Hexagonal Structure of Baby Skyrmion Lattices

0.67 MB·English

by  Itay Hen

Hexagonal Structure of Baby Skyrmion Lattices
Itay Hen∗ and Marek Karliner†
Raymond and Beverly Sackler School of Physics and Astronomy Tel-Aviv University, Tel-Aviv 69978, Israel
(Dated: February 2, 2008)
We study the zero-temperature crystalline structure of baby Skyrmions by applying a full-field
numerical minimization algorithm to baby Skyrmions placed inside different parallelogramic unit-
cellsandimposingperiodicboundaryconditions. Wefindthatwithinthissetup,theminimalenergy
is obtained for the hexagonal lattice, and that in the resulting configuration the Skyrmion splits
into quarter-Skyrmions. In particular, we find that the energy in the hexagonal case is lower than
the one obtained on the well-studied rectangular lattice, in which splitting into half-Skyrmions is
observed.
PACSnumbers: 12.39.Dc;11.27.+d
8
Keywords: BabySkyrmion;Skyrmion;Crystallinestructure
0
0
2
I. INTRODUCTION to ferromagnetic quantum Hall states are described, in

n terms of a gradient expansion in the spin density, a field
a withpropertiesanalogoustothepionfieldinthe3Dcase
J The Skyrme model [1] is a non-linear theory for pions
[18].
8 in (3+1) dimensions with topological soliton solutions
2 called Skyrmions. The model can be used to describe, The baby Skyrme model has also been studied in con-
with due care [2, 3, 4], systems of a few nucleons, and nection with baby Skyrmion lattices under various set-

] has also been applied to nuclear and quark matter. One tings [19, 20, 21, 22, 23] and in fact, it is known that
h
t ofthemostcomplicatedaspectsofthephysicsofhadrons the baby Skyrmions also split into half-Skyrmions when
- is the behavior of the phase diagram of hadronic matter placed inside a rectangular lattice [21]. However, to our
p
e at finite density at low or even zero temperature. Par- knowledge,to date ithasn’t beenknownif the rectangu-
h ticularly, the properties of zero-temperature Skyrmions lar periodic boundary conditions yield the true minimal
[ on a lattice are of interest, since the behavior of nuclear energyconfigurationsoverallpossiblelatticetypes. Con-

matter at high densities is now a focus of considerable versely,ifthereareothernon-rectangularbabySkyrmion
4
v interest. Within the standard zero-temperature Skyrme lattice configurations which have lower energy. Finding
7 modeldescription,acrystalofnucleonsturnsintoacrys- the answers to these questions is thus of particular im-
8 tal of half nucleons at finite density [5, 6, 7, 8, 9]. portancebothbecauseoftheirrelevancetoquantumHall
3 systemsintwo-dimensions,andalsobecausetheymaybe
To study Skyrmion crystals one imposes periodic
2 usedtoconjecturethecrystallinestructureofnucleonsin
boundary conditions on the Skyrme field and works
1. withinaunitcell(equivalently,3-torus)T ([10],p. 382). three-dimensions.
1 3 In two dimensions there are five lattice types as given
The first attempted construction of a crystal was by
7
Klebanov [5], using a simple cubic lattice of Skyrmions by the crystallographic restriction theorem [24], in all of
0
: whosesymmetriesmaximizetheattractionbetweennear- which the fundamental unit cell is a certain type of a
v est neighbors. Other symmetries were proposed which parallelogram. To find the crystalline structure of the
Xi lead to slightly lower, but not minimal, energy crystals baby Skyrmions, we place them inside different parallel-
[6, 7]. It is now understoodthat it is best to arrangethe ograms with periodic boundary conditions and find the
r
a Skyrmions initially as a face-centered cubic lattice, with minimal energy configurations over all parallelograms of
their orientations chosen symmetrically to give maximal fixedarea(thus keepingthe Skyrmiondensity fixed). As
attraction between all nearest neighbors [8, 9]. we show later, there is a certain type of parallelogram,
namely the hexagonal, which yields the minimal energy
The Skyrme model has an analogue in (2+1) dimen-
configuration. In particular, its energy is lower than the
sions known as the baby Skyrme model, also admitting
known‘square-cell’configurationsinwhichtheSkyrmion
stablefieldconfigurationsofasolitonicnaturecharacter-
splits into half-Skyrmions. As will be pointed out later,
ized by integral topological charges [11, 12]. Due to its
the hexagonal structure revealed here is not unique to
lowerdimension, itservesasasimplificationofthe origi-
thepresentmodel,butalsoarisesinothersolitonicmod-
nal model, but nonetheless it has a physical relevance in
els,suchasGinzburg-Landauvortices[25],quantumHall
its own right, as a variant of the model arises in ferro-
systems[14,15],andeveninthecontextof3DSkyrmions
magneticquantumHallsystems[13,14,15,16,17]. This
[26]. Thereasonforthiswillbediscussedintheconclud-
effective theory is obtained when the excitations relative
ing section.
Inthe followingsectionwe review the setup ofour nu-
merical computations, introducing a systematic way for
∗Electronicaddress: [email protected] the detectionof the minimal energycrystalline structure
†Electronicaddress: [email protected] ofbabySkyrmions. Insubsequentsectionsweoutlinethe
2
numericalprocedurethroughwhichthefullfieldminimal π/2isthe anglebetweenthe ‘sL’sideandthe verticalto
configurationsareobtained. InsectionIVwepresentthe the ‘L’ side (as sketched in Fig. 1). Each parallelogram
main results of our study and in section V, a somewhat isthusspecifiedbyaset L,s,γ andtheSkyrmionden-
more analytical analysis of the problem is presented. In sityinsideaparallelogram{ isρ=}B/(sL2cosγ),whereB
the last section we make some remarks with regards to is the topological charge of the Skyrmion. The periodic
future research. boundary conditions are taken into account by identify-
ing each of the two opposite sides of a parallelogram as
equivalent:
II. BABY SKYRMIONS INSIDE A
PARALLELOGRAM
φ(x)=φ(x+n (L,0)+n (sLsinγ,sLcosγ)), (4)
1 2
The target manifold in the baby Skyrme model is de- with (n ,n Z). We are interested in static finite-
scribed by a three-dimensional vector φ with the con- 1 2 ∈
energysolutions,whichinthelanguageofdifferentialge-
straintφ·φ=1(i.e., φ∈S2)anditsLagrangiandensity ometry are T2 S2 maps. These are partitioned into
is: 7→
homotopysectorsparameterizedby aninvariantintegral
1 κ2 topological charge B, the degree of the map, given by:
= ∂ φ ∂µφ (∂ φ ∂µφ)2 (1)
µ µ
L 2 · − 2 ·
1
− (∂µφ·∂νφ)·(∂µφ(cid:2)·∂νφ) −µ2(1−φ3) B = 4π dxdy(φ·(∂xφ×∂yφ)) . (5)
ZΛ
consisting of a kinetic term, a 2D S(cid:3)kyrme term, and a
potential term. The static solutions of the model are The static energy E can be shown to satisfy
those field configurations which minimize the static en-
ergy functional: E 4πB (6)
with equality possible on≥ly in the ‘pure’ O(3) case (i.e.,
1
E = dxdy (∂ φ)2+(∂ φ)2 +κ2(∂ φ ∂ φ)2 when both the Skyrme and the potential terms are ab-
x y x y
2ZΛ (cid:16)+2µ2(1 φ ) × (2) soennRt)2[2w1i]t.hWfixeendotbeotuhnadtawryhciloenidnittihoensbathbye pSoktyernmtieamltoedrmel
3
− is necessary to prevent the solitons from expanding in-
(cid:17)
definitely, in our setup it is not required, due to the pe-
within each topological sector, where, in our setup, the
riodic boundary conditions [21]. In this work we study
integration is over parallelogramsdenoted here by Λ:
the model both with and without the potential term.
Λ= α (L,0)+α (sLsinγ,sLcosγ):0 α ,α <1 .(3) The problem in question can be simplified by a linear
1 2 1 2
{ ≤ } mapping of the parallelogramsΛ into the unit-area two-
Here L is the length of one side of the parallelogram,sL torusT (seeFig. 1). Inthenewcoordinates,theenergy
2
with0<s 1isthelengthofitsothersideand0 γ < functional becomes
≤ ≤
1 κ2ρ µ2B
E = dxdy s2(∂ φ)2 2ssinγ∂ φ∂ φ+(∂ φ)2 + dxdy(∂ φ ∂ φ)2+ dxdy(1 φ ) .(7)
x x y y x y 3
2scosγ T − 2B T × ρ T −
Z 2 Z 2 Z 2
(cid:0) (cid:1)
Note that the dependence of the energy on the Skyrme III. THE NUMERICAL MINIMIZATION
parameters κ and µ and the Skyrmion density ρ is only PROCEDURE
through κ2ρ and µ2/ρ.
TheEuler-Lagrangeequationsderivedfromtheenergy
In order to find the minimal energy configuration of functional (7) are non-linear PDE’s, so in general one
Skyrmionsoverallparallelogramswithfixedarea(equiv- must resort to numerical techniques. The minimal en-
alently,aspecifiedρ),wescantheparallelogramparame- ergy configuration of a Skyrmion with charge B and a
ter space s,γ and find the parallelogramfor which the givenparallelogram(definedbyaset s,γ )withaspec-
{ } { }
resultantenergyisminimalovertheparameterspace. An ifiedSkyrmiondensityρisfoundbyafull-fieldrelaxation
alternative approachto this problem, which is of a more methodona100 100rectangulargridonthe two-torus
analyticalnature,hasalsobeenimplemented, andis dis- T , where at eac×h point a field triplet φ(x ,y ) is de-
2 m n
cussed in detail in section V. In the following section we fined. For the calculation of the energy and charge den-
describe the numerical minimization procedure. sities, weuse the procedure devisedby [27], in whichthe
3
y sure that the final configurations are independent of the
discretization and of the cooling scheme. As a further
check,someofthe minimizationswererepeatedwith nu-
L mericalderivativesofa higherprecision,usingeightfield
points for their evaluation. No apparent changes in the
L results were detected.
sL
IV. RESULTS
Γ T
2
Usingtheminimizationprocedurepresentedinthepre-
x vious section, we have found the minimal energy static
Skyrmion configurationsover all parallelograms,for var-
ious settings: The ‘pure’ O(3) case,in which both κ, the
FIG. 1: The parameterization of the fundamental unit-cell Skyrmeparameter,andµ,thepotentialcoupling,areset
parallelogram Λ(inblack)andthetwo-torusT2 intowhichit to zero, the Skyrme case for which only µ = 0, and the
is mapped (in gray). general case for which neither the Skyrme term nor the
potential term vanish.
Ineachofthesesettings,theparameterspaceofparal-
evaluationofthese quantitiesis performedatthe centers
lelograms was scanned, while the Skyrmion density ρ is
of the grid squares. Numerical derivatives are also eval-
heldfixed,yieldingforeachsetof s,γ aminimalenergy
uatedatthesepoints;the x-derivativesarecalculatedby { }
configuration. ThechoiceastohowmanySkyrmionsare
∂φ to be placed inside the unit cells was made after some
= (8)
preliminary testing in which Skyrmions of other charges
∂x(cid:12)(xm+12,yn+12) (up to B = 8) were also examined. The odd-charge
(cid:12)
1 φ((cid:12)xm+1,yn)+φ(xm+1,yn+1) minimal-energy configurations turned out to have sub-
∆x 2 normed stantially higher energies than even-charge ones, where
D E among the latter, the charge-two Skyrmion was found
φ(x ,y )+φ(x ,y ) to be the most fundamental, as it was observed that it
m n m n+1
,
− 2 normed! servesasa‘building-block’forthehigher-chargeones. In
D E what follows, a summary of the results is presented.
with the y-derivatives analogously defined, and the
“normed”subscriptindicatesthattheaveragedfieldsare
normalized to one. We find this procedure to work very A. The pure O(3) case (κ=µ=0)
well in practice. For a more detailed discussion of the
method, see [27]. The pure O(3) case corresponds to setting both κ and
The relaxation process begins by initializing the field µ to zero. In this case, analytic solutions in terms of
triplet φ to a rotationally-symmetric configuration Weierstrass elliptic functions may be found [21, 22, 23]
and the minimal energy configurations, in all parallelo-
φ =(sinf(r)cosBθ,sinf(r)sinBθ,cosf(r)) ,(9)
initial gram settings, saturate the energy bound in (6) giving
wheretheprofilefunctionf(r)issettof(r)=πexp( r) E = 4πB. Thus, comparison of our numerical results
with r and θ being the usual polar coordinates. The−en- withtheanalyticsolutionsservesasausefulcheckonthe
ergyofthebabySkyrmionisthenminimizedbyrepeating precision of our numerical procedure. The agreement is
the following steps: a point (x ,y ) on the grid is cho- to 6significantdigits. Contourplots ofthe chargedensi-
m n
sen at random, along with one of the three components tiesfordifferentparallelogramsettingsforthecharge-two
of the field φ(x ,y ). The chosen component is then Skyrmions are shown in Fig. 2, all of them of equal en-
m n
shifted by a value δ chosenuniformly from the segment ergy E/8π =1.
φ
[ ∆ ,∆ ] where ∆ = 0.1 initially. The field triplet is
φ φ φ
−
then normalized and the change in energy is calculated.
If the energy decreases, the modification of the field is B. The Skyrme case (κ6=0,µ=0)
accepted and otherwise it is discarded. The procedure
is repeated, while the value of ∆ is gradually decreased As pointed out earlier, in setting µ = 0, the depen-
φ
throughout the procedure. This is done until no further dence of the energy functional on the Skyrme parame-
decrease in energy is observed. ter κ is only through κ2ρ, so without loss of generality
To check the stability and reliability of our numerical we vary ρ and fix κ2 = 0.03 throughout (this particular
prescription, we set up the minimization scheme using choice for κ was made for numerical convenience). Min-
differentinitialconfigurationsandan80 80gridforsev- imization of the energy functional (7) over all parallelo-
×
eral ρ, s and γ values. This was done in order to make gramsyielded the following. For any fixed density ρ, the
4
functional), the Skyrmion structure is different at low
and at high densities and a phase transition is observed.
WhileatlowdensitiestheSkyrmionsstandisolatedfrom
oneanother,athighdensities theyfuse togetherforming
the quarter-Skyrmion crystal as in the Skyrme case re-
ported above. As the density ρ decreasesor equivalently
the value of µ increases, the size of the Skyrmions be-
comes small compared to the cell size. The exact shape
(a)s=1andγ=0 (b)s=0.9andγ=π/16 ofthelatticelosesitseffectsandthedifferencesinenergy
amongthe variouslattice types become verysmall. This
is illustrated in figures 4 and 5.
Furthermore, due to the finite size of the Skyrmion,
there is an optimal density for which the energy is min-
imal among all densities. Figure 6 shows the contour
plots of the charge density of the charge-two Skyrmion
for several densities with κ2 = 0.03 and µ2 = 0.1. The
energy of the Skyrmion is minimal for ρ 0.14 (Fig. 4).
(c)s=0.7andγ=0 (d)s=1andγ=π/6
≈
FIG. 2: Charge-two Skyrmions in the pure O(3) case: Con-
tour plots of the charge densities ranging from violet (low
density) to red (high density) for various parallelogram set- V. SEMI-ANALYTICAL APPROACH
tings,allofwhichsaturatetheenergyboundE =4πB=8π.
Theenergyfunctional(7)dependsbothontheSkyrme
field φ and on the parallelogram parameters γ and s.
Formally, the minimal energy configuration over all par-
minimalenergywasobtainedfors=1andγ =π/6. This allelograms may be obtained by functional differentia-
setofvaluescorrespondstothe‘hexagonal’or‘equilateral tion with respect to φ and regular differentiation with
triangular’ lattice. In this configuration, any three adja- respect to γ and s. However, since the resulting equa-
cent zero-energyloci (or ‘holes’) are the vertices of equi- tionsareverycomplicated,adirectnumericalsolutionis
lateraltriangles,andeightdistincthigh-densitypeaksare quite hard. Nonetheless, some analytical progress may
observed (as shown in Fig. 3b). This configuration can be achieved in the following way. As a first step, we dif-
thus be interpreted as the splitting of the two-Skyrmion ferentiate the energy functional (7) only with respect to
into eight quarter-Skyrmions. This result turned out to
be independentofthe Skyrmiondensityρ. Inparticular,
thewell-studiedsquare-cellminimalenergyconfiguration
(Fig. 3a),inwhichthetwo-Skyrmionsplitsintofourhalf-
Skyrmions turned out to have higher energy than the
hexagonal case. Figure 3 shows the total energies (di-
vided by 8π) and corresponding contour plots of charge
densities of the hexagonal, square and other configura-
tions (for comparison), all of them with ρ=2.
The total energy of the Skyrmions in the hexagonal
(a) s = 1, γ = 0, (b) s = 1, γ = π/6, and
settingturnedouttobelinearlyproportionaltotheden- andE/8π=1.446 E/8π=1.433
sity of the Skyrmions, reflecting the scale invariance of
themodel(Fig. 4). Inparticular,theglobalminimumof
E =4πB =8π is reached when ρ 0. This is expected
→
since setting ρ = 0 is equivalent to setting the Skyrme
parameterκtozero,inwhichcasethemodeliseffectively
pure O(3) and inequality (6) is saturated.
(c) s = 0.51, γ = 0, and (d) s = 1, γ = π/4, and
E/8π=1.587 E/8π=1.454
C. The general case (κ6=0, µ6=0)
FIG. 3: Charge-two Skyrmions in the Skyrme case with
κ2 = 0.03 and ρ = 2: Contour plots of the charge densities
The hexagonal setting turned out to be the energeti- for the hexagonal, square and other fundamental cells rang-
callyfavorableinthegeneralcaseaswell. Since,however, ing from violet (low density) to red (high density). As the
captions of the individual subfigures indicate, the hexagonal
inthiscase,theSkyrmionpossessesadefinitesize(ascan
configuration has thelowest energy.
be verified by looking at the ρ dependence in the energy
5
γ and s:
E
∂E 1
= sinγ( yy+s2 xx) 2s xy =0,
∂γ 2scos2γ E E − E
∂E 1 (cid:0) (cid:1)
1.40 = ( yy s2 xx)=0, (10)
∂s 2s2cosγ E − E
where ij = dxdy(∂ φ ∂ φ)andi,j x,y . Solving
E T2 i · j ∈{ }
these equations for γ and s yields
R
yy
s = E , (11)
1.25 xx
rExy
sinγ = E .
√ xx yy
E E
Substituting these expressionsintothe energyfunctional
(7), we arrive at a ‘reduced’ functional
1.10
κ2ρ µ2B
E = xx yy ( xy)2+ sk+ pot, (12)
E E − E 2BE ρ E
q
where = dxdy(∂ φ ∂ φ)2istheSkyrmeenergy
Ρ Esk T2 x × y
0.5 1 1.5 2 and Epot =R T2dxdy(1−φ3) is the potential energy.
Nowthatbothγ ands areeliminatedfromthe resultant
R
(a) expression, and the conditions for their optimization are
built into the functional, the numerical minimization is
E carried out. We note here, however, that the procedure
presented above should be treated with caution. This is
sincethe‘minimization’conditionsforγandsin(11)are
infactonlyextremumconditions,andmayturnouttobe
1.08 maximumorsaddle-pointconditions. Hence,itisimpor-
tant to confirm any results obtained using this method
by comparing them with corresponding results obtained
from the method described in the previous section.
It is therefore reassuring that numerical minimization
of the reduced functional (12) gives sinγ = 0.498 (γ
≈
1.05 π/6)ands=1(bothfortheSkyrmecaseandthegeneral
DEH%L
1.0
1.02
0.5
Ρ
0.1 0.2
(b)
Ρ
0.5 1 1.5 2
FIG. 4: Total energy E (divided by 8π) of the charge-two
Skyrmion in the hexagonal lattice ((cid:3) – Skyrme case and ♦
– general case) and in the square lattice ( (cid:4) – Skyrme case FIG. 5: Energy difference (in %) between the hexagonal
and(cid:7)–generalcase) asfunction oftheSkyrmiondensity(in settingandthesquaresetting,intheSkyrmecase((cid:3))andin
theSkyrmecase, κ2 =0.03 andin thegeneralcase κ2 =0.03 thegeneral case ((cid:7)),as function of theSkyrmiondensity. As
and µ2 = 0.1). Note the existence of an optimal density (at the density decreases, the energy difference between the two
ρ≈ 0.14) in the general case, for which the energy attains a lattice types is reduced.
global minimum. Figure (b) is an enlargement of the lower
left corner of figure(a).
6
we verified that there are no favorable lattices in which
theSkyrmionsorderthemselves,asallparallelogramset-
tings yielded the same minimal energy, saturating the
minimal energy bound given by inequality (6).
In the Skyrme case, without the potential term, the
resultsaredifferent. For anyfixedSkyrmiondensity, the
parallelogramfor which the Skyrmion energy is minimal
(a)ρ=1,E/8π=1.229 (b)ρ=0.5,E/8π=1.128 turnsouttobethehexagonallattice(forwhichs=1and
γ = π/6). In particular, the hexagonal lattice has lower
energythanthefourhalf-Skyrmionsconfigurationonthe
rectangularlattice. Forexample,atρ=2,E =1.433for
the hexagonal lattice vs. E = 1.446 for the rectangular
lattice.
Thehexagonalsettingturnsouttobetheenergetically
most favorable also in the general case with both the
(c)ρ=0.25,E/8π=1.087 (d)ρ=0.14,E/8π=1.08
Skyrme term and the potential term. In this case, how-
ever, the model is not scale invariant and the Skyrmion
has a definite size. This results in the existence of a
phasetransitionas afunction ofdensityandthe appear-
ance of an optimal density, for which the total energy
of the Skyrmion is minimal. This is analogous to the
Skyrmion behavior in the 3D Skyrme model, in which
the Skyrmions also possess a definite size [5].
(e)ρ=0.0625,E/8π=1.082 (f)ρ=0.03125,E/8π=1.084
As pointed out in the Introduction, the special role of
FIG. 6: Charge-two Skyrmions in the general case with the hexagonal lattice revealed here is not unique to the
µ2 =0.1 and κ2 = 0.03: Contour plots of the charge den- baby Skyrme model, but in fact arises in other solitonic
sities of the minimal-energy configurations in the hexagonal models. InthecontextofSkyrmemodels,theexistenceof
settingfordifferentdensities. Here,theenergeticallymostfa- a hexagonal two dimensional structure of 3D Skyrmions
vorabledensityisρ=0.14. Theplotcolors rangefrom violet has been found by [26], where it has already been noted
(low density)to red (high density). that energetically, the optimum infinite planar structure
of 3D Skyrmions is the hexagonal lattice, which resem-
bles the structure of a graphite sheet, the most stable
case), confirming the results presented in the previous
form of carbon thermodynamically ([10], p. 384). Other
section.
examplesinwhichthehexagonalstructureisrevealedare
In the general (µ = 0) case, the energy functional
Ginzburg-Landauvorticeswhichareknowntohavelower
6
(12) may be further differentiated with respect to the
energyinahexagonalconfigurationthaninasquarelat-
Skyrmion density ρ to obtain the optimal density for
tice configuration [25]. Thus, it should not come as a
which the Skyrmion energy is minimal. Differentiating
surprise that the hexagonal structure is found to be the
with respect to ρ, and substituting the obtained expres-
most favorable in the baby Skyrme model.
sion into the energy functional, results in the functional
As briefly noted in the Introduction, a certain type
ofbabySkyrmionsalsoariseinquantumHallsystemsas
E = xx yy ( xy)2+κµ 2 sk pot. (13) low-energyexcitationsofthegroundstatenearferromag-
E E − E E E
q q netic filling factors (notably 1 and 1/3)[14]. It has been
Numericalminimizationoftheaboveexpressionforκ2 = pointed out that this state contains a finite density of
0.03 and various µ values (0.1 µ2 10) yielded the Skyrmions [28], and in fact the hexagonal configuration
≤ ≤
hexagonal setting as in the Skyrme case. In particular, has been suggestedas a candidate for their lattice struc-
for µ2 = 0.1 the optimal density turned out to be ρ ture[15]. Ourresultsmaythereforeserveasasupporting
≈
0.14,inaccordwithresultspresentedinsub-sectionIVC. evidenceinthatdirection,althoughamoredetailedanal-
ysis is in order.
Our results also raise some interesting questions. The
VI. SUMMARY AND FURTHER REMARKS first is how the dynamical properties of baby Skyrmions
onthe hexagonallattice differ fromtheir behaviorin the
In this paper, we have studied the crystalline struc- usual rectangular lattice. Another question has to do
ture ofbabySkyrmionsintwo dimensionsby finding the with their behavior in non-zero-temperature settings.
minimalenergyconfigurationsofbabySkyrmionsplaced One may also wonder whether and how these results
inside parallelogramicfundamentalunit cells andimpos- can be generalized to the 3D case, once a systematic
ing periodic boundary conditions. In the pure O(3) case study like the one reported here is conducted. Is the
(whereboththeSkyrmeandpotentialtermsareabsent), face-centered cubic lattice indeed the minimal energy
7
crystalline structure of 3D Skyrmions among all paral- and suggestions. This work was supported in part by a
lelepiped lattices? If not, what would the minimal en- grant from the Israel Science Foundation administered
ergy structure be? and how would these results change by the Israel Academy of Sciences and Humanities.
when a mass term is present? We hope to answer these
questions in future research.
Acknowledgments
We thank Alfred S. Goldhaber, Moshe Kugler, Philip
Rosenau and Wojtek Zakrzewski for useful discussions
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