Magnetic skyrmion crystal at a topological insulator surface Stefan Divic, 1, ∗ Henry Ling, T. Pereg-Barnea, 2, † and Arun Paramekanti 3, ‡ 1 Department of Physics, University of California, Berkeley, CA 94720, USA 2 Department of Physics and the Centre for the Physics of Materials, McGill University, Montr´ eal, Qu´ ebec, H3A 2T8, Canada 3 Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7 (Dated: April 1, 2021) We consider a magnetic skyrmion crystal formed at the surface of a topological insulator. Incor- porating the exchange interaction between the helical Dirac surface states and the periodic N´ eel or Bloch skyrmion texture, we obtain the resulting electronic band structures. We discuss the prop- erties of the reconstructed skyrmion bands, namely the impact of symmetries on the energies and Berry curvature. We find substantive qualitative differences between the N´ eel and Bloch cases, with the latter generically permitting a low-energy tight-binding representation whose parameters are tightly constrained by symmetries. We explicitly construct the associated Wannier orbitals, which resemble the ring-like chiral bound states of helical Dirac fermions coupled to a single skyrmion in a ferromagnetic background. We construct a two-band tight-binding model with complex nearest- neighbor hoppings which captures the salient topological features of the low-energy bands. Our results are relevant to magnetic topological insulators (TIs), as well as to TI-magnetic thin film heterostructures, in which skyrmion crystals may be stabilized. I. INTRODUCTION Massless Dirac fermions emerge in condensed matter as low energy excitations of systems whose Fermi level lies close to a band crossing. Notable examples of this phenomenon in d> 1 include graphene, Weyl/Dirac semimetals, and the surface states of strong topologi- cal insulators (TIs) [1–7]. In certain cases, such band touchings may be protected by lattice or time-reversal symmetries, so that breaking these symmetries induces a Dirac mass gap, leading to physically observable con- sequences [8–12]. For instance, inducing a mass gap in graphene by breaking inversion symmetry leads to a val- ley Hall effect [13], while breaking time-reversal symme- try for a TI surface Dirac cone via a perpendicular Zee- man field leads to a gapped (half) quantum Hall state [14] with σ xy = e 2 /2h. Such symmetry breaking may be induced by proximity coupling with a symmetry-broken substrate, or by spontaneous ordering of magnetic mo- ments (for time-reversal symmetry breaking). Domain walls of such broken symmetries, where the Dirac mass changes sign, act as channels which support chiral edge modes [15–17]. Going beyond the impact of uniform symmetry break- ing orders or isolated domain walls, it is interesting to consider the effect of periodically modulated potentials on massless Dirac fermions. Such modulations have been extensively studied in the context of superlattices in graphene [18–23] and bilayer graphene [24–26], where they have been shown to produce new emergent 2d Dirac fermion excitations. While smooth periodic modulations * Electronic address: stefan divic@berkeley.edu † Electronic address: tamipb@physics.mcgill.ca ‡ Electronic address: arun.paramekanti@utoronto.ca are better understood from a band-theory perspective of reconstructing the low-energy Dirac theory, the impact of sharp superlattice potentials may be captured via the hybridization of the gapless domain wall states [24, 25]. The superlattice reconstruction of low-energy bands has also recently come to the fore in studies of twisted bi- layer graphene [27–40] and multi-layer transition metal dichalcogenides [41–48], where the Moir´ e pattern leads to an enlarged unit cell, as well as in recent work examining Moir´ e potential modulations on TI surface states [49, 50]. In this paper, inspired by these previous developments, we study magnetic skyrmion lattices on a TI surface and explore the resulting electronic states. Our work is also motivated by desire to understand the interplay of the momentum space topology of TIs, as reflected in their he- lical Dirac surface states, with the topological real space texture of magnetic skyrmions. For instance, materials such as topological Kondo insulators (TKIs) [51] can have Dirac surface states together with soft magnetic modes in the bulk due to strong correlation effects. Such materials might thus be prone to spontaneous magnetic ordering of the local moments at the surface. In SmB 6 , a candi- date TKI, surface magnetoresistance experiments show evidence of one-dimensional conducting channels within the surface gap [52, 53]. These have been attributed to magnetic domain wall bound states which were shown to give rise to a butterfly shaped hysteresis [54]. The inver- sion breaking at the TKI surface can also enhance the role of chiral Dzyaloshinskii-Moriya magnetic exchange interactions [55, 56], which could favor the formation of skyrmions at the surface. Magnetic topological materials such as MnBi 2 Te 4 are another proposed candidate for realizing skyrmions [57]. Further possibilities of realiz- ing magnetic skyrmions at TI surfaces include the order- ing of impurity magnetic moments of dopants induced by RKKY interactions [58], and proximity coupling to a magnetic substrate hosting these textures [59–63]. arXiv:2103.15841v2 [cond-mat.mes-hall] 31 Mar 2021