Magnon scattering off quantum Hall skyrmion crystals probes interplay of topology and symmetry breaking

We introduce a model to study magnon scattering in skyrmion crystals, sandwiched between ferromagnets, which act as the source of magnons. Thanks to recent experimental advances, such a set-up can be realised in quantum Hall heterojunctions, and it is interesting as skyrmions are topological objects while the skyrmion crystals break internal and translational symmetries, thus allowing to study the interplay of topological and symmetry breaking physics. Starting from a basis of holomorphic theta functions, we construct an appropriate analytical ansatz for such a junction with finite spatially modulating topological charge density in the central region and vanishing in the leads. We then construct a suitably defined energy functional for the junction in terms of these spinors and derive the resulting equations of motion, which take the form of a Bogoliubov-de Gennes-like equation. Using a combination of analytical techniques, field theory, heuristic models, and fully microscopic recursive transfer-matrix numerics, we calculate the spectra and magnon transmission properties of the skyrmion crystal. We find that magnon transmission can be understood via a combination of low-energy Goldstone modes and effective emergent Landau levels at higher energies. The presence of the former manifests in discrete low-energy peaks in the transmission spectrum and we show how the these features reflect the nature of the Goldstone modes arising from symmetry breaking. In turn, the effective Landau levels, which reflect the topology of the skyrmion crystal, lead to band-like transmission features, from the structure of which further details of the excitation spectrum of the skyrmion crystal can be inferred. Such characteristic transmission features are not present in competing phases of either the quantum Hall phase diagram or in metallic magnets, and hence provide direct and unique signatures of skyrmion crystal phases and their properties. We discuss experimental considerations regarding the realisation of our model, which most directly apply to heterojunctions in monolayer graphene with the central region doped slightly away from unit filling and the two ends exactly at unit filling, a $\nu =1:1\pm \delta \nu :1$ junction. Such physics is also relevant to junctions formed by metallic magnets, which host skyrmion crystal phases, or partly in junctions with artificially realized and periodically modulated gauge fields.

Figure 1

Results for the quantum Hall ferromagnet (QHFM)-skyrmion crystal junction. (a) Schematic picture of the scattering problem and the experimental setup—sharp interface drawn only for illustrative purposes. (b) Cartoon description of the spectrum of the skyrmion crystal comprising higher nonuniform Landau levels and the lowest Riemann-Goldstone Landau level. (c) High-energy band-like transmission features due to effective nonuniform Landau levels [generated due to spatially modulated topological charge density profile (with finite nonzero mean) in (f) and not uniform applied external field]. (d) Low-energy transmission features due to presence of Goldstone modes in the Riemann-Goldstone Landau level. Two sets of linearly separated peaks indicating linear dispersion and two distinct velocities of the modes. Small split in peaks with larger spacing (velocity), implies two modes are almost degenerate [see Fig. 9 for spectrum]. (e) Spin profile of the junction, from Eq. (6), generated by the truncated holomorphic theta function ansatz in Eq. (7). The tail of every arrow is a lattice site, its direction is the projection on the $z\text{−}x$ plane, and the color is the $y$ component. (f) Topological charge density profile for the junction, obtained from Eqs. (5) and (6). Parameters used for (c)–(f) are $g=0.8, J=1, N=140$, and $N^{\prime }=2$

Figure 2

Semiclassical analysis for heuristic model with constant magnetic field in the central region. All arguments and results in this panel are for the Landau gauge ($A_x=0$). [(a)–(c)] Qualitatively different effective potentials for different values of transverse momentum, $q_y\ge 0$ for (a), $-e{A}_y\left(\infty \right)/2<q_y<0$ for (b) and $-e{A}_y\left(\infty \right)<q_y<-e{A}_y\left(\infty \right)/2$ for (c), the $q_y\le -e{A}_y\left(\infty \right)$ case is a reflection of (a). Insets of (a) and (b) show the constant magnetic field profile with smooth decay away from central region and the corresponding vector potential in the Landau gauge respectively. d) Different qualitative regions of scattering in energy-transverse momenta parameter space with the labels in each region indicating the x support of the corresponding semiclassical trajectories—we get two regions of full reflection, $x\in ]-\infty ,x_1]$ or $x\in [x_0,\infty [$, one region with bound trajectories ($x\in [x_0,x_1]$) and one region with full transmission ($x\in ]-\infty ,\infty [$). [(e), (f)] Numerically obtained transmission and reflection coefficients respectively for the quantum problem with Hamiltonian in Eq. (1) showing great qualitative agreement with the semiclassical picture in (d). We only consider the case of a particle incident from the left which is why the reflection coefficient in (f) is not fully symmetric as in (d) which considers both left and right incident processes.

Figure 3

Qualitative arguments for heuristic model with spatially modulated magnetic field along $y$ axis. (a) Continuous spectrum for scattering and discrete spectrum for bound states. Modulation along transverse directions breaks $q_y$ conservation and periodic modulation implies $q_y$ is conserved modulo $2\pi /a$. Transmission below critical energy $E_{*}$ is possible if the energy of one of the channels coincides with the bound state. (b) Pictorial description of the possibility of tunneling into other propagating channels due to crystalline order. Each channel has its effective potential profile and regions of allowed transmission as in Fig. 2–2. The presence of multiple propagating modes allows transmission for an incoming magnon due to off-diagonal scattering. (c) Number of propagating modes in the $\omega -q_y$ plane in the unfolded zone scheme (for visual reasons). One can transfer this to the first Brillouin zone by standard folding techniques. Each color is for the two curves $2q_{yi}^2$ and $2(q_{yi}+A\left(\infty \right){)}^2$, such that in the region lying above both curves one gets an outgoing propagating mode for the $i\mathrm{th}$ channel, where $q_{yi}=q_y^{\left(0\right)}+2\pi (i-1)/a$. For this figure we use $B_0=2\pi /a^2$ and $L=20$. (d) Pictorial description of nonmonotonicity of channel resolved transmission from qualitative arguments presented in Sec. 3 and figure (b) in this panel [not real data, see Fig 5].

Figure 4

Results for heuristic model with spatially varying magnetic field. (a) Transmission spectra for a particle scattering off a region with periodic modulation, see main text, of magnetic field along $y$ axis, where $m$ is the parameter that controls the amplitude of modulation, $B=B_c(sin(4\pi y/a)/m+1)$, and $B_c$ is the magnetic field used in the last subsection. Small $m$ implies large modulation and vice versa. The case with $m=1$ resembles the same amplitude of modulation along $y$ axis as in the topological charge density profile in Fig. 1. (b) Transmission spectra for the case of a magnetic field modulated along $x$ and $y$ axes, where $m_1$ has the same properties as $m$ described, but for the $x$ axis in (a). The case with $m=1,m_1=1$ resembles the same modulation along both axes as in the topological charge density profile in Fig. 1. (c) Effect of modulation along y on Landau levels, adds bandwidth. (d) Effect of modulation along x on Landau levels, reduces gap. Black lines in (c) and (d) are for the unmodulated magnetic field case, red for the case with large modulations only along $y$ axis, and blue for the case with large modulations along both $x$ and $y$ axis.

Figure 5

Channel dependence and angular deviation of transmission for the heuristic model in Secs. 3. (a) Angular deviation of particle scattering off a constant magnetic field region, similar to magnon scattering off a single skyrmion. (b) Channel dependence of transmission coefficient for spatially modulated magnetic field ($B_0=8\pi /a^2, L=20, m=1, m_1=1, q_y^{\left(0\right)}=0$). Certain channels dominate in certain energy regions, and there is a nonmonotonic dependence of transmission coefficient on outgoing channel number $n$, where $q_y^{\left(n\right)}=q_y^{\left(0\right)}+2(n-1)\pi /a$. For low energies, below $E_{*}$, only one propagating channel is present $q_y^{\left(0\right)}$. On increasing energy the number of propagating channels increases as shown in Fig. 3, and the particle can scatter into these channels (the transmission matrix has nonzero off diagonal elements). This agrees with the qualitative picture of tunneling into other propagating channels, as presented in Sec. 3 and Fig. 3.

Figure 6

Results for the ferromagnet-antiferromagnet-ferromagnet quantum Hall junction heuristic model- parameters used $J_F=1,J_{AF}=1,L_I=18a_x$, where $a_x$ is the lattice spacing along $x$. (a) Lattice structure of the heuristic model with the $y$ spacing for the ferromagnet twice that of the $x$ spacing. Such a simplification is made to simplify the sublattice matching across the interface. (b) Fabry-Perot resonance peaks at normal incidence ($q_y=0,\theta =0$) in the total transmission. (c) Transmission as a function of angle of incidence and energy of incident magnon. At fixed angle of incidence, there is a critical energy for transmission following which there is a set of equally spaced peaks in the low-energy regime, reflecting the linear nature of the antiferromagnetic Goldstone modes. The cutoff energy at $\theta =0$, is a finite size effect, as we increase the length of the middle region, this value will go closer to zero.

Figure 7

(a) Change in solid angle between two geodesics on the sphere. (b) Plaquette-wise discretization of topological charge. Tight-binding model includes contributions of nearest and second-nearest neighbors (shown here for site 0).

Figure 8

(a) Slice-wise recursive transfer matrix procedure for the ferromagnet-skyrmion crystal-ferromagnet junction. The solid lines highlight the real-space discretization and their intersection points are the lattice sites. (b) Multiple scattering processes for propagating transmission and reflection matrices. Calculate the transfer matrix for the slab of length $L_c$ the usual way and then do successive slice-wise rotation to propagate the transmission matrices using the infinite series.

Figure 9

Transmission and energy spectra for the ferromagnet-skyrmion crystal-ferromagnet junction at normal incidence $q_y=0$ at g/J = 0.4 (red), 0.8 (blue), and 0 [magenta, only in (c), (d)]. The effect of increasing $g$ in the second term of the energy functional in Eq. (11) on (a) low-energy transmission, peaks shift and increase in height (b) Goldstone mode dispersion, modes become more dispersive and (c) high-energy transmission, slight shift in position of peaks but no significant change in height (d) High-energy dispersion, slight shift in energy.

Figure 10

Gauge fixing procedure to ensure zero vector potential in the left and right ends. (a) Illustration of the procedure explained in this section for the case $p=1,q=4$. (b) Sign convention and notation for components of vector potential.

Figure 11

Incoming and outgoing amplitudes for (a) a single interface problem and (b) a double interface problem.