Magnetic Topological Kagome Systems

The recently discovered quantum material Co$_3$Sn$_2$S$_2$ shows a quantum anomalous Hall (QAH) effect on the Kagome lattice in the proximity of a ferromagnetic-antiferromagnetic transition. The ferromagnetism occurs in the $z$ direction and the antiferromagnetic transition corresponds to a $120^o$ ordering of the spins in the $xy$ plane. We build a Kagome model to describe the Co-atoms taking into account both the presence of localized and conduction electrons which are coupled through a strong Hund's coupling along $z$-direction. We include an intrinsic spin-orbit coupling for the conduction electrons assuming 2/3 on-site occupancy. We find a topological transition from a QAH insulator in the ferromagnetic phase to a quantum spin Hall (QSH) phase in the antiferromagnetic phase. We generalize the analysis to describe Co$_3$Sn$_2$S$_2$ showing a non-zero ferromagnetic fraction in the antiferromagnetic phase. We show how the anomalous Hall conductivity can smoothly evolve with the ferromagnetic fraction due to fluctuations in the direction of the Hund's coupling.

The recently discovered quantum material Co3Sn2S2 shows a quantum anomalous Hall (QAH) effect on the Kagome lattice in the proximity of a ferromagnetic-antiferromagnetic transition. The ferromagnetism occurs in the z direction and the antiferromagnetic transition corresponds to a 120 o ordering of the spins in the xy plane. We build a Kagome model to describe the Co-atoms taking into account both the presence of localized and conduction electrons which are coupled through a strong Hund's coupling along z-direction. We include an intrinsic spin-orbit coupling for the conduction electrons assuming 2/3 on-site occupancy. We find a topological transition from a QAH insulator in the ferromagnetic phase to a quantum spin Hall (QSH) phase in the antiferromagnetic phase. We generalize the analysis to describe Co3Sn2S2 showing a non-zero ferromagnetic fraction in the antiferromagnetic phase. We show how the anomalous Hall conductivity can smoothly evolve with the ferromagnetic fraction due to fluctuations in the direction of the Hund's coupling.
Introduction.-When applying a magnetic field, the quantum Hall effect gives rise to an insulating behavior in the bulk of a material and is characterized by chiral edge states [1][2][3] which show a quantized Hall conductance. Bulk properties are described through a topological invariant, the Chern number [4]. The QAH effect, as originally introduced by Haldane [5], corresponds to a generalization of the quantum Hall effect on the honeycomb lattice with tunable Berry phases opening a gap for the Dirac fermions and breaking time-reversal symmetry, such that a unit cell yet shows a zero net flux. This model finds applications in quantum materials [6,7], light [8,9] and cold atom systems [10,11], and was developed in other geometries such as the Kagome lattice [12,13]. For practical realizations, it is important to find intrinsic ferromagnetic QAH systems with topologically non-trivial band gaps produced by spin-orbit coupling mechanisms [14]. The weyl-semimetal quantum material Co 3 Sn 2 S 2 has recently attracted a lot of attention experimentally in relation with the QAH effect [15,16]. The pure cobalt is known to have a Curie temperature of around 1388K associated to ferromagnetism. Here, a layered crystal structure with a Co-Kagome lattice in this material develops a perfectly out-of-plane ferromagnetic phase (along z direction) and an almost quantized Hall conductivity under 90K. Between 90K and 175K, the ferromagnetic fraction smoothly decreases while an in-plane antiferromagnetism (related to the xy plane) progressively develops and the anomalous Hall conductivity now evolves with the ferromagnetic fraction along z direction [15,16]. Motivated by these findings, we introduce magnetic topological Kagome systems comprising conduction electrons coupled to localized spins. The magnetic transition is described through localized spins and itinerant electrons will develop topological energy bands, as a result of the spin-orbit coupling, with occupancies that will depend on the magnetism.
We take into account the presence of 3d-electrons on the Co-atoms [17] which reveal both a localized and itin-erant character due to a structure with two orbitals t 2g and e g . While Kondo lattices have been shown to induce topological phases, referred to as topological Kondo insulators [18], here itinerant and localized electrons are coupled through a strong Hund's ferromagnetic mechanism. The presence of a Hund's coupling generally plays a key role in these multi-orbital electronic systems [13,19], tending to favor ferromagnetism. In our model, the itinerant electrons and the localized spins-1/2 (on each atom forming core spins-1/2) are coupled through a strong Hund's ferromagnetic coupling along z direction, which then induces an Ising J z ferromagnetic interaction between nearest-neighbor localized spins according to the double-exchange Zener mechanism. We also introduce an in-plane antiferromagnetic correlation J xy between the core spins which is produced by Mott physics and the Anderson-Goodenough-Kanamori rule for virtual electron-mediated interactions between the half-filled orbitals (associated to the localized spins) [20].
To study the magnetic transition, first we build a semi-classical approach and a quantum spin wave theory which reveals a direct transition when J * xy = 2J z , as shown in Fig. 1. In the spin-wave spectrum, a flat band touches the classical ferromagnetic state when approaching the transition, destabilizing the ferromagnetic alignement and stabilizing the antiferromagnetic 120 o spin ordering in the xy plane. The flat band then moves to higher energy because the azimuthal angle φ i associated to each spin on the Bloch sphere is now only a global quantity, since we fix φ i −φ j = 2π/3 in radians (or 120 o ), and the polar angle of each spin jumps to θ i = π/2. The magnetization along z direction jumps discontinuously to zero. It becomes continuous if we apply a small magnetic field. Then, we describe temperature effects in Co 3 Sn 2 S 2 below 175K by decreasing the ferromagnetic J z coupling or equivalently by increasing the antiferromagnetic coupling J xy if we set J z = 1. Taking into account fluctuations in the direction of the Hund's coupling produces a (Gaussian) distribution on the value of J z . We will then show that the (average) system's magnetization in z direction smoothly reduces to zero after the transition producing the progressive canting of the spins, such that the statistically averaged Chern number follows the ferromagnetic fraction. A similar effect could also be stabilized through a Dyaloshinkii-Moriya interaction [21].
Model.-The important mechanism leading to the anomalous Hall effect in our model is the intrinsic spinorbit coupling (SOC). Kane and Mele showed that the SOC can produce a QSH phase on the honeycomb lattice [22]. This phase (called a Z 2 topological insulator) is characterized by different edge modes for both electron species (up and down): each of electron species move in opposite direction producing a vanishing Hall conductance when the number of up and down electrons is identical. A QSH effect was also predicted and observed in two-dimensional Mercury [23,24] and in threedimensional Bismuth [25] quantum materials. In the Kane-Mele model, strong interaction effects in the Mott phase favor an in-plane antiferromagnetic phase [26], justifying that here we choose an antiferromagnetic XY spin coupling for the core spins in addition to the ferromagnetic coupling J z induced by the Hund's term. Below, we include the effect of the competition between the two magnetic channels J xy and J z , onto the probabilities of occupancies P ↑ and P ↓ for the spin up and spin down itinerant electrons in the canonical ensemble, assuming a strong Hund's coupling: where S z i refers to the z-magnetization of the localized spin-1/2 at site i and c † iα creates a conduction electron at site i with spin polarization α =↑, ↓. The J z spin coupling is formally induced by the Hund's coupling h c . Below, we address the case where the on-site occupancy for itinerant electrons is 2/3. If the system is spin-polarized with one electron species, the Fermi energy lies in the gap between the middle and the upper energy bands in Fig. 2, such that the system will show a quantized Hall response. To tackle the ferromagnetic-antiferromagnetic transition, we may introduce the number of particles associated to up and down species, as N ↑ = P ↑ N e and N ↓ = P ↓ N e with the number of electrons N e = N ↑ + N ↓ satisfying N e = 2 3 N a and N a being the number of atomic sites. When the number of up and down electrons is equal, then 2 of the 6 energy bands are filled, corresponding to the lowest energy band associated to each spin species. A link between SOC and QAH effect on the Kagome lattice was studied for Cs 2 LiMn 3 F 12 [14] and in relation with the occurrence of chiral spin states [13].
When J xy J z , the spins of the electrons adiabatically follow the polarization of the core spins due to the strong h c coupling, such that c i↑ = c i , and one can build an effective spin polarized electron model with only spinup electrons. The tight-binding model for the spin-up electrons takes the form: is the nearest-neighbor hopping amplitude on the Kagome lattice and λ the intrinsic spin-orbit coupling projected onto the spin-up electronic states [14]. Here, ν ij = +1(−1) if the electron jumps counterclockwise (clockwise) inside the triangle of the Kagome lattice containing sites i and j, and the symbol i; j refers to a coupling between nearest neighbors. We observe that the ferromagnetism should not modify the hopping amplitude of spin-up electrons compared to the case where S z i = 0, implying that t ↑ = t = t χ i |χ j , with |χ i representing a spin eigenstate at site i with θ i = 0.
To make a link with the Haldane model on the Kagome lattice, we can then re-write −(t + iν ij λ) = −re iΦνij /3 with r = √ t 2 + λ 2 and Φ = 3arg(t + iλ). In Fig. 1, in a triangle there is a flux Φ breaking time-reversal symmetry and in an hexagon (honeycomb cell) there is a flux −2Φ such that globally on a parallelogram unit cell represented by the vectors a 1 and a 2 the total net flux is zero. In wave-vector space, where occupancy 2/3 implies that the two lowest bands of the Kagome energy spectrum are occupied for spin-polarized electrons, we then check the presence of a QAH effect for an illustrative value of Φ = 3π/4, see Fig. 2(a). The 3 energy bands reflect the three distinct sites A, B, C in Fig. 1. The lowest energy band is described by a Chern number C l = sgn(sin Φ) = +1, the middle band has a Chern number zero, and the upper band shows a Chern number C u = −C l . It is important to remind that the middle band becomes perfectly flat for Φ = π/2 and it touches the bottom of the lowest band for Φ = 0, suppressing the QAH effect when λ = 0. In the Supplementary Ma- terial [27], we describe the method to evaluate the band structures, edge mode properties as well as the associated Chern numbers. We also present the evaluation of the local density of states of Fig. 2(b).
Changing the on-site occupancy it is possible to observe a metallic ferromagnetic topological phase and in that case the Chern number would not be quantized to unity [28,29].
Magnetic Transition.-Now, we study quantitatively the magnetic properties of the system in the presence of the couplings J z and J xy . The localized spins are described by the Hamiltonian: with (J z , J xy ) > 0, such that the classical energy on the Bloch sphere representation is: (3) To minimize the magnetic energy, we find that θ i = θ for all values of J xy /J z and in the antiferromagnetic phase φ i − φ j = 2π/3. Therefore, the classical energy takes the simple form E = 1 4 i;j [(−J z + 1 2 J xy ) cos 2 θ − Jxy 2 ]. For 2J z > J xy the energy reaches its minimum for θ = 0 or θ = π, corresponding to E = −N a J z and to a ferromagnetic state of the spins along z direction. For 2J z = J xy , the ground state energy takes the value E = − Na 2 J xy for all the values of θ. For J xy > 2J z , the ground state energy keeps the same value − Na 2 J xy if the spins now point in the xy plane with θ i = θ = π/2. Then, we study the effect of a small applied magnetic field h a along z direction which favors the classical minimum θ = 0 when 2J z + h a ≥ J xy , corresponding to an energy E(θ = 0) = −N a (J z + h a ). If J xy > 2J z + h a , then E is minimum for θ such that resulting in E = − Na shown in Fig. 1 (top right). While the SU(2) Heisenberg antiferromagnetic Hamiltonian shows a quantum spin liquid on the Kagome lattice [30], here magnetic ordered phases are stable classically through the form of H S .
To study quantum effects to the magnetic energy spectrum, we perform an Holstein-Primakoff transformation in the case of a spin-s and analyse the spin wave spectrum both in the ferromagnetic phase along z direction and in the antiferromagnetic phase with spins in the xy plane forming an angle of 120 o , adapting the calculation of Ref. [31] for the present situation. In the ferromagnetic phase, we check the presence of a quadratic dispersion relation in the vicinity of |k| = 0 with an energy of 2sJ z (2−γ F + γ F 2 |k| 2 ) where γ F = J xy /J z . This dispersive branch approaches the classical energy when J xy ∼ 2J z and corresponds to adiabatic deformations of the phase difference φ i − φ j for nearest neighbors around zero. In addition, we check the presence of a flat band corresponding to alternating 0 and π values of the phases φ i for the six sites forming an honeycomb cell [31]. The flat band energy also meets the classical energy at the phase transition. Taking into account the entropy at finite temperature, corresponding to degenerate states associated to the (free) angles φ i , then the free energy of this flat band should be lowered compared to the classical ferrromagnetic state when J xy = 2J z , justifying that the ferromagnetic ground state is not the correct classical ground state. The spin system rotates in the xy plane forming an antiferromagnetic phase where spin vectors order at 120 o . In the antiferromagnetic phase, the spins lock in the xy plane according to φ i −φ j = 2π/3 in radians, and the flat band now moves at higher energy as shown in Fig. 3. In this case, the flat band would rather correspond to outof-plane staggered spin excitations. The energy of the spin waves for the lowest dispersive band, for |k| 1, is given by 2J xy s √ 1 − 2γ AF |k| with J z = γ AF J xy and γ AF = 1/γ F . The linear dispersion of the spin waves corresponds to adiabatic deformations of φ i − φ j for nearest neighbors around the value 2π/3 in Eq. (3).
Topological Transition.-Here, we describe the effect of the magnetic transition on the conduction electrons for 2/3 on-site occupancy, first assuming that the fluctuations in the Hund's coupling direction are small. If the amplitude of h c is sufficiently large compared to t, we can write s z i = S z i [20], where s z i = (c † i↑ c i↑ − c † i↓ c i↓ )/2 represents the conduction electron's magnetization. Writing s z i = (P ↑ − P ↓ )/2 with P ↑ + P ↓ = 1, in that case we predict: P ↑ = 1 and P ↓ = 0 if J xy < 2J z corresponding to the ferromagnetic order along z direction and P ↑ = P ↓ = 1/2 if J xy > 2J z in the antiferromagnetic phase. When the spins align in the xy plane they do not modify the motion of the itinerant electrons when S z i = 0 in Eq. (1). This produces a QAH-QSH transition at the magnetic transition associated with a change of band topology. In the ferromagnetic case, as studied above, the lowest and middle bands associated with spin-up particles are filled, whereas in the QSH phase the lowest band associated to each spin species is completely filled, whereas middle and upper bands are empty (see Fig. 2). The QSH phase occurs because the spindown particles are described by an opposite phase −Φ compared to the spin-up particles on a triangle, if we generalize the form of the spin-orbit coupling as in the Kane-Mele model, , which is reminiscent of an atomic spin-orbit coupling L z s z with L being the angular momentum of electrons on a lattice. The spin-up and spin-down electrons are described by the same nearest-neighbor hopping amplitude in the antiferromagnetic phase such that t ↑ = t ↓ = t. The core spins act as a local magnetic field which breaks time-reversal symmetry if the net magnetization on a triangle is nonzero. In the antiferromagnetic phase, the sum of the three arrows describing the spins in a triangle is zero, therefore time-reversal symmetry is preserved if S z i = 0 and a Z 2 topological order can develop, where the Chern number of each lowest band is equal to C ↑ l = −C ↓ l = +sgn(sin Φ). For Co 3 Sn 2 S 2 , it is important to emphasize that the ferromagnetic fraction varies smoothly with temperature or here the ratio J xy /J z , which breaks time-reversal symmetry. In our approach, it produces a QAH conductance at the edges which is proportional to (e 2 /h)(P ↑ − P ↓ ) = 2(e 2 /h) s z i = 2(e 2 /h) S z i = (e 2 /h) cos θ (J z0 ); here, h corresponds to the Planck constant and e is the charge of an electron. In the last equality, we assume that we include the effect of fluctuations in the direction of the Hund's coupling. Such fluctuations induce a slightly disordered distribution of J z parameters that we study globally, with the same mean J z0 and with the same variance σ at each site. The symbol ... (J z0 ) refers to an ensemble averaged value. These variations on the value of h c could be produced by temperature effects generating a random (noisy) Hund's coupling along z direction. This variance could also be stabilized by a Dyaloshinskii-Moriya term D ij S i × S j producing weak ferromagnetism along z direction in the antiferromagnetic 120 o phase [21].
Then, we study the effect of such fluctuations on bulk properties. We take the distribution of J z couplings as Gaussian, P (J z ; J z0 ) = 1 √ 2πσ e −(Jz−Jz0) 2 /2σ 2 with a variance σ J z0 (or much smaller than k B T for Co 3 Sn 2 S 2 ). For 2J z0 = J xy , now the system can show a coexistence between ferromagnetism along z direction and antiferromagnetism in the xy plane. Introducing m z = 2S z i = cos θ, then m z = +1 if J xy < 2J z and m z = 0 if J xy > 2J z . The average value of m z including the Gaussian fluctuations is given by: where erfc corresponds to the complementary error function. For the conduction electrons, if J xy < 2J z we have a Chern number C = C ↑ l = +1 and for J xy > 2J z we have C = C ↑ l − C ↓ l = 0. Therefore, we introduce the averaged Chern number In Fig. 4, we show the behavior of the averaged Chern number and averaged magnetization. Eq. (6) relates the progressive evolution of the magnetization along z-axis in the bulk with the (averaged) Chern number, as observed in Refs. [15,16]. For Co 3 Sn 2 S 2 , the ensemble average could also refer to an average on different regions of a sample. We reproduce a bulk-edge correspondence where the conductance at the edges takes the form (e 2 /h) C (J z0 ). We observe that a transition from QSH to QAH effect was also reported in HgTe materials when doping with random magnetic Mn dopants [32], and in thin films of (Bi,Sb) 2 Te 3 doped with Cr-atoms [33].
To summarize, we have built a model taking into account both localized electrons giving rise to a magnetic transition and conduction electrons producing topology of Bloch bands on the Kagome lattice. We hope that this may participate to the understanding of the quantum material Co 3 Sn 2 S 2 . We observe that other interesting topological Kagome materials such as Fe 3 Sn 2 Kagome bilayer systems have been found these last years [34].
We acknowledge discussions with Joel Hutchinson, Philipp Klein, Alexandru Petrescu, Nicolas Regnault and Jakob Reichel. This work is funded through ANR BOCA and the Deutsche Forschungsgemeinschaft via DFG FOR2414 under Project No. 277974659.

Chern number calculation, edge modes, and local density of states
Here, we provide information on the calculations of Chern numbers, edge modes, and local density of states (LDOS).
Chern number and Hamiltonian.-The Chern number is evaluated from the formula (see [28]) The integral is taken over the Brillouin zone and the summation takes into account all the energy bands labelled by n = {1, 2, 3} and spin polarization σ = {↑, ↓}. It is restricted to the states with energy inferior to the Fermi energy E F . A n,σ (k) = −i n, σ, k| ∂ k |n, σ, k is the Berry gauge field with |n, σ, k the energy eigenstate associated to the n th energy band E n,σ (k) of the spin σ electron species. The Fermi energy E F is determined by the number of electrons N e = (N ↑ + N ↓ ) = N e (P ↑ + P ↓ ), where we take into account the magnetism from localized spins. Referring to the classical analysis on the magnetism in the Letter, we have the relations σ z i = (P ↑ − P ↓ )/2 = S z i with P ↑ + P ↓ = 1, such that P ↑ = (1/2)(1 + cos θ) and P ↓ = (1/2)(1 − cos θ). In the ferromagnetic phase, P ↑ = 1 and P ↓ = 0 and for the antiferromagnetic phase, if S z i = 0, then P ↑ = P ↓ = 1/2. The effect of the strong Hund's coupling is tackled through the on-site occupancies P ↑ and P ↓ . This is equivalent to include the effect of a sufficiently strong Hund's coupling −h c i S z i σ z i in the Hamiltonian written in the wave-vector basis such that σ z i = S z i , and this has the advantage of writing a common Fermi energy for the total system.
Edge modes and LDOS.-Here we evaluate the eigenenergies E Ψ and the eigenstates Ψ of the Hamiltonian, for a lattice cylinder geometry with periodicity along direction a 1 (see Fig. 5(a)), as in Ref. [28]. We consider the h a → 0 limit and we take into account the Hund's coupling as a constraint on N ↑ and N ↓ , as described above. We write the Hamiltonian H = σ H σ and we apply a partial Fourier transform on H σ . A numerical computation gives the eigenenergies associated to the spin σ electron species. In Fig. 5(b), we show the band structure associated to a given spin species for the case where the middle band is flat, for Φ σ = π/2.
The local density of states associated to the spin σ species given by ρ σ (E, r) ≡ Ψσ δ(E − E Ψ )| r|Ψ σ | 2 is sketched in Fig. 5(c) at E = −t and in Fig. 5(d) at E = +t, corresponding to the observed edge modes. The energy and the expression of these edge modes can be analytically determined. We write the eigenstates as a superposition of localized states on each atom : Ψ σ (q 1 ) ≡ α=A,B,C m φ α,m,σ (q 1 ) |α, m, σ . From the numerical evaluation, we assume that φ A,m (q 1 ) = 0 and we use the ansatz φ B,C (m) = λ m σ φ B,C (0). It gives the energies of the 2 edge modes E ± and the associated values of λ ±,σ , in accordance with the analytical solution: Whether |λ ±,σ | > 1 or |λ ±,σ | < 1 determines at which edge of the system the mode is localized. We notice that λ ±,σ (q 1 ) = 1/ λ ±,σ (−q 1 ) and that λ ±,σ (q 1 ) = 1/ λ ±,σ (q 1 ) with σ =↓ (↑) if σ =↑ (↓). This and the fact that the energy of each eigenmode is a cosine centered around q 1 = 0 predicts one eigenmode located at each edge of the system in the ferromagnetic phase and two counter-propagating eigenmodes at each edge of the system in the antiferromagnetic phase.