Intrinsic topological magnons in arrays of magnetic dipoles

We study a simple magnetic system composed of periodically modulated magnetic dipoles with an easy axis. Upon adjusting the geometric modulation amplitude alone, chains and two-dimensional stacked chains exhibit a rich magnon spectrum where frequency gaps and magnon speeds are easily manipulable. The blend of anisotropy due to dipolar interactions between magnets and geometrical modulation induces a magnetic phase with fractional Zak number in infinite chains and end states in open one-dimensional systems. In two dimensions it gives rise to topological modes at the edges of stripes. Tuning the amplitude in two-dimensional lattices causes a band touching, which triggers the exchange of the Chern numbers of the volume bands and switches the sign of the thermal conductivity.

show screenshots obtained from the energy minimization of the total energy consisting of magnetic dipolar interactions among all dipoles and an easy axis anisotropy energy (Eq.1 in the paper) in arrays made out of 50 point dipoles. We consider the case of chains with κ = π/2 and hight anisotropy K > K Λ c in (a) and (b), while in (c) we show the collinear state that result from minimizing chains at low anisotropy. (d) shows the case of a stripe at K > K Λ c and Λ ∼ 1. For small amplitudes Λ ∼ 0.1, fig. 1(a) shows the antiferromagnetic order. For Λ ∼ 1, figs. 1(b) and (d), a dimerized magnetic arrangement can be seen in these finite systems.
In Figure 2 we show chains with modulation of longer wavelenght. In cases with high anisotropy (a) and (b) show screenshots of chains with κ = π/4 and π/8 respectively. The collinear state at K < K Λ c and κ = π/8 is shown in (c). Figure 3 shows the total energy u, of modulated chains and stripes as a function of Λ at K > K Λ c . In all cases, at low Λ the magnetic configuration with the lowest energy consist in the antiferromagnetic state (AF ). For intermediate Λ (∼ 1) AF becomes energetically expensive and the two dimerized configurations 'up-up-down-down' and 'down-upup-down' (D1 and D2 ) are the lowest energy states. D1 and D2 are very close in energy.
B. Band spectrum of stripes for larger modulations. Figure 4 shows results for the band spectrum of chains and stripes with modulation having larger wavelengths κ = π/4 and κ = π/8 at K < K Λ c and K > K Λ c .
Depending on Λ and K, finite modulated chains of point dipoles minimize (in the paper ) Eq.1, in the magnetic configurations illustrated (in the paper) Fig.1. For small anisotropy K < K Λ c , the collinear magnetic order, Figs.1(c-d) is favored, while for K > K Λ c the parallel magnetic states (Figs.1(a-b)) minimize energy. One can find the critical anisotropy by balancing the total energy of the system in a parallel magnetic configuration with the total energy in the case of a collinear magnetic configuration, where J 1−3n 2 (n 2 +Λ 2 ) 3/2 . Solving for K yields where ψ is the PolyGamma function and ζ the Riemann zeta function.

Dirac magnons in 1D
The band touching at q 0 = Gx 2 for chains with Λ 1 can be seen as a single Dirac point around which the frequency dispersion for both bands can be approximated by a linear function. The singular structure of the frequency dispersion near the band touching can be studied using degenerate perturbation theory. For the magnon hamiltonian studied above it takes the form