Magnonic Weyl semimetal in pyrochlore ferromagnets
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 nodes gives rise to the Fermi arcs on sample surfaces due to the topologically protected surface states between each pair of Weyl nodes. Additional Weyl nodes and Fermi arcs can be generated in lower energy magnon bands when an anisotropic exchange interaction is introduced.
Magnetic materials are highly correlated spin systems that do not respect time-reversal symmetry. Their static states, such as domains, domain walls, and skyrmions, are the energy minimum spatial configurations of magnetization (vector order parameter) . The excitations of magnetic materials are spin waves whose quanta are magnons of spin-1 particles. Like electrons, magnons can carry, process and transmit information besides being a control knob of magnetization dynamics [2, 4, 3]. In fact, magnonics [5, 6, 7, 8, 9, 10, 11, 12] is a very active research field because of low energy consumption of magnonic devices and possible long spin coherence length [13, 14, 15]. One important issue in magnonics is the efficient transportation of magnons. Magnon (spin wave) flux normally decays fast during its propagation because it is difficult to confine magnons in the space. Finding materials or structures that can confine the motion of magnons in a restricted region under topological protection should open doors to new functional devices. Thus, the realization of topological states of matter in magnetic systems should be highly desirable [16, 17, 18].
In this work, we show that the pyrochlore ferromagnet LuVO,
which was recently shown to exhibit magnon Hall effect ,
is an intrinsic magnonic Weyl semimetal (MWS).
Two adjacent magnon bands in a MWS nontrivially cross each other at some
special points called Weyl nodes (WNs) in momentum space.
The WNs are monopoles of Berry curvature and are characterized
by integer topological charges. Because the net topological charges
in the entire Brillouin zone (BZ) must be zero, the WNs must appear
in pairs with opposite topological charges .
Like the electronic Weyl semimetal, the MWS has topologically
protected chiral surface states between each pair of WNs on the
sample surfaces [21, 22, 23].
energy contour of these surface states form arcs (called
Fermi arc), and
the number of Fermi arcs between two paired WNs equals to the number of topological charges carried by one of them.
Moreover, additional WNs and topologically protected surface states
can appear in lower energy magnon bands when anisotropic exchange
interaction, possibly induced by either doping or strain along the
 direction, is introduced.
ii.1 The effective spin model of LuVO
LuVO is an intrinsic ferromagnetic Mott-insulator in which each vanadium ion V carries spin [19, 24]. The magnetic properties of the material come purely from the vanadium ions that form a pyrochlore lattice consisting of four interpenetrating face-centered cubic (FCC) lattices with corner-sharing tetrahedrons, as shown in Fig. 1a. The primitive vectors are , , and , where the FCC lattice constant is set to unit. Three of the four FCC lattices are shifted by , , and , respectively. In each unit cell, there are four V ions as shown in Fig. 1b. Under an external magnetic field, the magnetic properties of the material is well described by a simple Heisenberg Hamiltonian with the Dzyaloshinskii-Moriya interaction (DMI) . The effective spin Hamiltonian reads
where denotes the nearest neighbor (NN) sites and is the spin of the V ion at site . h is the external magnetic field applied along the  direction in this study. The first term describes the NN exchange interaction with strengths . The second term represents the DMI with the DMI vectors . The last term is the Zeeman interaction.
As it was explained in ref. 19, the material is a collinear ferromagnet in spite of the DMI because the summation of six DMI vectors adjacent to each lattice site is zero. Under the Holstein-Primakoff transformation and by using the Bloch theorem, the Hamiltonian (1) is block diagonalized in momentum space as , where and are the creation and annihilation operators of magnons (see Methods). The four components correspond to the four different FCC sublattices. is the energy of zero magnon state (vacuum), where is the total number of lattice sites and denotes as the NN site of (see Methods). One can set to zero by choosing a proper energy reference. For a given k, is a matrix
and for isotropic exchange interaction.
The strength of the DMI is a constant .
Because only the components of parallel to the external
magnetic field contribute to Hamiltonian (1) (see Methods),
for h along the  direction [16, 19].
The magnitude of magnetic field is set as for simplicity (and without loss of generality) since the Zeeman interaction only shifts the magnon dispersion relation and does not affect the topological properties.
In the absence of DMI , the magnon spectrum contains two degenerate
flat bands () and two dispersive bands
(), where .
The magnon dispersion relation along the high symmetry path -K-W-X-U-L- (see Fig. 1c) is shown in Fig. 1d for
(blue curves) and for the experimental value  (red curves).
In comparison with the case of , the flat bands become dispersive and
band gaps are opened.
ii.2 Identification of Weyl nodes and Fermi arcs
Interestingly, a pair of WNs appears on the high symmetry line L--L as shown in Figs 1c and 1d. Two magnon bands of and linearly cross each other, giving rise to a MWS behavior. Along the L--L line, where , two magnon bands are flat with (), and the other two bands are dispersive with (). For , touches and at the point. For modest , and are split into two nondegenerate flat bands, and and cross at a pair of WNs at
for . Moreover, the flatness of along the L--L line means the magnon group velocity near the WNs vanishes along the  direction. According to a recent classification , this corresponds to the transition state from type-I to type-II Weyl semimetals with vanishing group velocity only in one direction. To visualize the magnon dispersion with vanishing group velocity along the  direction, we plot the magnon bands of and near one WN in a vertical plane (represented by the yellow plane in Fig. 1c) parallel to both direction and the diagonal - direction (termed as ). Obviously, the L--L line lies in the - plane, and band and band linearly cross each other at the WN of as shown in Fig. 1e.
Similar to the electronic Weyl semimetal, one fingerprint of the MWS
is the Fermi arcs on the sample surfaces. In order to illustrate this feature,
we consider a slab whose surfaces are perpendicular to the  direction.
The first BZ of the (001) surface is shown in Fig. 1c, where the
projection of the high symmetry points of the first bulk BZ onto the first
surface BZ are denoted by the barred symbols. The pair of WNs are
schematically represented by the red and blue dots (indicating they carry
opposite topological charges) in Fig. 1c.
The density plot of magnon spectral function on the top surface along the high
symmetry path of --- is shown in Fig. 2a where one WN can be identified.
The topologically protected surface states with high density on the top
surface are represented by red color. On the path of -- where both of the two WNs
lies in, the density plot of magnon spectral function on the top surface
is shown in Fig. 2b. Apparently, the pair of WNs are connected
by surface states. For fixed energies of , , and
around the WNs (see Fig. 2a), the corresponding density
plot of magnon spectral function on the top surface
in the first BZ are shown in Figs 2c-2e, respectively.
The Fermi arc formed by topologically protected surface states on the
top surface is clearly displayed.
For through the WNs, the Fermi arc is
terminated at the two WNs as shown in Fig. 2d.
ii.3 Anisotropic exchange interaction
We have shown that the pyrochlore ferromagnet LuVO is an intrinsic MWS. We would like to show now that more pairs of WNs and topologically protected surface states can come from the lower energy magnon bands of and in the presence of anisotropic exchange interaction. The pyrochlore lattice can be viewed as an alternative stack of kagome and triangular lattices along the  direction. In principle, the interlayer exchange interaction differs from the intralayer exchange interaction . can be tuned by either doping  or strain . The effective Hamiltonian with the anisotropic exchange interaction, under the same considerations as before, becomes
The anisotropic exchange interaction leads to different sublattice on-site potential because the one-site potential on each particular site is the sum of the exchange interaction strengths of all its NNs (see Methods).
For the isotropic exchange interaction, the energy gap minimum between and bands is at the X point as shown in Fig. 1d. The anisotropic exchange interaction can close the gap at the X point whenever equals to two critical values
where . The critical as functions of the
DMI strength are plotted as red and blue curves in Fig. 3a.
These are the phase boundaries between the normal magnonic insulator (without
topologically protected surface states in the gap) and the MWS from
and bands. The phase diagram of the MWS from and
bands in the - plane is shown in Fig. 3a.
In the shadowed regions of Fig. 3a where or ,
and bands always cross at three pairs of WNs due to the three-fold
rotation symmetry with respect to the L--L line (see Fig. 3b). For LuVO, and .
In the limits of and 0, all these WNs will merge at the
L point. The fact that the trivial region represented by white color shrinks
as decreases means that weak DMI is favorable for the existence of WNs
between and bands since only weak anisotropy (small difference
between the interlayer and intralayer exchange interactions) is required.
These results are applicable to other pyrochlore ferromagnets.
ii.4 Additional Weyl nodes and Fermi arcs
To visualize these additional WNs and topologically protected surface states existing in the MWS phase from and bands, the magnon spectral function of a slab with (111) surfaces is calculated. The density plot of magnon spectral function on the top surface for along the high symmetry path --- (marked by red solid lines in Fig. 3b) is shown in Fig. 3c. The energy gap minimum appears at the point to which the X point is projected. As the interlayer exchange interaction decreases to , three pairs of WNs are created from the linear crossing of and bands. The density plot of magnon spectral function on the top surface along various paths is shown in Fig. 3d. Along the path of ---, a WN is identified on the - segment. The topologically protected surface states is clearly visible within the energy gap with one end terminated at the WN. Along the path of --- (represented by red dash lines in Fig. 3b), a pair of WNs is connected by the surface states. Similar results for are shown in Fig. 3e along the --- and --- paths (marked by blue solid and dash lines, respectively, in Fig. 3b). In order to detect the Fermi arc feature, we fix the energy through the WNs for the two different interlayer exchange interaction strengths. The density plot of magnon spectral function on the top surface in the two-dimensional momentum space is shown in Figs 3f and 3g where the black hexagon encloses the first surface BZ. Apparently, the topologically protected surface states form three Fermi arcs of the three pairs of WNs.
The pair of WNs from and bands on the L--L line can remain for the anisotropic exchange interaction. Since the magnon dispersions on the L--L line with are and in the present case, and bands cross at a pair of WNs at
as long as .
Because the band is flat along the L--L line, the magnon
group velocity around the pair of WNs vanishes along the  direction.
The pyrochlore ferromagnet LuVO is an intrinsic topological material (called MWS) in the sense that two adjacent magnon bulk bands of and linearly cross each other at a special pair of points (called WNs) on the L--L line in momentum space. The distance between the paired WNs is determined by the strength of DMI. Similar to its electronic counterpart, the MWS has topologically protected chiral surface states whose equal energy contour yields the Fermi arc that connects the pair of WNs on the sample surfaces. By introducing different interlayer and intralayer exchange interaction strengths through either doping or strain along the  direction, three additional pairs of WNs can be generated from the lower energy magnon bands of and . On the surfaces of a slab perpendicular to the  direction, the three pairs of WNs are connected by three Fermi arcs in two-dimensional momentum space. Furthermore, the pair of WNs between and bands can remain on the L--L line whose distance is determined by both the DMI and interlayer exchange interaction. These results are applicable to other collinear pyrochlore ferromagnets with anisotropic exchange interaction.
The MWS featured by WNs and Fermi arcs can be detected by inelastic neutron
scattering which has been used to probe the magnon bands of a topological magnon
insulator . The topologically protected magnon surface states
can also be probed by the spin-polarized scanning tunneling microscopy through the second-order derivative of tunneling
current that contains the information of electron-magnon scattering .
iv.1 Holstein-Primakoff transformation
In this transformation , the spin-1/2 operators are mapped to the magnon creation and annihilation operators as
where the ladder operators are defined in the orthonormal coordinate with axis parallel to the external magnetic field. For the DMI, the local spin where and in the linear approximation. Thus, the Hamiltonian of DMI is
where and . Namely, the with vanishing component does not contribute to the Hamiltonian. Substitute these into the effective spin Hamiltonian (1), we get a tight-binding Hamiltonian of magnons as
Here denotes as the NN site of , and can be set to zero by choosing a proper energy reference, where is the total number of lattice sties. Moreover, the on-site potential of each lattice site is determined by the sum of all adjacent NN exchange interaction strengths such that the anisotropic exchange interaction can generate different sublattice on-site potential as shown in equation (4).
iv.2 Surface spectral function.
The spectral function of a specific layer is
Note added. Upon completion of this work, we became
aware of ref. 32, in which part of the results were obtained.
This work is supported by the NSF of China Grant (No. 11374249) and Hong Kong RGC Grants (No. 163011151 and No. 605413).
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