Magnetic Skyrmions coupled to fermions

TL;DR

This paper explores how fermionic states localized on magnetic Skyrmions influence their interactions, revealing stable Skyrmion configurations mediated by fermion-induced dipole forces.

Contribution

It demonstrates the existence of localized fermionic modes on Skyrmions and shows these modes lead to stable Skyrmion systems through attractive interactions.

Findings

Sequences of fermionic modes localized on Skyrmions

Existence of stable Skyrmion configurations

Fermion-mediated attractive dipole interactions

Abstract

The index theorem implies that there are fermionic states localized on a soliton. Presence of these modes may significantly alter the pattern of interaction between the solitons. As a particular example we investigate the chiral magnetic Skyrmions coupled to spin-isospin fermions. It is shown that there are sequences of fermionic modes localized on the Skyrmions. We investigate the pattern of interaction between the soltions with localized modes and proved the existence of stable system of magnetic Skyrmions bounded by the strong attractive dipole interaction mediated by the chargeless fermionic modes.

Figures

Equations

$${\mathcal{H}}_s=\frac{J}{2}{\left(\nabla \mathbf{\varphi }\right)}^2+D\mathbf{\varphi }\cdot \left(\nabla \times \mathbf{\varphi }\right)-\mathbf{B}\cdot \mathbf{\varphi }.$$

$${\mathcal{H}}_f={\Psi }^{†}{\hat{\gamma }}^3\left(-i{\hat{\gamma }}^k{\partial }_k+e{\hat{\gamma }}^k{A}_k+m+g\mathbf{\tau }\cdot \mathbf{\varphi }\right)\Psi .$$

$${\hat{\gamma }}^3\left(-i{\hat{\gamma }}^k{\partial }_k+e{\hat{\gamma }}^k{A}_k+m+g\mathbf{\tau }\cdot \mathbf{\varphi }\right)\psi =\epsilon \psi ,$$

$$\Delta \mathbf{\varphi }-2\nabla \times \mathbf{\varphi }+{\mathbf{\varphi }}_{\infty }-g{\psi }^{†}\widehat{{\gamma }_3}\mathbf{\tau }\psi =0.$$

$$\begin{array}{rl}x\to -x,& {\varphi }_y\to -{\varphi }_y,\psi \to {\hat{\gamma }}^3{\psi }^{\ast },\epsilon \to -\epsilon ;\\ y\to -y,& {\varphi }_x\to -{\varphi }_x,\psi \to {\tau }_3{\psi }^{\ast },\epsilon \to -\epsilon ;\end{array}$$

$$\mathbf{\varphi }=\left(\sin f(r)\cos \left(n\phi +\delta \right),\sin f(r)\sin \left(n\phi +\delta \right),\cos f(r)\right).$$

$$\psi=\mathcal{N}\left(\begin{array}[]{c}v_{1}e^{il\varphi}\\ iv_{2}e^{i(l+n)\varphi}\\ u_{1}e^{i(l+1)\varphi}\\ iu_{2}e^{i(l+n+1)\varphi}\end{array}\right)\,,$$

$$f^{\prime \prime }+\frac{f}{r}-\sin f\left(B+\frac{\cos f-2r\sin f}{r^2}\right)+g\sin f\left(u_1^2+v_2^2-u_2^2-v_1^2\right)+2g\cos f\left(v_1{v}_2-u_1{u}_2\right)=0.$$

$$f^{\prime \prime }+\frac{f^{\prime }}{r}-f\left(B+\frac{1}{r^2}\right)=0,$$

$${\epsilon }_k^2=M^2+B\left(|e|(2k+1)\pm e\right),$$

$$\begin{array}{rl}{u}_1& \sim K_l(\sqrt{(g+m{)}^2-{\epsilon }^2}r),\\ u_2& \sim K_{l+1}(\sqrt{(g-m{)}^2-{\epsilon }^2}r),\\ v_1& \sim K_{l+1}(\sqrt{(g+m{)}^2-{\epsilon }^2}r),\\ v_2& \sim K_{l+2}(\sqrt{(g-m{)}^2-{\epsilon }^2}r),\end{array}$$

Full text

*Introduction.  * The Skyrme model in 3+1 dimensions Skyrme:1961vq is a prototype example of a field theory which supports topological solitons, the Skyrmions. It was first proposed around 1960s to describe the strong interactions of hadrons, in this framework the Skyrmions are identified with nucleons. Later, it has been shown by Witten Witten:1983tw that the Skyrme model can be considered as a low energy effective theory of QCD in the limit of a large number of quark colours. Such an effective theory can be constructed after integration over the fermionic degrees of freedom, see e.g. Diakonov:1987ty , the quarks do not appear as fundamental physical fields. In this paradigm the contribution of the sea quarks to the baryon energy can be determined by considering the normalizable bounded fermionic mode Kahana:1984dx ; Kahana:1984be ; Ripka:1985am , which, according to the index theorem, always exist in the spectrum of fermions coupled to the Skyrmion.

The pioneering work by Skyrme has opened new avenues for many areas of physics, similar soliton configurations exist in various non-linear physical systems, see e.g. Manton:2004tk ; Shnir2018 . In particular, there is a simplified 2+1 dimensional analogue of the original Skyrme theory, so called baby Skyrme model BB ; Leese:1989gj ; Bsk . This model finds various physical realizations, for example, planar Skyrmions occur in description of the quantum Hall effect Hall ; Lee:1990td , such a model also arise in description of ferromagnetic structures Bogdanov , or in chiral nematic and anisotropic fluids Smalukh1 ; Smalukh2 . Very recently, there has been rapidly increasing interest both in theoretical and experimental study of magnetic Skyrmions, because of their possible use as information carriers in future magnetic storage devices, see Nago . In such a context the baby Skyrme model is no longer considered as an effective low energy theory, constructed via integration over the fermionic degrees of freedom. Further, general topological arguments based on the index theorem are given to support existence of quasi-zero fermionic modes localized on the soliton (see e.g. Witten:1982fp ; Novikov:1984ac ). One can expect the localized fermionic modes may exist on magnetic Skyrmion Bogdanov which has been experimentally observed in chiral magnets with Dzyaloshinskii-Moriya (DM) interaction skexp .

Since the magnetic chiral Skyrmions always repel each other, there is no multisoliton solution in the model with DM interaction term. However, an additional interaction mediated by the fermions, may balance the scalar repulsion.

The Dirac electrons coupled to the planar magnetic Skyrmions were qualitatively discussed recently in Hurst:2014tza ; Andrikop . Such a system is of particular interest because it was argued that localization of the fermions on Skyrmions may lead to a new mechanism of non-conventional superconductivity in two-dimensional systems Baskaran:2011wg ; Lu ; Khi . However, the symmetries of the DM interaction term must be properly taken into account in investigation of the fermion - magnetic Skyrmion system. A detailed analysis of the problem, is yet missing. It remains a major challenge to construct complete set of solutions of the corresponding full system of dynamical equations, especially for multisoliton configurations, which do not possess rotational invariance.

A main objective of this Letter is to examine this system consistently, taking into account the backreaction of the fermions. In our numerical simulations we find the spectrum of the corresponding Hamiltonian and show that, indeed, there are various spin-isospin fermionic modes localized on a chiral magnetic Skyrmion. Here we show, for the first time to our knowledge, that the these modes give rise to additional attractive interaction between the magnetic Skyrmions, which form bounded states. Notably, we do not find localized solutions for the chargeless Dirac fermions coupled to the DM Skyrmions in two spatial dimensions.

*The model. * We consider the Hamiltonian, which describes a chiral magnetic planar system with a violation of the inversion symmetry and a strong spin-orbit coupling of the compound:

[TABLE]

These three terms correspond to the Heisenberg interaction, the DM interaction energy and the symmetry breaking Zeeman energy of the interaction with an external magnetic field ${\bf B}=B\hat{\bf z}$, respectively. Here $J$ is the magnetic stiffness constant and $D$ is the strength of the DM interaction. The magnetization vector ${\boldsymbol{\phi}}$ is constrained to the surface of a sphere of unit radius: ${\boldsymbol{\phi}}\cdot{\boldsymbol{\phi}}=1$. Since on the boundary the magnetization vector is directed along the external magnetic field, ${{{\boldsymbol{\phi}}}}_{\infty}=(0,0,1)$, the field ${\boldsymbol{\phi}}$ is the map $S^{2}$ to $S^{2}$. The corresponding topological invariant is $Q=-\frac{1}{4\pi}\int{\boldsymbol{\phi}}\cdot(\partial_{x}{\boldsymbol{\phi}}\times\partial_{y}{\boldsymbol{\phi}})~{}dxdy$, see e.g. Manton:2004tk ; Shnir2018 .

The Lagrangian for the fermions, which are coupled to the magnetic Skyrmions, is

[TABLE]

Note that the fermion field $\Psi$ is a spin-isospin spinor, the isospin matrices are defined as $\boldsymbol{\tau}=\mathbb{I}\otimes\boldsymbol{\sigma}$, whereas the spin matrices are $\hat{\gamma_{\mu}}=\gamma_{\mu}\otimes\mathbb{I}$. Here $\mathbb{I}$ is two-dimensional identity matrix, $\not{\partial}=\hat{\gamma_{\mu}}\partial^{\mu}$ and $\boldsymbol{\sigma}$ are the usual Pauli matrices. The last term in (2) represents the Hund coupling, the constant $g$ parameterizes the strength of the fermion-Skyrmion interaction, $m$ is the fermion mass and $e$ is the electromagnetic coupling. The vector potential of the external magnetic field is $A_{\mu}=\frac{B}{2}\left(0,-y,x\right)$. The total Hamiltonian of the coupled system $H=H_{s}+H_{f}$ depends on six parameters, $J,D,B,g,m$ and $e$. An appropriate rescaling of the coordinates, fields and coupling constants allows us to reduce the number of independent parameters to three: $g,m$ and $e$.

Hereafter we consider stationary configurations, ${\boldsymbol{\phi}}={\boldsymbol{\phi}}(x^{k}),\Psi=\psi(x^{k})e^{-i\varepsilon t}$. The corresponding equation for the fermionic eigenfunctions is

[TABLE]

thus, the eigenvalues $\varepsilon$ correspond to the energy of the fermions. Further, the equation for the field ${\boldsymbol{\phi}}$ is

[TABLE]

Note that we neither impose the usual assumption that ${\boldsymbol{\phi}}$ is a fixed static background field, nor make an approximation of its profile.

The complete system of coupled equations (3),(4) must be solved numerically. This task can be simplified if we take into account the symmetry properties of this system. First, we notice that the DM interaction breaks the spatial and internal $\mathrm{O}(2)$ symmetries of the non-linear $\sigma$ model to the diagonal subgroup. Secondly, the system enjoys the following discrete symmetries

[TABLE]

In the limiting case of massless uncharged fermions with $m=e=0$ the system (3),(4) is enhanced by additional symmetry of the fermion field $\psi\to-i\sigma_{2}\otimes\sigma_{2}\psi^{\ast}\,,~{}\varepsilon\to-\varepsilon$.

First, let us consider $\mathrm{O}(2)$ invariant configuration, which is parameterized by the ansatz:

[TABLE]

Here $f(r)$ is some monotonically decreasing radial function, $\varphi$ is the usual azimuthal angle, $n\in\mathbb{Z}$ and the phase $\delta$ corresponds to the internal orientation of the Skyrmion. Notably, the energy of the magnetic Skyrmion depends on $\delta$, it is minimal for $\delta=\pi/2$, further, rotationally invariant configuration (6) exists only for $n=1$ (helical Bloch Skyrmions Nago ; magSkbook ). Since the field must approach the vacuum on the spatial asymptotic, it satisfies the boundary condition $\cos f(r)\to 1$ as $r\to\infty$, i.e., $f(\infty)\to 0$.

The corresponding rotationally invariant spin-isospin eigenfunctions with the eigenvalues $\varepsilon$ can be written as

[TABLE]

where the components $u_{i}$ and $v_{i}$ are functions of the radial coordinate only, $l\in\mathbb{Z}$ is the angular momentum of fermion and $\mathcal{N}$ is a normalization factor, which is defined from the usual condition $\int d^{2}x~{}{\psi}^{\dagger}\psi=2\pi{\mathcal{N}}^{2}\int_{0}^{\infty}rdr(v_{1}^{2}+v_{2}^{2}+u_{1}^{2}+u_{2}^{2})=1$.

The rotationally invariant fermionic Hamiltonian (2) commutes with the total angular momentum operator $K_{3}=-i\frac{\partial}{\partial\varphi}+\frac{\hat{\gamma_{3}}}{2}+\frac{\tau_{3}}{2}$. The corresponding half-integer eigenvalues $\kappa=1+l$ can be used to classify different field configurations. The ground state corresponds to $\kappa=0$ and thus $l=-1$.

First, we note that there are localized modes among fermionic eigenfunctions (7). Indeed, substitution of the ansatz (6) into the equation (4) yields

[TABLE]

Linearizing this equation in the asymptotic region, where both $f$ and the fermionic field profile functions approach zero, we obtain

[TABLE]

which is the usual modified Bessel equation, whose solution can be written in terms of the McDonald function, $f\sim K_{1}(r)$. Thus, as $r\to\infty$, $f\sim\frac{e^{-\sqrt{B}r}}{\sqrt{r}}$, the Skyrmion is exponentially localized for any form of the fermionic field and the asymptotic field of the magnetic Skyrmion may be thought of as generated by a pair of orthogonal dipoles.

Considering fermions, we notice that in the limit of vanishing Hund coupling the system is reduced to the usual Dirac fermions in the uniform magnetic field. The energy spectrum of the charged fermions is given by the Landau levels

[TABLE]

where $k\in\mathbb{Z}$. The levels are twofold degenerated (except $k=0$), we can expect this degeneration is lifted for non vanishing fermion-Skyrmion coupling. The charged modes then become exponentially localized on the Skyrmion, this effect gives rise to the electric charge of the configuration.

For the chargeless modes ($e=0$) the situation is different, this case is similar to that of the usual fermion-Skyrmion system considered in Perapechka:2018yux . The asymptotic expansion of the equations (3) with parametrization (7) at $r\to\infty$ yields

[TABLE]

and the exponentially localized modes exist as $|\varepsilon|<g-m$.

*Numerical results.  * The equations (4) and (3), together with constraint imposed by the normalization condition, yield a system of integro-differential equations, which can be solved numerically. In a general case the system is not rotationally invariant, then we impose an additional $\mathrm{O}(3)$ constraint on the scalar field ${\boldsymbol{\phi}}$. We make use of the 4th order Newton-Raphson method implemented in the CESDSOL package, relative numerical errors are no higher than $10^{-5}$.

First, we consider rotationally-invariant system of fermions coupled to the single Skyrmion.

The results for the fermionic energy spectrum are presented in Fig. 1. For the uncharged modes the general pattern is similar to what we found in our previous study of the fermions interacting with baby Skyrmions Perapechka:2018yux . In agreement with the index theorem, for a given value of $l$, there is one zero-crossing mode which runs from positive to negative continuum as the Hund coupling $g$ is increasing. Apart this mode there are localized states of two different types, which are linked to the negative and positive continuum. We refer to them as the modes of the types $A$ and $B$, respectively.

The spectral flow of the charged fermions is different, see Fig. 1, right plot. In the limiting case $g=0$ the fermions are decoupled from the Skyrmion, they occupy the Landau levels (10). For each value of $l\neq 0$ there are two modes on each level, the zero mode corresponds to $k=0,l=-1$. As the Hund coupling $g$ increases, the states start to deform, the energy of the lowest mode becomes positive, it has a maximum at some value of $g$. As the coupling increases further, the energy of the lowest mode is decreasing, it crosses zero at some critical value of the Hund coupling. This mode remains localized on the Skyrmion for all values of the coupling, while other charged modes with $k\neq 0$ are linked to the positive or negative energy continuum, approaching it at some set of critical values of the fermion-Skyrmion coupling $g$. Further, as $g$ increases, the fermion-Skyrmion interaction becomes stronger than the interaction between the fermions and magnetic field ${\bf B}$, thus there is a one-to-one correspondence between the corresponding localized modes and the uncharged modes of the types $A_{k}$ and $B_{k}$.

Localization of the fermionic modes may strongly affect the usual pattern of interaction between the magnetic Skyrmions. Notably, even chargeless fermionic modes may balance the repulsive interaction between the chiral Skyrmions.

Indeed, making use of the asymptotic equation (9) we can evaluate the potential of the repulsive scalar interaction between the Skyrmions separated by the distance $R$ as $V_{s}\sim K_{0}\left(\sqrt{B}R\right)$. Similarly, the asymptotic decay of the chargeless fermionic field (11) yields an additional channel of interaction. For solitons with two localized $A_{0}$-type modes the corresponding potential is $V_{f}\sim\cos\theta K_{0}\left(\sqrt{(g+m)^{2}-\varepsilon^{2}}R\right)-K_{0}\left(\sqrt{(g-m)^{2}-\varepsilon^{2}}R\right)$, where $\theta$ is an angle of relative orientations of the fermionic dipoles. Hence, this interaction can be attractive and bounded system of solitons may exist for a certain set of values of the parameters of the system, for example we can look for stable biskyrmion solutions, which represent two chiral Skyrmions coupled by the localized fermionic modes.

For the sake of simplicity we restrict our analysis to the massless uncharged fermions. The energy of interaction of two Skyrmions in such a system can be evaluated as $E_{\textrm{int}}=E-2M-g$, where $E$ is the total energy of the biskyrmion-fermion configuration, $M$ is the mass of a single Skyrmion and we take into account that for as $m=e=0$, the lower bound of continuum is $\varepsilon_{c}=g$.

Our numerical investigation shows that biskyrmion-fermion configurations exist for relatively large values of the coupling $g$, moreover, there are several branches of solutions and different configurations may exist for the same value of $g$. All solutions we found satisfy the symmetry restrictions (5).

In Fig. 2 we display the fermion energy (in unit of $g$) and interaction energy $E_{int}$ as functions of the coupling strength $g$. First, we observe that the bounded system of two charge one Skyrmions with two type $A_{0}$ modes localized on each of the solitons, appears as a local minimum as $g$ increases above $g_{cr}^{(1)}=2.17$, see the left plots in Fig. 3. The bounded solutions does not exist as $g<g_{cr}^{(1)}$. As the Hund coupling increases, the overlap between the modes becomes stronger and the solitons approach each other. The energy of the fermionic modes is decreasing, it crosses zero and the energy of interaction becomes negative although both Skyrmions remain separated, as shown in plots 2, Fig. 3. This branch of solutions terminates at some upper critical value of the coupling $g_{cr}^{(2)}\approx 5.22$, here it bifurcates with the second branch, which extends backwards as the coupling $g$ is decreasing. Along this branch the maximum of the fermionic density distribution is located at the center of the elongated soliton configuration, see plots 4 in Fig. 3. The energy of the fermionic modes is increasing as $g$ is decreasing, in accordance with the index theorem it crosses zero for a second time and tends towards the positive continuum as the solitons merge forming a rotationally invariant configuration of topological degree 2, see plots 5, Fig. 3. At some lower critical value of the coupling $g_{cr}^{(3)}\approx 0.75$ the symmetric combination of two fermionic modes approaches the continuum as it bifurcates with the lowest rotationally invariant mode of the charge 2 configuration. Along the corresponding third branch the coupling becomes stronger, the total energy of the configuration is decreasing. No indication is found for termination of this branch, it exist for arbitrary large values of the coupling.

Furthermore, we found other multisoliton configurations bounded by the fermionic modes. In Fig. 4, as particular examples, we display the solutions we found in sectors of degrees $Q=3,4$, there is an interesting pattern of transformations of the configurations as the Hund coupling varies.

In summary, we have shown that the localization of the spin-isospin fermionic modes on the magnetic chiral Skyrmions with DM interaction strongly affects the usual picture of interaction between the solitons. We found multisoliton solutions bounded by the localized fermionic modes and investigated the corresponding spectral flow. It must be stressed that two-component fermions do not localize on the magnetic Skyrmion, spin-isospin symmetry plays a crucial role in appearance of quasi-zero fermionic modes. Many issues remain for further study, in particular, it is of great interest to construct magnetic Skyrmions lattice bounded by fermions. Further, localization of the charged fermionic modes yields an electric charge of the soliton, thus there is a long range Coulomb electric interaction between two widely separated chiral Skyrmions with localized fermionic modes, which is supplemented by the short range scalar repulsion. We hope to address this problems in our future work. Finally, we expect similar bounded multisoliton solutions may exist in the conventional baby-Skyrme model coupled with fermions, as well as in various higher dimensional models, like for example in the Abelian Higgs model .

*Acknowledgements.  * We are grateful to Steffen Krusch, Nobuyuki Sawado and Paul Sutcliffe for helpful discussions. Ya.S. gratefully acknowledges the support of the Alexander von Humboldt Foundation and from the Ministry of Education and Science of Russian Federation, project No 3.1386.2017.