Review of Rotational Symmetry Breaking in Baby Skyrme Models

1.1 MB·English

by  Marek Karliner

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Review of Rotational Symmetry Breaking
in Baby Skyrme Models1 2
9
0 Marek Karliner and Itay Hen
0
Raymond and Beverly Sackler School of Physics and Astronomy
2
Tel-Aviv University, Tel-Aviv 69978, Israel.

n
[email protected]
a
J
Wediscussoneofthemostinterestingphenomenaexhibitedbybabyskyrmions–
2
breakingofrotationalsymmetry. Thetopicswewilldealwithhereincludetheap-
1
pearanceof rotationalsymmetrybreakingin thestaticsolutionsofbabySkyrme

] models, both in flat as well as in curved spaces, the zero-temperaturecrystalline
h
structureofbabyskyrmions,andfinally,theappearanceofspontaneousbreaking
t
- of rotational symmetry in rotating baby skyrmions.
p
e
h 1.1. Breaking of Rotational Symmetry in Baby Skyrme Models
[

1 The Skyrme model1,2 is an SU(2)-valued nonlinear theory for pions in (3+1) di-
v mensions with topological soliton solutions called skyrmions. Apart from a kinetic
9
term, the Lagrangianof the model contains a ‘Skyrme’ term which is of the fourth
8
order in derivatives, and is used to introduce scale to the model.3 The existence
4
1 of stable solutions in the Skyrme model is a consequence of the nontrivial topol-
1. ogy of the mapping of the physical space into the field space at a given time,
M
0 : S3 SU(2) = S3, where the physical space R3 is compactified to S3
9 M → ∼
by requiring the spatial infinity to be equivalent in each direction. The topology
0
which stems from this one-point compactification allows the classification of maps
:
v
into equivalence classes, each of which has a unique conserved quantity called the
i
X topological charge.
r The Skyrme model has an analogue in (2+1) dimensions known as the baby
a
Skyrmemodel,alsoadmittingstablefieldconfigurationsofasolitonicnature.4 Due
to its lower dimension, the baby Skyrme model serves as a simplification of the
original model, but nonetheless it has a physical significance in its own right, hav-
ing several applications in condensed-matter physics,5 specifically in ferromagnetic
quantum Hall systems.6–9 There, baby skyrmions describe the excitations relative
to ferromagnetic quantum Hall states, in terms of a gradient expansion in the spin
density, a field with properties analogous to the pion field in the 3D case.10
Thetargetmanifoldinthebabymodelisdescribedbyathree-dimensionalvector
φ=(φ ,φ ,φ )withtheconstraintφ φ=1. Inanalogywiththe(3+1)Dcase,the
1 2 3
·
domain of this model R2 is compactified to S2, yielding the topology required for
1toappearin: G.BrownandM.Rho,Eds.,MultifacetedSkyrmions,(WorldScientific,Singapore,
2009).
2A version of this manuscript with higher-resolution figures is available at
www.tau.ac.il/∼itayhe/SkReview/SkReview.rar.
January12,2009 1:7 WorldScientificReviewVolume-9.75inx6.5in text
2 M. Karliner and I. Hen
the classification of its field configurations into classes with conserved topological
charges. The Lagrangiandensity of the baby Skyrme model is given by:
1 κ2
= ∂ φ ∂µφ (∂ φ ∂µφ)2 (∂ φ ∂ φ) (∂µφ ∂νφ) U(φ ),
µ µ µ ν 3
L 2 · − 2 · − · · · −
(cid:2) (cid:3) (1.1)
and consists of a kinetic term, a Skyrme term and a potential term.
While in (3+1) dimensions the latter term is optional,11 its presence in the
(2+1)D model is necessary for the stability of the solutions. However, aside from
therequirementthatthepotentialvanishesatinfinityforagivenvacuumfieldvalue
(normally taken to be φ(0) =(0,0,1)), its exact form is arbitrary and gives rise to
a rich family of possible baby-Skyrme models, several of which have been studied
in detail in the literature. The simplest potential is the ‘holomorphic’ model with
U(φ )=µ2(1 φ )4.12–14 It is known to have a stable solution only in the charge-
3 3
−
one sector (the name refers to the fact that the stable solution has an analytic
form in terms of holomorphic functions). The model with the potential U(φ ) =
3
µ2(1 φ ) (commonly referred to as the ‘old’ model) has also been extensively
3
−
studied. This potential gives rise to very structured non-rotationally-symmetric
multi-skyrmions.4,15 Another model with U(φ ) = µ2(1 φ2) produces ring-like
3 − 3
multi-skyrmions.16 Other double-vacuum potentials which give rise to other types
of solutions have also been studied.17
Clearly, the form of the potential term has a decisive effect on the properties
of the minimal energy configurations of the model. It is then worthwhile to see
how the multisolitons of the baby Skyrme model look like for the one-parametric
family of potentials U =µ2(1 φ )s which generalizes the ‘old‘ model (s=1) and
3
−
the holomorphic model (s = 4).18 As it turns out, the value of the parameter s
has dramatic effects on the static solutions of the model, both quantitatively and
qualitatively,inthesensethatitcanbe viewedasa‘control’parameterresponsible
fortherepulsionorattractionbetweenskyrmions,whichinturndetermineswhether
or not the minimal-energy configuration breaks rotational symmetry.
The Lagrangiandensity is now:
1 κ2
= ∂ φ ∂µφ (∂ φ ∂µφ)2 (∂ φ ∂ φ) (∂µφ ∂νφ) µ2(1 φ )s,
µ µ µ ν 3
L 2 · − 2 · − · · · − −
(cid:0) (cid:1) (1.2)
and contains three free parameters, namely κ,µ and s. Since either κ or µ may be
scaledaway,theparameterspaceofthismodelisinfactonlytwodimensional. Our
main goal here is to study the effects of these parameters on the static solutions of
the model within each topological sector.
Themulti-skyrmionsofourmodelarethosefieldconfigurationswhichminimize
thestaticenergyfunctionalwithineachtopologicalsector. Inpolarcoordinatesthe
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energy functional is given by
1 1 κ2(∂ φ ∂ φ)2
E = rdrdθ (∂ φ ∂ φ+ ∂ φ ∂ φ)+ r × θ +µ2(1 φ )s .
2 r · r r2 θ · θ 2 r2 − 3
Z (cid:18) (cid:19)
(1.3)
TheEuler-Lagrangeequationsderivedfromtheenergyfunctional(1.3)arenon-
linear PDE’s, so in most cases one must resort to numerical techniques in order to
solve them. In our approach,the minimal energy configurationof a baby skyrmion
ofchargeBandagivensetofvaluesµ,κ,sisfoundbyafull-fieldrelaxationmethod,
which we describe in more detail in the Appendix.
1.1.1. Results
Applyingtheminimizationprocedure,weobtainthestaticsolutionsofthemodelfor
1 B 5. Since the parameter space of the model is effectively two dimensional
≤ ≤
(as discussed earlier), without loss of generality we fix the potential strength at
µ2 =0.1throughout,andthes-κparameterspaceisscannedintheregion0<s 4,
≤
0.01 κ2 1.
≤ ≤
1.1.1.1. Charge-one skyrmions
In the charge-one sector, the solutions for every value of s and κ are stable
rotationally-symmetricconfigurations. Figure1.1a showsthe obtainedprofile func-
tions of the B = 1 solution for different values of s with κ fixed at κ2 = 0.25.
Interestingly,the skyrmionenergyas a functionofs is notmonotonic,butacquires
a minimum at s 2.2, as is shown in Fig. 1.2.
≈
Fig. 1.1. Profile functions of the B = 1 (left) and B = 2 (right) skyrmions for s = 0.5 (solid),
s=1(dotted) ands=2(dot-dashed). Hereκisfixedatκ2=0.25.
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Fig.1.2. Totalenergies(dividedby4πB)ofthecharge-one((cid:7))charge-two((cid:4))andcharge-three
(♦) skyrmionsas afunction ofthe parameter sforκ2 =0.05. Eachof theenergy graphs attains
a minimal value at some s. At s ≈ 2 the energy-per-topological-charge of the charge-two and
charge-three solutions reaches the charge-one energy (from below), and stable solutions are no
longerobserved.
1.1.1.2. Charge-two skyrmions
Stable solutions also exist in the B = 2 sector, but only for s < 2. They are
rotationally-symmetric and ring-like, corresponding to two charge-one skyrmions
ontop of eachother. Figure 1.1b shows the profile functions ofthe stable solutions
for different values of s and κ2 =0.25.
As in the B = 1 case, the energy of the charge-two skyrmion as a function of
s is non-monotonic and has a minimum around s = 1.3. As shown in Fig. 1.2, at
s 2 the energyofthe ring-likeconfigurationreachesthe valueof twice the energy
≈
of the charge-one skyrmion and stable configurations cease to exist. At this point,
the skyrmionbreaksapartinto its constituentcharge-oneskyrmions,whichin turn
start drifting away from each other, thus breaking the rotational symmetry of the
solution. Contourplotsofthe energydistributionoftheB =2skyrmionareshown
in Fig. 1.3 for κ2 = 1 and for two s values. While for s = 1.5 a ring-like stable
configuration exists (Fig. 1.3a), for s = 2.6 the skyrmion breaks apart. The latter
case is shown in Fig. 1.3b which is a “snapshot” taken while the distance between
the individual skyrmions kept growing.
These results are in accord with corresponding results from previously known
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studies of both the ‘old’ (s = 1) model in which ring-like configurations have been
observed,4,15 and the holomorphic model for which no stable solutions have been
found.12,13
Fig. 1.3. Contour plots of the energy distributions (ranging from violet – low density to red –
high density) of the B = 2 skyrmion for κ2 = 1. In the s < 2 regime, ring-like rotationally-
symmetric configurations exist, corresponding to two charge-one skyrmions on top of each other
(left), whereas for s > 2, the charge-two skyrmion splits into two one-charge skyrmions drifting
infinitelyapart(right).
Rotationally-symmetriccharge-twolocallystablesolutionsmayalsobeobserved
in the large s regime, including the ‘holomorphic’ s = 4 case, in which case the
global minimum in this regime corresponds to two infinitely separated charge-one
skyrmions. The total energy of the rotationally symmetric solutions is larger than
twice the energy of a charge-oneskyrmion, indicating that the split skyrmion is an
energetically more favorable configuration. We discuss this issue in more detail in
the section 1.2.2.
1.1.1.3. Charge-three and higher-charge skyrmions
AswiththeB =2skyrmion,theexistenceofstablecharge-threeskyrmionswasalso
found to be s dependent. For any tested value of κ in the range 0.01 κ2 1, we
≤ ≤
have found that above s 2, no stable charge-three solutions exist; in this region
≈
the skyrmion breaks apart into individual skyrmions drifting further and further
away from each other.
In the s < 2 region, where stable solutions exist, the energy distribution of the
charge-three skyrmionturns out to be highly dependent on both s and κ. Keeping
κ fixed and varying s, we find that in the small s regime, ring-like rotationally-
symmetric configurations exist. Increasing the value of s results in stable minimal
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6 M. Karliner and I. Hen
energy configurations with only Z(2) symmetry, corresponding to three partially-
overlapping charge-one skyrmions in a row, as shown in Figs 1.4b and 1.4c. The
energyofthecharge-threeskyrmionalsohasaminimumins,atarounds=1.5(as
shown in Fig. 1.2). At s 2 the energy of the skyrmionbecomes larger than three
≈
times the energy of a charge-one skyrmion and stable configurations are no longer
obtainable. This is illustrated in Fig. 1.4 which shows contour plots of the energy
distributionoftheB =3 skyrmionfor differentvaluesofs andfixedκ. Fors=0.5
(Fig. 1.4a), the solution is rotationally symmetric and for s=0.75 and s=1 (Figs
1.4b and 1.4c respectively) the rotational symmetry of the solution is broken and
only Z(2) symmetry remains. At s = 3, no stable solution exists. The latter case
is shown in Fig. 1.4d which is a “snapshot” taken while the distance between the
individual skyrmions kept growing.
The dependence of the skyrmion solutions on the value of κ with fixed s show
the following behavior: While for small κ the minimal energy configurations are
rotationally-symmetric, increasing its value results in an increasingly larger rota-
tional symmetry breaking. This is illustrated in Fig. 1.5.
Fig.1.4. Energydensitiesandcorrespondingcontour plots(rangingfromviolet–lowdensityto
red – highdensity) of the B =3 skyrmionfor fixed κ (κ2 =0.01) and varying s. In the s=0.5
case, the minimal energy configuration is rotationally symmetric, corresponding the three one-
skyrmionsontopofeachother. Fors=0.75ands=1thesolutionsexhibitonlyZ(2)symmetry,
correspondingtopartially-overlappingone-skyrmions. Fors=3nostablesolutionexistsandthe
individualskyrmionsaredriftingapart.
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Fig.1.5. Energydensitiesandcorrespondingcontour plots(rangingfromviolet–lowdensityto
red – high density) of the B = 3 skyrmion for fixed s (s = 0.5) and varying κ. At low κ, the
minimalenergy configuration isrotationallysymmetric. Asκisincreased, breakingofrotational
symmetryappears,andonlyZ(2)symmetryremains.
The B = 4 and B = 5 skyrmion solutions behave similarly to the B = 3
solutions. This is illustratedinFig. 1.6, whichshowsthe stablesolutionsthathave
been obtained in the s = 0.9 case and the splitting of these skyrmions into their
constituents in the s=4 case.
1.2. The Lattice Structure of Baby Skyrmions
TheSkyrmemodel1 mayalsobeusedtodescribesystemsofafewnucleons,andhas
also been applied to nuclear and quark matter.19–21 One of the most complicated
aspects of the physics of hadrons is the behavior of the phase diagram of hadronic
matteratfinitedensityatloworevenzerotemperature. Particularly,theproperties
of zero-temperature skyrmions on a lattice are interesting, since the behavior of
nuclear matter at high densities is now a focus of considerableinterest. Within the
standard zero-temperature Skyrme model description, a crystal of nucleons turns
into a crystal of half nucleons at finite density.22–26
To study skyrmion crystals one imposes periodic boundary conditions on the
Skyrme field and works within a unit cell.11 The first attempted construction of a
crystalwasbyKlebanov,22 using asimplecubic lattice ofskyrmionswhosesymme-
tries maximize the attraction between nearest neighbors. Other symmetries were
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Fig. 1.6. Contour plots of the energy distributions (ranging from violet – low density to red –
high density) of the B =4 and B =5 skyrmions for s=0.9 and s=4 (κ2 =0.1). In the lower
sregion stablesolutions exist; the upper figures show a B=4skyrmioninabound state oftwo
charge-two skyrmions(left), andaB=5skyrmioninatwo-one-twoconfiguration. Forvalues of
shigherthan2,themulti-skyrmionssplitintoindividualone-skyrmionsconstantlydriftingapart
(lowerfigures).
proposedwhichleadto crystalswith slightly lower,but notminimal energy.23,24 It
isnowunderstoodthatitisbesttoarrangetheskyrmionsinitiallyasaface-centered
cubic lattice, with their orientations chosen symmetrically to give maximal attrac-
tion between all nearest neighbors.25,26
ThebabySkyrmemodeltoohasbeenstudiedundervariouslatticesettings27–31
andinfact,itisknownthatthebabyskyrmionsalsosplitintohalf-skyrmionswhen
placed inside a rectangular lattice.29 However, as we shall see, the rectangular
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periodic boundary conditions do not yield the true minimal energy configurations
overallpossiblelatticetypes.32 Thisfactisparticularlyinterestingbothbecauseof
its relevance to quantum Hall systems in two-dimensions, and also because it may
be used to conjecture the crystalline structure of nucleons in three-dimensions.
In two dimensions there are five lattice types, as given by the crystallographic
restrictiontheorem.33 Ininallofthemthefundamentalunitcellisacertaintypeof
a parallelogram. To find the crystalline structure of the baby skyrmions, we place
theminsidedifferentparallelogramswithperiodicboundaryconditionsandfindthe
minimalenergyconfigurationsoverallparallelogramsoffixedarea(thuskeepingthe
skyrmiondensity fixed). As we show later, there is a certaintype of parallelogram,
namelythehexagonal,whichyieldstheminimalenergyconfiguration. Inparticular,
itsenergyislowerthantheknown‘square-cell’configurationsinwhichtheskyrmion
splits into half-skyrmions. As will be pointed out later, the hexagonal structure
revealed here is not unique to the present model, but also arises in other solitonic
models,suchasGinzburg-Landauvortices,34 quantumHallsystems,6,7 andevenin
the context of 3D skyrmions.35 The reason for this will also be discussed later.
Inwhatfollowswereviewthesetupofournumericalcomputations,introducinga
systematicapproachfortheidentificationoftheminimalenergycrystallinestructure
ofbaby skyrmions. In section1.2.2we presentthe main results ofour study and in
section 1.2.3, a somewhat more analytical analysis of the problem is presented.
1.2.1. Baby skyrmions inside a parallelogram
Wefindthestaticsolutionsofthemodelbyminimizingthestaticenergyfunctional:
1
E = dxdy (∂ φ)2+(∂ φ)2 +κ2(∂ φ ∂ φ)2+2µ2(1 φ ) , (1.4)
x y x y 3
2 × −
ZΛ (cid:16) (cid:17)
within each topological sector. In this example, we use the ‘old’ model potential
term. In our setup, the integration is over parallelograms,denoted here by Λ:
Λ= α (L,0)+α (sLsinγ,sLcosγ):0 α ,α <1 . (1.5)
1 2 1 2
{ ≤ }
Here L is the length of one side of the parallelogram, sL with 0 < s 1 is the
≤
length of its other side and 0 γ <π/2 is the angle between the ‘sL’ side and the
≤
vertical to the ‘L’ side (as sketched in Fig. 1.7).
Eachparallelogramis thus specified by a set L,s,γ and the skyrmiondensity
{ }
insideaparallelogramisρ=B/(sL2cosγ),whereB isthetopologicalchargeofthe
skyrmion. The periodic boundary conditions are taken into account by identifying
each of the two opposite sides of a parallelogramas equivalent:
φ(x)=φ(x+n (L,0)+n (sLsinγ,sLcosγ)), (1.6)
1 2
with n ,n Z. We are interested in static finite-energy solutions, which in the
1 2
∈
language of differential geometry are T S maps. These are partitioned into
2 2
7→
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10 M. Karlinerand I. Hen
y
L
L
sL
Γ T
2
x
Fig. 1.7. The parameterization of the fundamental unit-cell parallelogram Λ (in black) and the
two-torusT2 intowhichitismapped(ingray).
homotopy sectors parameterized by an invariant integral topological charge B, the
degree of the map, given by:
1
B = dxdy(φ (∂ φ ∂ φ)) . (1.7)
x y
4π · ×
ZΛ
The static energy E can be shown to satisfy
E 4πB, (1.8)
≥
with equality possible only in the ‘pure’O(3) case(i.e., whenboth the Skyrme and
the potential terms are absent).29 We note that while in the baby Skyrme model
on R2 with fixed boundary conditions the potential term is necessary to prevent
the solitons from expanding indefinitely, in our setup it is not required, due to the
periodic boundary conditions.29 We study the model both with and without the
potential term.
The problem in question can be simplified by a linear mapping of the paral-
lelograms Λ into the unit-area two-torus T . In the new coordinates, the energy
2
functional becomes
1
E = dxdy s2(∂ φ)2 2ssinγ∂ φ∂ φ+(∂ φ)2
x x y y
2scosγ T −
Z 2
κ2ρ (cid:0) µ2B (cid:1)
+ dxdy(∂ φ ∂ φ)2+ dxdy(1 φ ) . (1.9)
x y 3
2B T × ρ T −
Z 2 Z 2
We note thatthe dependence ofthe energyonthe Skyrme parametersκ andµand
the skyrmion density ρ is only through κ2ρ and µ2/ρ.