A Effective field theory

Spin textures and spin-wave excitations in doped Dirac-Weyl semimetals


We study correlations and magnetic textures of localized spins, doped in three-dimensional Dirac semimetals. An effective field theory for magnetic moments is constructed by integrating out the fermionic degrees of freedom. The spin correlation shows a strong anisotropy, originating from spin-momentum locking of Dirac electrons, in addition to the conventional Heisenberg-like ferromagnetic correlation. The anisotropic spin correlation allows topologically nontrivial magnetic excitation textures such as a transient hedgehog state, as well as the ferromagnetic ground state. The spin-wave dispersion in ferromagnetic Weyl semimetal also becomes anisotropic, being less dispersed perpendicular to the magnetization.

75.30.Hx, 75.30.Ds, 71.70.Ej, 11.30.Qc

Introduction. – Spin-spin correlation gives rise to magnetism in materials. In magnetically-doped metals and semiconductors, the correlation among localized magnetic moments is mediated by the motion of carriers, known as Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction (1); (2); (3), which can give rise to a ferromagnetic order. In the presence of strong spin-orbit coupling, so-called Van Vleck paramagnetism, in which mixing of the valence and conduction bands contributes to the magnetic susceptibility (4), can also play an important role in inducing a magnetic order. In magnetically-doped topological insulators, it has been theoretically and experimentally verified that a ferromagnetic order is spontaneously induced by the Van Vleck mechanism, showing the quantum anomalous Hall effect (5); (6); (7); (8); (9); (10); (11).

One of the candidate materials that may show both of the mechanisms discussed above is a magnetically-doped Dirac semimetal. A three-dimensional (3D) Dirac semimetal manifests gapless linear dispersions, namely the “Dirac cone structure”, doubly degenerate with time-reversal and spatial inversion symmetries (12). A Weyl semimetal is a new type of topologically protected gapless quantum state, with either time-reversal or spatial inversion symmetries broken in a Dirac semimetal (13); (14). Recent realizations of 3D Dirac and Weyl semimetals (15); (16); (17); (18) have opened a new way to access various nontrivial transport properties, such as the anomalous Hall effect (19); (20); (21), the chiral magnetic effect (22); (24); (23); (25), the negative magnetoresistance (26); (27); (28); (29), the charge-induced spin torque (30), and the anomalous magnon electrodynamics (31); (32), arising from the chiral anomaly of the Dirac-Weyl Hamiltonian.

In Dirac and Weyl semimetals, it has been proposed that the RKKY interaction becomes anisotropic, showing Ising- and Dzyaloshinskii–Moriya-like interaction terms (33); (34). Since spin-momentum locking in the Dirac-Weyl Hamiltonian correlates electron spin degrees of freedom to its motion in real space, it is possible that the correlation among localized magnetic moments can depend on the spatial configuration. Such an anisotropic correlation may give rise to nontrivial magnetic textures.

In this Letter, we construct an effective field theory for localized magnetic moments in doped Dirac-Weyl semimetals, to clarify the properties of local spin correlations. Based on this effective field theory, we investigate the consequent magnetic textures and spin-wave dispersion, paying attention to the global rotational symmetry of the system. We show that the spin correlation gains a spatial anisotropy due to spin-momentum locking, and that the Van Vleck mechanism strongly contributes to the anisotropy. Localized magnetic moments become strongly correlated parallel to the local magnetization, while the correlation is rather weak in the perpendicular direction. We find that the additional correlation allows several types of topologically nontrivial magnetic textures as excitations from the ferromagnetic ground state, such as a “hedgehog” around a single point or a “radial vortex” around an axis. We also investigate the properties of spin-wave (magnon) excitation in the ferromagnetic phase. The magnon excitation discussed here is not a conventional Nambu–Goldstone (NG) mode, since it accompanies the real-space symmetry as well as the spin symmetry due to spin-momentum locking. Consequently, the spin-wave dispersion becomes anisotropic, less dispersed in the direction transverse to the magnetization direction. We see that the anisotropy is strongly enhanced when the Fermi level is close to the Dirac or Weyl nodes, which is a characteristic feature of Van Vleck mechanism (35).

Model. – We consider here a three-dimensional Dirac semimetal, described by the effective Hamiltonian


in the continuum limit, where the electron spin is locked to the direction of the momentum due to strong spin-orbit coupling. Here and are annihilation and creation operators of Dirac electrons with chirality , is Pauli matrix in the spin space, is the Fermi velocity, and is the Fermi energy measured from the Dirac points.

Localized magnetic moments are introduced by the spin operator . Correlation between the Dirac electrons and the localized magnetic moments is given by the - exchange interaction,


where is the exchange coupling constant, and is the spin operator of Dirac electrons. We should note that the total Hamiltonian is no longer symmetric under a global SU(2) transformation for both itinerant and localized spins. However, it is still symmetric under a simultaneous rotation in both the O(3) real space and the SU(2) spin space, generated by the total angular momentum , with the orbital angular momentum of Dirac electrons. Here we call this continuous symmetry “spin-orbital rotational” symmetry. This symmetry is broken either by a real-space anisotropy, such as an anisotropy of Dirac cone, or a magnetic anisotropy, i.e. an easy magnetization axis/plane or (spontaneous) formation of a ferromagnetic order.

Effective field theory. – Let us construct an effective field theory for the localized magnetic moments by integrating out the fermionic degrees of freedom. Here we use a classical field for local magnetization, in place of the quantum spin operator . The free energy for the local magnetization is obtained by path integral formalism as


where is the bare free energy that accounts for the magnetic susceptibility . The imaginary-time action corresponds to the total Hamiltonian , with the field variables accompanied with arguments . The imaginary time runs from to the inverse temperature . The single-particle action can be diagonalized in the Fourier space, with the Dirac propagator , where is the fermionic Matsubara frequency.

Figure 1: (a) Feynman diagram of the susceptibility tensor defined in Eq.(5). (b)(c) Schematic pictures of the relationship between the electron momenta and the exchanged (spin) angular momentum . becomes relatively larger for smaller , as depicted in (c).

Taking the local magnetization perturbatively, we can evaluate with the unperturbed Dirac propagator as


Contribution from the itinerant electrons starts from , with the “spin susceptibility” tensor defined by


whose Feynman diagram is given by Fig.1(a). Here denotes the wavenumber of spin wave in the real space, and is the volume of the system. In order to regularize the integration by , here we introduce a spherical cutoff with the radius , where is the characteristic microscopic length scale such as the lattice constant. Since the quantitative behavior of the ultraviolet (UV) part depends on the shape of the cutoff, we discuss only the qualitative dependence on , with the equivalence sign (36).

Due to the spin-orbital rotational symmetry in 3D, can be decomposed into uniform, longitudinal, and transverse parts as


Using this form, we can split the free energy in terms of gradient expansion, with the homogeneous part


and the remaining gradient part . The homogeneous part fixes the amplitude of local magnetization , while the gradient part determines the configuration of , namely the magnetic texture. Since we are interested in the magnetic texture at a long wavelength, here we evaluate by power series expansion in .

Substituting up to into Eq.(4), we obtain the real-space form of the gradient part,


This form consists of the isotropic term proportional to the square of gradient, , with the coefficient , and the anisotropic term depending on the square of vorticity, , with the coefficient . The coefficients are given at temperature (37) as


The logarithmic factor

comes from the integration over all the occupied states in the valence and conduction bands. Such an interband contribution is generated by Van Vleck mechanism, in which the interband matrix element contributes to the magnetic susceptibility due to the strong spin-orbit coupling. On the other hand, the non-logarithmic part in both and comes from the carriers at the Fermi surface. Such an intraband contribution corresponds to Pauli paramagnetism, namely the RKKY interaction mediated by carriers.

The dominance of the interband (Van Vleck) contribution depends on the Fermi level (35). It becomes stronger at a smaller Fermi momentum, due to the large angular momentum exchange: the angular momentum exchanged between two magnetic impurities via an electron is equal to the difference between the incoming electron spin and the outgoing one in the scattering process at a magnetic impurity, which becomes larger when is small (see Fig.1(b)(c)). This argument does not apply if the Fermi level is extremely close to the Dirac nodes, i.e. ; here the -expansion is not applicable due to the non-analyticity at , and in the interband contribution gets replaced by (37).

Magnetic textures. – What does the structure of this effective field theory imply? The isotropic term in Eq.(8) tends to suppress the gradient , leading to a Heisenberg-like uniform ferromagnetic order, which comes only from the intraband part. The anisotropic term, on the other hand, tends to suppress the vorticity as long as , which comes from both the interband and intraband contributions. The total free energy prefers the ferromagnetic ground state at zero temperature.

If the interband contribution is dominant over the intraband contribution, i.e. the Fermi level is close to the Dirac nodes, the anisotropic term restricts the magnetic textures of possible excited states. This term favors vortex-free configurations, where the local magnetization can be written by using a certain “scalar potential” as . It strongly requires the local magnetic moments to be aligned along a line, while it does not correlate the magnetic moments located perpendicular to the local magnetization, due to spin-momentum locking. Such kind of correlation can be regarded as “Ising-type” (33); (34), in the sense that the quantization axis is taken parallel to the relative coordinate between two magnetic moments. There can be many types of configurations that satisfy this requirement; the uniform order is the simplest case, while we can take a “hedgehog” configuration around a point , given by , or a “radial vortex” around an axis , given by (see Fig. 2).

The intraband contribution gives rise to the isotropic correlation, which requires a finite excitation energy for the topologically nontrivial textures discussed above, compared to the ferromagnetic ground state. As for a hedgehog structure, for instance, the energy cost per one hedgehog is , while it gains the entropy . Hence the entropy contribution to the free energy cannot overcome the energy cost as long as for . In a finite-size system, on the other hand, we can expect a crossover at finite temperature, between the ferromagnetic ground state and the excited state with topological defects satisfying .

Figure 2: Schematic pictures of possible magnetic textures satisfied by the effective field theory: (a)uniform, (b)hedgehog, and (c)radial vortex structures. All of them are degenerate under the interband contribution, while the intraband contribution splits the degeneracy and chooses (a) as the ground state.

Spin-wave excitations. – Let us go back to the ferromagnetic ground state at zero temperature. In conventional ferromagnets, spin SU(2) symmetry is spontaneously broken by the order, and a spin-wave mode arises as a fluctuation around the mean field. In Dirac semimetals, on the other hand, the order breaks the spin-orbital rotational symmetry, which accompanies the real-space symmetry as well as the spin space. Thus the spin-wave mode in this system is conceptually different from conventional ones. Here we search for any consequence of this difference.

Provided that the local spins are fully polarized in the -direction, the magnetization serves as a mean field through the exchange coupling, with the splitting energy . The quantum dynamics of the magnon modes is described by the bosonic field operators , introduced by the Holstein–Primakoff transformation . Here the approximation with is justified as long as the fluctuation is sufficiently small.

The effective field theory for the magnon modes can again be obtained, by integrating out the fermionic degrees of freedom in the path integral formalism, as done in Ref. (38) for dilute magnetic semiconductors. Here we embed the exchange coupling between the fermions and the mean-field magnetization, given by , into the single-particle Hamiltonian, leaving the spin wave fluctuation part as a perturbation. Extracting the contribution up to the bilinear in the fluctuation field , we obtain an effective action for the magnons,


where , , and . The dispersion relation is given by the pole of the propagator as


Taking the exchange splitting perturbatively, the effective Hamiltonian can be constructed from the susceptibility tensor defined in Eq.(5). Limiting the magnon momentum smaller enough than the Fermi momentum , the coefficients can be evaluated as

using the effective exchange coefficients and in Eq.(9) (37).

The magnon dispersion obtained here is gapless, i.e. . Although the magnon mode discussed here is not a traditionally-known Nambu–Goldstone (NG) mode related to spontaneous breaking of internal (e.g. spin) symmetries, as discussed above, it still keeps the gapless structure. The emergence of such unconventional NG modes, related to real-space symmetries, has recently been discussed in several literatures (39); (40). An intrinsic real-space anisotropy, such as the Fermi velocity anisotropy or the cutoff anisotropy, can open a gap in the magnon spectrum, which possibly occurs in realistic crystalline systems.

The consequence of spin-momentum locking appears in the spin-wave dispersion, starting from the quadratic order in . The dispersion of longitudinally-propagating mode becomes , while the transverse one reads with . If the interband (Van Vleck) contribution to the magnetism is larger than the Pauli contribution, i.e. if , there arises a strong anisotropy,


Here the spin-wave dispersion becomes almost flat in the transverse direction, i.e. . In other words, as long as the longitudinal correlation is kept rigid by , the energy cost to modulate the magnetization in the transverse direction is relatively small, as shown schematically in Fig. 3(c), while the longitudinal spin wave violates the longitudinal correlation, as shown in Fig. 3(b). This is the straightforward consequence from the effective field theory obtained in Eq.(8), since all the longitudinally correlated configurations are almost degenerate for .

(a) (b) (c)
Figure 3: Schematic pictures of the spin wave excitations, with red arrows representing the local magnetization. (a) The uniform ferromagnetic phase. (b) Longitudinal spin-wave configuration, which is against the longitudinal spin rigidity required by the effective Hamiltonian. (c) Transverse spin-wave configuration, which does not violate the longitudinal spin rigidity.

Summary and Discussions. – We have studied the characteristics of correlation among localized magnetic moments in Dirac and Weyl semimetals, in terms of the effective field theory obtained by integrating out the fermion fields. Spin-momentum locking leads to various unconventional magnetic properties, namely the strong longitudinal spin correlation, possibility of topologically nontrivial excitations, and the anisotropic dispersion of spin waves in the ferromagnetic phase. The emergence of such unconventional effects depends on the magnitude of interband (Van Vleck) contribution characterized by the logarithmic factor , which can be tuned through carrier doping. If the Fermi level is close to the Dirac point, the interband contribution becomes dominant, which makes the correlation anisotropies found here quite strong.

Tuning of the chemical potential in Dirac semimetals is realized by introducing dopants; in CdAs observed in Ref. (16), for instance, is successfully tuned by alkaline metal doping such as K, up to beyond the Dirac point. Using the in-plane lattice constant and the Fermi velocity , the lower bound of can be estimated as , where and become quite comparable. The upper bound , which becomes infinite in the ideal case , can be reduced by the typical long-range length scales, such as the size of the sample, temperature, the spin coherence length under disorder, etc. It can be suppressed by the intrinsic breakdown of the spin-orbital rotational symmetry as well, due to the lattice structure of the system. We still expect that the upper bound is quite larger than the lower bound , which may enable us to observe the crossover between the interband-dominant regime and the trivial regime by carrier doping.

The authors are thankful to Joseph Barker, Takahiro Chiba, Yusuke Nishida, and Akihiko Sekine for fruitful discussions and comments. This work was supported by Grant-in-Aid for Scientific Research (Grants No. 15H05854, No. 26107505, and No. 26400308) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.

Supplemental Material

In this Supplemental Material, we show some detailed calculation of the effective field theory for background magnetization and the spin-wave dispersion.

Appendix A Effective field theory

The effective action for the background magnetization can be obtained by evaluating the spin susceptibility tensor


where we have shifted the momentum by from the definition in Eq.(5). Here is the single-particle Green’s function (propagator) of the Dirac electron with chirality , with the matrix structure


Separating the denominator and the numerator of the Green’s function, we can evaluate the trace in the susceptibility tensor,


where we have used the relations for Pauli matrices,


Here we limit ourselves to the case , and evaluate the susceptibility tensor by power series expansion by up to . Since the -expansion of the denominator reads


the -expansion of the susceptibility tensor up to is given by


where we have averaged the angular part by the spherical symmetry in the denominator. Thus term can be decomposed into the longitudinal and transverse terms,


with the coefficients


Let us evaluate the sum over the Matsubara frequency and the momentum to estimate the coefficients and . All we need to evaluate are three types of integrals appearing in Eqs.(S.14)(S.15). By introducing a function with a dimensionless parameter ,


those three types of integrals can be expressed as


We can perform the Matsubara sum and the momentum integration in , yielding


Here, the first (logarithmic) term in Eq.(S.27) or Eq.(S.28) comes from both valence and conduction bands, corresponding to the Van Vleck contribution, while the second (constant) term picks up the Fermi surface, corresponding to the Pauli contribution. Thus we obtain the integrals


leading to the longitudinal and transverse coefficients


We can rearrange them into the “isotropic” and “anisotropic” exchange coefficients,


a.1 Logarithmic divergence at

If the Fermi level is at the Dirac point , we can no longer rely on the power series expansion by to evaluate the susceptibility tensor . Here we have


where we have changed the variable for the second term in Eq.(S.36). By performing the Matsubara sum, we obtain