Bound states of skyrmions and merons near the Lifshitz point

TL;DR

This paper investigates topological defects like skyrmions and merons in anisotropic ferromagnets near the Lifshitz point, revealing their stability, interactions, and potential for forming bound states, with implications for frustrated magnets.

Contribution

It demonstrates the stability and interactions of skyrmions and merons near the Lifshitz point, including the formation of bound states and deconfinement of fractional charges.

Findings

Skyrmions attract and form ring-shaped bound states in zero magnetic field.

Merons become deconfined at the Lifshitz point.

Topological excitations can exist in weakly frustrated magnets.

Abstract

We study topological defects in anisotropic ferromagnets with competing interactions near the Lifshitz point. We show that skyrmions and bi-merons are stable in a large part of the phase diagram. We calculate skyrmion-skyrmion and meron-meron interactions and show that skyrmions attract each other and form ring-shaped bound states in a zero magnetic field. At the Lifshitz point merons carrying a fractional topological charge become deconfined. These results imply that unusual topological excitations may exist in weakly frustrated magnets with conventional crystal lattices.

Equations

$$E=$$

$$E$$

$$\frac{E_Q}{Q}=\frac{2\pi }{q\varkappa }\left[\rho \left(q^2+{\varkappa }^2\right)+b_1\left(q^4+{\varkappa }^4\right)+\frac{b_2}{3}{q}^2{\varkappa }^2+K\right].$$

$$\frac{E_Q}{|Q|}=4\pi \left(\rho +\sqrt{K\left(2b_1+\frac{1}{3}{b}_2\right)}\right).$$

$$U_C(r)=2\pi \rho \ln (r/r_0),$$

Full text

*Introduction: * Some fifty years ago Tony Skyrme identified topologically stable “hedgehog”-like configurations of the meson field with baryons, such as proton and neutron Skyrme1962 . The ensuing theoretical work showed that skyrmions indeed provide a semiquantitative description of physical properties of baryons and their interactions Zahed1986 . Multi-skyrmion bound states describe ground states and low-energy excitations of atomic nuclei Battye2016 . Periodic crystals of skyrmions and half-skyrmions were used to model nuclear matter Klebanov1985 ; Goldhaber1987 ; Kugler1989 .

Two-dimensional analogues of Skyrme’s skyrmions are relevant topological excitations in many condensed matter systems Nagaosa2013 , such as Quantum Hall magnets Sondhi1993 ; Barett1995 , spinor Bose-Einstein condensates Ho1998 , chiral liquid crystals Fukuda2011 , and chiral magnets Bogdanov1989 , which provide the playground for experimental studies of skyrmions. Skyrmion crystals and isolated skyrmions in chiral magnets can be observed by neutron scattering and Lorentz microscopy Muhlbauer2009 ; Yu2010 , and controlled by ultralow electric currents Jonietz2010 ; Yu2012 , applied electric fields White2014 ; Seki2012 , and thermal gradients Mochizuki2014 , which opened a new active field of research on skyrmion-based magnetic memories Fert2013 ; Iwasaki2013 ; Zhou2014 ; Jiang2015 ; Moreau2016 ; Woo2016 . Half-skyrmions (or merons) carying half-integer topological charge were also discussed theoretically in the context of quantum Hall systems Brey1996 , bilayer graphene Cote2010 and chiral magnets Ezawa2011 ; Lin2015 , but so far they eluded experimental detection.

Here, we are interested in magnetic multi-skyrmion and multi-meron configurations with a large topological charge, $Q$. An example is the skyrmion crystal in chiral magnets. However, skyrmions in the crystal can hardly be considered as independent particle-like objects, since to a good approximation this state is a superposition of three spin spirals plus a uniform magnetization Muhlbauer2009 . Isolated skyrmions appear under an applied magnetic field that suppresses modulated spiral and skyrmion crystal phases and induces a collinear ferromagnetic (FM) state. Skyrmions in chiral magnets repel each other Roessler2011 , so that multi-skyrmion states are merely a gas of elementary skyrmions with $Q=\pm 1$.

It was recently suggested that skyrmion crystals and isolated skyrmions can also exist in frustrated magnets with conventional centrosymmetric lattices, where they are stabilized by competing ferromagnetic and antiferromagnetic (AFM) exchange interactions Okubo2012 ; Leonov2015 ; Lin2016 . In frustrated magnets, the skyrmion-skyrmion interaction potential changes sign as a function of the distance between skyrmions, which makes possible formation of skyrmion clusters as well as rotationally symmetric skyrmions with the topological charge $Q=\pm 2$ Leonov2015 .

We note that topological excitations in frustrated magnets can be stable even in zero magnetic field. Consider a magnet in which the degree of frustration described by the parameter $f$ that can be varied, e.g. by an applied pressure or a chemical substitution. The phase diagram of such magnets often contains the so-called Lifshitz point (LP), $f=f_{\ast}$, which separates the uniform FM state ($f<f_{\ast}$) from periodically modulated phases ($f>f_{\ast}$). The behavior close to the LP was recently discussed in the context of Bose condensation of multi-magnon bound states in quantum low-dimensional systems Balents2016 . Skyrmions, which can be considered as bound states of a large number of magnons, so far were studied in the strongly frustrated regime $(f>f_{\ast})$. In this Letter, we focus on the “underfrustrated” side of the LP and show that skyrmions are stable in a large interval of $f<f_{\ast}$. We show that elementary skyrmions attract each other and can form bound states with an arbitrarily large $Q$. Surprisingly, despite the attraction, skyrmions do not aggregate into clusters. Instead, they form topological ring-shaped domain walls.

The aforementioned unusual multi-$Q$ states appear in easy-axis magnets. An easy-plane anisotropy forces skyrmion to transform into a bound pair of merons with opposite vorticities, each carrying topological charge $Q=\frac{1}{2}$. The lowest-energy multi-meron configuration is a square lattice of merons with alternating vorticities. Our results show that stability of skyrmions and merons does not require strong magnetic frustration, implying that these exotic topological excitations with interesting physical properties can exist in already known magnetic materials. In addition, we find a number of striking similarities between multi-Q skyrmions in condensed matter and nuclear physics.

*The Model: * We consider classical spins on a square lattice with competing exchange interactions and magnetic anisotropy. The energy of the model is

[TABLE]

where $\bm{S}_{i}$ is the spin of unit length at the lattice site $i$ and the first, second and third terms describe, respectively, FM nearest-neighbor and AFM second- and third-neighbor exchange interactions ($J_{1},J_{2},J_{3}>0$). The $z$ axis is normal to the lattice plane (the $xy$ plane) and $K$ is the strength of the single-ion magnetic anisotropy of easy axis ($K>0$) or easy plane $(K<0)$ type. In what follows, energy is measured in units of $J_{1}=1$ and distances are measured in units of the lattice constant.

For slowly varying spin textures, Eq.(1) is equivalent to the coninuum model (see e.g. Ref. [Milstein2011, ]),

[TABLE]

where $\rho=J_{1}-2J_{2}-4J_{3}$, $b_{1}=\frac{1}{12}(-J_{1}+2J_{2}+16J_{3})$, $b_{2}=J_{2}$ and $i=x,y$. The first term in Eq. (2) is the $O(3)$ nonlinear sigma model of an isotropic two-dimensional ferromagnet. The spin stiffness, $\rho$, plays the role of $f_{\ast}-f$: in the FM state $\rho>0$, at the Lifshitz point $\rho$ vanishes and for $\rho<0$, the system has either a spiral or a columnar antiferromagnetic (CAF) ground state Seabra2015 , as shown in Fig. 1(a) footnote1 . The fourth-order terms in gradients of $\bm{S}$ stabilize the spiral state and determine its wave vector provided that $b_{1}>0$ and $b_{1}+b_{2}/2>0$. These terms also stabilize skyrmions and merons in the FM state.

*Skyrmions: * The nonlinear sigma model with $\rho>0$ allows for analytic expression for skyrmions with an arbitrary $Q$ found by Belavin and Polyakov Belavin1975 . In the conformally invariant sigma model skyrmions have no internal length scale: the energy, $E_{Q}$, of the skyrmion with topological charge $Q$ is $4\pi\rho|Q|$ independent of the skyrmion size.

The radius, $R$, of the skyrmion with $Q=\pm 1$ (the elementary skyrmion) in frustrated magnets is determined by the competition between the fourth-order terms favoring infinite $R$ and the easy-axis anisotropy that tends to shrink the skyrmion. The dimensional analysis shows that $R\sim\left[\frac{b}{K}\right]^{1/4}$, where $b$ is a linear combination of $b_{1}$ and $b_{2}$ footnote2 . Skyrmion stability requires $\rho>0$ and $b>0$. In particular, skyrmions in the Heisenberg model with the nearest-neighbor interactions only are unstable. Figure 1(b) shows the stability region of the elementary skyrmion in the $J_{2}-K$ plane calculated numerically for $J_{3}=0$ footnote2 . Note that skyrmions are stable quite far from the LP $J_{2}=1/2$.

The skyrmion shape is controlled by the parameters $b_{1}$ and $b_{2}$, as shown in Figs. 2(a,b,c) and the corresponding contour plots of the topological charge density, $\rho_{Q}(x,y)=\frac{1}{4\pi}\bm{S}\cdot[\partial_{x}\bm{S}\times\partial_{y}\bm{S}]$ (Figs. 2(d,e,f)). The square-shaped skyrmions are observed close to the LPs $(J_{2},J_{3})=(1/2,0)$ and $(J_{2},J_{3})=(0,1/4)$. For $J_{2}>2J_{3}$, the FM phase transforms into the spiral state with the wave vector, $\bm{q}$, parallel to the square lattice axes, in which case the skyrmion has the shape shown in Figs. 2(a,d). For $J_{2}<2J_{3}$, $\bm{q}$ is along the diagonals of squares and the skyrmion has the shape shown in Figs. 2(c,f).

An important difference between skyrmions for positive and negative spin stiffness is the form of the skyrmion-skyrmion interaction potential, $U_{12}(r)$. For $\rho<0$, the potential oscillates, which leads to repulsion or attraction depending on the distance, $r$, between skyrmions Leonov2015 ; Lin2016 . Similar considerations show that for $\rho>0$, $U_{12}(r)$ remains positive and decreases monotonically at large $r$. This is also the case for skyrmions in chiral magnets which repel each other Roessler2011 ; Rozsa2016 because they all have the same helicity angle describing the direction of the in-plane spin components Nagaosa2013 . In easy-axis magnets with competing interactions, the skyrmion helicity is arbitrary and the repulsion for equal helicities changes to attraction for opposite helicities (see Fig. 3(a)).

Because of the attraction, a multi-$Q$ skyrmion has a lower energy than $Q$ elementary skyrmions and can be considered as their bound state. The fact that $U_{12}(r)$ has minimum at $r=r_{0}$ (see Fig. 3(a)) suggests that the skyrmion with a large $Q$ occupies the area $\sim Q\pi r_{0}^{2}$, so that the skyrmion radius $R\sim r_{0}Q^{1/2}$. Surprisingly, this is not the case: the topological charge and energy densities of the multi-$Q$ skyrmion are concentrated in a ring of radius $R\sim r_{0}Q$ (see Figs. 4(a,c)).

Figure 3b shows that the energy per skyrmion, $E_{Q}/Q$, decreases with increasing $Q$ and approaches a constant, because the width of the ring and the length of the ring segment occupied by one skyrmion become $Q$-independent. The energy of skyrmion in the ring is significantly lower than that of the elementary skyrmion. This “mass defect” drives the fusion of skyrmions, which increases the magnitude of the skyrmion magnetic moment, $M_{z}=\sum_{i}(S_{i}^{z}-1)<0$, counted from the positive magnetic moment of the FM state: for $Q$ elementary skyrmions $M_{z}\propto-Q$, whereas for the ring with topological charge $Q$, $M_{z}\propto-Q^{2}$. A magnetic field applied in the positive $z$ direction would lead to fission of multi-$Q$ skyrmions into skyrmions with smaller topological charges.

For $Q\gg 1$, we can neglect the ring curvature and consider a straight domain wall with the spiral spin structure, $\bm{S}=(\sin\theta(x)\cos qy,\sin\theta(x)\sin qy,\cos\theta(x))$, separating the $S_{z}=-1$ FM state at $x<0$ from the $S_{z}=+1$ FM state at $x>0$. The length of the wall in the $y$ direction is $L_{y}=2\pi R=\frac{2\pi Q}{q}$. Using a variational Ansatz for the wall shape, $\cos\theta(x)=-\tanh(\varkappa x)$, where $\varkappa$ is the inverse domain wall width, we obtain

[TABLE]

Minimization with respect to $q$ and $\kappa$ gives $\kappa=|q|=\left[\frac{K}{2b_{1}+\frac{1}{3}b_{2}}\right]^{\frac{1}{4}}$ and

[TABLE]

Note that the first term in Eq.(4) is the lower bound for the energy of the multi-$Q$ skyrmion in the nonlinear sigma model Belavin1975 and that $\varkappa$, $q$ and $E_{Q}/|Q|$ are indeed independent of $Q$. The domain wall stability requires $6b_{1}+b_{2}>0$ (or $4J_{2}+16J_{3}>J_{1}$) and this result can be shown to be independent of the orientation of the wall with respect to the crystal axes. The binding energy makes multi-$Q$ skyrmions more stable than elementary skyrmions footnote2 .

*Merons: * So far we discussed magnets with an easy-axis anisotropy. For an easy-plane anisotropy ($K<0$), the topological defect with $Q=1$ is a bound state of vortex and antivortex (see Fig. 5). The sign of the out-of-plane magnetization in the core of the vortex is opposite to that in the antivortex core, so that each half of the skyrmion, called meron, has $Q=\frac{1}{2}$.

The emergence of vortices and antivortices is related to the spontaneous breaking of $O(2)$ rotational symmetry by the uniform in-plane magnetization. The bi-meron configuration in the FM state with $S_{x}=+1$ can to a good approximation be obtained from the skyrmion configuration for the $S_{z}=+1$ state by $\pi/2$-rotation around the $y$ axis: $(S_{x},S_{y},S_{z})\rightarrow(S_{z},S_{y},-S_{x})$ (see Supplementary material for details), which explains the similarity between the stability diagrams for skyrmions and meron pairs in the $J_{2}-J_{3}$ and $J_{2}-K$ planes (Figs. 1(a,b) and Fig. 2 of Supplemental material).

Near a LP the distance between merons is large giving rise to two distinct peaks in the distribution of topological charge density (Figs. 5 (a,d,c,f)). The fractionalization of skyrmion occurs because the two-dimensional Coulomb potential that confines vortex to an antivortex Kosterlitz1973 ,

[TABLE]

$r_{0}$ being the meron radius, vanishes at the LP, $\rho=0$.

Figure 6(a) shows that at zero temperature the deconfinement of merons is incomplete: at the LP the optimal distance between merons, $R_{12}$, remains finite, because the meron-meron interaction energy, $U_{12}(r)$, has a minimum even at $\rho=0$ (Figs. 6(b,c)). The bi-meron molecules will, however, dissociate at $T\cancel{=}0$.

At large distances bi-merons interact via two-dimensional dipole-dipole interactions Gross1978 resulting in formation of multimeron bound states. For $Q\gg 1$, the minimal-energy configuration is the square lattice formed by merons (Figs. 4 (c,d)), analogous to the simple cubic lattice of half-skyrmions in nuclear physics Kugler1989 .

*Conclusions: * We showed that metastable skyrmions and merons can exist in two-dimensional ferromagnets with conventional centrosymmetric lattices. Magnetic frustration required for stabilization of these topological excitations is considerably weaker than the one that destabilizes the FM state. Skyrmions in easy-axis magnets attract each other and in absence of magnetic field form long lines or rings facilitating their observation. In easy-plane magnets, skyrmions transform into pairs of merons, which dissociate at the Lifshitz point. Our results are directly relevant for square-lattice ferromagnets Shannon2004 , and, qualitatively, they hold for other lattice types including layered antiferromagnets with a weak AFM coupling between FM layers. In particular, bi-merons can exist in the collinear phase of the easy-plane triangular antiferromagnet NiBr2 Day1976 . The bound states of magnetic skyrmions and merons show similarities to the skyrmions with large baryon numbers and crystals of half skyrmions in the original Skyrme model.

The authors are grateful to L. Balents for fruitful discussions. MM acknowledges the hospitality of UNSW and the FOM financial support.