Unconventional topological Hall effect in high-topological-number skyrmion crystals

Skyrmions with the topological number $Q$ equal an integer larger than 1 are called high-topological-number skyrmions or high-$Q$ skyrmions. In this work, we theoretically study the topological Hall effect in square-lattice high-$Q$ skyrmion crystals (SkX) with $Q=2$. As a result of the emergent magnetic field, Landau-level-like electronic band structure gives rise to quantized Hall conductivity while the Fermi energy is within the gaps between adjacent single band or multiple bands intertwined. We found that different from conventional ($Q=1$) SkX the Hall quantization number increases by $1/Q$ in average while the elevating Fermi energy crosses each band. We attribute the result to the fact that the Berry phase ${\cal{C}}$ is measured in the momentum space and the quantum number of a single skyrmion $Q$ is measured in the real space. The reciprocality does not affect the conventional SkX because $Q=1=1/Q$.


I. INTRODUCTION
Magnetic skyrmion 1-7 is the spin vortex structure in ferromagnets with a nontrivial topological number Q of the two-dimensional spinor quantum field n (r), with Q equal to the surface integral of the solid angle of n (r), i.e. Q = qd 2 r with q = 1 4π n · ∂n ∂x × ∂n ∂y . Concerning the symmetry of the skyrmion, one can write the spin field of the skyrmion in a general form of n (r) = [sin Θ (r) cos Φ (ϕ) , sin Θ (r) sin Φ (ϕ) , cos Θ (r)], with r and ϕ the polar coordinate in the real space, Θ and Φ polar and azimuthal angles of the local spin.
The topological number of a single skyrmion can be obtained as While the center spin points upward and the edge spin points downward, or vice versa, we can have the vorticity of the skyrmion m = [cos Θ (r)] r=0 r=∞ = ±1. While the skyrmion whirls in the pattern of Φ (ϕ) = ξϕ + γ, the topological number is explicitly expressed by Q = mξ and γ determines the helicity of the skyrmion. Conventional magnetic skyrmions have Q = ±1, with the sign distinguishing skyrmions and antiskyrmions. By varying the whirling period ξ of the azimuthal angle of the local spin Φ, high-topological-number skyrmions with Q > 1 are theoretically predicted [8][9][10][11] . While identical skymions form a spatially periodic array, we obtain a skyrmion crystal (SkX). Top view of the square-lattice high-topological-umber SkX with Q = 2 and that of the conventional SkX with Q = 1 are shown in Figs. 1 and 2, respectively.
The conventional SkX with Q = 1 spin vortices forming two-dimensional hexagonal, triangular, or square crystal structures were recently discovered in magnetic metal alloys, insulating multiferroic oxides, and the doped semiconductors 2-6,12-23 . Their spin structures can be detected by neutron scattering 3 and Lorentz transmission electron microscopy 4 . The Hall effect measurements in the SkX metals establish the physics of emergent electrodynamics [24][25][26] .
Since its discovery, the SkX has attract intensive interest due to its fundamental meaning and potential application in topological computers. The generation, deleting, and dynamics of isolated skyrmion and SkX have been investigated by theory, numerical simulation, and experiments. Among these, the topological Hall effect of the SkX resulting from the emergent magnetic field of the nontrivial spin field has attract a lot of attention.
Recently, various works have considered the creation and manipulation of hightopological-number skyrmions or SkX with Q = 2. Some of the works we happen to come conventional SkX. They found that the Hall conductivity quantized at even-integer times of e 2 /h and attributed the result to the topology of the crystal. This is because the topological Hall conductivity is a combined result of the topology of a single skyrmion and that of the crystal. In the following sections, we show results and discussions of the approach. The other parts of the work is organized as follows. Sec. II is the theory and technique. Sec. III is numerical results and discussions. A conclusion is given in Sec. IV.

II. MODEL AND FORMALISM
We consider the free-electron system coupled with the background spin texture n i by Hund's coupling. n i is the atomic-lattice-discretized version of the magnetic spin field n (r) introduced in the previous section. Hamiltonian of the electron is given by the doubleexchange model 6 , where c i = (c i↑ , c i↓ ) T is the two-component annihilation operator at the i site, c † i is its creation counterpart. t is the hopping integral between nearest-neighbor sites. We assume it the same at all lattice sites. J is the strength of the Hund's coupling between the electron spin and background spin texture. σ denotes the Pauli matrix.
In the limit that J ≫ t, the spin of the hopping electron is forced to align parallel to the spin texture. Because there is no other spin-flipping mechanism, hopping can only occur between electrons with parallel spins. We can arrive at a "tight-binding" model with the effective transfer energy site dependent and equal to t multiplied by the magnitude of the spin overlap 6 . Strength of the spin overlap between sites i and j can be obtained by χ i | χ j with |χ i the wave function of the conduction electron at site i corresponding to the localized spin n i . Using spherical coordination in the spin space of the electron The effective Hamiltonian can be expressed as 6 with Here is the spinless creation (annihilation) operator at the i site. Considering the periodic structure of SkX, Eq. (4) can be taken as a "tight-binding" model of electrons on a lattice of giant unit cells. Each unit cell corresponds to a single skyrmion. We can rewrite the effective model as where the summation of i goes through the complete atomic lattice, the summation of δ goes through the four nearest-neighbor sites of the i-th site, site i locates on the s-type sublattice and site i + δ locates on the s ′ -type sublattice, and t s,s ′ = t ij eff with j = i + δ. Schematics of the model is shown in Fig. 3. Four examples of t s,s ′ , i.e., t BA , t UA , t EA , and t FA are shown in the figure with the corresponding δ are −aê x , −aê y , aê x , aê y , respectively. There are two lattice constants: one is a measuring the distance between adjacent atoms; the other is 2λ measuring the size and interval of the skyrmions. Without loss of generosity, radius of the skyrmion λ is set to be 2.5a and a single skyrmion consists of 5 × 5 = 25 atoms. By Fourier transformation of Eq. (6), we can obtain which is diagonal in the k-space and a 25 × 25 matrix in the sublattice space. We consider a background spin texture n (r) made of a high-topological-number square-lattice SkX. Each skyrmion has a nontrivial topological number Q = 2. The skyrmion profile is well assumed as Θ (r) = π (1 − r/λ) for r < λ, Θ (r) = 0 for r > λ, and Φ (ϕ) = 2ϕ + γ. It is obvious from Eqs. (4) and (5) The topological Hall conductivity at zero-temperature calculated from the Kubo formula is in which Ω is the first Brillouin zone. Eq. (10) is actually equal to the total Berry phase below the Fermi energy in the unit of e 2 /h. A brief proof is provided in the appendix. In direct diagonalization of the Hamiltonian, the phase factor of the eigenspinor is not definite, which is also called gauge of the state. Different gauge induces a sign difference in the Berry phase of a particular band. Usually, the gauge sets the n-th row of the eigenspinor to be unity if one directly calculate the Berry phase from geometry of the eigenstate. However, this procedure is not necessary because the Hall conductivity formula always has the bra and ket of the eigenspinor in pair. One should only be careful to use the same gauge throughout one work.

III. RESULTS AND DISCUSSIONS
Numerical results of the high-topological-number SkX with Q = 2 are given in Fig. 1.
For comparison, those of the conventional SkX with Q = 1 are given in Fig. 2. We can see that n x and n y of the Q = 2 SkX have a four-leaf structure and n y is n x clockwisely rotating by π/4; n x and n y of the Q = 1 SkX have a double-leaf structure and n y is

IV. CONCLUSIONS
In the limit of large Hund's interaction, the free-electron system coupled with the background spin texture of the SkX can be approximated to a spinless "tight-binding" model with the local hopping energy determined by the spin field of the SkX. By taking each skyrmion as a giant unit cell, the model is periodic and can be exactly diagonalized. From the bands and giant spinor states obtained, the Berry phase and topological Hall conductivity can be obtained, which are actually the same at zero temperature. The technique does not depend on the detailed structure of each skyrmion. Therefore, we extend the previous approach on the conventional SkX to the high-topological-number SkX and found that the Berry phase C of a single band varies between 0 and 1 for all the bands except a 3 for E 8 and a −2 for E 9 , which averages to be 1/Q and hence the Hall conductivity increases with a step smaller than the conventional SkX. We attribute the result to the fact that the Berry phase C is measured in the momentum space and the quantum number of a single skyrmion Q is measured in the real space. The reciprocality does not affect the conventional SkX because Because m = n, nk| mk = 0. Hence, we have Using m(m =n) The second term on the right hand side of Eq. (15) It is equal to zero because the expectation value of the conjugate of any operator is equal to that of the operator itself. The first term of Eq. (15) is which multiplied by − i 2π and integrated over the first Brillouin zone is just the total Berry phase of the bands below the Fermi energy defined by Eqs. (8) and (9). Because each skyrmion has a topological number Q = 2, the four-skyrmion array has the total topological number Q t = 8. (e) (f) Band structure of the lowest ten bands in the vicinity of the M (π/λ, π/λ) point in the first Brillouin zone. Serial number of the bands is labeled by E 1 to E 10 counting from the lowest to the highest. The Berry phase C of each band is given beside its serial number. (g) Zero-temperature topological Hall conductivity as a function of the Fermi energy, which demonstrates plateaus in the energy gap between adjacent single band or intertwined multiple bands. It can be seen that averagely each band contributes a Berry phase of 1/Q to the topological Hall conductivity.