Spin anisotropy and quantum Hall effect in thekagomélattice: Chiral spin state based on a ferromagnet

Abstract

A ferromagnet with spin anisotropies on the 2D Kagome lattice is theoretically studied. This is a typical example of the flat-band ferromagnet. The Berry phase induced by the tilting of the spins opens the band gap and quantized Hall conductance σxy = ±e 2 /h is realized without external magnetic field. This is the most realistic chiral spin state based on the ferromagnetism. We also discuss the implication of our results to anomalous Hall effect observed in the metallic pyrochlore ferromagnets R2Mo2O7 (R =Nd, Sm, Gd).75.10. Lp, 75.30.Gw, 72.15.Eb The spin Berry phase plays an important role in the quantum transport of strongly correlated electronic systems. Consider an electron hopping from site i to j coupled to a spin at each site with Hund's coupling J H [1]. When J H is strong enough the spin of the hopping electron is forced to align parallel to S i and S j at each site, with the spin wave function being |χ i > and |χ j >, respectively. The spin wavefunction is explicitly given bywhere we have introduced the polar coordinates asThe overall phase b i corresponds to the gauge degrees of freedom and does not appear in physical quantities. Therefore, the effective transfer integral t ij is given by [1]where θ ij is the angle between the two spins S i and S j . The phase a ij is the vector potential generated by the spin, and corresponds to the Berry phase felt by the hopping electron. Let us consider an electron hopping along a loop 1 → 2 → 3 → 1. The total phase that the electron obtains is the gauge flux by a ij , which corresponds to the solid angle subtended by the three spins S i (i = 1, 2, 3). This is called the spin chirality and is one of the key concept in the physics of strongly correlated electronic systems [2][3][4][5][6]. One can easily see that the spin chirality is absent for collinear spin alignment, and the spin chirality has been intensively discussed in the context of quantum spin liquid where the spins and hence a ij fluctuate quantum mechanically [2][3][4][5][6]. This a ij is the leading actor in the gauge theory of strongly correlated electronic systems [3,6]. Among them the proposal of chiral spin liquid with broken time-reversal symmetry in a triangular lattice [2], and later the anyon superconductivity [4] attracted great interests at the early stage of the high-T c research. Wen et al.[5] constructed a mean field theory for a chiral spin liquid on a square lattice. They start from the π-flux state, and break the time-reversal symmetry by introducing the next-nearest-neighbor hopping with a phase. However, it turned out to be rather difficult to find physical realization of the (chiral) spin liquid in real materials, even in frustrated lattices.The spin Berry phase has been discussed also in the context of anomalous Hall effect (AHE) in manganites [7]. It is proposed that the spin-orbit interaction H ′ leads to the coupling between the magnetization M and the spin chirality, i.e., the gauge flux, b as expressed by H ′ = λbM . At finite temperature T , Skyrmions are thermally excited...

Topological protection of bound states against the hybridization

“…In this magnetic state, the [111] surface or any of its symmetry equivalents supports a kagome-lattice plane with a non-coplanar spin ordering. As the spin chirality induced by noncoplanar spins can give rise to a QHI on the surface 28 while the bulk insulating state is topologically trivial, pyrochlore iridates are an ideal platform to realize intriguing localization phenomena protected by topology.…”

Noncommutative geometry and non-Abelian Berry phase in the wave-packet dynamics of Bloch electrons

“…[22] Recently, the Berry phase in AHE has attracted a renewed attention, revealing its rich topological structures. [3,4,6,7,23,24,25,26] The Berry phase has also been generalized to a non-Abelian case. [27] The equations of motion (EOM) for a wave packet of Bloch functions 1 is instrumental in all the analyses done in the paper.…”

“…B(k) encodes information on the topological nature of band structure, in particular, that of band crossings. [7] Indeed, a degeneracy point corresponds to a momopole of B(k), [5] which has played a crucial role in the understanding of anomalous Hall effect (AHE) [6,23,24,25]. In this paper we study the wave-packet dynamics of Bloch electrons subject to a perturbation β(x, t) varying slowly in space and time.…”

On the Stability of Quantum Hall Kagome-Ice Insulator

“…Although the origin of the gap opening is unclear yet, the Hall conductivity in the kagome-ice insulator becomes quantized at a nonzero integer value despite the absence of the magnetic order. This Hall response is induced by the spin Berry phase mechanism [20], and the sign of the Hall conductivity is controlled by the net spin scalar chirality from the triangles in the kagome lattice [15]. The realization of the kagome-ice insulator at finite temperature was shown [15] and the localized state at a defect was also studied [16].…”