# Spin susceptibility of three-dimensional Dirac semimetals

###### Abstract

We theoretically study the spin susceptibility of Dirac semimetals using the linear response theory. The spin susceptibility is decomposed into an intraband contribution and an interband contribution. We obtain analytical expressions for the intraband and interband contributions of massless Dirac fermions. The spin susceptibility is independent of the Fermi energy while it depends on the cutoff energy, which is introduced to regularize the integration. We find that the cutoff energy is appropriately determined by comparing the results for the Wilson-Dirac lattice model, which approximates the massless Dirac Hamiltonian around the Dirac point. We also calculate the spin susceptibility of massive Dirac fermions for the model of topological insulators. We discuss the effect of the band inversion and the strength of spin-orbit coupling.

## I Introduction

Topological semimetals, such as Dirac semimetals Young et al. (2012); Wang et al. (2012), Weyl semimetals Murakami (2007); Burkov and Balents (2011); Wan et al. (2011), and nodal line semimetals Burkov et al. (2011); Phillips and Aji (2014); Kim et al. (2015); Yu et al. (2015), possess exotic electronic band structure, which is significantly different from conventional metals and insulators. They exhibit fascinating physical properties originating from their topologically nontrivial band structure. There are many theoretical proposals to realize topological semimetals, some of which were experimentally confirmed Liu et al. (2014); Neupane et al. (2014); Borisenko et al. (2014); Xu et al. (2015); Lu et al. (2015); Lv et al. (2015). A Dirac semimetal has band touching points and the energy bands are doubly degenerate. By breaking either inversion symmetry or time-reversal symmetry, the degeneracy is lifted and a Dirac semimetal becomes a Weyl semimetal. The inversion broken Weyl semimetals are experimentally confirmed Xu et al. (2015); Lu et al. (2015); Lv et al. (2015) and there are several materials including type II Weyl semimetals Deng et al. (2016). On the other hand, there are few experimental indications for the Weyl semimetals with broken time-reversal symmetry, i.e. the magnetic Weyl semimetals Nakatsuji et al. (2015); Nayak et al. (2016); Liu et al. (2017), though there are many theoretical predictions Burkov et al. (2011); Wan et al. (2011); Kurebayashi and Nomura (2014); Wang et al. (2016); Ito and Nomura (2017); Yang et al. (2017); Jin et al. (2017); Xu et al. (2017); Cho (2011).

One of the theoretical predictions to realize the magnetic Weyl semimetals is magnetically doped topological insulators Yu et al. (2010); Kurebayashi and Nomura (2014); Cho (2011); Liu et al. (2013). Ferromagnetic ordering in topological insulators is experimentally observed Chen et al. (2010); Wray et al. (2011); Liu et al. (2012); Zhang et al. (2012, 2013); Chang et al. (2013a, b). In these systems, the ferromagnetic Weyl phase can emerge if the exchange coupling is sufficiently strong to overcome the energy gap. The magnetic properties and the topological phase transition induced by magnetic doping are characterized by the spin susceptibility of band electrons. Within the mean field theory, a condition to exhibit the ferromagnetic ordering is given by Yu et al. (2010), where is the exchange coupling constant, is the spin susceptibility of local magnetic moments, and is the spin susceptibility of band electrons. obeys the Curie law and is proportional to inverse of temperature (). Therefore, the ferromagnetic ordering can be observed at sufficiently low temperature as long as is finite. The investigation of in topological semimetals and insulators is an important issue to discuss the magnetic and topological phase transition in these systems.

In this paper, we study the spin susceptibility of three-dimensional Dirac semimetals within the linear response theory. The spin susceptibility is composed of the intraband contribution and the interband contribution . In the presence of strong spin-orbit coupling, gives large contribution. The interband effect is important in the orbital diamagnetism of the Dirac fermions Fukuyama and Kubo (1970); Fuseya et al. (2009, 2012); Koshino and Hizbullah (2016). We obtain analytical expressions for the spin susceptibility of the massless Dirac fermions. The spin susceptibility is independent of Fermi energy while it depends on the cutoff energy, which is introduced by hand to regularize the integration. We calculate the spin susceptibility of the Wilson-Dirac lattice model, which reduces to the massless Dirac Hamiltonian around the point. We find that the cutoff energy can be related to some parameters of the lattice model and that the Fermi energy dependence of the spin susceptibility exhibits quantitatively the same behavior in the two models. We also calculate the spin susceptibility of massive Dirac fermions, which are models of band electrons in topological insulators. The spin susceptibility is finite even in the energy gap because of strong spin-orbit coupling.

The paper is organized as follows. In Sec. II, we formulate the spin susceptibility and briefly review qualitative behavior of the spin susceptibility in the presence of spin-orbit coupling. In Sec. III and IV, we introduce a continuum model and a lattice model which describe electronic states in a Dirac semimetal. The spin susceptibility of them is calculated. In Sec. V, we calculate the spin susceptibility of massive Dirac fermions. The conclusion is given in Sec. VI.

## Ii spin susceptibility

To calculate the spin susceptibility, we introduce the Zeeman coupling between the electrons and an external magnetic field. The Hamiltonian is given by

(1) |

where is an unperturbed Hamiltonian and the Zeeman term is given by

(2) |

where is the factor, is the Bohr magneton, and is the triplets of Pauli matrices acting on the real spin degree of freedom.

We apply an external magnetic field with infinitely slow spatial variation

(3) |

The slow spatial variation of the field is controlled by the wave vector , which will tend to zero at the end of the calculation. Within the linear response, the induced magnetization is given by

(4) | |||

(5) |

where the spin susceptibility is obtained as

(6) |

where is the volume of the system, is the Fermi distribution function, is a Bloch state of the unperturbed Hamiltonian and is its energy eigenvalue.

Taking the long wavelength limit , we obtain

(7) |

where is the intraband contribution,

(8) |

and is the interband contribution,

(9) |

At the zero temperature, only electronic states on the Fermi surface contribute to . On the other hand, all electronic states below the Fermi energy can contribute to . In order to get a finite , the commutation relation between the Hamiltonian and the spin operator has to be non-zero,

(10) |

If the commutation relation is zero, the matrix elements in Eq. (9) vanish and becomes zero. In the presence of the strong spin-orbit coupling, gives a large contribution.

## Iii Massless Dirac fermions

We consider a model Hamiltonian for electrons in Dirac semimetals,

(11) |

where is the velocity, and are the triplets of Pauli matrices acting on the real spin and the pseudo spin (chirality) degrees of freedom. We calculate the spin susceptibility of the above model. In the present model, the chirality is a good quantum number, so that the chirality degrees of freedom just double the spin susceptibility. The eigenstates of the Hamiltonian with positive chirality are given by

(12) | ||||

(13) |

where is the eigenstate with the energy,

(14) |

where and . and are the zenith and azimuth angles of the wave vector .

The intraband and interband matrix elements are calculated as

(15) | |||

(16) |

We obtain an analytical expression for ,

(17) |

where is the Fermi energy. is proportional to the density of states ,

(18) |

and corresponds to Pauli paramagnetism. The interband contribution is also calculated analytically,

(19) |

where is a cutoff energy. This corresponds to the Van Vleck paramagnetismYu et al. (2010); Zhang et al. (2013). In the present model, there are infinite states below the Fermi energy, so that we introduce a spherical cutoff with the radius in order to regularize the integration by .

The spin susceptibility , which is the sum of and , is obtained as

(20) |

There are two important features. The spin susceptibility is independent of the Fermi energy Koshino and Hizbullah (2016), because the Fermi-energy dependent term of and exactly cancel each other. The spin susceptibility is proportional to . In the present model, the cutoff energy is introduced by hand. Therefore the net value of the spin susceptibility can not be determined. At the first glance, this result is unreasonable, but we can appropriately determine the cutoff energy as we discuss in the next section.

## Iv Lattice Model

In this section, we calculate the spin susceptibility of the Wilson-Dirac type cubic lattice model,

(21) |

where in Eq. (11) is simply replaced by with the hopping energy and the lattice spacing , and these parameters are related as

(22) |

The second term, , is introduced to gap out the point nodes other than the origin . In the vicinity of the origin, Eq. (21) approximates the continuum model, Eq. (11), within the first order of . The eigenstates of the lattice model are given by

(23) | ||||

(24) |

where and . and correspond to the eigenstates of the continuum model with positive and negative chiralities.

The intraband matrix elements are calculated as

(25) |

and the interband matrix elements are

(26) |

Using these matrix elements, we numerically calculate Eqs. (8) and (9).

Figure 1 shows the spin susceptibility as a function of the Fermi energy . Around the zero energy where the dispersion relation is linear, the qualitative behavior of the spin susceptibility of the lattice model is the same as the continuum model. The interband contribution has a peak structure at the zero energy. The width of the peak is related to the structure of the Hamiltonian. The Hamiltonian is composed of two terms, the first sin term, which does not commute with the spin operator,

(27) | |||

(28) |

and the second cos term, which commutes with the spin operator,

(29) | |||

(30) |

Around the Dirac point, the electronic states are approximately described by , and the interband matrix element is finite. Far from the Dirac point, on the other hand, the electronic states are approximately described by , and the interband matrix element is negligibly small. Therefore, the interband contribution has the peak structure and finite value near the zero energy. The peak decays when the cos term is comparable to the sin term .

Here, we relate the peak width of and the cutoff energy , which is introduced in the previous section. In the continuum model, the interband contribution vanishes at the cutoff energy, while in the lattice model, the interband contribution decays far from the Dirac point. Therefore, we assume that the cutoff energy corresponds to the peak width and is determined by

(31) |

which means the sin term and the cos term is comparable. In the above equation, we introduce a numerical factor to fit the spin susceptibility of the continuum and lattice model as discussed following. Solving the above equation, we obtain

(32) |

In Fig. 2, we compare the spin susceptibility of the continuum model and the lattice model. Using Eqs. (22) and (32), the two spin susceptibilities are compared in the same unit. The numerical factor is determined as

(33) |

to get quantitative agreement between the two spin susceptibilities at . In the vicinity of the zero energy, they are good agreement with each other. On the other hand, we see the deviation apart from the zero energy because of the deviation from the liner dispersion relation.

Figure 3 compares the spin susceptibility of the continuum model and that of the lattice model at as a function of . Again we see the quantitative agreement between the two spin susceptibilities. In a condition that , we can derive an approximate analytical expression for the spin susceptibility of the lattice model. In this condition, the interband matrix elements Eq. (26) is expanded as

(34) |

and the spin susceptibility of the lattice model is calculated as

(35) |

In the present approximation, Eq. (32) becomes . Consequently, we obtain . This is consistent with the above agreement.

## V Massive Dirac Fermions

In this section, we calculate the spin susceptibility of the massive Dirac Hamiltonian, which can describes an electronic state of topological insulators. A magnetically doped topological insulator is one of the candidate materials for magnetic Weyl semimetals Yu et al. (2010); Cho (2011); Liu et al. (2013); Kurebayashi and Nomura (2014). Therefore, to clarify the properties of the spin susceptibility of topological insulators is an important issue to realize magnetic Weyl semimetals.

The electronic state is described by the effective Hamiltonian Zhang et al. (2009); Liu et al. (2010),

(36) |

where , , and . In the following calculation, the parameters are taken as , and , which are the parameters for the topological insulator Liu et al. (2010). The above Hamiltonian describes ordinary insulators, Dirac semimetals, and topological insulators by tuning the parameter , which is related to the strength of the spin-orbit coupling. In the presence of a magnetic field, the Zeeman coupling is given by

(37) |

where the spin operators, , are written as

(38) | ||||

(39) | ||||

(40) |

We set the effective factors as , , , and , which are also the parameters for the topological insulator Liu et al. (2010). In this model, there are two kinds of Zeeman terms, ”orbital-independent” term () and ”orbital-dependent” term () Nakai and Nomura (2016). This originates from the non-equality of the effective factors in the two orbitals. The eigenstates of the above Hamiltonian are given by

(41) | ||||

(42) | ||||

(43) | ||||

(44) |

where , and the energy for is given by

(45) |

Based on the symmetry, the spin susceptibility along the axis and the axis exhibit the same behavior. Therefore, we calculate the spin susceptibility along the axis and the axis. The intraband matrix elements are calculated as

(46) |

and

(47) |

The interband matrix elements are

(48) |

and

(49) |

The spin susceptibility is numerically calculated in a similar manner to the previous sections. Figure 4 shows the density of states and the spin susceptibility as a function of the Fermi energy . The top panels in Fig. 4 show the energy bands. We calculate them for three parameters (a) (Ordinary insulator), (b) (Dirac semimetal), and (c) (Topological insulator). Even in the current effective model, which includes the anisotropy and the two types of the Zeeman term, the qualitative behavior of the interband contribution is similar to that of the previous models. The interband contribution takes the maximum value in the energy gap or at the band touching point, where the density of states vanishes. Away from the zero energy, the interband contribution monotonically decreases in a similar manner to the previous model. On the other hand, the intraband contribution behaves in a slightly different manner from the precious model. In the previous models, the intraband contribution is proportional to the density of states. In the current model, the density of states of the valence band is larger than the conduction band, but the intraband contributions for in the valence and conduction bands are comparable. This originates from the cross term of and in Eq. (47). The cross term gives positive contribution in the conduction band and negative in the valence band. Consequently, the intraband contribution in the valence and conduction bands are comparable. On the other hand, the intraband contributions for in the valence and conduction bands are not comparable. This is because the effective factors and have opposite signs, so that the cross term does not work as the case of , where and have same signs. In Fig. 4 (c) the topological insulator case, there is another important feature. The intraband contribution for exhibits a peak structure in the valence band. The peak width corresponds to the band inverted region. On the other hand, there is no peak structure in .

In Fig. 5, we plot the spin susceptibility in the energy gap as a function of . The spin susceptibility increases with the decrease of , which means the increase of the spin-orbit coupling Yu et al. (2010); Zhang et al. (2013). The strong spin-orbit coupling gives the large interband contribution. is much larger than , because the effective factors for the direction are much larger than the direction.

## Vi Conclusion

We have studied the spin susceptibility of the Dirac semimetals. The spin susceptibility is calculated for the massless Dirac continuum model and the Wilson-Dirac lattice model. In the massless Dirac continuum model, we have to introduce the cutoff energy in order to regularize the integration. The spin susceptibility is independent of the Fermi energy and proportional to . We find that the cutoff energy is appropriately determined and related to the some parameters of the lattice model. The cutoff energy corresponds to the energy where the band dispersion deviates from the linear dispersion relation. The spin susceptibility of the lattice model is in quantitatively good agreement with the massless Dirac continuum model. We also calculate the spin susceptibility of massive Dirac fermions with the Zeeman coupling including the orbital dependent term and orbital independent term. The spin susceptibility along the axis is enhanced in the conduction band because of the existence of two types of the Zeeman term and has the peak structure in the band inverted region, which are not observed in the spin susceptibility along the axis.

## Acknowledgment

The authors thank Yasufumi Araki and Masaki Oshikawa for helpful discussions. This work was supported by Kakenhi Grants-in-Aid (Nos. JP15H05854 and JP17K05485) from the Japan Society for the Promotion of Science (JSPS).

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