Magnonic Weyl semimetal and chiral anomaly in pyrochlore ferromagnets

Ying Su, X. S. Wang, and X. R. Wang

Phys. Rev. B 95, 224403 – Published 1 June 2017

Abstract

Topological states of matter have been a subject of intensive studies in recent years because of their exotic properties such as the topologically protected edge and surface states. The initial studies were exclusively for electron systems. It is now known that topological states can also exist for other particles. Indeed, topologically protected edge states have already been found for phonons and photons. In spite of active searching for topological states in many fields, the studies in magnetism are relatively rare although topological states are apparently important and useful in magnonics. Here we show that the pyrochlore ferromagnets with the Dzyaloshinskii-Moriya interaction are intrinsic magnonic Weyl semimetals. Similar to the electronic Weyl semimetals, the magnon bands in a magnonic Weyl semimetal are nontrivially crossing in pairs at special points (called Weyl nodes) in momentum space. The equal energy contour around the Weyl node energy is made up of the magnon arcs on sample surfaces due to the topologically protected surface states between each pair of Weyl nodes. Additional Weyl nodes and magnon arcs can be generated in lower energy magnon bands when an anisotropic exchange interaction is introduced. Finally, magnonic chiral anomaly is realized under mutually perpendicular inhomogeneous electric and magnetic fields. The magnonic chiral anomaly results in linear electric spin and heat conductances that are experimentally detectable.

Received 23 September 2016
Revised 30 March 2017

DOI:https://doi.org/10.1103/PhysRevB.95.224403

Physics Subject Headings (PhySH)

Ferromagnetism Magnons Spin waves

Ferromagnets

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Ying Su, X. S. Wang, and X. R. Wang^*

Physics Department, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong and HKUST Shenzhen Research Institute, Shenzhen 518057, China

^*Corresponding author: phxwan@ust.hk

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Issue

Vol. 95, Iss. 22 — 1 June 2017

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Images

Figure 1
(a) Pyrochlore structure of four interpenetrating face-centered cubic lattices with corner-sharing tetrahedrons. The orange dots represent $V^{4 +}$ ions. (b) Four $V^{4 +}$ ions in a unit cell lie in a corner-sharing cubic. The arrows represent the DMI vectors $D_{i j}$ perpendicular to the bond $j \to i (i > j)$ and parallel to the surfaces of the cubic. $D_{i j} = - D_{j i}$ . (c) The first bulk Brillouin zone (BZ) and the first (001) surface BZ of the pyrochlore lattice. The projection of the high symmetry points of the bulk BZ onto the surface BZ are denoted by the barred symbols. The red and blue dots schematically represent the pair of WNs with opposite topological charges. (d) The magnon energy spectrum along the high symmetry path shown in (c) with DMI (red curves) and without DMI (blue curves). (e) The magnon dispersion around the WN shown in (d) in the $k_{z} - k_{d}$ plane which is represented by the yellow plane in (c).
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Figure 2
Density plots of magnon spectral function on the top (001) surface along (a) $\bar{Γ} \bar{X} \bar{L} \bar{Γ}$ and (b) $\bar{L} \bar{Γ} \bar{L}$ . (c)–(e) Density plot of magnon spectral function on the top (001) surface in the first BZ for fixed energies of $E_{c}, E_{d}$ , and $E_{e}$ denoted in (a).
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Figure 3
(a) Phase diagram of the MWS in the $D - J^{'}$ plane. The MWS, together with WNs and magnon arcs, exists in the shadowed regions. The white region between the two critical interlayer exchange interactions $J_{\pm}^{'} (D)$ denoted by red and blue curves is the normal magnonic insulator without topologically protected surface states in the energy gap. (b) The first bulk Brillouin zone (BZ) and the first (111) surface BZ of the pyrochlore lattice. The red and blue dots schematically represent the three pairs of WNs with opposite topological charges between the bands of $E_{3}$ and $E_{4}$ . (c) The density plot of magnon spectral function on the top surface for $J^{'} = J$ along $\bar{Γ} \bar{K} \bar{M} \bar{Γ}$ . (d) The density plot of magnon spectral function on the top surface for $J^{'} = 0.6 J$ along $\bar{Γ} \bar{K} \bar{M} \bar{Γ}$ (upper panel) and $\bar{Γ} {\bar{M}}^{'} \bar{M} \bar{Γ}$ (lower panel) that are presented by red solid and dashed lines in (b). (e) The density plot of magnon spectral function on the top surface for $J^{'} = 1.6 J$ along $\bar{Γ} {\bar{K}}^{'} {\bar{M}}^{'} \bar{Γ}$ (upper panel) and $\bar{Γ} \bar{M} {\bar{M}}^{'} \bar{Γ}$ (lower panel) that are presented by blue solid and dashed lines in (b). (f)–(g) The magnon arcs on the top (111) surface for energies through the WNs for $J^{'} = 0.6 J < J_{-}^{'}$ (f) and $J^{'} = 1.6 J > J_{+}^{'}$ (g). The black hexagon encloses the first BZ.
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Figure 4
(a)–(c) The Berry curvature $Ω_{3} (k)$ of magnon band $E_{3} (k)$ on the $k_{z} - k_{d}$ plane for $J^{'} = J, 2 J$ , and $1.3 J$ , respectively. The arrows represent the directions of Berry curvature vectors projected onto the plane, and the color encodes the divergence of Berry curvature $\nabla_{k} \cdot Ω_{3} (k)$ . $S_{β}$ ( $β = 1, 2, 3, 4$ ) denote the four spherical surfaces enclosing the four WNs (the red and blue dots) in (a) and (b). The insets show the three-dimensional view of the spherical surfaces and the Berry curvature vectors on the surfaces.
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Figure 5
(a) Schematic diagram of the MLLs. The red and blue curves are the zeroth and higher MLLs, respectively. The black dots represent the magnons occupying the zeroth MLL. The arrows indicate the magnon motion in momentum space driven by the inhomogeneous magnetic field $B$ . (b) Schematic diagram of the two-terminal setup along the $z$ direction. The MWS is connected to two magnon reservoirs. The inhomogeneous magnetic field is applied along the $z$ direction.
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