#### Fractional Hopfions in the Faddeev-Skyrme model with a symmetry breaking potential

Received: July
Fractional Hop ons in the Faddeev-Skyrme model
A. Samoilenka 0
Ya. Shnir 0
Bobruiskaya Street
Minsk
Belarus
Joliot-Curie Street
Dubna
Russia
0 Department of Theoretical Physics and Astrophysics , BSU
We construct new solutions of the Faddeev-Skyrme model with a symmetry breaking potential admitting S1 vacuum. It includes, as a limiting case, the usual SO(3) symmetry breaking mass term, another limit corresponds to the potential m2 2, which 1 gives a mass to the corresponding component of the scalar eld. However we nd that the spacial distribution of the energy density of these solutions has more complicated structure, than in the case of the usual Hop ons, typically it represents two separate linked tubes with di erent thicknesses and positions. In order to classify these con gurations we de ne a counterpart of the usual position curve, which represents a collection of loops C1; C 1 corresponding to the preimages of the points ~ = ( 1; 0; 0), respectively. Then the Hopf invariant can be de ned as Q = link(C1; C 1). In this model, in the sectors of degrees Q = 5; 6; 7 we found solutions of new type, for which one or both of these tubes represent trefoil knots. Further, some of these solutions possess di erent types of curves C1 and C 1.
Solitons Monopoles and Instantons; Topological States of Matter
1 Introduction
rations, whose topology is de ned by the rst Hopf map S3 7! S2, provide a rst, and the
best known example of knot solitons in eld theory.
It was suggested later that stable knotted vortex con gurations of that type may
exist in a system of multi-component superconductors [
3, 4
], in two-condensate
GinzburgLandau model with oppositely charged components [5], or in the single-component
BoseEinstein condensate with trapping potential [
6
]. Recently, there was a signi cant interest
in construction of the Hop on-like solutions in liquid crystals [7]. It was pointed out that
the knotted Hop on con guration may exist in frustrated magnets [8]. In low energy QCD
the solutions which represent knots of the gluon
elds, may describe glueballs [
9, 10
]. It
was pointed out that the Faddeev-Skyrme model emerges as a low-energy limit of scalar
QED with a certain type potential [
11
]. Although knot solutions not always enjoy the
topological stability [
12
], extended solitons which are formed from vortex lines taking the
form of knots or links, are of considerable interest.
The structure of the Faddeev-Skyrme model looks similar to the usual Skyrme
model [13, 14], in 3+1 dimensions the corresponding Lagrangian includes the usual
model term, the Skyrme term, which is quartic in derivatives of the eld, and an optional
potential term, which does not contain the derivatives of the eld. A very interesting
feature of the Skyrme model is that the asymptotic form of the eld of the Skyrmion solutions
{ 1 {
and therefore, the character of interaction between them, strongly depends to the form of
the potential. Thus, a particular choice of the potential, which in the original Skyrme
model was introduced to provide a mass to the pion eld, may strongly a ect the structure
of multisoliton con gurations [15, 16]. The explicit form of the potential becomes even
more important in the self-dual modi cations of the Skyrme model, proposed to construct
the soliton solutions which satisfy the rst-order Bogomol'nyi-type equation [17{19].
In the last few years a number of modi ed versions of the Skyrme model were proposed,
in particular the model with an additional potential term that is quartic in the pion eld
was considered in [20]. These modi cations are mainly motivated by the fact that there is
a large disagreement between the experimental data for binding energies of hadrons and
eld term and/or the higher-order potential may signi cantly
reduce the binding energy of the solitons [
21, 22
]. On the other hand, various symmetry
breaking potentials were considered in the Skyrme model to construct half-Skyrmions [24{
28] or various vortex strings [23]. Clearly, all these modi cations preserve the topological
properties of the Skyrme model.
Despite the qualitative similarity between the Skyrme model and the Faddeev-Skyrme
model, the e ect of the potential term on the structure of the multisoliton con gurations not
yet been studied in detail. In the paper by Foster [29] the Faddeev-Skyrme model with an
SO(3) symmetry breaking potential, which is similar to the usual pion mass potential of the
Skyrme model, was considered in the limit of the in nite mass. The e ect of this term for
the axially symmetric con gurations of lowest degrees was investigated in earlier work [
30
]
Note that potential of that type, was also used to stabilize the classically isospinning
Hop ons [31, 32]. The double vacuum potential in the Faddeev-Skyrme model with a
potential term with two discrete vacua was considered in [33].
In this paper we investigate the e ect of a symmetry breaking potential admitting
S1 vacuum on the soliton solutions of the Faddeev-Skyrme model. We perform full 3d
numerical computations to
nd the corresponding eld con gurations in the sectors of
degrees up to Q = 7.
2
The model
The Lagrangian density of the Faddeev-Skyrme model in (3+1)-dimensional at space can
be written in terms of the real triplet of scalar elds ~ = ( 1; 2; 3) as [
1, 2
]
L =
Z
c4
2
(2.1)
uum value at spacial boundary, i.e. ~
where c2; c4 are some real positive parameters and g
= diag(1; 1; 1; 1).
As in the usual non-linear O(3) sigma model, the scalar eld is constrained to the unit
sphere S2: j~j = 1. The topological restriction on the eld ~ is that it approaches its
vac1 = (0; 0; 1). This allows a one-point compacti cation
{ 2 {
the homotopy group 3(S2) = Z.
of the domain space R3 to S3 and the eld of the nite energy solutions of the model, the
Hop ons, is a map a : R3 7! S2 which belongs to an equivalence class characterized by
Mathematically, the de nition of the Hopf invariant can be given by de ning an area
form !, which is a generator of the second cohomology group H2(S3) on the target space
S2. Then the Hopf map
: S3 7! S2 has an induced pullback of the cohomology group
: H2(S2) 7! H2(S3). Since the second cohomology group of S3 is trivial H2(S3) = 0,
the pullback F =
! of the area two-form ! by
is exact, F = dA where A is a one-form.
Explicitly, the corresponding Hopf invariant is de ned as the integral of a Chern-Simons
three-form over the space S3
where F = 12 F dx ^ dx is a 2-form with components
1 Z
Note that the Hopf invariant cannot be written in terms of
as a local density. More
simple geometrical de nition of this invariant is as the linking number of two loops on the
domain space S3, which are the preimages of two distinct points on the target space S2.
The static energy functional of the model (2.1) is
E =
Z
d3xE =
Z
+ V
2
The soliton solutions of the Faddeev-Skyrme model, correspond to the stationary points of
this functional.
Similarity of the Lagrangian (2.1) to the planar Skyrme model [34{37] suggests that
the model can support extended stringlike solutions, which can be constructed from baby
Skyrmions located in the plane transverse to the direction of the string [39]. Intuitively,
the topological charge of such a con guration can be given by the product of the winding
number of the planar Skyrmions and the number of the twists of the string in the extra
spatial direction. Thus, the solutions we expect to
nd may in some way resemble the
elastic rods that can bend, twist and stretch. Physically, eld con gurations of that type
can be considered as a vortex, which is bended and twisted a few times [38]. Then the
identi cation of the end points of the vortex yields the loop, which can transform itself
into a knot to minimise its energy.
Another peculiar feature of the Faddeev-Skyrme model is that for a given degree Q,
there are usually several di erent stable static soliton solutions of rather similar energy [40].
The number of solutions seems to grow with Q, thus the identi cation of a global minimum
of the energy functional in a given sector becomes rather involved.
In the absence of the potential term the corresponding solutions are well known, they
are constructed in the sectors of degrees up to Q = 29 or even higher [40, 41]. In the model
with the simple potential term, which breaks global SO(3) symmetry [29{32], the structure
of the solutions of the massive Faddeev-Skyrme model is not very much di erent from the
{ 3 {
(2.2)
(2.3)
(2.4)
HJEP09(217)
massless case. Note also the modi cation of the model via inclusion other 4th order on
derivative terms does not change this pattern [42].
It is known that the energy functional of the Faddeev-Skyrme model is bounded from
below by the Vakulenko-Kapitanskii inequality [43]
E
N jQj3=4
where N is a positive constant. However, this bound is modi ed if a potential term is
included in the model [44].
Note that the relation between the topological charge of the Hop ons and the energy
2 [0; ] is an angular parameter. Depending on the
value of this parameter the potential interpolates between the usual term, which explicitly
breaks global SO(3) symmetry [29{32], as
= 0, and the Heisenberg type potential
V = m2 12
(2.5)
(2.6)
(2.7)
as
= =2.
Potential of this form in the usual Skyrme model allows us to construct half-Skyrmion
solutions [25{27], as we will see in the Faddeev-Skyrme model it will also signi cantly a ect
the structure of the Hop ons. Note that the vacuum of the model with the potential (2.6)
is a circle S1 on the target space S2 and the vacuum boundary condition remains the
same, since V
! 0 as ~
!
~
1 = (0; 0; 1). Another remark is that the potential (2.6)
can be equivalently written as V = m2( 1
c)2 via an appropriate rotation of the elds
in the internal space and by setting c = cos . Also the corresponding vacuum boundary
condition should be imposed.
Unlike Skyrmions, the location of the Hop ons cannot be identi ed with a maximum of
the topological charge density distribution. Instead, it is convenient to look on the position
of the maxima of the energy density. Since it costs a lot of energy to deviate from the
vacuum ~1, the corresponding curve usually follows the positions of the preimage of antipodal
point ~0 = (0; 0; 1). This curve is usually referred to as the position curve [40], considering
the Hop ons most authors make use of it to visualize the shape of the con guration.
It is instructive to visualize the linking invariant via plotting not just the position curve
but also another curve in its neighborhood, which allows us to see the Hopf charge as the
{ 4 {
linking number of these two curves. Actually, any other preimage curve of ~ will be linked
with the preimage of ~0 Q times. In the absence of the symmetry breaking potential, or in
the model with usual pion-mass type potential [29], it is convenient to de ne this linking
curve as preimage of the vector ~ = ( 1; 0; 0). The linking number then is just the number
of times the linking curve wraps around the position curve.
The situation is di erent however in the model (2.1) with potential (2.6). Then, for the
relatively large values of the mass parameter m the location of the soliton can be identi ed
as collection of curves of maximal energy, it interpolates between the preimages of antipodal
point ~ = (0; 0; 1) as above for
distinct points ~ = ( 1; 0; 0), as
= 0 to the loops, which follow the preimages of two
= =2: C1 = ~ 1(1; 0; 0) and C 1 = ~ 1
( 1; 0; 0). Since
these loops are linked Q times, the de nition of the linking number now can be related
with the positions of the preimages of these points: Q = link(C1; C 1).
Note that the rst two terms of the energy functional (2.4) are invariant with respect
to global SO(3) transformations. Thus, the internal rotations of the triplet ~ allow us to
consider the limiting symmetry breaking potentials V
12,
22 and
32 all on equal
footing.
3
Initial approximation
variables which parameterize the sphere S3 [40]
The peculiarity of the Faddeev-Skyrme model is that the Hopf index is not a winding
number, the boundary conditions are identical for con gurations in all topological sectors.
As usual, the energy minimization scheme needs an appropriate initial con guration in a
given sector. The most e ective approach here is related with the generalization of the
rational map approximation, suggested by Sutcli e [40]. This construction can be nicely
described in terms of a degree one map R3 7! S3 2 C2. One can consider the complex
(Z1; Z0) = sin f (r) sin ei'; cos f (r) + i sin f (r) cos
(3.1)
a curve Q(Z1; Z0) with a unit sphere S3.
where f (r) is a monotonically decreasing function with the boundary values f (0) =
and
f (1) = 0. Clearly, the coordinates Z1; Z0 are restricted to the unit sphere S3, jZ1j2 +
jZ2j2 = 1. Then a torus knot on the domain space can be described as the intersection of
Then the components of the eld ~, which are coordinates on the Riemann sphere S2,
are given by the rational map W : S3 2 C2 7! CP 1:
projective line, W : S3 7! CP 1.
Here the polynomials P (Z1; Z0) and Q(Z1; Z0) have no common factors and have no
common roots on the two-sphere S2. Thus, the rational map ansatz (3.2) produces a curve in
: R3 7! S2 is equivalent to the map from a three-sphere to the complex
{ 5 {
According to the classi cation of [40], there are three di erent types of input con
gurations. The axially symmetric Hop ons are produced by the rational map
in this case the position curve, which is de ned as solution of the equation Z0 = 0, is just
a circle in the x-y plane centered at the origin. This Hop on is constructed via embedding
of a two-dimensional charge n planar Skyrmion con guration as a slice of a circle in
threedimensional space. Since the baby Skyrmions possess an internal phase, the con guration
can be twisted by the angle 2 m as it travels along the circle. Thus, the total topological
charge of the three dimensional con guration is given by the product of the winding number
in the plane and the number of twists: Q = mn. Following [40], we can label the axially
symmetric Hop ons of that type as QAmn, here the rst subscript gives the number of
twists and the second is the winding number of the two-dimensional planar solitons. In
terms of the mathematical knot theory the axially symmetric Hop ons are trivial knots, so
called unknots. They are closed eld con gurations with two independent winding numbers
along two fundamental circles of the torus.
Note that numerical simulations reveal that in general con gurations of higher degree
do not possess the axial symmetry, since increase of number of twists per unit resolution
may break the axial symmetry making the position curve not planar. This happened,
for example, for the charge three Hop on, the energy minimization transforms the
corresponding axially symmetric initial con guration A31 into the pretzel-like loop, bending
toward the third direction. It was suggested to label con gurations of that type as Ae31 to
emphasize the deformation [40].
As the value of the Hopf invariant increases, some new possibilities arise. First, we
can construct the Hop ons with two or more interlinked and disconnected position curves.
These con gurations are referred to as links. The Hopf charge of this con guration is
not just a simple sum of the Hopf indexes associated with each individual unknot, it also
includes in addition the sum of their secondary linking numbers due to the inter-linking with
the other components. Con gurations of that type are labeled as Lan;;bm, here the subscripts
label the Hopf indexes of the unknots and the superscript above each subscript counts the
secondary linking number, which appears due to inter-linking with the other components.
The total charge of the Hop on of that type is just the sum of all four indices [40].
Rational map approximation (3.2) can be used to produce various links in a given
topological sector. Now the denominator of the map must be reducible to give rise to the
linked position curves. For example, the link of the type L1n;;1n is generated by the map
W =
Zn+1
1
Z2
1
Z2 =
0
Zn
1
Zn
1
+
2(Z1
Z0)
2(Z1 + Z0)
:
(3.4)
The Hopf index of this con guration is Q = 2n + 2, it corresponds to the two An1 unknots
linked once.
{ 6 {
In order to construct con gurations of another type, the knots, Sutcli e suggested to
consider the rational map of the following form [40]
where
is a positive integer and
is a non-negative integer. This Hop on is denoted as
Ka;b, in order to produce a torus knot a and b must be co-prime1 and a > b. Thus, this
con guration possesses a(b
1) crossings, for example the trefoil knot K3;2 is also the (3; 2)
torus knot. The Hopf index is given by the crossing number and the number of times the
linking curve wraps around the position curve,
HJEP09(217)
To nd the stationary points of the energy functional (2.4) with the potential (2.6) we make
use of the numerical minimization technique described in [45, 46]. The elds are discretized
on the grid with 1503, or 2003 point with spatial grid spacings
x = 0:1, 0:06 or
x = 0:05.
The initial con guration were produced via the rational map approximation as described
above. As a consistency check, we verify that our algorithm correctly reproduces the known
results for the Hop on con guration of the usual rescaled massless Faddeev-Skyrme model
at m = 0, it agrees with previously known value within 0:5% accuracy. For each solution
we evaluated the value of the Hopf invariant and checked that the corresponding virial
in (2.4), holds. The estimated errors are of order of 10 2 or smaller.
relation E2 + 3V = E4 between the potential, quadratic, and quartic in derivatives terms
4.1
Q=1
First, we considered simplest unknot Hop on in the sector of degree one. We set in (2.6)
the angular parameter
=
=2, thus this is the Heisenberg type potential which explicitly
breaks the symmetry as m 6= 0. In
gure 1 we present isosurfaces of the energy density
of the Q = 1 solutions of the model (2.1) with the potential V = m2 21 at c2 = 1, c4 = 1
for m = 0; 1; 2; 4. As m = 0 we reproduce the usual axially symmetric A11 con guration.
However, as the mass parameter m starts to increase, the con guration deforms taking the
form of two linked rings.
Clearly, the position curve 3 =
1, which is a single loop, does not de ne the
maximum of the energy density distribution alone, as seen in
gure 1. Further, the increase of
m in the model with potential (2.6) does not localize the energy distribution, however it
makes the eld con guration more compact, see gure 1. Note that, as the mass parameter
m increases, the axial symmetry of the position curve is violated, see the plots in the third
1If they are not, the rational map (3.5) is degenerated producing a link.
{ 7 {
at m = 0 ( rst row), m = 1 (second row), m = 2 (third row), m = 4 (fourth row) and c2 = 1,
c4 = 1. Each plot corresponds to a value of the energy density E = 15 in unrescaled units.
column in gure 1. Thus, the contribution of the potential term now becomes critical, the
curves, which are the preimages of the 1 =
1, actually de ne the position of the maxima
of the energy density distribution of the con guration, as can be seen by comparing the
left and the middle columns in gure 1. Here, in order to visualize the location curves, we
plot tubes, which correspond to the isosurfaces 1 =
0:9.
It is seen in the left plot of gure 2, which displays pro les of the components 1(x; 0; 0)
and
2(0; y; 0) for some set of values of mass parameter m, that, as m increases, both
components become more localized. However the asymptotic behavior of these components
is di erent, while the eld 1 decays exponentially, the component 2 remains massless.
~
' ( 1; 2; 1
1
2 ( 12 + 22)) as 1; 2 ! 0. Then the energy density takes the form
Indeed, let us consider the asymptotic form of the elds in the limit r ! 1, when
)
where we take into account the explicit form of the potential (2.6). Thus, the linearized
{ 8 {
ϕ1,ϕ2 0.0
0.5
-0.5
-1.00
V = m2 21 and c2 = 1, c4 = 1. Left: eld components 1(x; 0; 0) at m = 0 (purple solid), 1(x; 0; 0)
(blue dashed) and
2(0; y; 0) (red long-dashed) at m = 4. Right:
eld component
3(x; 0; 0) at
m = 0 (black dotted), m = 1 (green dotted-dashed), m = 2 (cyan dashed) and m = 4 (orange
long-dashed).
Euler-Lagrange equations on the spacial asymptotic are
Hence, as r ! 1,
(m sin )
2
c2
1
exp r mpscin2
r
;
Note that the eld component 3 also possesses the asymptotic Yukawa massive interaction
tail, see the right plot in gure 2, increase of the mass parameter m also e ectively decreases
the characteristic size of this component.
It is instructive to plot the energy isosurface of the Q = 1 Hop on together with the
location curves of the eld components, see gure 3. Clearly, the shape of the energy
density distribution is similar to the location curve, which correspond to the isosurfaces of the
preimages of the vectors ~ = ( 1; 0; 0), it is visibly distinguished from the curve, which
corresponds to the isosurface of ~ 1(0; 0; 1). Further, it is seen in the right upper plot in
gure 3, which displays the location curves of the preimages of the points ~ = ( 1; 0; 0)
together with the energy isosurface, the maximum of the energy density distribution
corresponds to the domain, where these curve interlace. Indeed, as the value of the mass
parameter m increases, the contribution of the potential (2.6) into the total energy becomes more
signi cant. However, in the limit m ! 1 the virial condition for the Hop ons becomes E4 =
3V , where E4 is the energy contribution of the quartic in derivatives term in (2.4). Thus,
it does not allow us to neglect the corresponding contribution of the gradients of the elds.
Let us now consider the energy functional (2.4) with the potential2
V = m2 ( 1
1=3)2
(4.2)
2Note that the vacuum boundary conditions on the eld components ~1 are now di erent. However,
{ 9 {
together with the energy isosurface at E = 45 (left) and E = 100 (right). Bottom row: location
curves of 2 =
0:9 (left) and 3 =
0:9 together with the energy isosurface at E = 45.
This potential represents an intermediate form between the limiting cases of the Heisenberg
type potential, and the usual pion mass potential. Note that potential term of similar form
was suggested in the usual Skyrme model in 3+1 dimensions to construct charge one
Skyrmion solution, which consists of two components with fractional topological charges
1=3 and 2=3, respectively [
27
].
It is seen in
gure 4, which displays energy isosurfaces of the corresponding Q = 1
con guration, that similar to our consideration above, the Hop on is composed of two
linked loops. However, one of these tubes is much thinner and has a smaller size than
the other, see left plot on
gure 4. Right plot of gure 4 presents location curves of the
preimages of the points ~ = ( 1; 0; 0) together with the corresponding energy isosurface (cf.
gure 3). Note that the blue curve, which corresponds to the isosurface of the 1 =
1 +
has bigger size and it is a bit thicker than the red curve, which corresponds to the isosurface
of the 1 = 1
.
This is consistent with our observation above that in the model (2.1) the shape of the
energy density distribution follows the maxima of the potential although the virial relation
holds as before.
an appropriate rotation of the components
1 ! 1 sin + 3 cos
with cos
= 1=3, allows us to recover the general form of the potential (2.6) with the usual boundary
conditions on the eld ~1 = (0; 0; 1).
the Faddeev-Skyrme model (2.4) with the potential V = m2
1
for E = 20 (left) and E = 100 together with the location curves 1 =
Skyrme model (2.4) with the potential V = m2 12 at c2 = 0:5; c4 = 1; m = 4 for E = 52 (left) and
E = 90 (right) together with the location curves 1 =
0:9.
4.2
Q=2
Let us consider now the Hop on solutions in the topological sector of degree two.
We can see that, as in previously considered case of the Q = 1 Hop on, for relatively
large value of the mass parameter m the spacial distribution of the energy density follows
the location curves C1 = ~ 1(1; 0; 0) and C 1 = ~ 1
( 1; 0; 0). Indeed, in gure 5, left plot,
we display the energy density isosurfaces of the Q = 2 con guration together with the
isosurfaces of 1 =
0:9. Clearly, for higher values of the energy density, left plot, the energy
density isosurface has a form of twisted bead necklace with four beads. This structure is
similar to the energy density isosurface of the corresponding four-component solution of
the Skyrme model with Heisenberg type potential [
27
], which however, is not twisted.
We can consider now the Q = 2 Hop on solution in the model (2.1) with the
potential (4.2). As it is seen in gure 6, in this case the energy density isosurfaces also represent
two buckled tubular loops twisted two times around each other. However, similar to the
corresponding Q = 1 solution above, cf. gure 4, the energy is not equally distributed
among these loops (see table 2), one of them is thinner than another, it disappears as
the value of the energy density increases, see left plot in gure 6. Interestingly, for higher
model (2.4) with the potential V = m2
1
The energy isosurfaces at E = 90 together with the location curves 1 =
values of the energy density the corresponding tube is not decomposed into four isolated
beads, as it happens in the model with Heisenberg type potential above.
To classify solutions of the Faddeev-Skyrme model with symmetry breaking potential
we introduce new notations, which generalize the above-mentioned classi cation by
Sutcli e [40]. Since the spacial position of maxima of the energy distribution corresponds to
the location curves C1 and C 1, which could be of di erent types, the con guration with
linking number Q can be denoted as
Q (C1 G C 1)C
(4.3)
where the subindex corresponds to the type of the loop C = ~ 1(0; 0; 1), which is the
antipodal to the vacuum. In this notations, for example, the con guration of degree two is
denoted as 2(Ae1 G Ae1)A2;1 . Further remark is that for larger values of the mass parameter
m, the loop C in the sector of degree Q becomes symmetric with respect to the dihedral
group DQ. This is a symmetry of the truncated c = 0 model.
Note that in our numerical simulation we nd another solution of the Faddeev-Skyrme
model with potential m2 12 in this sector, 2(Ae1 G Ae1)Ae1;2 , see table 1. This con guration
represents a local minimum, its energy is about 16% higher than the energy of the global
minimum, see table 3. However, as the parameter
in (2.6) varies, this con guration
becomes a saddle point, our algorithm is unable to
nd this solution in the model with
potential (4.2) (see table 3). Indeed, it was noticed that the Hop on of that type, which in
the usual m = 0 Faddeev-Skyrme model represents an unstable con guration of two charge
one solitons stacked one above the other [49, 50], it has the energy around 13% above the
global minimum [40].
4.3
4.3.1
Multisoliton con gurations
Degrees 3 and 4
A peculiar feature of the Faddeev-Skyrme model is that for a given degree Q, there are
usually several di erent stable static soliton solutions of similar energy, furthermore, the
number of solutions grows with Q. We considered both the con gurations with minimal
model (2.4) with the potential V = m2 12 at c2 = 0:5; c4 = 1; m = 4 and E = 80 together with the
location curve of the preimage of the 3 =
0:9 (upper plot). Bottom row displays these isosutfaces
together with the isosurfaces of 1 =
0:9 for E = 50 (left) and E = 80 (right).
energies and the local minima in a given sector. Results of our numerical simulations are
summarized in table 1.
For Hopf degree Q = 3 the minimum-energy con guration in the model with the
symmetry breaking potential m2 12 is presented in gure 7. Similar to the corresponding
solutions in the sectors of degrees one and two, cf. gures 5, 3, we displayed there the energy
isosurface of the Q = 3 con guration 3(Ae1 G Ae1)A1 together with the location curves of
the eld components. Clearly, the energy density distribution follows the location curves
C1 = ~ 1(1; 0; 0) and C 1 = ~ 1
( 1; 0; 0), as above. Note that for the higher values of the
energy, the energy density isosurface has a form of a bead necklace with six beads. The
antipodal to the vacuum curve C = ~ 1(0; 0; 1), which is bounding these beads, is planar
but it is not exactly circular. Interestingly it has the hexagonal shape possessing dihedral
symmetry D3, see gure 7, bottom-left plot.
As said above, the Hop on con guration can be constructed from planar Skyrmions
located in the plane transverse to the direction of the loop in 3d [39]. It is known, however
that the structure of the multisoliton solution of the planar Skyrme model is very sensitive
to the choice of the potential [47, 48, 51{53]. In particular, it was proposed to consider the
potential [54, 55], which explicitly breaks the O(3) symmetry of the planar Skyrme model
to the dihedral group D3. In the baby Skyrme model with the potential V = m2( 1
c)2
a single Skyrmion is splitted into two partons with the same topological charge Q ' 1=2,
as c = 0 and into constituents with fractional charges Q = 1=3; 2=3, as c = 1=3. Hence the
1(A1 G A1)A1;1
2(A1 G A1)A2;1
2(Ae1 G Ae1)A1;2
3(A1 G A1)A3;1
3(Ae1 G Ae1)Ae3;1
4(L1;1 G L1;1)A2;2
4(Ae1 G Ae1)Ae4;1
4(A1 G A1)A4;1
5(L1;2 G L1;2)L1;2
5(A1 G A1)Ae5;1
6(K3;2 G K3;2)A3;2
6(L1;3 G K3;2)L11;;13
6(Ae1 G Ae1)Ae6;1
7(L12;;22 G K3;2)K3;2
7(K2;3 G K2;3)K2;3
7(Ae1 G Ae1)A7;1
1(A1 G A1)A1;1
2(A1 G A1)A2;1
3(A1 G A1)A3;1
3(Ae1 G Ae1)Ae3;1
4(Ae1 G L1;1)Ae4;1
4(L1;1 G L1;1)A2;2
4(A1 G A1)A4;1
5(L1;2 G K3;2)L1;2
5(A1 G A1)Ae5;1
6(K3;2 G K3;2)K3;2
6(L1;3 G K3;2)K3;2
6(Ae1 G Ae1)Ae6;1
7(K3;2 G K3;2)K3;2
7(K2;3 G K2;3)K2;3
7(Ae1 G Ae1)Ae7;1
1=3)2.
7 Hop ons in the model (2.1) with c2 = 0:5; c4 = 1; m = 4 and the potential V =
appearance of two tubes in the energy density distribution of the Faddeev-Skyrme model
with the potential (4.2), can be related with decomposition of the planar solitons.
Nonequal distribution of the topological charge between these lumps is re ected in di erent
thickness of the corresponding location curves C1 and C 1, see table 2.
In the sector of degree Q = 3 we also found another solution 3(Ae1 G Ae1)Ae1 (table 1).
Since the energies of both solutions are very close, we are not able to identify which
con guration represents a global minimum, our algorithm suggests that the con guration
with hexagonal symmetry of the location curve of the preimage of the
3 =
0:9 has
slightly lower energy.
Considering the structure of the Q = 3 Hop on solutions in the (2.1) with the
potential (4.2), we found that the energy density isosurfaces again represents two buckled
Q
1
2
3
4
5
6
7
K2;1; A1;3 ! (Ae1 G Ae1)Ae3;1
A2;2; L1;1; Ae4;1 ! (L1;1 G L1;1)A2;2
Con guration
A1;1 ! (A1 G A1)A1;1
A2;1 ! (A1 G A1)A2;1
A1;2 ! (Ae1 G Ae1)A1;2
A3;1 ! (A1 G A1)A3;1
K2;1 ! (Ae1 G Ae1)Ae4;1
A4;1 ! (A1 G A1)A4;1
L1;2 ! (L1;2 G L1;2)L1;2
A5;1 ! (A1 G A1)Ae5;1
L1;3 ! (L1;3 G K3;2)L11;;13
K5;1 ! (Ae1 G Ae1)Ae6;1
K3;2 ! (L12;;22 G K3;2)K3;2
K2;3 ! (K2;3 G K2;3)K2;3
K5;1 ! (Ae1 G Ae1)Ae7;1
E
tubular loops with interlinking number three, see table 2. Thus, as before, the location
curves C1 = ~ 1(1; 0; 0) and C 1 = ~ 1
( 1; 0; 0) de ne the spacial distribution of the energy
density. Similar to the corresponding Q = 1; 2 solutions we discussed above, the energy is
not equally distributed among these tubes. This observation holds for all con gurations in
the sectors of higher degrees.
At degree four in the model (2.1) with potential V = m2 12 we found solution of
L1;1)Ae2;2
three di erent types, see table 1. The minimal energy con guration is of the type 4(L1;1 G
. The energy density isosurfaces of this very symmetric Hop on represent four
tubular loops interlinked with each other, see the corresponding plot in the table 1.
The con guration of the form 4(Ae1;1 G Ae1;1)Ae4;1 has less symmetry, it is a local
minimum of the energy functional. In this sector we also found the dihedrally symmetric Hop on
4(A1;1 G A1;1)A4;1 , its energy is even higher. Similar to the corresponding solutions in the
sectors of lower degrees, for large values of the energy, the energy density isosurface forms
a bead necklace with 8 beads and the bounding planar curve C = ~ 1(0; 0; 1) possessing
dihedral symmetry D4.
Q = 4 Hop ons in the model (2.1) with mixed potential (4.2) are of similar forms, see
. More symmetric solution 4(A1 G A1)A4;1 then has almost the same energy as
the con guration 4(L1;1 G L1;1)A2;2 , so it is di cult to identify which state is the rst local
minimum. Remind that the former con guration represents the global minimum energy
solution in the massless model [40].
4.3.2
The Q = 5 minimal energy solution is 5(L1;2 G L1;2)L1;2 , it is a link, which is similar
to the usual minimal energy charge ve Hop on in the massless Faddeev-Skyrme model.
However, the structure of the energy density of this solution in the model with Heisenberg
type potential is di erent, one can clearly identify four linked tubes, see table 1. Con rming
with energy about 5% above the global minimum.
the general pattern we also found the dihedrally symmetric con guration 5(A1 G A1)Ae5;1
As expected, variations of the parameter
in the potential (2.6) a ects the solutions.
In the model with the potential V = m2( 1
1=3)2 the ground state con gurations is of
very unusual type, the location curves C1 = ~ 1(1; 0; 0) and C 1 = ~ 1
( 1; 0; 0) correspond
to the knot K3;2 and to the link L1;2, respectively, see table 2. At the same time, the curve
C = ~ 1(0; 0; 1) is a link of two bend tubes L1;2, thus this new Hop on with mixed types
of location curves can be labeled as 5(L1;2 G K3;2)L1;2 . The local minimum, which we found
in this sector, represent the deformation of the D5 symmetric con guration 5(A1 G A1)Ae5;1
Considering the Faddeev-Skyrme model with the Heisenberg type potential at m = 4,
.
we found that in the sector of degree six the minimal energy has the double trefoil knot
con guration 6(K3;2 G K3;2)A3;2 . This is the solution with dihedral symmetry, see the
corresponding plots in table 1. Recall that in the massless Faddeev-Skyrme model the link
L12;;12 is the minimal energy charge six solution [40] whereas in the presence of the usual
mass term the global minimum is the axially symmetric Hop on A3;2 [31, 32].
Making use of variety of initial con gurations in the model with the potential V =
m2 12, we found two other solutions, which represent local minima. The solution 6(L1;3 G
K3;2)L11;;13 yields another example of mixed type con guration, here the location curves C1
and C 1 form the trefoil knot and the link, respectively, see table 1. The curve C in this case
is the usual link L11;;31, which corresponds to a local minimum in the model with the usual
pion-mass type potential. The bent con guration 6(Ae1 G Ae1)Ae6;1 has even higher energy.
Last solution also exist as a local minimum in the model with potential V = m2( 1
1=3)2, see table 2. But global minimum now is represented by trefoil knot Hop on 6(K3;2 G
K3;2)K3;2 , the next by energy con guration is 6(L1;3 G K3;2)K3;2 , which di ers from the
corresponding c = 0 solution by con guration of the position curve. We should mention
that for these solutions, curves of the types K3;2; L1;3; L1;2; A3;2 are looking very similar to
each other, they are almost indistinguishable in some cases, moreover they possess almost
the same energy, so we cannot exclude a possibility that further more precise numerical
minimization scheme may reduce the number of local minima.
Finally, in the sector of degree seven we identify 7(L12;;22 G K3;2)K3;2 as the lowest
energy solution in the model with the potential V = m2 12 at m = 4. The curve C, which
in the massless limit corresponds to the position curve, has the form of the trefoil knot,
which becomes less symmetric as the mass parameter m increases. Another double knot
con guration (K2;3 G K2;3)K2;3 in that case represents a local minimum, the double bend
gurations of the last two types also exist in the model with potential V = m2( 1
axial con guration (Ae1 G Ae1)Ae7;1 is another metastable local minimum, see table 1.
Con1=3)2,
however the global minimum now has the form of usual trefoil knot, see table 2.
( rst, second and third row respectively) in the model (2.1) with potential V = m2 21 at c2 = 0,
c4 = 1 and m = 4.
5
Compact solutions
It is known that, according to the scaling arguments of the Derrick's theorem, stable soliton
solitons may exist in the truncated c2 = 0 submodel of (2.1), which contains only Skyrme
term and a potential [29, 56]. This submodel e ectively corresponds to the in nite mass
limit m ! 1. Interestingly, these Hop ons usually are solitons with compact support, the
con guration possess no exponentially decaying tail, instead the eld components attain
the vacuum values outside some nite domain in space.
In our simulations we considered the c2 = 0 submodel of (2.1) with the symmetry
breaking potential (2.6). We x the scale by setting m = 4. Since there are usually severe
di culties with the numerical simulations of the compactons, we make use of the small c2
parameter as a regulator in our numerical scheme.
In gure 8 we present the solutions of this submodel with Heisenberg type symmetry
breaking potential. As in the case of the usual pion-mass type potential, these solurions are
compactons. Note that the interior domain, where the elds are located, is not spherically
symmetric.
We found that the solutions of the c2 = 0 submodel are similar to the corresponding
minimal energy con gurations in the full Faddeev-Skyrme with potential (2.6). On the
other hand, the planar Skyrmions in the cross sections of such compact Hop ons have
more prominent dihedral symmetry. The spacial distribution of the energy density possess
much \sharper" structure, as expected.
C 1 = ~ 1
As seen in the second column of gure 8, the location curves C1 = ~ 1(1; 0; 0) and
( 1; 0; 0) de ne the spacial distribution of the energy density again. The curve
C = ~ 1(0; 0; 1) of these Hop ons becomes very geometrical, see the third column of
gure 8. Note that 2
; 3 components of the solutions of the c2 = 0 submodel have no
exponentially decaying tail.
6
The main purpose of this work was to construct Hop on solutions of the Faddeev-Skyrme
model with symmetry breaking potential V = m2 ( 1 sin
(1
3) cos )2, which
interpolates between the usual form of the \pion mass" potential as
= 0 and the Heisenberg
type potential as
=2, hence the vacuum of the model is a circle S1. Further, we
found that, in contrast to the usual massive Faddeev-Skyrme model [29], increase of the
mass parameter in the model with Heisenberg type potential, makes the energy density
distribution more extended, although it still localizes the elds.
We show that in the general case, the structure of the solutions is de ned by the location
( 1; 0; 0), which correspond to the maximum of the
potential. The Hopf index then is the linking number of these curves.
The usual classi cation scheme of the Hop ons, related to the possible types of the
curve C = ~ 1(0; 0; 1), which is the antipodal to the vacuum, becomes less useful in such
a case, the spacial distribution of the energy density follows the location curves C1 and
C 1, which form various structures of two linked knots and links. Relative thickness of the
tubes plotted around these location curves depends on the value of the angular parameter
, both tubes are of the same thickness as
decreases, one of the tubes becomes
thinner, it disappears as
= 0. The remaining tube C 1 = ~ 1
( 1; 0; 0) written in the
internally rotated eld components, which appear in the potential (2.6), coincides with the
position curve C = ~ 1(0; 0; 1). Further, the energy density isosurfaces of the Hop on
con gurations in the model with Heisenberg type potential may possess dihedral symmetry.
In this model, in the sectors of degrees Q = 5; 6; 7 we found solutions of new type, for which
one or both of these tubes represent trefoil knots. Further, some of these solutions possess
di erent types of curves C1 and C 1. Considering the c2 = 0 submodel with Heisenberg type
symmetry breaking potential, we found new compacton solutions with dihedral symmetry.
Our nal remark is that, since the Hop on solution can be thought as an embedding of
a two-dimensional planar Skyrmion con guration as a slice of a circle in three-dimensional
space and consequent twisting of the con guration along this loop, the appearance of these
tubes in the model with symmetry breaking potential can be related with decomposition
of the planar solitons into two components with fractional topological charge. Non-equal
distribution of the topological charge between these lumps is re ected in di erent thickness
of the corresponding location curves.
Acknowledgments
Y.S. gratefully acknowledges support from the Russian Foundation for Basic Research
(Grant No. 16-52-12012), the Ministry of Education and Science of Russian Federation,
project No 3.1386.2017, and DFG (Grant LE 838/12-2). The parallel computations were
performed on the cluster HIBRILIT at LIT, JINR, Dubna.
HJEP09(217)
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with broken time reversal symmetry, Phys. Rev. Lett. 107 (2011) 197001 [arXiv:1107.0995]
HJEP09(217)
[13] T.H.R. Skyrme, A non-linear eld theory, Proc. Roy. Soc. Lond. 260 (1961) 127.
[14] T.H.R. Skyrme, A uni ed eld theory of mesons and baryons, Nucl. Phys. 31 (1962) 556
[15] R. Battye and P. Sutcli e, Skyrmions and the pion mass, Nucl. Phys. B 705 (2005) 384
[16] R.A. Battye, N.S. Manton, P.M. Sutcli e and S.W. Wood, Light nuclei of even mass number
in the Skyrme model, Phys. Rev. C 80 (2009) 034323 [arXiv:0905.0099] [INSPIRE].
[17] C. Adam, J. Sanchez-Guillen and A. Wereszczynski, A Skyrme-type proposal for baryonic
matter, Phys. Lett. B 691 (2010) 105 [arXiv:1001.4544] [INSPIRE].
Rev. Lett. 82 (1999) 1624 [hep-th/9807069] [INSPIRE].
D 70 (2004) 114033 [hep-ph/0308240] [INSPIRE].
599 [arXiv:0812.1493] [INSPIRE].
[18] C. Adam, J. Sanchez-Guillen and A. Wereszczynski, A BPS Skyrme model and baryons at
HJEP09(217)
Phys. Rev. D 94 (2016) 065018 [arXiv:1606.02981] [INSPIRE].
(2016) 025008 [arXiv:1606.00336] [INSPIRE].
Phys. 57 (2016) 072903 [arXiv:1603.01305] [INSPIRE].
[28] A. Lukacs, \Half a proton" in the Bogomol'nyi-Prasad-Sommer eld Skyrme model, J. Math.
485 [INSPIRE].
[INSPIRE].
[33] M. Kobayashi and M. Nitta, Toroidal domain walls as Hop ons, arXiv:1304.4737 [INSPIRE].
[34] A.A. Bogolubskaya and I.L. Bogolubsky, Stationary topological solitons in the
two-dimensional anisotropic Heisenberg model with a Skyrme term, Phys. Lett. A 136 (1989)
[35] A.A. Bogolyubskaya and I.L. Bogolyubsky, On stationary topological solitons in
two-dimensional anisotropic Heisenberg model, Lett. Math. Phys. 19 (1990) 171 [INSPIRE].
[36] B.M.A.G. Piette, W.J. Zakrzewski, H.J.W. Mueller-Kirsten and D.H. Tchrakian, A modi ed
Mottola-Wipf model with sphaleron and instanton
elds, Phys. Lett. B 320 (1994) 294
[37] B.M.A.G. Piette, B.J. Schroers and W.J. Zakrzewski, Multi-solitons in a two-dimensional
Skyrme model, Z. Phys. C 65 (1995) 165 [hep-th/9406160] [INSPIRE].
HJEP09(217)
pions, Phys. Lett. B 728 (2014) 518 [arXiv:1311.2403] [INSPIRE].
Rev. D 95 (2017) 045002 [arXiv:1610.01300] [INSPIRE].
[hep-th/9811238] [INSPIRE].
451 (1999) 60 [hep-th/9811053] [INSPIRE].
[arXiv:1304.6021] [INSPIRE].
[1] L.D. Faddeev , Quantization of solitons, preprint- 75 -0570, IAS, Princeton U.S.A., ( 1975 ) [2] L.D. Faddeev and A.J. Niemi , Knots and particles, Nature 387 ( 1997 ) 58 [ hep -th/9610193] [3] E. Babaev , Non-Meissner electrodynamics and knotted solitons in two-component superconductors , Phys. Rev. B 79 ( 2009 ) 104506 [arXiv: 0809 .4468] [INSPIRE].
[4] J. Garaud , J. Carlstrom and E. Babaev , Topological solitons in three-band superconductors [5] E. Babaev , L.D. Faddeev and A.J. Niemi , Hidden symmetry and knot solitons in a charged two-condensate Bose system , Phys. Rev. B 65 ( 2002 ) 100512 [cond-mat/0106152] [INSPIRE].
[6] Y.V. Kartashov , B.A. Malomed , Y. Shnir and L. Torner , Twisted toroidal vortex-solitons in inhomogeneous media with repulsive nonlinearity , Phys. Rev. Lett . 113 ( 2014 ) 264101 [7] P.J. Ackerman and I.I. Smalyukh , Diversity of knot solitons in liquid crystals manifested by linking of preimages in torons and Hop ons , Phys. Rev. X 7 ( 2017 ) 011006 [8] P. Sutcli e , Skyrmion knots in frustrated magnets , Phys. Rev. Lett . 118 ( 2017 ) 247203 [9] L.D. Faddeev and A.J. Niemi , Partially dual variables in SU(2) Yang-Mills theory , Phys.
[10] L. Faddeev , A.J. Niemi and U. Wiedner , Glueballs, closed ux tubes and (1440), Phys . Rev.
[11] A. Gorsky , M. Shifman and A. Yung , Revisiting the Faddeev-Skyrme model and Hopf solitons , Phys. Rev. D 88 ( 2013 ) 045026 [arXiv: 1306 .2364] [INSPIRE].
[12] J.M. Speight , Supercurrent coupling in the Faddeev-Skyrme model , J. Geom. Phys . 60 ( 2010 ) large- Nc , Phys. Rev. D 82 ( 2010 ) 085015 [arXiv: 1007 .1567] [INSPIRE].
[19] P. Sutcli e , Skyrmions, instantons and holography, JHEP 08 ( 2010 ) 019 [arXiv: 1003 .0023] [20] M. Gillard , D. Harland and M. Speight , Skyrmions with low binding energies , Nucl. Phys. B [21] S.B. Gudnason , B. Zhang and N. Ma, Generalized Skyrme model with the loosely bound potential , Phys. Rev. D 94 ( 2016 ) 125004 [arXiv: 1609 .01591] [INSPIRE].
[22] S.B. Gudnason and M. Nitta , Modifying the pion mass in the loosely bound Skyrme model , [23] S.B. Gudnason and M. Nitta , Skyrmions con ned as beads on a vortex ring , Phys. Rev. D 94 [24] J. Jaykka and M. Speight , Easy plane baby skyrmions , Phys. Rev. D 82 ( 2010 ) 125030 [25] M. Kobayashi and M. Nitta , Vortex polygons and their stabilities in Bose-Einstein condensates and eld theory , J. Low. Temp. Phys . 175 ( 2014 ) 208 [arXiv: 1307 .1345] [26] M. Kobayashi and M. Nitta , Fractional vortex molecules and vortex polygons in a baby Skyrme model , Phys. Rev. D 87 ( 2013 ) 125013 [arXiv: 1307 .0242] [INSPIRE].
[27] S.B. Gudnason and M. Nitta , Fractional Skyrmions and their molecules , Phys. Rev. D 91 [29] D. Foster , Massive Hop ons, Phys. Rev. D 83 ( 2011 ) 085026 [arXiv: 1012 .2595] [INSPIRE].
[30] A. Kundu and Yu. P. Rybakov , Closed vortex type solitons with Hopf index , J. Phys. A 15 [31] D. Harland , J. Jaykka, Y. Shnir and M. Speight , Isospinning hop ons, J. Phys. A 46 ( 2013 ) [32] R.A. Battye and M. Haberichter , Classically isospinning Hopf solitons , Phys. Rev. D 87 [40] P. Sutcli e, Knots in the Skyrme-Faddeev model , Proc. Roy. Soc. Lond. A 463 ( 2007 ) 3001 [41] P. Jennings , Cabling in the Skyrme-Faddeev model , J. Phys. A 48 ( 2015 ) 315401 [38] D. Harland , M. Speight and P. Sutcli e, Hopf solitons and elastic rods , Phys. Rev. D 83 [39] M. Kobayashi and M. Nitta , Torus knots as Hop ons , Phys. Lett. B 728 ( 2014 ) 314 [42] J. Gladikowski and M. Hellmund , Static solitons with nonzero Hopf number , Phys. Rev . D [43] A.F. Vakulenko and L.V. Kapitansky , Stability of solitons in S2 in the nonlinear -model , [45] A. Samoilenka and Ya. Shnir, Gauged multisoliton baby Skyrme model , Phys. Rev. D 93 Sov. Phys. Dokl . 24 ( 1979 ) 433 [INSPIRE].