Second-order topological solitonic insulator in a breathing square lattice of magnetic vortices

Abstract

Recent acoustic and electrical-circuit experiments have reported the third-order (or octupole) topological insulating phase, while its counterpart in classical magnetic systems is yet to be realized. Here we explore the collective dynamics of magnetic vortices in three-dimensional breathing cuboids, and find that the vortex lattice can support zero-dimensional corner states, one-dimensional hinge states, two-dimensional surface states, and three-dimensional bulk states, when the ratio of alternating intralayer and interlayer bond lengths goes beyond a critical value. We show that only the corner states are stable against external frustrations because of the topological protection. Full micromagnetic simulations verify our theoretical predictions with good agreement.

Topological insulators and semimetals in classical magnetic systems

“…All these phases, however, are first order by nature. In this section, we move on to the higher-order topological phase in magnetic soliton crystals, by presenting thorough calculations details in the breathing kagome [96], honeycomb [97], and square [98] lattices of magnetic vortex.…”

“…1(b) and 1(c), respectively. Due to their novel properties, the HOTIs have been investigated extensively in the broad community of photonics [56][57][58][59][60][61][62][63][64][65][66][67], acoustics [68][69][70][71][72][73][74][75][76][77][78][79][80][81][82], mechanics [83][84][85], electronics [86][87][88][89][90][91][92][93][94], and magnetics [95][96][97][98] in the last few years. The topological description of HOTIs goes beyond the conventional bulk-boundary correspondence and is characterized by a few new topological invariants, such as the bulk polarization (Wannier center) [51,70,99,100], Green's function zeros…”

“…The HOTIs are broadening our understanding on topological insulating phases of matter. Notablely, the practical applications of higher-order topological edge states (corner states for instance) are also stimulating significant research enthusiasm in the community [77,98].…”

“…Li et al [236] and Go et al [237] studied the Su-Schrieffer-Heeger (SSH) [238] states in one-dimensional magnetic soliton lattice. Li et al predicted theoretically the second-order topological phases (corner states) in two-dimensional breathing kagome [96], honeycomb [97], and square [98] lattice of magnetic vortices and showed that the emerging corner states are very robust against disorder and defects because of the generalized chiral symmetry. The collective motion of magnetic solitons in two dimensions is described by the generalized Thiele's equation, which results in a wavelike equation in the artificial crystal, and this equation differs from the wave equations of its electronic, photonic, and acoustic counterparts in the following respects: (i) The nonvanishing topological charge induces a gyration term that is analogous to an effective magnetic field acting on a quasiparticle, thus breaking time-reversal symmetry.…”

“…(f) Dependence of Chern number and ℤ 4 Berry phase on the ratios 2 ∕ 1 with 1 being fixed to 3 . Source: The figures are taken from Ref [98].…”