Isolated Skyrmions in the $CP^{2}$ nonlinear $\sigma$-model with a Dzyaloshinskii-Moriya type interaction

In the 1960s, Skyrme introduced a (3+1)-dimensional $O(4)$ nonlinear (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 identified 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 NL$\sigma$-model, the so-called baby Skyrme model [4, 5, 6]. This (2+1)-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 [7, 4, 5]. Then, it was pointed out that Skyrmion configurations naturally arise in condensed matter systems with intrinsic and induced chirality [8, 9, 10, 11, 12]. These baby Skyrmions, often referred to as magnetic Skyrmions, were experimentally observed in non-centrosymmetric or chiral magnets [13, 14, 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 Dzyaloshinskii-Moriya (DM) interaction [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

where $\bm{m}(\bm{r})=\left(m^{1},m^{2},m^{3}\right)$ is a three component unit magnetization vector which corresponds to the spin expectation value at position $\bm{r}$. The first term in Eq. (1.1) is the continuum limit of the Heisenberg exchange interaction, i.e., the kinetic term of the $O(3)$ NL$\sigma$-model, 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 $A\left\{|\bm{m}|^{2}+\left(m^{3}\right)^{2}\right\}$ represents the uniaxial anisotropy.

It is remarkable that in the limiting case $A=\kappa^{2}/2J,B=0$, the Hamiltonian (1.1) can be written as the static version of the $SU(2)$ gauged $O(3)$ NL$\sigma$-model [19, 20]

with a background gauge field $\bm{A}_{1}=(-\kappa/J,\leavevmode\nobreak\ 0,\leavevmode\nobreak\ 0),\leavevmode\nobreak\ \bm{A}_{2}=(0,-\kappa/J,\leavevmode\nobreak\ 0)$. Though the DM term is usually introduced phenomenologically, a mathematical derivation of the Hamiltonian (1.2) with arbitrary $\bm{A}_{k}$ has 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

where the sum $\langle ij\rangle$ is taken over the nearest-neighbor sites, $S^{a}_{i}$ denotes the $a$-th component of spin operators at site $i$ and ${\cal W}_{i}\in SU(2)$. 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 $\bm{A}_{k}$ can 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 which 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 NL$\sigma$-model with various form of Lifshitz invariants which are mathematically derived by choice of the background field $\bm{A}_{k}$, 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 NL$\sigma$-model. 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 $CP^{1}$ complex field on the compactified domain space $S^{2}$ [6]. Further, there are various generalizations of this model; for example, two-dimensional $CP^{2}$ Skyrmions have been studied in the pure $CP^{2}$ NL$\sigma$-model [26, 27, 28] and in the Faddeev-Skyrme type model [29, 30].

Remarkably, the two dimensional $CP^{2}$ NL$\sigma$-model can be obtained as a continuum limit of the $SU(3)$ ferromagnetic (FM) Heisenberg model [31, 32] on a square lattice defined by the Hamiltonian

where $J$ is a positive constant, and $T^{m}_{i}$ ($m=1,...,8$) stand for the $SU(3)$ spin operators of the fundamental representation at site $i$ satisfying the commutation relation

Here, the structure constants are given by $f_{lmn}=-\frac{i}{2}\mathrm{Tr}\left(\lambda_{l}\left[\lambda_{m},\lambda_{n}\right]\right)$, where $\lambda_{m}$ are the usual Gell-Mann matrices.

The $SU(3)$ FM 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 constants, so-called FM $SU(3)$ point, see, e.g., Ref. [35]. The $SU(3)$ spin operators can be defined in terms of the $SU(2)$ spin operators $S^{a}$ ($a=1,2,3$) as

Using the $SU(2)$ commutation relation $\left[S^{a}_{i},S^{b}_{i}\right]=i\varepsilon_{abc}S^{c}_{i}$ where $\varepsilon_{abc}$ denotes the anti-symmetric tensor, one can check that the operators (1.6) satisfy the $SU(3)$ commutation relation (1.5).

In the present paper, we study baby Skyrmion solutions of an extended $CP^{2}$ NL$\sigma$-model composed of the $CP^{2}$ 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 $SU(3)$ tilted Heisenberg model. Below we will implement this approach in our derivation of the Lifshitz invariant. In the extended $CP^{2}$ NL$\sigma$-model, 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 $SU(3)$ gauged $CP^{2}$ NL$\sigma$-model from the $SU(3)$ tilted Heisenberg model. Similar to the $SU(2)$ case described as Eq. (1.2), the term linear in a background field can be viewed as a Lifshitz invariant term. In Sec. 3, we study exact Skyrmionic solutions of the $SU(3)$ gauged $CP^{2}$ NL$\sigma$-model 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. 4. Our conclusions are given in Sec. 5.

To find Lifshitz invariant terms relevant for the $CP^{2}$ NL$\sigma$-model, we begin to derive an $SU(3)$ gauged $CP^{2}$ NL$\sigma$-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 non-dynamical background gauge potential of the gauged $CP^{2}$ model.

Following the procedure to obtain a gauged NL$\sigma$-model from a spin system, as discussed in Ref. [19], we consider a generalization of the $SU(3)$ Heisenberg model defined by the Hamiltonian

where $\hat{U}_{ij}$ 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 $SU(3)$ group in the adjoint representation. As in the $SU(2)$ case [19], the field $\hat{U}_{ij}$ 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 $\tilde{T}^{m}_{i}={\cal W}_{i}T^{m}_{i}{\cal W}^{-1}_{i}$ where ${\cal W}_{i}\in SU(3)$, because one can write ${\cal W}_{j}T^{m}_{j}{\cal W}^{-1}_{j}=(R_{j})_{mn}T_{j}^{n}$ where $R_{j}$ is an element of $SU(3)$ in the adjoint representation. Clearly, $\hat{U}_{ij}=R_{i}^{\rm T}R_{j}$, where $\rm T$ 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

where $\left|{m}\right>=\left|{S=1,m}\right>$ ($m=0$, $\pm 1$). In terms of the Cartesian basis, an arbitrary spin-1 state at a site $j$ can be expressed as a linear combination $\left|{Z}\right>_{j}=Z^{a}(\bm{r}_{j})\left|{x^{a}}\right>_{j}$ where ${\bm{r}}_{j}$ stands for the position of the site $j$, and $\bm{Z}=\left(Z^{1},Z^{2},Z^{3}\right)^{\rm T}$ is a complex vector of unit length [36, 31]. Since the state $\left|{Z}\right>_{j}$ satisfies an over-completeness relation, one can obtain the classical Hamiltonian using the state

Since $\bm{Z}$ is normalized and has the gauge degrees of freedom corresponding to the overall phase factor multiplication, it takes values in $S^{5}/S^{1}\approx CP^{2}$. In terms of the basis (2.2), the $SU(3)$ spin operators can be defined as

where $\lambda_{m}$ is the $m$-th component of the Gell-Mann matrices. One can check that they satisfy the $SU(3)$ commutation relation (1.5). The expectation values of the $SU(3)$ operators in the state (2.3) are given by

where $\bar{Z}^{a}$ denotes the complex conjugation of $Z^{a}$. In the context of QCD, the field $n^{m}$ is usually termed a color (direction) field [37]. The color field satisfies the constraints

where $d_{mpq}=\frac{1}{4}\mathrm{Tr}\left(\lambda_{m}\left\{\lambda_{p},\lambda_{q}\right\}\right)$. 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 $d_{mpq}n^{m}n^{p}n^{q}=8/9$.

In terms of the color field, the classical Hamiltonian is given by

Let us write the position of a site $j$ next to a site $i$ as $\bm{r}_{j}=\bm{r}_{i}+a\epsilon\bm{e}_{k}$ where $\bm{e}_{k}$ is the unit vector in the $k$-th direction, $\epsilon=\pm 1$, and $a$ stands for the lattice constant. For $a\ll 1$, the field $\hat{U}_{ij}$ can be approximated by the exponential expansion

where ${\mathbb{1}}$ is the unit matrix and $\hat{l}_{m}$ are the generators of $SU(3)$ in the adjoint representation, i.e., $(\hat{l}_{m})_{pq}=if_{mpq}$. 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

Replacing the sum over the lattice sites in Eq. (2.7) by the integral $\displaystyle{a^{-2}\int{\mathrm{d}}^{2}x}$, we obtain a continuum Hamiltonian, except for a constant term, of the form

where $A_{k}=A^{m}_{k}\lambda_{m}$ and $\mathfrak{n}=n^{m}\lambda_{m}$. Similar to its $SU(2)$ counterpart expressed as Eq. (1.2), this Hamiltonian can also be written as the static energy of an $SU(3)$ gauged $CP^{2}$ NL$\sigma$-model

where $D_{k}\mathfrak{n}={\partial}_{k}\mathfrak{n}-i\left[A_{k},\mathfrak{n}\right]$ is the $SU(3)$ covariant derivative. Since the Hamiltonian is given by the $SU(3)$ covariant derivative, Eq. (2.11) is invariant under the $SU(3)$ gauge transformation

where $g\in SU(3)$. 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 $SU(2)$ case. 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 $CP^{2}$ NL$\sigma$-model 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 $\mathfrak{n}$ approaches to a constant value $\mathfrak{n}_{\infty}$ at spatial infinity so that the physical space $\mathbb{R}^{2}$ can be topologically compactified to $S^{2}$. Therefore, they are characterized by the topological degree of the map $\mathfrak{n}:\mathbb{R}^{2}\sim S^{2}\mapsto CP^{2}$ given by

Combining with the assumption that the gauge is fixed, it is reasonable to identify this quantity (2.13) with the topological charge in our model¹¹1 If one extends the model (2.11) with a dynamical gauge field, the topological charge is defined by the $SU(3)$ gauge invariant quantity which is directly obtained by replacing the partial difference in Eq. (2.13) with the covariant derivative..

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 $SU(2)$ case, a surface term, which appears in the process of variation, cannot be ignored if the physical space is non-compact and the gauge potential $A_{k}$ does not vanish at the spatial infinity like the DM term. This problem can be cured by introducing an appropriate boundary term, like [20]

where $\rho=J/8$. Here the gauge potential $A_{k}$ satisfies

where $\mathfrak{n}_{\infty}$ is the asymptotic value of $\mathfrak{n}$ at 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 (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 $A_{k}$ intact under the scaling, i. e., $E_{1}+2E_{0}=0$ where $E_{1}$ denotes the energy contribution from the first derivative terms including the boundary term (3.1) and $E_{0}$ from no derivative terms.