The quantum skyrmion Hall effect in $f$ electron systems

From fundamental physical processes to application-oriented information storage and processing, the stability of a physical effect is paramount. Some physical effects, especially quantum Hall effects, accommodate observables that are topologically protected, i.e., they are robust against a continuous deformation of selected parameters. Recently, experimental and theoretical studies have found topologically protected classical magnetic structures in thin films or effectively two-dimensional systems, which have been coined magnetic skyrmions Bogdanov (1995); Bogdanov and Hubert (1999); Mühlbauer et al. (2009); Yu et al. (2010); Heinze et al. (2011), connected to earlier ideas in particle physics Skyrme (1962). The stability of these objects and the possibility of creating them by electrical currents or time-controlled magnetic boundary conditions Everschor-Sitte et al. (2017); Stier et al. (2017); Schäffer et al. (2020); Siegl et al. (2022a) promote the idea of using them in spintronics and as information carriers Parkin et al. (2008); Tomasello et al. (2014); Fert et al. (2017). Furthermore, in sight of the ongoing miniaturization of magnetic skyrmions, they have also been proposed as ingredients in quantum computing Psaroudaki and Panagopoulos (2021).

Classical magnetic skyrmions experience an additional drag transversal to the direction of an applied electric current, which leads to an accumulation of skyrmions at one side Thiele (1973); Everschor et al. (2011, 2012); Iwasaki et al. (2013a, b); Sampaio et al. (2013); Moon et al. (2022); Jiang et al. (2016); Litzius et al. (2016). The angle between the direction of motion and the direction of the current is called the Magnus angle of the skyrmion Hall effect. The question arises if there is a quantum version of the skyrmion Hall effect and, if so, which characteristics of the electronic Hall effects transfer, including hypothetical skyrmion edge channels with their quantized conductance. A previous study treating the quantum skyrmion as a product state and calculating an effective action for the quantum skyrmion demonstrated the existence of the Magnus force Takashima et al. (2016). Yet, the challenge in describing general quantum skyrmions comes with the large Hilbert spaces that need to be considered in two-dimensional spin systems carrying quantum skyrmions of a size of minimally $3\times 3$ spins Sotnikov et al. (2021); Lohani et al. (2019); Siegl et al. (2022b), which demand special theoretical techniques like density matrix renormalization group Haller et al. (2022) or artificial neural networks Yoshi et al. (2023) to investigate them numerically.

Another method to analyze quantum skyrmions is the use of localized spins from interacting electrons, like $f$ electrons, to represent the skyrmions in a correlated electronic system Kobayashi and Hayami (2022). Representing the skyrmions as electronic degrees of freedom has the advantage that we can treat considerably large quantum spin systems with established advanced numerical techniques for correlated electronic systems and that we can apply an electric current within the model without further assumptions. Interestingly, skyrmions in $f$-electron systems have been experimentally detected in EuAl₄, in which the skyrmions have been treated classically Takagi et al. (2022). Yet, the quantum nature of skyrmions in strongly correlated electronic systems is not well studied. Ultimately, a quantum skyrmion Hall effect could have direct practical applications extenting the manifold of suggested technical applications of magnetic skyrmions Back et al. (2020) to the quantum world. Moreover, fundamental questions about the topological nature of quantum skyrmions, which, strictly speaking, gets lost because of quantum spin slip processes Kim and Tserkovnyak (2016); Posske and Thorwart (2019); Siegl et al. (2022b); Vijayan et al. (2023), could be answered when quantum skyrmions are connected to quantum Hall effects and their unambiguous topological origin.

In this paper, we numerically study a square lattice of localized $f$ electrons that are coupled to two-dimensional itinerant conduction ($c$) electrons in the presence of spin-orbit coupling and a small magnetic field perpendicular to the two-dimensional plane. Using dynamical mean-field theory, we reliably identify regions in parameter space that host quantum nano skyrmions. We subsequently study the effect of a current in the itinerant electrons on the quantum skyrmion in linear response theory and find a strong initial drag into the direction perpendicular to the current, marking the onset of the quantum skyrmion Hall effect. The shift of the skyrmion is accompanied by an Edelstein effect and a magnetoelectric effect,Edelstein (1990); Culcer and Winkler (2007); Chernyshov et al. (2009); Manchon and Zhang (2008); Garate and MacDonald (2009); Peters and Yanase (2018); Fiebig (2005) which leads to a rotation of the localized $f$-electron spins. Our study stimulates further investigation of the quantum skyrmion Hall effect, especially its steady state, and possibly quantized skyrmion edge channels.

The remainder of this paper is structured as follows: In Sec. II, we introduce the model and the method. In Sec. III, we analyze the stability and structure of the quantum skyrmions for different model parameters. This is followed by Sec. IV, where we demonstrate the onset of the skyrmion Hall effect using linear response theory. Finally, in Sec. V, we discuss our results and conclude the paper.

Motivated by the discovery of magnetic skyrmions in Eu-compounds Takagi et al. (2022), including partially filled $f$ electrons, we focus here on magnetically ordered ground states and low-energy metastable states in $f$-electron systems on a square lattice with a lattice constant of $a$ on the order of half a nanometer. In particular, we study the ground states of a noncentrosymmetric $f$-electron system described by a periodic Anderson model Yanase and Sigrist (2008); Peters and Yanase (2018); Michishita and Peters (2019). It is important to note that we explicitly start with an electronic Hamiltonian instead of a quantum spin model. Thus, charge fluctuations and other effective interactions besides the effective Heisenberg and Dzyaloshinskii–Moriya (DM) interaction generally affect the ground state. Furthermore, due to the hybridization between the itinerant conduction ($c$) electrons and the $f$ electrons, the magnetic moments generated by the $f$ electrons are intrinsically coupled to the $c$ electrons. Such a coupling, which is necessary to observe skyrmion Hall and skyrmion drag effects, does hence not need to be inserted manually but is naturally included.

Our model Hamiltonian can be split into a single-particle part, $H_{0}$, and an interaction part, $H_{U}$. The single-particle Hamiltonian is

where $\bm{c}_{\bm{k}}=\left(c_{k_{x},k_{y},\uparrow},f_{k_{x},k_{y},\uparrow},c_{k_{x},k_{y},\downarrow},f_{k_{x},k_{y},\downarrow}\right)$ is the spinor containing the momentum space annihilation operators of the itinerant electrons and $f$ electrons, respectively, corresponding to the real-space operators $c_{i,j,\sigma}$ and $f_{i,j,\sigma}$ at site $\left(i,j\right)$ of a square lattice with spin $\sigma$. The matrices $\sigma^{\lambda}=s^{\lambda}\otimes s^{0}$ and $\tau=s^{0}\otimes s^{\lambda}$ denote the Pauli matrices on the spin and sublattice space, respectively, where $s$ are the bare Pauli matrices. The particle number operators are $n^{c}_{i,j,\sigma}=c^{\dagger}_{i,j,\sigma}c_{i,j,\sigma}$ and $n^{f}_{i,j,\sigma}=f^{\dagger}_{i,j,\sigma}f_{i,j,\sigma}$. The strength of the nearest neighbor hopping of the $c$ electrons on the square lattice is denoted by $t$. Throughout this paper, we use $t$ as the unit of energy. $\mu_{c}$ and $\mu_{f}$ are local energies of the $c$ and $f$ electrons, respectively. $V$ is a local hybridization between the $c$ and $f$ electrons as commonly used in the periodic Anderson model. $B$ corresponds to a small magnetic field applied in the $z$ direction. Finally, we include a spin-orbit coupling between the $c$ and $f$ electrons with hopping amplitude $\alpha_{c}$. This spin-orbit coupling corresponds to a Rashba-type spin-orbit interaction as present in noncentrosymmetric $f$-electron systems Yanase and Sigrist (2008). The interaction part of the Hamiltonian is

corresponding to a density-density interaction between $f$ electrons on the same lattice site. The full Hamiltonian is

The calculations are performed on a finite lattice $L_{x}\times L_{y}=11\times 11$ with periodic boundary conditions.

To find the ground state of this quantum model, we use the real-space dynamical mean-field theory (RDMFT)Georges et al. (1996); Potthoff and Nolting (1999); Vahedi et al. (2021); Peters and Kawakami (2015, 2014). RDMFT maps each atom of a unit cell (finite lattice) on its own quantum impurity model by calculating the local Green’s function

where $\tilde{h}_{0}$ is the single-particle matrix of the Fourier transform of $H_{0}$ in Eq. (1) into real space, i.e., $\tilde{H}_{0}=\sum_{n,m}c^{\dagger}_{n}\tilde{h}_{n,m}c_{m}$. Here, $n$ and $m$ are super indices including the lattice positions, the $f$-$c$ sublattice, and the spin. Furthermore, $\Sigma(z)$ is a matrix including the local self-energies of each lattice site in the finite lattice, where, by the defining approximation of RDMFT, $\Sigma_{n,m}(z)$ vanishes when the spatial components of $n$ and $m$ differ. Writing the local Green’s function as

we can map each lattice site on a quantum impurity model, where $\Delta_{n,m}(z)$ is the local hybridization of the impurity model. This hybridization function describes the environment for one lattice site created by the rest of the lattice. Here, the self-energy differs for each lattice site, and hence this hybridization function is different for each lattice site. Summarizing the numerical procedure, the local hybridization functions define quantum impurity models, which are solved to obtain the local self-energy for each lattice site. These updated self-energies are then used in Eq. (4), which defines a self-consistency problem. To calculate the self-energy of each lattice site, we use the numerical renormalization group (NRG) Wilson (1975); Bulla et al. (2008); Peters et al. (2006), which can calculate accurate Green’s functions and self-energies at low temperatures.

The magnetic properties of the periodic Anderson model without Rashba spin-orbit interaction are well understood within the DMFT approximation Georges et al. (1996). At half-filling, $\langle n^{c}_{i,j}\rangle=\langle n^{f}_{i,j}\rangle=1$, on a square lattice, the periodic Anderson model orders antiferromagnetically for weak hybridization strengths $V$ Peters and Kawakami (2015). For large hybridization strengths, the periodic Anderson model at half-filling becomes a Kondo insulator. On the other hand, when the number of $c$ electrons is small and the $f$ electrons are nearly half-filled, the system orders ferromagnetically Peters et al. (2012). This paper aims to study the existence and properties of magnetic skyrmions in a ferromagnetic periodic Anderson model, including Rashba spin-orbit interaction. We thus look for parameters where the $f$ electrons are nearly half-filled, and the $c$-electron filling is about $\langle n^{c}\rangle\approx 0.2$.

An exhaustive search of the parameter space of the periodic Anderson model for stable quantum skyrmions in the ground state is numerically unfeasible. In advance to our fully quantum-mechanical calculations, we therefore first identify candidate parameter regions where the ground state or low-energy metastable states accommodate magnetic skyrmions. We do so by mapping Eq. (1) to a classical Heisenberg spin model with nearest-neighbor coupling by integrating out the $c$ electrons using second-order perturbation theory, which obtains the RKKY spin-spin interactions Ruderman and Kittel (1954); Kasuya (1956); Yosida (1957). We then use classical Monte Carlo methods to find the ground states of these spin models Neuhaus-Steinmetz et al. (2022). In particular, we have varied in this procedure the local hybridization $V$, the spin-orbit coupling $\alpha_{c}$, and the $c$-electron level position, $\mu_{c}$. We subsequently transfer parameter configurations where we find a classical skyrmion to the quantum model and conduct RDMFT calculations to obtain the system’s ground state. Here, the presence of magnetic skyrmions in the corresponding classical model generally is a good indicator for quantum skyrmions in the quantum model. Setting $U/t=6$ and $\mu_{f}/t=-3$, corresponding to half-filling of the $f$ electrons, we find quantum skyrmions in a ferromagnetic background for $V=t$, $\mu_{c}/t\approx 3.6$, and a finite spin-orbit coupling in combination with a magnetic field, in agreement with previous results on classical and quantum magnetic skyrmions Sotnikov et al. (2021); Lohani et al. (2019); Siegl et al. (2022b); Haller et al. (2022). In the RDMFT calculations, we vary the strength of the spin-orbit interaction, $\alpha_{c}$, and the strength of the magnetic field, $B$, in the region according to the results of the classical calculations.

To unambiguously identify a magnetic skyrmion, we break the spin translation symmetry of the model in the first DMFT iteration. By this, we select a specific state of the translationally invariant space of ground states. We use two different strategies in our calculations. We either start with a ferromagnetic solution where all $f$ electrons point downwards and flip a single $f$ electron upwards. Alternatively, we directly start with a magnetic skyrmion solution obtained for a different set of parameters. Then, by iterating the DMFT self-consistency equation, we find possible, stable magnetic skyrmion solutions when the algorithm converges. In the skyrmion phase, both initial states lead to identical DMFT solutions. We show the convergence of the self-energies for a typical skyrmion solution in Appendix B.

To verify the existence of a magnetic skyrmion, we calculate the spin expectation values of the $c$ and $f$ electrons for each lattice site,

where $\bm{r}=(i,j)$ corresponds to the coordinates of a lattice site, and $\bm{\sigma}=\left(\sigma^{x},\sigma^{y},\sigma^{z}\right)$ is the vector containing the spin space Pauli matrices. Using these spin expectation values, we calculate the local lattice skyrmion density for the $f$ and $c$ electrons based on the solid angle spanned by three vectors as

where $d$ either stands for $f$ or $c$ electrons, $\bm{r}_{1}$, $\bm{r}_{2}$, and $\bm{r}_{3}$ are nearest-neighbor lattice sites spanning an elemental triangle $\langle\bm{r}_{1},\bm{r}_{2},\bm{r}_{3}\rangle$ in the densest triangular tessellation of the lattice. The sum of this skyrmion density over all triangles spanning the square lattice yields the skyrmion number

Unlike in a classical calculation, the spin expectation values in a quantum model do not need to be $\hbar/2$ in magnitude. In fact, these expectation values are usually smaller due to quantum fluctuations, $|\bm{S}|<\hbar/2$. We thus calculate two types of skyrmion densities: One is the quantum skyrmion density/number using unnormalized spin expectation values. The second type is a classical skyrmion density, where we normalize all spin expectation values to $\hbar/2$ before using them in Eq. (8). The skyrmion number is an integer when using normalized spin expectation values. When using unnormalized spin expectation values, the skyrmion number is not quantized, and instead, its magnitude is an indicator of the skyrmion stability Siegl et al. (2022b), similar to the scalar chirality defined in Ref. Sotnikov et al. (2021).

A representative magnetic skyrmion solution is shown in Fig. 1 calculated for a spin-orbit coupling $\alpha_{c}/t=0.3$ and a magnetic field $B/t=0.002$. We note that within the accuracy of our calculations, we cannot find discernible energy differences between the ferromagnetic configuration and the magnetic skyrmion. The described skyrmions can, therefore, be metastable excitations on a ferromagnetic background with almost vanishing excitation energy or present in the ground state itself. Such an almost degenerate situation is favorable for applications in racetrack devices. If skyrmions were energetically strongly favorable, a skyrmion lattice would form instead of individual skyrmions. Figure 1 shows the spin texture of the $f$ and $c$ electrons underlaid with the local skyrmion density for normalized spin expectation values as 2D color plot, see Eq. (8). Due to the local hybridization, $V$, which leads to an effective antiferromagnetic interaction between the $c$ and $f$ electrons, the spins of the $c$ and $f$ electrons mostly point in opposite directions, with deviations in the skyrmion’s center, where the Rashba interaction and the itinerant character of the $c$ electrons play a stronger role. The combined state corresponds to a bound skyrmion-antiskyrmion pair where the $f$ electrons form a magnetic skyrmion with skyrmion number $N^{f}=1$ and the $c$ electrons form a magnetic antiskyrmion with skyrmion number $N^{c}=-1$. However, in this system, the spin expectation values of the $c$ and $f$ electrons are of very distinct origins and magnitudes. Because the $f$ electrons are strongly interacting, they form localized magnetic moments, and their spin expectation values in this calculation are approximate $|\langle S^{f}\rangle|_{a}\approx 0.75\frac{\hbar}{2}$. They are not perfectly polarized due to the entanglement between the $c$ and the $f$ electrons. On the other hand, the $c$ electrons are noninteracting, and their spin expectation values vary around $|\langle S^{c}\rangle|_{a}\approx 0.03\frac{\hbar}{2}$. The $c$ electrons’ polarization is a direct cause of the hybridization with the magnetized $f$ electrons and, thus, a secondary effect. In this situation, the skyrmion of the $f$ electrons and the antiskyrmion of the $c$ electrons do not annihilate. This is revealed by the finite total skyrmion number calculated with unnormalized spin expectation values. This is indeed different from classical skyrmions, where the magnitude of the spin vectors is normalized. The reduced polarization decreases the topological protection of the magnetic skyrmion. The smaller the spin expectation value, the easier the spin can be flipped, and the magnetic skyrmion is destroyed Siegl et al. (2022b). On the other hand, this facilitates manipulating them as necessary for technical applications.

Next, we analyze the stability of the magnetic skyrmion for different magnetic field strengths, as shown in Fig. 2. As stated above, we generally apply a small magnetic field which helps to stabilize the magnetic skyrmion against spin spiral solutions Sotnikov et al. (2021); Haller et al. (2022). In Fig. 2(a), we show the skyrmion number using normalized and unnormalized spins, respectively. We observe that, while the skyrmion number (normalized) is constantly one for $B/t\lesssim 0.0056=B_{c}$, the skyrmion number using unnormalized spin expectation values is $N^{f}\approx 0.4$ and gradually drops for an increased magnetic field until $B_{c}$ is reached. The difference between these numbers demonstrates the relevance of quantum effects to the system at hand. We furthermore show the average spin expectation values of the $c$ and $f$ electrons, indicating that the $f$ electrons are considerably more stronger polarized than the $c$ electrons. Furthermore, for magnetic fields stronger than $B_{c}$, we only find ferromagnetic solutions. Figures 2(b) and (c) give representative spin textures of the $f$ electrons for the corresponding parameter regimes, i.e., small and large magnetic fields.

We next analyze the structure of the skyrmion depending on the strength of the Rashba spin-orbit coupling $\alpha_{c}$. We show the skyrmion number using normalized spin expectation values, the skyrmion number using unnormalized spin expectation values, and the average spin expectation values ($\langle S^{f}\rangle$ and $\langle S^{c}\rangle$) in Fig. 3(a). Increasing the Rashba interaction, the $f$-electron spin expectation value is slightly suppressed, while the $c$-electron spin expectation value slightly increases. This increase in the $c$ electron magnetization can be explained by the stronger coupling between the $c$ and $f$ electrons. While we need $\alpha_{c}/t>0$ to create a finite DM interaction that stabilizes the magnetic skyrmion, we see that for increasing $\alpha_{c}$ the skyrmion gets destabilized and for $\alpha_{c}/t>0.4$, magnetic skyrmions become unstable indicated by the vanishing skyrmion number. To analyze this transition further, we calculate the average size of the skyrmion. First, the center of the skyrmion created by the $f$ electrons is

where $\bm{r}_{1}$, $\bm{r}_{2}$, and $\bm{r}_{3}$ are the coordinates of the lattice sites spanning the elemental triangle as explained below Eq. (8). The extension of the skyrmion in the $x$ and the $y$-direction $\bm{L}=(L_{x},L_{y})$ is then given as

where $x_{i}$ ($y_{i}$) is the $x$ ($y$) component of the position $\bm{r}_{i}$ and the center of the skyrmion is $\bm{R}_{s}=(x_{s},y_{s})$. In Fig. 3(b), we show the extension of the skyrmion in the $x$ and $y$-direction depending on the Rashba spin-orbit interaction. We see that while the average extension of the skyrmion in the $x$ direction remains unchanged when increasing $\alpha_{c}$, the magnetic skyrmion is strongly elongated in the $y$ direction. At $\alpha_{c}/t\approx 0.4$, the magnetic skyrmion changes into a spiral phase, again consistent with findings for quantum skyrmions on nonelectronic spin lattices Sotnikov et al. (2021); Haller et al. (2022). Representative spin textures of the $f$ electrons are shown in Fig. 3(c)-(e), depicting a skyrmion for small $\alpha_{c}$ (c), an elongated skyrmion close to the phase transition (d), and a spiral wave for large $\alpha_{c}$ (e). For larger values of the spin-orbit coupling, we do not find stable skyrmion solutions.

Finally, we note that we have confirmed the stability of the quantum skyrmion phase for smaller lattice sizes, such as 7x7. Magnetic skyrmions remain stable as long as their elongation is smaller than the lattice width. Furthermore, we do not find an even/odd effect in the lattice width, which can be understood by the fact that all spins are ferromagnetically aligned far away from the magnetic skyrmion, irrespective of changes of the lattice sizes once it exceeds the size of the quantum skyrmion.

Finally, we study the response of the identified stable skyrmion textures to an applied charge current in the itinerant $c$ electrons. To do this, we calculate the change in the spin expectation values of all lattice sites in linear response theory. We focus on describing the time-transient behavior of the system. A description of the nonequilibrium steady state poses considerable numerical challenges, as discussed in the concluding remarks.

In linear response, the change in an expectation value of operator $A$ resulting from a perturbation $B$ is given by

where $\Theta(\tau)$ is the Heaviside step function. Because we are interested in the linear response of the spin expectation values of the $f$ electrons to a charge current in the itinerant electrons, we use

where $A$ corresponds exactly to the local spin of an $f$ electron, and $B$ is the charge current operator in the $c$ electrons.

For these operators, Eq. (14) corresponds to a nonlocal two-particle Green’s function. Using the DMFT approximation, where vertex corrections in nonlocal Green’s functions vanish, we write these two-particle Green’s functions as the convolution of two single-particle Green’s functions. Then, we calculate the time evolution of the spin expectation values of all spins. Because the self-energy depends on the lattice site, also the time evolution of the spin expectation value depends on the lattice site. This is shown in Fig. 4, where we show the change of the $x$, $y$, and $z$ component of the spin expectation values $\Delta\langle S^{x}\rangle(\tau)$, $\Delta\langle S^{y}\rangle(\tau)$, and $\Delta\langle S^{z}\rangle(\tau)$ along the $x$-direction of the lattice across the center of the skyrmion solution for $\alpha_{c}/t=0.3$ (shown in Fig. 1). Specifically, the spin expectation values are shown for lattice sites $(x_{S}+x,y_{S})$, where $\bm{R}_{S}=(x_{S},y_{S})$ is the center of the skyrmion, see Eq. (10). $\Delta\langle S_{x}\rangle$ and $\Delta\langle S_{z}\rangle$ show a strong dependence on the position close to the center of the skyrmion. $\Delta\langle S_{z}\rangle$ changes even its sign when changing the position from the left of the center to the right of the center. On the other hand, $\Delta\langle S_{y}\rangle$ is nearly independent of the lattice site. Also, while $\Delta\langle S_{z}\rangle$ becomes small for spins far away from the skyrmion center, $\Delta\langle S_{x}\rangle$ and $\Delta\langle S_{y}\rangle$ are nonzero. Thus, even in the ferromagnetic region away from the skyrmion center, $\langle S_{x}\rangle$ and $\langle S_{y}\rangle$ change. This rotation of the spin in the ferromagnetic state when a charge current is applied is explained by the Edelstein and the magnetoelectric effect Peters and Yanase (2018); in a system where the Fermi surface is split due to the Rashba spin-orbit coupling, a charge current results in an accumulation of spin. This can be seen here as a rotation of the spin expectation values in the $x$ and the $y$ direction, even far away from the magnetic skyrmion. Furthermore, we emphasize that the linear-response results only remain valid within sufficiently small times $\tau$. In Fig. 4, we see that the initial linear trend in $\tau$ is reduced, and, as an expected artifact from linear response theory, all spin expectation values start oscillating after a certain individual time. The change of all spin expectation values for $\tau\cdot t/\hbar=2$ and $\tau\cdot t/\hbar=4$ are visualized in Fig. 5 as arrows. Each arrow corresponds to the direction of the local change in the spin expectation values $\Delta\langle\bm{S}\rangle(\tau)$. The actual length of each change is $|\Delta\left\langle\bm{S}\right\rangle(\tau)|\approx 0.04\frac{\hbar}{2}J$ and $|\Delta\left\langle\bm{S}\right\rangle(\tau)|\approx 0.07\frac{\hbar}{2}J$ for $\tau\cdot t/\hbar=2$ and $\tau\cdot t/\hbar=4$,respectively. We see that in the ferromagnetic region, all spins are rotated in the same direction. Only close to the magnetic skyrmion, the change in the spin expectation values significantly depends on the lattice site.

Finally, we take the time evolution of each spin on the lattice and calculate the skyrmion density and the time-dependent size and position of the skyrmion according to Eqs. (10-11). By Eq. (14), we find that the center of the skyrmion moves almost perpendicularly to the applied current, as shown in Fig. 6(a). While the current is applied in the $x$ direction, the skyrmion moves in the positive $y$ direction. Thus, our results demonstrate the onset of a quantum skyrmion Hall effect with a Magnus angle close to $90$ degrees. Notably, the size of the skyrmion effectively remains constant during the motion, shown in Fig. 6(b).

In conclusion, we show that noncentrosymmetric $f$-electron systems with spin-orbit coupling in the presence of a small external magnetic field can host nano quantum skyrmions in the ground state, and we demonstrate the onset of the quantum skyrmion Hall effect upon applying a charge current, which is accompanied by an Edelstein and magnetoelectric effect.

The reason for the stability of the quantum skyrmion is an effective DM interaction generated by the spin-orbit interaction and a local density-density interaction. Despite the itinerant $c$ electrons being magnetized like an antiskyrmion, the quantum skyrmions of the $f$ electrons remain stable and dominate the physical behavior of the system because of its considerably stronger polarization due to strong correlations. Concerning the quantum skyrmion Hall effect, we observe a Magnus angle close to $90\deg$. This is consistent with the behavior of classical skyrmions, where the Magnus angle increases when the size of the skyrmions is smaller or when dissipative effects are small Litzius et al. (2016); Jiang et al. (2016). Both is the case for the observed nano quantum skyrmions. Furthermore, no quantum skyrmion pinning is visible in our study. We note that our method can only describe the onset of the skyrmion motion. In particular, in a full nonequilibrium calculation, time-dependent spin expectation values would lead to time-dependent self-energies. The system would adapt to the changed spin expectation values and backaction effects would alter our conclusions when the linear-response regime is left. For example, linear response theory can permanently decrease the polarization locally, ultimately resulting in a site with vanishing spin polarization. However, this situation is energetically unfavorable due to the strong density-density interaction. Thus, in a full nonequilibrium calculation, self-energies will change in a way that an atom with vanishing spin polarization is prevented, rendering the quantum magnetic skyrmion stable and letting it continue its motion perpendicular to an applied current. Yet, a full nonequilibrium calculation, as well as a steady-state analysis, goes beyond the scope of the current paper and is left for future work.

We note that other forms of spin-orbit interaction also lead to stable nano quantum skyrmions in the $f$-electron system at hand. We show the results for a different form of the spin-orbit interaction, where the momenta couple to the same spin direction, in Appendix A. Also in these systems, the spin-orbit interaction results in a spin accumulation when a current is applied, which leads to a site-dependent change of the spin expectation values, and to a skyrmion Hall effect. These results emphasize that the existence of magnetic skyrmions in strongly correlated $f$-electron systems with spin-orbit coupling and the skyrmion Hall effect is a general effect.