctuations and
like structures behave as standard structures as r in- should be further studied to be stabilized.
creases to larger and larger values; however, for r small, On the other hand, the topological structures are sta-
thespinstructuresquicklybecomeuniform,unveilingits ble under spherically symmetric fluctuations, and have
semi-compact character around its inner core, in a way skyrmionnumber1/2and1,sotheyarecapableofmod-
8
eling vortex- and skyrmion-like textures, respectively. seem to be of current interest since it provides a way
They have asymptotic behavior similar to the standard to control the magnetic profile inside the localized struc-
vortex and skyrmion configurations. However, since the tures. In this sense, the two topological models unveil a
associated fields quickly vanish as one approaches the route that helps us tayloring the core of the vortex- and
center of the configurations, they indicate a compact be- skyrmion-like textures at the nanometric scale.
havior at the corresponding cores. Thus, they behave as
semi-compact structures. Interestingly, they engender a ThisworkispartiallysupportedbyCNPq,Brazil. DB
singleparameter,whichcanbeusedthemodeltheinter- acknowledges support from grants 455931/2014-3 and
nal configurations and describe how the associated spin 306614/2014-6, and EIBR acknowledges support from
textures behave around their central region. The results grant 160019/2013-3.
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