Isolated skyrmions in the C P 2 nonlinear sigma model with a Dzyaloshinskii-Moriya type interaction

Abstract

We study two dimensional soliton solutions in the CP2 nonlinear sigma model with a Dzyaloshinskii-Moriya type interaction. First, we derive such a model as a continuous limit of the SU(3) tilted ferromagnetic Heisenberg model on a square lattice. Then, introducing an additional potential term to the derived Hamiltonian, we obtain exact soliton solutions for particular sets of parameters of the model. The vacuum of the exact solution can be interpreted as a spin nematic state. For a wider range of coupling constants, we construct numerical solutions, which possess the same type of asymptotic decay as the exact analytical solution, both decaying into a spin nematic state.

Full text

Isolated skyrmions in the CP2nonlinear sigma model
with a Dzyaloshinskii-Moriya type interaction
Yutaka Akagi ,1Yuki Amari ,2,3 Nobuyuki Sawado ,4and Yakov Shnir 2
1Department of Physics, Graduate School of Science, The University of Tokyo,
Bunkyo, Tokyo 113-0033, Japan
2BLTP, JINR, Dubna 141980, Moscow Region, Russia
3Department of Mathematical Physics, Toyama Prefectural University,
Kurokawa 5180, Imizu, Toyama 939-0398, Japan
4Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
(Received 2 February 2021; accepted 10 February 2021; published 22 March 2021)
We study two dimensional soliton solutions in the CP2nonlinear sigma model with a Dzyaloshinskii-
Moriya type interaction. First, we derive such a model as a continuous limit of the SUð3Þtilted
ferromagnetic Heisenberg model on a square lattice. Then, introducing an additional potential term to the
derived Hamiltonian, we obtain exact soliton solutions for particular sets of parameters of the model. The
vacuum of the exact solution can be interpreted as a spin nematic state. For a wider range of coupling
constants, we construct numerical solutions, which possess the same type of asymptotic decay as the exact
analytical solution, both decaying into a spin nematic state.
DOI: 10.1103/PhysRevD.103.065008
I. INTRODUCTION
In the 1960s, Skyrme introduced a (3þ1)-dimensional
Oð4Þnonlinear (NL) sigma model [1,2], which is now well
known as a prototype of a classical field theory that
supports topological solitons (See Ref. [3], for example).
Historically, the Skyrme model has been proposed as a low-
energy effective theory of atomic nuclei. In this framework,
the topological charge of the field configuration is iden-
tified with the baryon number.
The Skyrme model, apart from being considered a good
candidate for the low-energy QCD effective theory, has
attracted much attention in various applications, ranging
from string theory and cosmology to condensed matter
physics. One of the most interesting developments here is
related to a planar reduction of the NLσmodel, the so-
called baby Skyrme model [4–6]. This (2þ1)-dimensional
simplified theory resembles the basic properties of the
original Skyrme model in many aspects.
The baby Skyrme model finds a number of physical
realizations in different branches of modern physics.
Originally, it was proposed as a modification of the
Heisenberg model [4,5,7]. Then, it was pointed out that
skyrmion configurations naturally arise in condensed
matter systems with intrinsic and induced chirality [8–12].
These baby skyrmions, often referred to as magnetic
skyrmions, were experimentally observed in noncentro-
symmetric or chiral magnets [13–15]. This discovery
triggered extensive research on skyrmions in magnetic
materials. This direction is a rapidly growing area both
theoretically and experimentally [16].
A typical stabilizing mechanism of magnetic skyrmions
is the existence of the Dzyaloshinskii-Moriya (DM) inter-
action [17,18], which stems from the spin-orbit coupling. In
fact, the magnetic skyrmions in chiral magnets can be well
described by the continuum effective Hamiltonian
H¼Zd2xJ
2ð∇mÞ2þκm·ð∇×mÞ−Bm3
þAfjmj2þðm3Þ2g;ð1:1Þ
where mðrÞ¼ðm1;m
2;m
3Þis a three component unit
magnetization vector which corresponds to the spin expect-
ation value at position r. The first term in Eq. (1.1) is the
continuum limit of the Heisenberg exchange interaction,
i.e., the kinetic term of the Oð3ÞNLσmodel, which is often
referred to as the Dirichlet term. The second term there
is the DM interaction term, the third one is the Zeeman
coupling with an external magnetic field B, and the last,
symmetry breaking term Afjmj2þðm3Þ2grepresents the
uniaxial anisotropy.
Published by the American Physical Society under the terms of
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Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
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PHYSICAL REVIEW D 103, 065008 (2021)
2470-0010=2021=103(6)=065008(13) 065008-1 Published by the American Physical Society

It is remarkable that in the limiting case A¼κ2=2J;
B¼0, the Hamiltonian (1.1) can be written as the static
version of the SUð2Þgauged Oð3ÞNLσmodel [19,20]
H¼J
2Zd2xð∂kmþAk×mÞ2;k¼1;2;ð1:2Þ
with a background gauge field A1¼ð−κ=J; 0;0Þ;A2¼
ð0;−κ=J; 0Þ. Though the DM term is usually introduced
phenomenologically, a mathematical derivation of the
Hamiltonian (1.2) with arbitrary Akhas been developed
recently [19]; i.e., it has been shown that the Hamiltonian
can be derived mathematically in a continuum limit of the
tilted (quantum) Heisenberg model
H¼−JX
hijiðWiSa
iW−1
iÞðWjSa
jW−1
jÞ;ð1:3Þ
where the sum hijiis taken over the nearest-neighbor sites,
Sa
idenotes the ath component of spin operators at site iand
Wi∈SUð2Þ. It was reported that the tilting Heisenberg
model can be derived from a Hubbard model at half-filling
in the presence of spin-orbit coupling [21]. Therefore, the
background field Akcan still be interpreted as an effect of
the spin-orbit coupling.
There are two advantages of utilizing the expression
(1.2) for the theoretical study of baby skyrmions in the
presence of the so-called Lifshitz invariant, an interaction
term that is linear in a derivative of an order parameter
[22,23], like the DM term. The first advantage of the form
Eq. (1.2) is that one can study a NLσmodel with various
forms of Lifshitz invariants which are mathematically
derived by choice of the background field Ak, although
Lifshitz invariants have, in general, a phenomenological
origin corresponding to the crystallographic handedness
of a given sample. The second advantage of the model (1.2)
is that it allows us to employ several analytical techniques
developed for the gauged NLσmodel. It has been recently
reported in Ref. [20] that the Hamiltonian (1.2) with a
specific choice of the potential term exactly satisfies the
Bogomol’nyi bound, and the corresponding Bogomol’nyi-
Prasad-Sommerfield (BPS) equations have exact closed-
form solutions [20,24,25].
Geometrically, the planar skyrmions are very nicely
described in terms of the CP1complex field on the
compactified domain space S2[6]. Further, there are
various generalizations of this model; for example, two-
dimensional CP2skyrmions have been studied in the
pure CP2NLσmodel [26–28] and in the Faddeev-
Skyrme type model [29,30].
Remarkably, the two-dimensional CP2NLσmodel can
be obtained as a continuum limit of the SUð3Þferromag-
netic (FM) Heisenberg model [31,32] on a square lattice
defined by the Hamiltonian
H¼−J
2X
hiji
Tm
iTm
j;ð1:4Þ
where Jis a positive constant, and Tm
i(m¼1;…;8) stand
for the SUð3Þspin operators of the fundamental represen-
tation at site isatisfying the commutation relation
½Tl
i;Tm
i¼iflmnTn
i:ð1:5Þ
Here, the structure constants are given by flmn ¼
−i
2Trðλl½λm;λnÞ, where λmare the usual Gell-Mann
matrices.
The SUð3ÞFM Heisenberg model may play an important
role in diverse physical systems ranging from string theory
[33] to condensed matter, or quantum optical three-level
systems [34]. It can be derived from a spin-1 bilinear-
biquadratic model with a specific choice of coupling con-
stants, so-called FM SUð3Þpoint; see, e.g., Ref. [35]. The
SUð3Þspin operators can be defined in terms of the SUð2Þ
spin operators Sa(a¼1,2,3)as
0
B
@
T7
T5
T2
1
C
A¼0
B
@
S1
−S2
S3
1
C
A;
0
B
B
B
B
B
B
@
T3
T8
T1
T4
T6
1
C
C
C
C
C
C
A
¼−
0
B
B
B
B
B
B
@
ðS1Þ2−ðS2Þ2
1
ffiffi3
p½S·S−3ðS3Þ2
S1S2þS2S1
S3S1þS1S3
S2S3þS3S2
1
C
C
C
C
C
C
A
:ð1:6Þ
Using the SUð2Þcommutation relation ½Sa
i;S
b
i¼iεabcSc
i
where εabc denotes the antisymmetric tensor, one can check
that the operators (1.6) satisfy the SUð3Þcommutation
relation (1.5).
In the present paper, we study baby skyrmion solutions
of an extended CP2NLσmodel composed of the CP2
Dirichlet term, a DM type interaction term, i.e., the Lifshitz
invariant, and a potential term. The Lifshitz invariant,
instead of being introduced ad hoc in the continuum
Hamiltonian, can be derived in a mathematically well-
defined way via consideration of a continuum limit of the
SUð3Þtilted Heisenberg model. Below we will implement
this approach in our derivation of the Lifshitz invariant. In
the extended CP2NLσmodel, we derive exact soliton
solutions for specific combinations of coupling constants
called the BPS point and solvable line. For a broader range
of coupling constants, we construct solitons by solving the
Euler-Lagrange equation numerically.
The organization of this paper is the following: In the
next section, we derive an SUð3Þgauged CP2NLσmodel
from the SUð3Þtilted Heisenberg model. Similar to the
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SUð2Þcase described as Eq. (1.2), the term linear in a
background field can be viewed as a Lifshitz invariant term.
In Sec. III, we study exact skyrmionic solutions of the
SUð3Þgauged CP2NLσmodel in the presence of a
potential term for the BPS point and solvable line using
the BPS arguments. The numerical construction of baby
skyrmion solutions off the solvable line is given in Sec. IV.
Our conclusions are given in Sec. V.
II. GAUGED CP2NLσMODEL FROM
A SPIN SYSTEM
To find Lifshitz invariant terms relevant for the CP2NLσ
model, we begin to derive an SUð3Þgauged CP2NLσ
model, a generalization of Eq. (1.2), from a spin system on
a square lattice. By analogy with Eq. (1.2), the Lifshitz
invariant, in that case, can be introduced as a term linear in a
nondynamical background gauge potential of the gauged
CP2model.
Following the procedure to obtain a gauged NLσmodel
from a spin system, as discussed in Ref. [19], we consider a
generalization of the SU(3) Heisenberg model defined by
the Hamiltonian
H¼−J
2X
hiji
Tm
iðˆ
UijÞmn Tn
j;ð2:1Þ
where
ˆ
Uij is a background field which can be recognized as
a Wilson line operator along with the link from the point i
to the point j, which is an element of the SUð3Þgroup in the
adjoint representation. As in the SUð2Þcase [19], the field
ˆ
Uij may describe effects originated from spin (nematic)-
orbital coupling, complicated crystalline structure, and so
on. This Hamiltonian can be viewed as the exchange
interaction term for the tilted operator
˜
Tm
i¼WiTm
iW−1
i,
where Wi∈SUð3Þ, because one can write WjTm
jW−1
j¼
ðRjÞmnTn
jwhere Rjis an element of SUð3Þin the adjoint
representation. Clearly,
ˆ
Uij ¼RT
iRj, where T stands for the
transposition.
Let us now find the classical counterpart of the quantum
Hamiltonian (2.1). It can be defined as an expectation value
of Eq. (2.1) in a state possessing over completeness,
through a path integral representation of the partition
function. In order to construct such a state for the spin-1
system, it is convenient to introduce the Cartesian basis
jx1i¼ i
ffiffiffi
2
pðj þ 1i−j−1iÞ;
jx2i¼ 1
ffiffiffi
2
pðj þ 1iþj−1iÞ;
jx3i¼−ij0i;ð2:2Þ
where jmi¼jS¼1;mi(m¼0,1). In terms of the
Cartesian basis, an arbitrary spin-1 state at a site jcan
be expressed as a linear combination jZij¼ZaðrjÞjxaij
where rjstands for the position of the site j, and Z¼
ðZ1;Z
2;Z
3ÞTis a complex vector of unit length [31,36].
Since the state jZijsatisfies an over-completeness relation,
one can obtain the classical Hamiltonian using the state
jZi¼⊗jjZij¼⊗jZaðrjÞjxaij:ð2:3Þ
Since Zis normalized and has the gauge degrees of freedom
corresponding to the overall phase factor multiplication, it
takes values in S5=S1≈CP2. In terms of the basis (2.2), the
SUð3Þspin operators can be defined as
Tm¼ðλmÞabjxaihxbj;m¼1;2;…;8;ð2:4Þ
where λmis the mth component of the Gell-Mann matrices.
One can check that they satisfy the SUð3Þcommutation
relation (1.5). The expectation values of the SUð3Þoper-
ators in the state (2.3) are given by
hTm
ji≡nmðrjÞ¼ðλmÞab ¯
ZaðrjÞZbðrjÞ;ð2:5Þ
where ¯
Zadenotes the complex conjugation of Za. In the
context of QCD, the field nmis usually termed a color
(direction) field [37]. The color field satisfies the
constraints
nmnm¼4
3;n
m¼3
2dmpqnpnq;ð2:6Þ
where dmpq ¼1
4Trðλmfλp;λqgÞ. Consequently, the number
of degrees of freedom of the color field reduces to four.
Note that, combining the constraints (2.6), one can get the
Casimir identity dmpqnmnpnq¼8=9.
In terms of the color field, the classical Hamiltonian is
given by
H≡hZjHjZi¼−J
2X
hiji
nlðriÞð ˆ
UijÞlm nmðrjÞ:ð2:7Þ
Let us write the position of a site jnext to a site ias
rj¼riþaϵek, where ekis the unit vector in the kth
direction ϵ¼1, and astands for the lattice constant.
For a≪1, the field
ˆ
Uij can be approximated by the
exponential expansion
ˆ
Uij ≈eiaϵAm
kðriÞ
ˆ
lm
¼1þiaϵAm
kðriÞˆ
lm−a2
2Am
kðriÞAn
kðriÞˆ
lm
ˆ
lnþOða3Þ;
ð2:8Þ
where 1is the unit matrix and
ˆ
lmare the generators of
SUð3Þin the adjoint representation, i.e., ðˆ
lmÞpq ¼ifmpq.
In addition, since the model (2.1) is ferromagnetic, it is
natural to assume that nearest-neighbor spins are oriented
in the almost same direction, which allows us to use the
Taylor expansion
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nmðrjÞ¼nmðriÞþaϵ∂knmðriÞþOða2Þ:ð2:9Þ
Replacing the sum over the lattice sites in Eq. (2.7) by the
integral a−2Rd2x, we obtain a continuum Hamiltonian,
except for a constant term, of the form
H¼J
8Zd2x½Trð∂kn∂knÞ−2iTrðAk½n;∂knÞ
−Trð½Ak;n2Þ;ð2:10Þ
where Ak¼Am
kλmand n¼nmλm. Similar to its SUð2Þ
counterpart expressed as Eq. (1.2), this Hamiltonian can
also be written as the static energy of an SUð3Þgauged CP2
NLσmodel
H¼J
8Zd2xTrðDknDknÞ;ð2:11Þ
where Dkn¼∂kn−i½Ak;nis the SUð3Þcovariant deriva-
tive. Since the Hamiltonian is given by the SUð3Þcovariant
derivative, Eq. (2.11) is invariant under the SUð3Þgauge
transformation
n→gng−1;A
k→gAkg−1þig∂kg−1;ð2:12Þ
where g∈SUð3Þ. Note that, however, since the
Hamiltonian (2.11) does not include kinetic terms for
the gauge field, like the Yang-Mills term, or the Chern-
Simons term, the gauge potential is just a background field,
not the dynamical one. We suppose that the gauge field is
fixed beforehand by the structure of a sample and give the
value by hand, like the SUð2Þcase. The gauge fixing allows
us to recognize the second term in Eq. (2.10) as a Lifshitz
invariant term.
We would like to emphasize that we do not deal with
Eq. (2.11) as a gauge theory. Rather, we deem it the CP2
NLσmodel with a Lifshitz invariant, and show the
existence of the exact and the numerical solutions. For
the baby skyrmion solutions we shall obtain, the color
field napproaches to a constant value n∞at spatial infinity
so that the physical space R2can be topologically com-
pactified to S2. Therefore, they are characterized by the
topological degree of the map n∶R2∼S2↦CP2given by
Q¼−i
32πZd2xεjkTrðn½∂jn;∂knÞ:ð2:13Þ
Combining with the assumption that the gauge is fixed, it is
reasonable to identify this quantity (2.13) with the topo-
logical charge in our model.1
III. EXACT SOLUTIONS OF THE SUð3Þ
GAUGED CP2NLσMODEL
In this section, we derive exact solutions of the
model with the Hamiltonian (2.11) supplemented by a
potential term. We first remark on the validity of the
variational problem. As discussed in Refs. [20,25] for
the SUð2Þcase, a surface term, which appears in the
process of variation, cannot be ignored if the physical space
is noncompact and the gauge potential Akdoes not vanish
at the spatial infinity like the DM term. This problem can
be cured by introducing an appropriate boundary term,
like [20]
HBoundary ¼∓4ρZd2xεjk∂jTrðnAkÞ;ð3:1Þ
where ρ¼J=8. Here the gauge potential Aksatisfies
½n∞;A
ji
2εjk½n∞;½n∞;A
k ¼ 0;ð3:2Þ
where n∞is the asymptotic value of nat spatial infinity.
Note that Eq. (3.2) corresponds to the asymptotic form of
the BPS equation, which we shall discuss in the next
subsection. Hence, all field configurations we consider in
this paper satisfy this equation automatically.
Since Eq. (3.1) is a surface term, it does not contribute to
the Euler-Lagrange equation, i.e., the classical Heisenberg
equation. Note that the solutions derived in the following
sections satisfy Derrick’s scaling relation with the boundary
term, which is obtained by keeping the background field Ak
intact under the scaling, i.e., E1þ2E0¼0, where E1
denotes the energy contribution from the first derivative
terms including the boundary term (3.1) and E0from no
derivative terms.
A. BPS solutions
Recently, it has been proved that the SUð2Þgauged
CP1NLσmodel (1.2) possesses BPS solutions in the
presence of a particular potential term [20,24]. Here, we
show that BPS solutions also exist in the SUð3Þgauged
CP2model with a special choice of the potential term,
which is given by
Hpot ¼4ρZd2xTrðnF12Þ;ð3:3Þ
where Fjk ¼∂jAk−∂kAj−i½Aj;A
k. As we shall see in
the next subsection, the potential term can possess a
natural physical interpretation for some background
gauge field. It follows that the Hamiltonian we study here
reads
1If one extends the model (2.11) with a dynamical gauge field,
the topological charge is defined by the SUð3Þgauge invariant
quantity which is directly obtained by replacing the partial
difference in Eq. (2.13) with the covariant derivative.
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H¼ρZd2xTrðDknDknÞ4ρZd2xTrðnF12Þ
∓4ρZd2xεjk∂jTrðnAkÞ;ð3:4Þ
where the double-sign corresponds to that of Eq. (3.1).
First, let us show that the lower energy bound of
Eq. (3.4) is given by the topological charge (2.13). The
first term in Eq. (3.4) can be written as
ρZd2xTrðDknDknÞ
¼ρ
2Zd2xTrðDknDknÞþi
22
Trð½n;D
kn2Þ
¼ρ
2Zd2xTrDjni
2εjk½n;D
kn2
iρ
2Zd2xεjkTrðn½Djn;D
knÞ
≥iρ
2Zd2xεjkTrðn½Djn;D
knÞ:ð3:5Þ
It follows that the equality is satisfied if
Djni
2εjk½n;D
kn¼0;ð3:6Þ
which reduces to Eq. (3.2) at the spatial infinity. Therefore,
one obtains the lower bound of the form
H≥ρ
2Zd2x½iεjkTrðn½Djn;D
knÞ þ 8TrðnF12Þ
−8εjk∂jTrðnAkÞ
¼iρ
2Zd2xεjkTrðn½∂jn;∂knÞ
¼∓16πρQ; ð3:7Þ
where the corresponding BPS equation is given by
Eq. (3.6). Note that, unlike the energy bound of the
CPNself-dual solutions [7,27], the energy bound (3.7)
can be negative, and it is not proportional to the absolute
value of the topological charge.
As is often the case in two-dimensional BPS equations
[7,20], solutions can be best described in terms of the
complex coordinates z¼x1ix2. Further, we make use
of the associated differential operator and background field
defined as ∂¼1
2ð∂1∓i∂2Þand A¼1
2ðA1∓iA2Þ.
Then, the BPS equation (3.6) can be written as
Dn−1
2½n;D
n¼0:ð3:8Þ
Similar to the SUð2Þcase [20], Eq. (3.8) with a plus sign
can be solved if the background field has the form
Aþ¼ig−1∂þg; ð3:9Þ
where g∈SLð3;CÞ. Note that Eq. (3.9) is not necessarily a
pure gauge. Similarly, Eq. (3.8) with the minus sign on the
right-hand side can be solved if A−¼ig−1∂−g. For the
background field (3.9), one finds that the BPS equa-
tion (3.8) is equivalent to
∂þ
˜
n−1
2½˜
n;∂þ
˜
n¼0;˜
n¼gng−1;ð3:10Þ
because, under the SLð3;CÞgauge transformation, the
fields are changed as n→
˜
n¼gng−1and Aþ→
˜
Aþ¼
gAþg−1þig∂þg−1¼0. In the following, we only consider
Eq. (3.9) to simplify our discussion.
In order to solve the equation (3.10), we introduce a
tractable parametrization of the color field
n¼−2
ffiffiffi
3
pUλ8U†;ð3:11Þ
with U¼ðY1;Y2;ZÞ∈SUð3Þ, where Zis the continuum
counterpart of the vector Zin Eq. (2.3) and Y1,Y2are
vectors forming an orthonormal basis for C3with Z.Upto
the gauge degrees of freedom, the components Yican be
written as
Y1¼ð−¯
Z3;0;¯
Z1ÞT
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−jZ2j2
p;Y2¼ð−¯
Z2Z1;1−jZ2j2;−¯
Z2Z3ÞT
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−jZ2j2
p:
ð3:12Þ
Therefore, the vector Zfully defines the color field n.
Accordingly, we can write
˜
n¼−2
ffiffiffi
3
pWλ8W−1;ð3:13Þ
with W¼gU ¼ðW1;W2;W3Þ∈SLð3;CÞ. It follows that
the field Z, which is the fundamental field of the model, is
given by Z¼g−1W3. Substituting the field (3.13) into the
equation (3.10), one finds that Eq. (3.10) reduces to the
coupled equation
W−1
1∂þW3¼0
W−1
2∂þW3¼0ð3:14Þ
with W−1
l¼Y†
lg−1(l¼1, 2). Since the three vectors
fY1;Y2;Zgform an orthonormal basis, Eq. (3.14) implies
∂þW3¼βW3where the function βis given by
β¼βW−1
3W3¼W−1
3∂þW3. Therefore, the Eq. (3.10) is
solved by any configuration satisfying
DþW3¼0;ð3:15Þ
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where DþΦ¼∂þΦ−ðΦ−1∂þΦÞΦfor arbitrary nonzero
vector Φ. Moreover, we write
W3¼ffiffiffiffiffiffiffiffiffiffiffi
jW3j2
qw;ð3:16Þ
where wis a three component unit vector, i.e.,
jwj2¼w†w¼1. Then, Eq. (3.15) can be reduced to
Dþw≡∂
μw−ðw†∂μwÞw¼0;ð3:17Þ
which is the very BPS equation of the standard CP2NLσ
model. Thus, a general solution of Eq. (3.15), up to the
gauge degrees of freedom, is given by
w¼P
jPj;P¼ðP1ðz−Þ;P
2ðz−Þ;P
3ðz−ÞÞT;ð3:18Þ
where Phas no overall factor, and Pais a polynomial in z−.
Therefore, we finally obtain the solution for the Zfield
Z¼g−1W3¼χg−1w¼χg−1P;ð3:19Þ
where χis a normalization factor.
B. Properties of the BPS solutions
As the BPS bound (3.7) indicates, the lowest energy
solution among Eq. (3.19) with a given background
function gpossesses the highest topological charge. In
terms of the explicit calculation of the topological charge,
we discuss the conditions for the lowest energy solutions.
The topological charge (2.13) can be written in terms
of Zas
Q¼−i
2πZd2xεijðDiZÞ†DjZ:ð3:20Þ
We employ the constant background gauge field Aþfor
simplicity. Then, the matrix gin Eq. (3.9) becomes
g¼exp ð−iAþzþÞ;ð3:21Þ
so that the components of g−1are given by power series in
zþ. It allows us to write Eq. (3.20) as a line integral along
the circle at spatial infinity
Q¼1
2πZS1
∞
C; ð3:22Þ
with C¼−iZ†dZ[27,38], since the one-form Cbecomes
globally well defined. To evaluate the integral in Eq. (3.22),
we write explicitly
Z¼χ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jP1j2þjP2j2þjP3j2
pX
a
0
B
B
@
g−1
1aðzþÞPaðz−Þ
g−1
2aðzþÞPaðz−Þ
g−1
3aðzþÞPaðz−Þ
1
C
C
A
;
ð3:23Þ
where g−1
ab is the ða; bÞcomponent of the inverse matrix g−1.
Let Na(Kab) be the highest power in Pa(g−1
ab ). Note that
though g−1
ab are formally represented as power series in zþ,
the integers Kba are not always infinite; especially, if a
positive integer power of Aþis zero, all of Kba become
finite because g−1reduces to a polynomial of finite degree
in zþ. Using the plane polar coordinates fr; θg, one can
write g−1
ba ðzþÞPaðz−Þ∼rNaþKba exp½−iðNa−KbaÞθat the
spatial boundary and find that only the components of the
highest power in rcontribute to the integral (3.22). Since
we are interested in constructing topological solitons, we
consider the case when the physical space R2can be
compactified to the sphere S2, i.e., the field Ztakes some
fixed value on the spatial boundary. Such a compactifica-
tion is possible if there is only one pair fNa;K
baggiving
the largest sum NaþKba or any pairs fNa;K
bag, sharing
the largest sum, have the same value of the difference. For
such configurations, the topological charge is given by
Q¼−NaþKba;ð3:24Þ
where the combination fNa;K
bagyields the largest sum
among any pairs fNc;K
dcg. This equation (3.24) indicates
that the highest topological charge configuration is given
by the choice Na¼0for a particular value of awhich gives
the biggest Kba.
We are looking for the lowest energy solutions with an
explicit background field. As a particular example, let us
consider
A1¼κðλ1þλ4þλ5Þ;A
2¼κðλ2þλ4−λ5Þ;ð3:25Þ
where κis a constant. Clearly, this choice yields the
potential term
FIG. 1. Topological charge density of the axial symmetric
solution (3.28) with κ¼1.
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V¼4TrðnF12Þ¼−16 ffiffiffi
3
pκ2n8¼16κ2ð2−3hðS3Þ2iÞ;
ð3:26Þ
which can be interpreted as an easy-axis anisotropy, or
quadratic Zeeman term, which naturally appears in con-
densed matter physics. In this case, the solution (3.19) can
be written as
Z¼χ
ffiffiffiffi
Δ
p0
B
B
@
P1ðz−Þþ ffiffiffi
2
pκzþeπi
4P3ðz−Þ
P2ðz−ÞþiκzþP1ðz−Þþκ2z2
þ
ffiffi2
pe3πi
4P3ðz−Þ
P3ðz−Þ
1
C
C
A
:
ð3:27Þ
Therefore, the solution with the highest topological charge
is given by P1¼α1,P2¼α2z−þα3with αi∈C, and P3
being a nonzero constant. Choosing P1¼P2¼0, one can
obtain the axially symmetric solution
Z¼1
ffiffiffiffi
Δ
p0
B
B
@
ffiffiffi
2
pκzþeπi
4
κ2z2
þ
ffiffi2
pe3πi
4
1
1
C
C
A
;Δ¼1þ2κ2zþz−þκ4
2z2
þz2
−;
ð3:28Þ
which possesses the topological charge Q¼2. Note that
this configuration also satisfies the BPS equation of
the pure CP2NLσmodel [26,27,31]. Figure 1shows the
distribution of the topological charge (3.20) of this solution
(3.28) with κ¼1. We find that the topological charge
density has a single peak, although higher charge topo-
logical solitons with axial symmetry are likely to possess a
volcano structure, see, e.g., Ref. [39]. These highest charge
solutions give the asymptotic values at spatial infinity of the
color field
ðn1
∞;n
2
∞;n
3
∞;n
4
∞;n
5
∞;n
6
∞;n
7
∞;n
8
∞Þ
¼ð0;0;−1;0;0;0;0;1=ffiffiffi
3
pÞ:ð3:29Þ
It indicates that ntakes the vacuum value in the Cartan
subalgebra of SUð3Þ. Hence, the vacuum of the model
corresponds to a spin nematic, i.e., hS1i¼hS2i¼hS3i¼0
and hðS2Þ2i¼0;hðS1Þ2i¼hðS3Þ2i¼1. Unlike the pure
CP2model, there is no degeneracy between the spin
nematic state and ferromagnetic state in our model because
the SUð3Þglobal symmetry is broken. As shown in Fig. 2,
the spin nematic state is partially broken around the soliton
because the expectation values hSaibecome finite. Figure 3
shows that hðSaÞ2iof the solution (3.28) are axially
symmetric, although the expectation values hSaihave
angular dependence.
C. Exact solutions off the BPS point
Note that the Hamiltonian (1.1) with B¼2Aadmits
closed-form analytical solutions [40]. Further, the CP1BPS
truncation corresponds to the restricted choice of the
parameters, B¼2A¼κ2. The relation B¼2Ais referred
to as the solvable line, whereas the restriction B¼2A¼κ2
is called the BPS point [25]. Here we show that similar
restrictions occur in our model. For this purpose, we
consider the generalized Hamiltonian
H¼HDþHLþHBoundary þν2Hani þμ2Hpot;ð3:30Þ
where νand μare real coupling constants. Here, HD
indicates the CP2Dirichlet term, i.e., the first term in the
right-hand side (r.h.s.) of Eq. (2.10), and HLdoes the
Lifshitz invariant term which is the second term of that.
Explicitly, these and other terms read
HD¼ρZd2xTrð∂kn∂knÞ;ð3:31Þ
FIG. 2. The expectation values hSaifor the solution (3.28) with κ¼1.
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HL¼−2iρZd2xTrðAk½n;∂knÞ;ð3:32Þ
Hani ¼−ρZd2x½Trð½Ak;n2Þ−Trð½Ak;n∞2Þ;ð3:33Þ
Hpot ¼4ρZd2x½TrðnF12Þ−Trðn∞F12Þ;ð3:34Þ
where Akis a constant background field, as before. Finally,
the boundary term HBoundary is defined by Eq. (3.1) with the
negative sign in the r.h.s., the same as before. Note that we
also introduced constant terms in Eqs. (3.33) and (3.34) in
order to guarantee the finiteness of the total energy. Clearly,
the Hamiltonian (3.30) is reduced to Eq. (3.4) as we
set ν2¼μ2¼1.
The existence of exact solutions of the Hamiltonian
(3.30) with ν2¼μ2can be easily shown if we rescale
the space coordinates as ⃗
x→r0
⃗
x, where r0is a positive
constant, while the background gauge field Akremains
intact. By rescaling, the Hamiltonian (3.30) becomes
H¼HDþr0ðHLþHBoundaryÞþr2
0ðν2Hani þμ2HpotÞ:
ð3:35Þ
Setting ν2¼μ2and choosing the scale parameter r0¼ν−2,
one gets
Hr0¼ν−2
ν2¼μ2¼HDþν−2ðHLþHBoundary þHani þHpotÞ:
ð3:36Þ
Notice that since the solutions (3.19) with Pibeing
arbitrary constants are holomorphic maps from S2to
CP2, they satisfy not only the variational equations
δHν2¼μ2¼1¼0, but also the equations δHD¼0, where δ
denotes the variation with respect to nwith preserving
the constraint (2.6). Therefore, the solutions also satisfy
the equations δHr0¼ν−2
ν2¼μ2¼0. This implies that, in the limit
μ2¼ν2, the Hamiltonian (3.30) supports a family of exact
solutions of the form
Zðν2Þ¼exp ½iν2Aþzþc;ð3:37Þ
where cis a three-component complex unit vector.
Since the solution (3.37) is a BPS solution of the pure
CP2model with the positive topological charge Q, one gets
HD½Zðν2Þ ¼ 16πρQ. In addition, the lower bound at
the BPS point (3.7) indicates that Hν2¼μ2¼1½Zðν2¼1Þ ¼
−16πρQ. Combining these bounds, we find that the total
energy of the solution (3.37) is given by
Hν2¼μ2½Zðν2Þ ¼ 16πρ1−2
ν2Q: ð3:38Þ
Since the energy becomes negative if ν2<2, we can expect
that for small values of the coupling ν2, the homogeneous
vacuum state becomes unstable, and then separated 2D
skyrmions (or a skyrmion lattice) emerge as a ground state.
IV. NUMERICAL SOLUTIONS
A. Axial symmetric solutions
In this section, we study baby skyrmion solutions of the
Hamiltonian (3.30) with various combinations of the
coupling constants. Apart from the solvable line, no exact
solutions could find analytically, and then we have to solve
the equations numerically. Here, we restrict ourselves to the
case of the background field given by Eq. (3.25).
For the background field (3.25), by analogy with the case
of the single CP1magnetic skyrmion solution, we can look
for a configuration described by the axially symmetric
ansatz
Z¼0
B
@
sin FðrÞcos GðrÞeiΦ1ðθÞ
sin FðrÞsin GðrÞeiΦ2ðθÞ
cos FðrÞ
1
C
A;ð4:1Þ
FIG. 3. The expectation values hðSaÞ2ifor the solution (3.28) with κ¼1.
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where Fand G(Φ1and Φ2) are real functions of the plane
polar coordinates r(θ).
The exact solution on the solvable line ν2¼μ2with axial
symmetry can be written in terms of the ansatz with the
functions
F¼tan−1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ν4κ2r2þν8κ4r4
2
r;G¼tan−1ν2κr
2;
Φ1¼θþπ
4;Φ2¼2θþ3π
4:ð4:2Þ
Further, the solution (3.28) is given by Eq. (4.2)
with ν2¼1. This configuration is a useful reference
point in the configuration space as we discuss below
some properties of numerical solutions in the extended
model (3.30).
For our numerical study, it is convenient to introduce
the energy unit 8ρand the length unit κ−1, in order
to scale the coupling constants. Then, the rescaled
components of the Hamiltonian with the ansatz (4.1)
become
HD¼Zd2xF02þsin2FG02
þsin2F
r2f_
Φ2
1cos2Gþ_
Φ2
2sin2Gg
−sin4F
r2ð_
Φ1cos2Gþ_
Φ2sin2GÞ2;ð4:3Þ
HL¼−2Zd2x
rffiffiffi
2
pcos θþπ
4−Φ1rcos GF0−sin 2Fsin GG0
2þsin 2Fcos G
_
Φ1
2
−sin 2Fsin2Fcos Gðcos2G_
Φ1þsin2G_
Φ2Þ−sin ðθþΦ1−Φ2Þrsin2FG0þ1
2sin2Fsin 2Gð_
Φ1þ_
Φ2Þ
−sin4Fsin 2Gðcos2G_
Φ1þsin2G_
Φ2Þ;ð4:4Þ
Hani ¼1
2Zd2x16sin2Fcos2Gcos2F−1
ffiffiffi
2
pcos 2Φ1−Φ2þπ
4sin 2Fsin Gþsin2Fsin G2
þsin22Fð1þ2sin2GÞþ8ðcos2F−cos2Gsin2FÞ2þ4cos22Gsin4F−4;ð4:5Þ
Hpot ¼2Zd2xð1−ffiffiffi
3
pn8Þ¼6Zd2xcos2F; ð4:6Þ
where the prime 0and the dot _stands for the derivatives with respect to the radial coordinate rand angular coordinate θ,
respectively. The system of corresponding Euler-Lagrange equations for Φican be solved algebraically for an arbitrary set
of the coupling constants, and the solutions are
Φ1¼θþπ
4;Φ2¼2θþ3π
4þmπ;ð4:7Þ
where mis an integer. Without loss of generality, we choose m¼0by transferring the corresponding multiple windings of
the phase Φ2to the sign of the profile function G. Then, the system of the Euler-Lagrange equations for the profile functions
with the phase factor (4.7) reads
δHD
δFþδHL
δFþν2δHani
δFþμ2δHpot
δF¼0;
δHD
δGþδHL
δGþν2δHani
δGþμ2δHpot
δG¼0;ð4:8Þ
with
δHD
δF¼2rF00 þ2F0−sin 2FrG02þ1þ3sin2G
r−2sin2F
rð1þsin2GÞ2;ð4:9Þ
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δHL
δF¼−22ffiffiffi
2
psin2Ff−rsin GG0þcos Gþcos Gð1þsin2GÞð4cos2F−1Þg
−rsin 2FG0−3
2sin 2Fsin 2Gþ4cos Fsin3Fsin 2Gð1þsin2GÞ;ð4:10Þ
δHani
δF¼2r½4ffiffiffi
2
psin Gcos2Gsin2Fð3−4sin2FÞ−4cos Fsin3Fcos22G
þ4sin 2Ffcos2F−sin2Fcos2Gð1þsin2GÞg −sin 2Fcos 2Fð1þ2sin2GÞ;ð4:11Þ
δHpot
δF¼6rsin 2F; ð4:12Þ
δHD
δG¼2rsin F2G00 þ2rsin 2FF0G0þ2sin2FG0−sin2Fsin 2G
rf3−2sin2Fð1þsin2GÞg;ð4:13Þ
δHL
δG¼−2½ffiffiffi
2
psin2Fsin Gf2rF0þsin 2Fð1−3sin2GÞg
þrsin 2FF0þsin2Fð1−3cos 2GÞþsin4Fð1þ3cos 2G−2cos22GÞ;ð4:14Þ
δHani
δG¼r½8ffiffiffi
2
pcos Fsin3Fcos Gð1−3sin2GÞþ16sin4Fcos3Gsin G−sin22Fsin 2G;ð4:15Þ
δHpot
δG¼0:ð4:16Þ
We solve the equations for ν2≠μ2numerically with the
boundary condition
Fð0Þ¼Gð0Þ¼0;lim
r→∞FðrÞ¼lim
r→∞GðrÞ¼π=2;ð4:17Þ
which the exact solution (4.2) satisfies. This vacuum
corresponds to the spin nematic state (3.29).
Let us consider the asymptotic behavior of the solutions
of the equations (4.8). Near the origin, the leading terms in
the power series expansion are
F≈cFr; G ≈cGr; ð4:18Þ
where cFand cGare some constants implicitly depending
on the coupling constants of the model. To see the behavior
of solutions at large r, we shift the profile functions as
F¼π
2−F;G¼π
2−G:ð4:19Þ
Then, one obtains linearized asymptotic equations on the
functions Fand Gof the forms
F00 þF0
r−4F
r2þ2ffiffiffi
2
pG0−G
r−2ðν2þ3μ2ÞF¼0;
G00 þG0
r−G
r2−2ffiffiffi
2
pF0þ2F
r¼0:
ð4:20Þ
FIG. 4. Plot of the profile functions fF; Gg(left) and the topological charge density (right) of numerical solutions for changing the
coupling constant ν2at μ2¼1.5. The gray line indicates the quantities of the exact solution (4.2) on the solvable line.
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Unfortunately, Eqs. (4.20) may not support an analytical
solution. However, these equations imply that the
asymptotic behavior of the profile functions is similar to
that of the functions (4.2), by a replacement ν2κwith
ðν2þ3μ2Þ=4. Indeed, the asymptotic equations (4.20)
depend on such a combination of the coupling constants,
and there may exist an exact solution on the solvable line
with the same character of asymptotic decay as the
localized soliton solution of the equation (4.8).
To implement a numerical integration of the coupled
system of ordinary differential equations (4.8), we intro-
duce the normalized compact coordinate X∈ð0;1via
r¼1−X
X:ð4:21Þ
The integration was performed by the Newton-Raphson
method with the mesh point NMESH ¼2000.
In Fig. 4, we display some set of numerical solutions for
different values of the coupling ν2at μ2¼1.5and their
topological charge density Qdefined through Q¼
2πRrQdr. The solutions enjoy Derrick’s scaling relation
and possess a good approximated value of the topological
charge, as shown in Table I. One observes that as the value
of the coupling ν2becomes relatively small, the function G
is delocalizing while the profile function Fis approaching
its vacuum value everywhere in space except for the origin.
This is an indication that any regular nontrivial solution
does not exist ν2¼0.
B. Asymptotic behavior
Asymptotic interaction of solitons is related to the
overlapping of the tails of the profile functions of well-
separated single solitons [3]. Bounded multisoliton con-
figurations may exist if there is an attractive force between
two isolated solitons.
Considering the above-mentioned soliton solutions of
the gauged CP2NLσmodel, we have seen that the exact
solution (4.2) has the same type of asymptotic decay as any
solution of the general system (4.8). Therefore, it is enough
to examine the asymptotic force between the solutions
on the solvable line (4.2) to understand whether or not
the Hamiltonian (3.30) supports multisoliton solutions of
higher topological degrees. Thus, without loss of general-
ity, we can set μ2¼ν2.
Following the approach discussed in Ref. [3], let us
consider a superposition of the two exact solutions
above. This superposition is no longer a solution of the
Euler-Lagrange equation, except for in the limit of infinite
separation, because there is a force acting on the solitons.
The interaction energy of two solitons can be written as
EintðRÞ¼Hsp ðRÞ−2Hexact;ð4:22Þ
where HspðRÞis the energy of two BPS solitons separated
by some large but finite distance Rfrom each other, and
Hexact stands for the static energy of a single exact solution.
Notice that the lower bound of the Hamiltonian (3.30) with
μ2¼ν2is given
H¼ν−2Hν2¼μ2¼1þð1−ν2ÞHD≥2πð1−2ν−2ÞQ; ð4:23Þ
where the equality is enjoyed only by holomorphic sol-
utions. Therefore, we immediately conclude
HspðRÞ≥2Hexact ;ð4:24Þ
where the equality is satisfied only at the limit R→∞.
It follows that the interaction energy is always positive
for finite separation, and the interaction is repulsive. Since
the exact solution has the topological charge Q¼2,it
implies that there are no isolated soliton solutions with the
topological charge Q≥4in this model. Note that, however,
as the BPS solution (3.19) suggests, there can exist soliton
solutions with an arbitrary negative charge, which are
topological excited states on top of the homogeneous
vacuum state.
V. CONCLUSION
In this paper, we have studied two-dimensional sky-
rmions in the CP2NLσmodel with a Lifshitz invariant term
which is an SUð3Þgeneralization of the DM term. We have
shown that the SUð3Þtilted FM Heisenberg model turns out
to be an SUð3Þgauged CP2NLσmodel in which the term
linear in a background gauge field can be viewed as a
Lifshitz invariant. We have found exact BPS-type solutions
TABLE I. The Hamiltonian and topological charge for the numerical solutions with μ2¼1.5where “Derrick”denotes the value
ðHLþHBoundaryÞ=ðν2Hani þμ2Hpot Þ, which is expected to be −2by the scaling argument. For ν2¼1.5, we used the exact solution (4.2)
so that the Derrick and topological charge for ν2¼1.5are exact values.
ν2HH
DHLν2Hani μ2Hpot HBoundary Derrick Q
0.1 −117.47 13.51 −136.48 125.49 5.67 −125.67 −2.00 2.00
0.3 −34.02 13.41 −53.60 41.37 6.69 −41.89 −1.99 2.00
0.8 −8.46 13.06 −29.37 14.73 8.82 −15.71 −1.91 2.00
1.5 −4.19 12.57 −16.76 1.09 15.66 −16.76 −22
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of the gauged CP2model in the presence of a potential term
with a specific value of the coupling constant. The least
energy configuration among the BPS solutions has been
discussed. We have reduced the gauged CP2model to the
(ungauged) CP2model with a Lifshitz invariant by choos-
ing a background gauge field. In the reduced model, we
have constructed an exact solution for a special combina-
tion of coupling constants called the solvable line and
numerical solutions for a wider range of them.
For numerical study, we chose the background field,
generating a potential term that can be interpreted as
the quadratic Zeeman term or uniaxial anisotropic term.
One can also choose a background field generating the
Zeeman term; if the background field is chosen as A1¼
−κλ7and A2¼κλ5, the associated potential term is
proportional to hS3i. The Euler-Lagrange equation for
the extended CP2model with this background field is
not compatible with the axial symmetric ansatz (4.1).
Therefore, a two-dimensional full simulation is required
to obtain a solution with this background field. This
problem, numerical simulation for nonaxial symmetric
solutions in the CP2model with a Lifshitz invariant, is
left to future study. In addition, the construction of a CP2
skyrmion lattice is a challenging problem. The physical
interpretation of the Lifshitz invariants is also an impor-
tant future task. The microscopic derivation of the SUð3Þ
tilted Heisenberg model [21] may enable us to understand
the physical interpretation and physical situation where
the Lifshitz invariant appears. Other future work would
be the extension of the present study to the SUð3Þ
antiferromagnetic Heisenberg model where soliton or
sphaleron solutions can be constructed [41–43].
We restricted our analysis on the case that the additional
potential term μ2Hpot is balanced or dominant against the
anisotropic potential term ν2Hani, i.e., ν2≤μ2. We expect
that a classical phase transition occurs outside of the
condition, and it causes instability of the solution. At the
moment, the phase structure of the model (3.30) is not clear,
and we will discuss it in our subsequent work.
Moreover, it has been reported that in some limit of a
three-component Ginzburg-Landau model [44,45], and of a
three-component Gross-Pitaevskii model [46,47], their
vortex solutions can be well described by planar CP2
skyrmions. We believe that our result provides a hint to
introduce a Lifshitz invariant to the models, and that our
solutions find applications not only in SUð3Þspin systems
but also in superconductors and Bose-Einstein condensates
described by the extended models, including the Lifshitz
invariant.
ACKNOWLEDGMENTS
This work was supported by JSPS KAKENHI Grants
No. JP17K14352, No. JP20K14411, and JSPS Grant-in-
Aid for Scientific Research on Innovative Areas “Quantum
Liquid Crystals”(KAKENHI Grant No. JP20H05154).
Y. S. gratefully acknowledges support by the Ministry of
Education of Russian Federation, project FEWF-2020-
0003. Y. Amari would like to thank Tokyo University of
Science for its kind hospitality.
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127 (1961).
[2] T. Skyrme, A unified field theory of mesons and baryons,
Nucl. Phys. 31, 556 (1962).
[3] N. Manton and P. Sutcliffe, Topological Solitons,
Cambridge Monographs on Mathematical Physics
(Cambridge University Press, Cambridge, England, 2004).
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